Optimal. Leaf size=27 \[ \frac{2 x (c x)^m}{(2 m-n+2) \sqrt{b x^n}} \]
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Rubi [A] time = 0.0059093, antiderivative size = 27, 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} \[ \frac{2 x (c x)^m}{(2 m-n+2) \sqrt{b x^n}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 20
Rule 30
Rubi steps
\begin{align*} \int \frac{(c x)^m}{\sqrt{b x^n}} \, dx &=\frac{x^{n/2} \int x^{-n/2} (c x)^m \, dx}{\sqrt{b x^n}}\\ &=\frac{\left (x^{-m+\frac{n}{2}} (c x)^m\right ) \int x^{m-\frac{n}{2}} \, dx}{\sqrt{b x^n}}\\ &=\frac{2 x (c x)^m}{(2+2 m-n) \sqrt{b x^n}}\\ \end{align*}
Mathematica [A] time = 0.0047464, size = 26, normalized size = 0.96 \[ \frac{x (c x)^m}{\left (m-\frac{n}{2}+1\right ) \sqrt{b x^n}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 26, normalized size = 1. \begin{align*} 2\,{\frac{x \left ( cx \right ) ^{m}}{ \left ( 2+2\,m-n \right ) \sqrt{b{x}^{n}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02594, size = 36, normalized size = 1.33 \begin{align*} \frac{2 \, c^{m} x x^{m}}{\sqrt{b}{\left (2 \, m - n + 2\right )} \sqrt{x^{n}}} \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(-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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c x\right )^{m}}{\sqrt{b x^{n}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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