Evaluate: `int(a x^3+b x)/(x^4+c^2)dx`

Answer

`(a)/(4)log(x^(4)+c^(2))+(b)/(2)(1)/(c)"tan"^(-1)(x^(2))/(c)+k`

Explanation

`I=int(ax^(3)+bx)/(x^(4)+c^(2))dx `
`=int[(ax^(3))/(x^(4)+c^(2))+(bx)/(x^(4)+c^(2))]dx`
`=(a)/(4)int(4x^(3))/(x^(4)+c^(2))dx+(b)/(2)int(2x)/(x^(4)+c^(2))dx `
` " " I_(1) " "+ " "I_(2)`
`=(a)/(4)log(x^(4)+c^(2))+(b)/(2)int(dt)/(t^(2)+c^(2)) " " (" in " I_(2), " put " x^(2)=t)`
` =(a)/(4)log(x^(4)+c^(2))+(b)/(2)(1)/(c)"tan"^(-1)(x^(2))/(c)+k`