Optimal. Leaf size=73 \[ \frac{1}{3} x^3 \left (a+b \left (c x^q\right )^n\right )^p \left (\frac{b \left (c x^q\right )^n}{a}+1\right )^{-p} \, _2F_1\left (-p,\frac{3}{n q};1+\frac{3}{n q};-\frac{b \left (c x^q\right )^n}{a}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0293813, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {370, 365, 364} \[ \frac{1}{3} x^3 \left (a+b \left (c x^q\right )^n\right )^p \left (\frac{b \left (c x^q\right )^n}{a}+1\right )^{-p} \, _2F_1\left (-p,\frac{3}{n q};1+\frac{3}{n q};-\frac{b \left (c x^q\right )^n}{a}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 370
Rule 365
Rule 364
Rubi steps
\begin{align*} \int x^2 \left (a+b \left (c x^q\right )^n\right )^p \, dx &=\operatorname{Subst}\left (\int x^2 \left (a+b c^n x^{n q}\right )^p \, dx,x^{n q},c^{-n} \left (c x^q\right )^n\right )\\ &=\operatorname{Subst}\left (\left (\left (a+b c^n x^{n q}\right )^p \left (1+\frac{b c^n x^{n q}}{a}\right )^{-p}\right ) \int x^2 \left (1+\frac{b c^n x^{n q}}{a}\right )^p \, dx,x^{n q},c^{-n} \left (c x^q\right )^n\right )\\ &=\frac{1}{3} x^3 \left (a+b \left (c x^q\right )^n\right )^p \left (1+\frac{b \left (c x^q\right )^n}{a}\right )^{-p} \, _2F_1\left (-p,\frac{3}{n q};1+\frac{3}{n q};-\frac{b \left (c x^q\right )^n}{a}\right )\\ \end{align*}
Mathematica [A] time = 0.0813474, size = 73, normalized size = 1. \[ \frac{1}{3} x^3 \left (a+b \left (c x^q\right )^n\right )^p \left (\frac{b \left (c x^q\right )^n}{a}+1\right )^{-p} \, _2F_1\left (-p,\frac{3}{n q};1+\frac{3}{n q};-\frac{b \left (c x^q\right )^n}{a}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.845, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( a+b \left ( c{x}^{q} \right ) ^{n} \right ) ^{p}\, 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*} \int{\left (\left (c x^{q}\right )^{n} b + a\right )}^{p} x^{2}\,{d x} \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 ({\left (\left (c x^{q}\right )^{n} b + a\right )}^{p} x^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \left (a + b \left (c x^{q}\right )^{n}\right )^{p}\, dx \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{\left (\left (c x^{q}\right )^{n} b + a\right )}^{p} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]