Optimal. Leaf size=84 \[ \frac{1}{3} x^3 \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac{d x^2}{c}+1\right )^{-q} F_1\left (\frac{3}{2};-p,-q;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0700929, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {511, 510} \[ \frac{1}{3} x^3 \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac{d x^2}{c}+1\right )^{-q} F_1\left (\frac{3}{2};-p,-q;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 511
Rule 510
Rubi steps
\begin{align*} \int x^2 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx &=\left (\left (a+b x^2\right )^p \left (1+\frac{b x^2}{a}\right )^{-p}\right ) \int x^2 \left (1+\frac{b x^2}{a}\right )^p \left (c+d x^2\right )^q \, dx\\ &=\left (\left (a+b x^2\right )^p \left (1+\frac{b x^2}{a}\right )^{-p} \left (c+d x^2\right )^q \left (1+\frac{d x^2}{c}\right )^{-q}\right ) \int x^2 \left (1+\frac{b x^2}{a}\right )^p \left (1+\frac{d x^2}{c}\right )^q \, dx\\ &=\frac{1}{3} x^3 \left (a+b x^2\right )^p \left (1+\frac{b x^2}{a}\right )^{-p} \left (c+d x^2\right )^q \left (1+\frac{d x^2}{c}\right )^{-q} F_1\left (\frac{3}{2};-p,-q;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\\ \end{align*}
Mathematica [A] time = 0.0482629, size = 86, normalized size = 1.02 \[ \frac{1}{3} x^3 \left (a+b x^2\right )^p \left (\frac{a+b x^2}{a}\right )^{-p} \left (c+d x^2\right )^q \left (\frac{c+d x^2}{c}\right )^{-q} F_1\left (\frac{3}{2};-p,-q;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.059, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( b{x}^{2}+a \right ) ^{p} \left ( d{x}^{2}+c \right ) ^{q}\, 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 (b x^{2} + a\right )}^{p}{\left (d x^{2} + c\right )}^{q} 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 (b x^{2} + a\right )}^{p}{\left (d x^{2} + c\right )}^{q} x^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 (b x^{2} + a\right )}^{p}{\left (d x^{2} + c\right )}^{q} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]