Optimal. Leaf size=36 \[ \frac{2 x^{1-n} (c x)^m}{a (2 m-3 n+2) \sqrt{a x^n}} \]
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Rubi [A] time = 0.0070287, antiderivative size = 36, normalized size of antiderivative = 1., 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} \[ \frac{2 x^{1-n} (c x)^m}{a (2 m-3 n+2) \sqrt{a x^n}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 20
Rule 30
Rubi steps
\begin{align*} \int \frac{(c x)^m}{\left (a x^n\right )^{3/2}} \, dx &=\frac{x^{n/2} \int x^{-3 n/2} (c x)^m \, dx}{a \sqrt{a x^n}}\\ &=\frac{\left (x^{-m+\frac{n}{2}} (c x)^m\right ) \int x^{m-\frac{3 n}{2}} \, dx}{a \sqrt{a x^n}}\\ &=\frac{2 x^{1-n} (c x)^m}{a (2+2 m-3 n) \sqrt{a x^n}}\\ \end{align*}
Mathematica [A] time = 0.0059399, size = 26, normalized size = 0.72 \[ \frac{x (c x)^m}{\left (m-\frac{3 n}{2}+1\right ) \left (a x^n\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 26, normalized size = 0.7 \begin{align*} 2\,{\frac{x \left ( cx \right ) ^{m}}{ \left ( 2+2\,m-3\,n \right ) \left ( a{x}^{n} \right ) ^{3/2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01017, size = 36, normalized size = 1. \begin{align*} \frac{2 \, c^{m} x x^{m}}{a^{\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{\left (c x\right )^{m}}{\left (a x^{n}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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