Optimal. Leaf size=27 \[ \frac{x^{2 (p+1)} \left (b+c x^2\right )^{p+1}}{2 (p+1)} \]
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Rubi [A] time = 0.0089422, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.037, Rules used = {449} \[ \frac{x^{2 (p+1)} \left (b+c x^2\right )^{p+1}}{2 (p+1)} \]
Antiderivative was successfully verified.
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Rule 449
Rubi steps
\begin{align*} \int x^{-1+2 (1+p)} \left (b+c x^2\right )^p \left (b+2 c x^2\right ) \, dx &=\frac{x^{2 (1+p)} \left (b+c x^2\right )^{1+p}}{2 (1+p)}\\ \end{align*}
Mathematica [C] time = 0.078724, size = 97, normalized size = 3.59 \[ \frac{x^{2 p+2} \left (b+c x^2\right )^p \left (\frac{c x^2}{b}+1\right )^{-p} \left (2 c (p+1) x^2 \, _2F_1\left (-p,p+2;p+3;-\frac{c x^2}{b}\right )+b (p+2) \, _2F_1\left (-p,p+1;p+2;-\frac{c x^2}{b}\right )\right )}{2 (p+1) (p+2)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 26, normalized size = 1. \begin{align*}{\frac{{x}^{2+2\,p} \left ( c{x}^{2}+b \right ) ^{1+p}}{2+2\,p}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09265, size = 47, normalized size = 1.74 \begin{align*} \frac{{\left (c x^{4} + b x^{2}\right )} e^{\left (p \log \left (c x^{2} + b\right ) + 2 \, p \log \left (x\right )\right )}}{2 \,{\left (p + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.06487, size = 72, normalized size = 2.67 \begin{align*} \frac{{\left (c x^{3} + b x\right )}{\left (c x^{2} + b\right )}^{p} x^{2 \, p + 1}}{2 \,{\left (p + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [B] time = 1.10196, size = 70, normalized size = 2.59 \begin{align*} \frac{{\left (c x^{2} + b\right )}^{p} c x^{3} e^{\left (2 \, p \log \left (x\right ) + \log \left (x\right )\right )} +{\left (c x^{2} + b\right )}^{p} b x e^{\left (2 \, p \log \left (x\right ) + \log \left (x\right )\right )}}{2 \,{\left (p + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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