Optimal. Leaf size=18 \[ \frac{x^3 \left (b x^2\right )^p}{2 p+3} \]
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Rubi [A] time = 0.0043131, antiderivative size = 18, 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} \[ \frac{x^3 \left (b x^2\right )^p}{2 p+3} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin{align*} \int x^2 \left (b x^2\right )^p \, dx &=\left (x^{-2 p} \left (b x^2\right )^p\right ) \int x^{2+2 p} \, dx\\ &=\frac{x^3 \left (b x^2\right )^p}{3+2 p}\\ \end{align*}
Mathematica [A] time = 0.0020921, size = 18, normalized size = 1. \[ \frac{x^3 \left (b x^2\right )^p}{2 p+3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 19, normalized size = 1.1 \begin{align*}{\frac{{x}^{3} \left ( b{x}^{2} \right ) ^{p}}{3+2\,p}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05078, size = 26, normalized size = 1.44 \begin{align*} \frac{b^{p} x^{3} x^{2 \, p}}{2 \, p + 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.97141, size = 34, normalized size = 1.89 \begin{align*} \frac{\left (b x^{2}\right )^{p} x^{3}}{2 \, p + 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14206, size = 24, normalized size = 1.33 \begin{align*} \frac{\left (b x^{2}\right )^{p} x^{3}}{2 \, p + 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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