Optimal. Leaf size=84 \[ \frac{x (b c-a d)^2 \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{c d^2}-\frac{b x (b c (n+1)-a d (2 n+1))}{d^2 (n+1)}+\frac{b x \left (a+b x^n\right )}{d (n+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0940034, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {416, 388, 245} \[ \frac{x (b c-a d)^2 \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{c d^2}-\frac{b x (b c (n+1)-a d (2 n+1))}{d^2 (n+1)}+\frac{b x \left (a+b x^n\right )}{d (n+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 416
Rule 388
Rule 245
Rubi steps
\begin{align*} \int \frac{\left (a+b x^n\right )^2}{c+d x^n} \, dx &=\frac{b x \left (a+b x^n\right )}{d (1+n)}+\frac{\int \frac{-a (b c-a d (1+n))-b (b c (1+n)-a d (1+2 n)) x^n}{c+d x^n} \, dx}{d (1+n)}\\ &=-\frac{b (b c (1+n)-a d (1+2 n)) x}{d^2 (1+n)}+\frac{b x \left (a+b x^n\right )}{d (1+n)}+\frac{(b c-a d)^2 \int \frac{1}{c+d x^n} \, dx}{d^2}\\ &=-\frac{b (b c (1+n)-a d (1+2 n)) x}{d^2 (1+n)}+\frac{b x \left (a+b x^n\right )}{d (1+n)}+\frac{(b c-a d)^2 x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{c d^2}\\ \end{align*}
Mathematica [A] time = 0.0444925, size = 82, normalized size = 0.98 \[ \frac{a^2 x}{c}+\frac{x (a d-b c)^2 \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{c d^2}-\frac{x (b c-a d)^2}{c d^2}+\frac{b^2 x^{n+1}}{d (n+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.353, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{x}^{n} \right ) ^{2}}{c+d{x}^{n}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \int \frac{1}{d^{3} x^{n} + c d^{2}}\,{d x} + \frac{b^{2} d x x^{n} -{\left (b^{2} c{\left (n + 1\right )} - 2 \, a b d{\left (n + 1\right )}\right )} x}{d^{2}{\left (n + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}{d x^{n} + c}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 3.31903, size = 170, normalized size = 2.02 \begin{align*} \frac{a^{2} x \Phi \left (\frac{d x^{n} e^{i \pi }}{c}, 1, \frac{1}{n}\right ) \Gamma \left (\frac{1}{n}\right )}{c n^{2} \Gamma \left (1 + \frac{1}{n}\right )} - \frac{2 a b x \Phi \left (\frac{c x^{- n} e^{i \pi }}{d}, 1, \frac{e^{i \pi }}{n}\right ) \Gamma \left (\frac{1}{n}\right )}{d n^{2} \Gamma \left (1 + \frac{1}{n}\right )} + \frac{2 b^{2} x x^{2 n} \Phi \left (\frac{d x^{n} e^{i \pi }}{c}, 1, 2 + \frac{1}{n}\right ) \Gamma \left (2 + \frac{1}{n}\right )}{c n \Gamma \left (3 + \frac{1}{n}\right )} + \frac{b^{2} x x^{2 n} \Phi \left (\frac{d x^{n} e^{i \pi }}{c}, 1, 2 + \frac{1}{n}\right ) \Gamma \left (2 + \frac{1}{n}\right )}{c n^{2} \Gamma \left (3 + \frac{1}{n}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{n} + a\right )}^{2}}{d x^{n} + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]