Optimal. Leaf size=19 \[ \frac{x^2 \left (b x^2\right )^p}{2 (p+1)} \]
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Rubi [A] time = 0.0046985, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {15, 30} \[ \frac{x^2 \left (b x^2\right )^p}{2 (p+1)} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin{align*} \int x \left (b x^2\right )^p \, dx &=\left (x^{-2 p} \left (b x^2\right )^p\right ) \int x^{1+2 p} \, dx\\ &=\frac{x^2 \left (b x^2\right )^p}{2 (1+p)}\\ \end{align*}
Mathematica [A] time = 0.001993, size = 18, normalized size = 0.95 \[ \frac{x^2 \left (b x^2\right )^p}{2 p+2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 18, normalized size = 1. \begin{align*}{\frac{{x}^{2} \left ( b{x}^{2} \right ) ^{p}}{2+2\,p}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82108, size = 36, normalized size = 1.89 \begin{align*} \frac{\left (b x^{2}\right )^{p} x^{2}}{2 \,{\left (p + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.24322, size = 22, normalized size = 1.16 \begin{align*} \begin{cases} \frac{b^{p} x^{2} \left (x^{2}\right )^{p}}{2 p + 2} & \text{for}\: p \neq -1 \\\frac{\log{\left (x \right )}}{b} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15935, size = 26, normalized size = 1.37 \begin{align*} \frac{\left (b x^{2}\right )^{p + 1}}{2 \, b{\left (p + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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