Optimal. Leaf size=140 \[ \frac{B x^{m-1} \left (b x^2+c x^4\right )^{p+1}}{c (m+4 p+3)}-\frac{x^{m+1} \left (\frac{c x^2}{b}+1\right )^{-p} \left (b x^2+c x^4\right )^p (b B (m+2 p+1)-A c (m+4 p+3)) \, _2F_1\left (-p,\frac{1}{2} (m+2 p+1);\frac{1}{2} (m+2 p+3);-\frac{c x^2}{b}\right )}{c (m+2 p+1) (m+4 p+3)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.136641, antiderivative size = 126, normalized size of antiderivative = 0.9, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2039, 2032, 365, 364} \[ x^{m+1} \left (\frac{c x^2}{b}+1\right )^{-p} \left (b x^2+c x^4\right )^p \left (\frac{A}{m+2 p+1}-\frac{b B}{c (m+4 p+3)}\right ) \, _2F_1\left (-p,\frac{1}{2} (m+2 p+1);\frac{1}{2} (m+2 p+3);-\frac{c x^2}{b}\right )+\frac{B x^{m-1} \left (b x^2+c x^4\right )^{p+1}}{c (m+4 p+3)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2039
Rule 2032
Rule 365
Rule 364
Rubi steps
\begin{align*} \int x^m \left (A+B x^2\right ) \left (b x^2+c x^4\right )^p \, dx &=\frac{B x^{-1+m} \left (b x^2+c x^4\right )^{1+p}}{c (3+m+4 p)}-\left (-A+\frac{b B (1+m+2 p)}{c (3+m+4 p)}\right ) \int x^m \left (b x^2+c x^4\right )^p \, dx\\ &=\frac{B x^{-1+m} \left (b x^2+c x^4\right )^{1+p}}{c (3+m+4 p)}-\left (\left (-A+\frac{b B (1+m+2 p)}{c (3+m+4 p)}\right ) x^{-2 p} \left (b+c x^2\right )^{-p} \left (b x^2+c x^4\right )^p\right ) \int x^{m+2 p} \left (b+c x^2\right )^p \, dx\\ &=\frac{B x^{-1+m} \left (b x^2+c x^4\right )^{1+p}}{c (3+m+4 p)}-\left (\left (-A+\frac{b B (1+m+2 p)}{c (3+m+4 p)}\right ) x^{-2 p} \left (1+\frac{c x^2}{b}\right )^{-p} \left (b x^2+c x^4\right )^p\right ) \int x^{m+2 p} \left (1+\frac{c x^2}{b}\right )^p \, dx\\ &=\frac{B x^{-1+m} \left (b x^2+c x^4\right )^{1+p}}{c (3+m+4 p)}+\left (\frac{A}{1+m+2 p}-\frac{b B}{c (3+m+4 p)}\right ) x^{1+m} \left (1+\frac{c x^2}{b}\right )^{-p} \left (b x^2+c x^4\right )^p \, _2F_1\left (-p,\frac{1}{2} (1+m+2 p);\frac{1}{2} (3+m+2 p);-\frac{c x^2}{b}\right )\\ \end{align*}
Mathematica [A] time = 0.0978766, size = 135, normalized size = 0.96 \[ \frac{x^{m+1} \left (x^2 \left (b+c x^2\right )\right )^p \left (\frac{c x^2}{b}+1\right )^{-p} \left (A (m+2 p+3) \, _2F_1\left (-p,\frac{1}{2} (m+2 p+1);\frac{1}{2} (m+2 p+3);-\frac{c x^2}{b}\right )+B x^2 (m+2 p+1) \, _2F_1\left (-p,\frac{1}{2} (m+2 p+3);\frac{1}{2} (m+2 p+5);-\frac{c x^2}{b}\right )\right )}{(m+2 p+1) (m+2 p+3)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.313, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( B{x}^{2}+A \right ) \left ( c{x}^{4}+b{x}^{2} \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 (B x^{2} + A\right )}{\left (c x^{4} + b x^{2}\right )}^{p} x^{m}\,{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 )}{\left (c x^{4} + b x^{2}\right )}^{p} x^{m}, 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 )}{\left (c x^{4} + b x^{2}\right )}^{p} x^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]