(i) `int(2x-5x) (3+2x) (1-x) dx`, (ii) `int sqrt(x) (ax^(2) + bx + c) dx`
(iii) `int (sqrtx - 3sqrt(x^(4)) + 7/(3sqrt(x^(2))) - 6e^(x) + 1) dx`

Answer

Let's solve the given integrals step by step.

### (i) Integral: \(\int (2x - 5x)(3 + 2x)(1 - x) \, dx\)

1. **Simplify the expression inside the integral:**
\[ 2x - 5x = -3x \]
So, we rewrite the integral:
\[ \int (-3x)(3 + 2x)(1 - x) \, dx \]

2. **Expand the expression:**
\[ (-3x)(3 + 2x)(1 - x) = -3x[(3 + 2x)(1 - x)] \]
Now, expand \((3 + 2x)(1 - x)\):
\[ = 3(1 - x) + 2x(1 - x) = 3 - 3x + 2x - 2x^2 = 3 - x - 2x^2 \]
Thus,
\[ -3x(3 - x - 2x^2) = -9x + 3x^2 + 6x^3 \]

3. **Rewrite the integral:**
\[ \int (-9x + 3x^2 + 6x^3) \, dx \]

4. **Integrate term by term:**
\[ = \int -9x \, dx + \int 3x^2 \, dx + \int 6x^3 \, dx \]
\[ = -\frac{9}{2}x^2 + x^3 + \frac{6}{4}x^4 + C \]
\[ = -\frac{9}{2}x^2 + x^3 + \frac{3}{2}x^4 + C \]

### (ii) Integral: \(\int \sqrt{x}(ax^2 + bx + c) \, dx\)

1. **Rewrite \(\sqrt{x}\) as \(x^{1/2}\):**
\[ \int x^{1/2}(ax^2 + bx + c) \, dx \]

2. **Distribute \(\sqrt{x}\):**
\[ = \int (ax^{5/2} + bx^{3/2} + cx^{1/2}) \, dx \]

3. **Integrate term by term:**
\[ = a \int x^{5/2} \, dx + b \int x^{3/2} \, dx + c \int x^{1/2} \, dx \]
\[ = a \cdot \frac{x^{7/2}}{7/2} + b \cdot \frac{x^{5/2}}{5/2} + c \cdot \frac{x^{3/2}}{3/2} + C \]
\[ = \frac{2a}{7} x^{7/2} + \frac{2b}{5} x^{5/2} + \frac{2c}{3} x^{3/2} + C \]

### (iii) Integral: \(\int (\sqrt{x} - 3\sqrt{x^4} + \frac{7}{3\sqrt{x^2}} - 6e^x + 1) \, dx\)

1. **Rewrite the terms:**
\[ = \int (x^{1/2} - 3x^2 + \frac{7}{3}x^{-1} - 6e^x + 1) \, dx \]

2. **Integrate term by term:**
\[ = \int x^{1/2} \, dx - 3 \int x^2 \, dx + \frac{7}{3} \int x^{-1} \, dx - 6 \int e^x \, dx + \int 1 \, dx \]
\[ = \frac{x^{3/2}}{3/2} - 3 \cdot \frac{x^3}{3} + \frac{7}{3} \ln|x| - 6e^x + x + C \]
\[ = \frac{2}{3} x^{3/2} - x^3 + \frac{7}{3} \ln|x| - 6e^x + x + C \]

### Summary of Results:
1. \(\int (2x - 5x)(3 + 2x)(1 - x) \, dx = -\frac{9}{2}x^2 + x^3 + \frac{3}{2}x^4 + C\)
2. \(\int \sqrt{x}(ax^2 + bx + c) \, dx = \frac{2a}{7} x^{7/2} + \frac{2b}{5} x^{5/2} + \frac{2c}{3} x^{3/2} + C\)
3. \(\int (\sqrt{x} - 3\sqrt{x^4} + \frac{7}{3\sqrt{x^2}} - 6e^x + 1) \, dx = \frac{2}{3} x^{3/2} - x^3 + \frac{7}{3} \ln|x| - 6e^x + x + C\)