Optimal. Leaf size=19 \[ \frac{x^4 \left (b x^2\right )^p}{2 (p+2)} \]
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Rubi [A] time = 0.0050369, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {15, 30} \[ \frac{x^4 \left (b x^2\right )^p}{2 (p+2)} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin{align*} \int x^3 \left (b x^2\right )^p \, dx &=\left (x^{-2 p} \left (b x^2\right )^p\right ) \int x^{3+2 p} \, dx\\ &=\frac{x^4 \left (b x^2\right )^p}{2 (2+p)}\\ \end{align*}
Mathematica [A] time = 0.0022863, size = 18, normalized size = 0.95 \[ \frac{x^4 \left (b x^2\right )^p}{2 p+4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 18, normalized size = 1. \begin{align*}{\frac{{x}^{4} \left ( b{x}^{2} \right ) ^{p}}{4+2\,p}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02323, size = 24, normalized size = 1.26 \begin{align*} \frac{b^{p}{\left (x^{2}\right )}^{p} x^{4}}{2 \,{\left (p + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.05338, size = 36, normalized size = 1.89 \begin{align*} \frac{\left (b x^{2}\right )^{p} x^{4}}{2 \,{\left (p + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.551244, size = 24, normalized size = 1.26 \begin{align*} \begin{cases} \frac{b^{p} x^{4} \left (x^{2}\right )^{p}}{2 p + 4} & \text{for}\: p \neq -2 \\\frac{\log{\left (x \right )}}{b^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15558, size = 39, normalized size = 2.05 \begin{align*} \frac{x^{2} e^{\left (p \log \left (b x^{2}\right ) + \log \left (b x^{2}\right )\right )}}{2 \, b{\left (p + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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