`int (dx )/(""^(3) sqrt(x ^(5//2) (x+1)^(7//2)))` is equal to:

Options

A. `-((x-1)/(x))^(1//6)+C`
B. `6 ((x+1)/(x )) ^(-1//6)+C`
C. `((x)/(x+1)) ^(5//6) +C`
D. `-((x)/(x+1)) ^(5//6) +C`

Answer

To solve the integral

\[ \int \frac{dx}{\sqrt[3]{x^{\frac{5}{2}}(x+1)^{\frac{7}{2}}}} \]

we can follow these steps:

### Step 1: Rewrite the Integral
We start by rewriting the integral in a more manageable form:

\[ \int \frac{dx}{\sqrt[3]{x^{\frac{5}{2}}(x+1)^{\frac{7}{2}}}} = \int \frac{dx}{(x^{\frac{5}{2}}(x+1)^{\frac{7}{2}})^{\frac{1}{3}}} \]

This simplifies to:

\[ \int \frac{dx}{x^{\frac{5}{6}}(x+1)^{\frac{7}{6}}} \]

### Step 2: Substitution
Next, we will use the substitution \( t = \frac{x}{x+1} \). This implies:

\[ x = \frac{t}{1-t} \]

Differentiating both sides gives:

\[ dx = \frac{1}{(1-t)^2} dt \]

### Step 3: Change of Variables
Now we need to express \( x^{\frac{5}{6}} \) and \( (x+1)^{\frac{7}{6}} \) in terms of \( t \):

1. \( x + 1 = \frac{t}{1-t} + 1 = \frac{t + (1-t)}{1-t} = \frac{1}{1-t} \)
2. Therefore, \( (x+1)^{\frac{7}{6}} = \left(\frac{1}{1-t}\right)^{\frac{7}{6}} = (1-t)^{-\frac{7}{6}} \)

Now we can express \( x^{\frac{5}{6}} \):

\[ x^{\frac{5}{6}} = \left(\frac{t}{1-t}\right)^{\frac{5}{6}} = \frac{t^{\frac{5}{6}}}{(1-t)^{\frac{5}{6}}} \]

### Step 4: Substitute Back into the Integral
Substituting these back into the integral, we have:

\[ \int \frac{dx}{x^{\frac{5}{6}}(x+1)^{\frac{7}{6}}} = \int \frac{\frac{1}{(1-t)^2} dt}{\frac{t^{\frac{5}{6}}}{(1-t)^{\frac{5}{6}}} (1-t)^{-\frac{7}{6}}} \]

This simplifies to:

\[ \int \frac{(1-t)^{\frac{7}{6}} dt}{t^{\frac{5}{6}}} \]

### Step 5: Simplify the Integral
This can be simplified to:

\[ \int t^{-\frac{5}{6}} (1-t)^{\frac{7}{6}} dt \]

### Step 6: Integration
Now we can integrate using the Beta function or standard integral techniques. The integral can be expressed in terms of the Beta function \( B(x,y) \):

\[ \int t^{-\frac{5}{6}} (1-t)^{\frac{7}{6}} dt = B\left(\frac{1}{6}, \frac{8}{6}\right) \]

### Step 7: Back Substitute
After integrating, we will need to substitute back for \( t \):

\[ t = \frac{x}{x+1} \]

### Final Result
The final result will be:

\[ 6 \left(\frac{x}{x+1}\right)^{\frac{1}{6}} + C \]

Where \( C \) is the constant of integration.