Optimal. Leaf size=32 \[ \frac{2 x^{m-n+1}}{b (2 m-3 n+2) \sqrt{b x^n}} \]
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Rubi [A] time = 0.0061174, antiderivative size = 32, 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} \[ \frac{2 x^{m-n+1}}{b (2 m-3 n+2) \sqrt{b x^n}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin{align*} \int \frac{x^m}{\left (b x^n\right )^{3/2}} \, dx &=\frac{x^{n/2} \int x^{m-\frac{3 n}{2}} \, dx}{b \sqrt{b x^n}}\\ &=\frac{2 x^{1+m-n}}{b (2+2 m-3 n) \sqrt{b x^n}}\\ \end{align*}
Mathematica [A] time = 0.0057825, size = 25, normalized size = 0.78 \[ \frac{x^{m+1}}{\left (m-\frac{3 n}{2}+1\right ) \left (b x^n\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 25, normalized size = 0.8 \begin{align*} 2\,{\frac{{x}^{1+m}}{ \left ( 2+2\,m-3\,n \right ) \left ( b{x}^{n} \right ) ^{3/2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01226, size = 32, normalized size = 1. \begin{align*} \frac{2 \, x x^{m}}{b^{\frac{3}{2}}{\left (2 \, m - 3 \, n + 2\right )}{\left (x^{n}\right )}^{\frac{3}{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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m}}{\left (b x^{n}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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