Optimal. Leaf size=31 \[ \frac{2 \left (b x^n\right )^{5/2} (c x)^{m+1}}{c (2 m+5 n+2)} \]
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Rubi [A] time = 0.0133699, antiderivative size = 36, normalized size of antiderivative = 1.16, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {15, 20, 30} \[ \frac{2 b^2 x^{2 n+1} \sqrt{b x^n} (c x)^m}{2 m+5 n+2} \]
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^n\right )^{5/2} \, dx &=\left (b^2 x^{-n/2} \sqrt{b x^n}\right ) \int x^{5 n/2} (c x)^m \, dx\\ &=\left (b^2 x^{-m-\frac{n}{2}} (c x)^m \sqrt{b x^n}\right ) \int x^{m+\frac{5 n}{2}} \, dx\\ &=\frac{2 b^2 x^{1+2 n} (c x)^m \sqrt{b x^n}}{2+2 m+5 n}\\ \end{align*}
Mathematica [A] time = 0.0095282, size = 26, normalized size = 0.84 \[ \frac{x \left (b x^n\right )^{5/2} (c x)^m}{m+\frac{5 n}{2}+1} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 26, normalized size = 0.8 \begin{align*} 2\,{\frac{x \left ( cx \right ) ^{m} \left ( b{x}^{n} \right ) ^{5/2}}{2+2\,m+5\,n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03179, size = 36, normalized size = 1.16 \begin{align*} \frac{2 \, b^{\frac{5}{2}} c^{m} x x^{m}{\left (x^{n}\right )}^{\frac{5}{2}}}{2 \, m + 5 \, n + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15526, size = 42, normalized size = 1.35 \begin{align*} \frac{2 \, b^{\frac{5}{2}} x x^{\frac{5}{2} \, n} e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )}}{2 \, m + 5 \, n + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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