Optimal. Leaf size=22 \[ \frac{x \left (b x^2\right )^p (c x)^m}{m+2 p+1} \]
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Rubi [A] time = 0.0054969, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {15, 20, 30} \[ \frac{x \left (b x^2\right )^p (c x)^m}{m+2 p+1} \]
Antiderivative was successfully verified.
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Rule 15
Rule 20
Rule 30
Rubi steps
\begin{align*} \int (c x)^m \left (b x^2\right )^p \, dx &=\left (x^{-2 p} \left (b x^2\right )^p\right ) \int x^{2 p} (c x)^m \, dx\\ &=\left (x^{-m-2 p} (c x)^m \left (b x^2\right )^p\right ) \int x^{m+2 p} \, dx\\ &=\frac{x (c x)^m \left (b x^2\right )^p}{1+m+2 p}\\ \end{align*}
Mathematica [A] time = 0.0029196, size = 22, normalized size = 1. \[ \frac{x \left (b x^2\right )^p (c x)^m}{m+2 p+1} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 23, normalized size = 1.1 \begin{align*}{\frac{x \left ( cx \right ) ^{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 = 1.09021, size = 36, normalized size = 1.64 \begin{align*} \frac{b^{p} c^{m} 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 = 2.06913, size = 77, normalized size = 3.5 \begin{align*} \frac{\left (c x\right )^{m} x e^{\left (2 \, p \log \left (c x\right ) + p \log \left (\frac{b}{c^{2}}\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 = 1.17046, size = 39, normalized size = 1.77 \begin{align*} \frac{x e^{\left (p \log \left (b\right ) + m \log \left (c\right ) + 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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