Optimal. Leaf size=28 \[ \frac{2 b \sqrt{b x^n} x^{m+n+1}}{2 m+3 n+2} \]
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Rubi [A] time = 0.0077379, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {15, 30} \[ \frac{2 b \sqrt{b x^n} x^{m+n+1}}{2 m+3 n+2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin{align*} \int x^m \left (b x^n\right )^{3/2} \, dx &=\left (b x^{-n/2} \sqrt{b x^n}\right ) \int x^{m+\frac{3 n}{2}} \, dx\\ &=\frac{2 b x^{1+m+n} \sqrt{b x^n}}{2+2 m+3 n}\\ \end{align*}
Mathematica [A] time = 0.0063787, size = 25, normalized size = 0.89 \[ \frac{x^{m+1} \left (b x^n\right )^{3/2}}{m+\frac{3 n}{2}+1} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 25, normalized size = 0.9 \begin{align*} 2\,{\frac{{x}^{1+m} \left ( b{x}^{n} \right ) ^{3/2}}{2+2\,m+3\,n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.990538, size = 32, normalized size = 1.14 \begin{align*} \frac{2 \, b^{\frac{3}{2}} x x^{m}{\left (x^{n}\right )}^{\frac{3}{2}}}{2 \, m + 3 \, 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 = 1.21096, size = 32, normalized size = 1.14 \begin{align*} \frac{2 \, b^{\frac{3}{2}} x x^{m} x^{\frac{3}{2} \, n}}{2 \, m + 3 \, n + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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