Optimal. Leaf size=27 \[ \frac{\left (-a+b x^2+c x^4\right )^{p+1}}{2 (p+1)} \]
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Rubi [A] time = 0.0195965, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {1247, 629} \[ \frac{\left (-a+b x^2+c x^4\right )^{p+1}}{2 (p+1)} \]
Antiderivative was successfully verified.
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Rule 1247
Rule 629
Rubi steps
\begin{align*} \int x \left (b+2 c x^2\right ) \left (-a+b x^2+c x^4\right )^p \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int (b+2 c x) \left (-a+b x+c x^2\right )^p \, dx,x,x^2\right )\\ &=\frac{\left (-a+b x^2+c x^4\right )^{1+p}}{2 (1+p)}\\ \end{align*}
Mathematica [A] time = 0.0140538, size = 27, normalized size = 1. \[ \frac{\left (-a+b x^2+c x^4\right )^{p+1}}{2 (p+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 26, normalized size = 1. \begin{align*}{\frac{ \left ( c{x}^{4}+b{x}^{2}-a \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 = 1.16771, size = 50, normalized size = 1.85 \begin{align*} \frac{{\left (c x^{4} + b x^{2} - a\right )}{\left (c x^{4} + b x^{2} - a\right )}^{p}}{2 \,{\left (p + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.07853, size = 74, normalized size = 2.74 \begin{align*} \frac{{\left (c x^{4} + b x^{2} - a\right )}{\left (c x^{4} + b x^{2} - a\right )}^{p}}{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 = 1.15072, size = 93, normalized size = 3.44 \begin{align*} \frac{{\left (c x^{4} + b x^{2} - a\right )}^{p} c x^{4} +{\left (c x^{4} + b x^{2} - a\right )}^{p} b x^{2} -{\left (c x^{4} + b x^{2} - a\right )}^{p} a}{2 \,{\left (p + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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