Optimal. Leaf size=21 \[ \frac{x^{m+1} \left (b x^2\right )^p}{m+2 p+1} \]
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Rubi [A] time = 0.0055522, antiderivative size = 21, 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} \[ \frac{x^{m+1} \left (b x^2\right )^p}{m+2 p+1} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin{align*} \int x^m \left (b x^2\right )^p \, dx &=\left (x^{-2 p} \left (b x^2\right )^p\right ) \int x^{m+2 p} \, dx\\ &=\frac{x^{1+m} \left (b x^2\right )^p}{1+m+2 p}\\ \end{align*}
Mathematica [A] time = 0.003798, size = 21, normalized size = 1. \[ \frac{x^{m+1} \left (b x^2\right )^p}{m+2 p+1} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 22, normalized size = 1.1 \begin{align*}{\frac{{x}^{1+m} \left ( b{x}^{2} \right ) ^{p}}{1+m+2\,p}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01495, size = 32, normalized size = 1.52 \begin{align*} \frac{b^{p} x e^{\left (m \log \left (x\right ) + 2 \, p \log \left (x\right )\right )}}{m + 2 \, p + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.05308, size = 63, normalized size = 3. \begin{align*} \frac{x x^{m} e^{\left (p \log \left (b\right ) + 2 \, p \log \left (x\right )\right )}}{m + 2 \, p + 1} \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 = 1.18249, size = 32, normalized size = 1.52 \begin{align*} \frac{x x^{m} e^{\left (p \log \left (b\right ) + 2 \, p \log \left (x\right )\right )}}{m + 2 \, p + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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