Optimal. Leaf size=27 \[ \frac{x^{p+1} \left (b x+c x^3\right )^{p+1}}{2 (p+1)} \]
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Rubi [A] time = 0.0230009, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {1590} \[ \frac{x^{p+1} \left (b x+c x^3\right )^{p+1}}{2 (p+1)} \]
Antiderivative was successfully verified.
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Rule 1590
Rubi steps
\begin{align*} \int x^{1+p} \left (b+2 c x^2\right ) \left (b x+c x^3\right )^p \, dx &=\frac{x^{1+p} \left (b x+c x^3\right )^{1+p}}{2 (1+p)}\\ \end{align*}
Mathematica [C] time = 0.0753241, size = 97, normalized size = 3.59 \[ \frac{x^{p+2} \left (x \left (b+c x^2\right )\right )^p \left (\frac{c x^2}{b}+1\right )^{-p} \left (2 c (p+1) x^2 \, _2F_1\left (-p,p+2;p+3;-\frac{c x^2}{b}\right )+b (p+2) \, _2F_1\left (-p,p+1;p+2;-\frac{c x^2}{b}\right )\right )}{2 (p+1) (p+2)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 31, normalized size = 1.2 \begin{align*}{\frac{{x}^{2+p} \left ( c{x}^{2}+b \right ) \left ( c{x}^{3}+bx \right ) ^{p}}{2+2\,p}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.2988, 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 = 1.42504, size = 72, normalized size = 2.67 \begin{align*} \frac{{\left (c x^{3} + b x\right )}{\left (c x^{3} + b x\right )}^{p} x^{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 = 1.18203, size = 73, normalized size = 2.7 \begin{align*} \frac{c x^{3} e^{\left (p \log \left (c x^{2} + b\right ) + 2 \, p \log \left (x\right ) + \log \left (x\right )\right )} + b x e^{\left (p \log \left (c x^{2} + b\right ) + 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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