Optimal. Leaf size=53 \[ -\frac{2 \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{d+e x}}-\frac{4 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{\sqrt{d} e} \]
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Rubi [A] time = 0.0321486, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {2319, 63, 208} \[ -\frac{2 \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{d+e x}}-\frac{4 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{\sqrt{d} e} \]
Antiderivative was successfully verified.
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Rule 2319
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^{3/2}} \, dx &=-\frac{2 \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{d+e x}}+\frac{(2 b n) \int \frac{1}{x \sqrt{d+e x}} \, dx}{e}\\ &=-\frac{2 \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{d+e x}}+\frac{(4 b n) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{e^2}\\ &=-\frac{4 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{\sqrt{d} e}-\frac{2 \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{d+e x}}\\ \end{align*}
Mathematica [A] time = 0.0416505, size = 53, normalized size = 1. \[ -\frac{2 \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{d+e x}}-\frac{4 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{\sqrt{d} e} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.522, size = 0, normalized size = 0. \begin{align*} \int{(a+b\ln \left ( c{x}^{n} \right ) ) \left ( ex+d \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43749, size = 382, normalized size = 7.21 \begin{align*} \left [\frac{2 \,{\left ({\left (b e n x + b d n\right )} \sqrt{d} \log \left (\frac{e x - 2 \, \sqrt{e x + d} \sqrt{d} + 2 \, d}{x}\right ) -{\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )} \sqrt{e x + d}\right )}}{d e^{2} x + d^{2} e}, \frac{2 \,{\left (2 \,{\left (b e n x + b d n\right )} \sqrt{-d} \arctan \left (\frac{\sqrt{e x + d} \sqrt{-d}}{d}\right ) -{\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )} \sqrt{e x + d}\right )}}{d e^{2} x + d^{2} e}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 13.7146, size = 66, normalized size = 1.25 \begin{align*} \frac{- \frac{2 a}{\sqrt{d + e x}} + 2 b \left (\frac{2 n \operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{- d}} \right )}}{\sqrt{- d}} - \frac{\log{\left (c \left (- \frac{d}{e} + \frac{d + e x}{e}\right )^{n} \right )}}{\sqrt{d + e x}}\right )}{e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30374, size = 77, normalized size = 1.45 \begin{align*} \frac{4 \, b n \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right ) e^{\left (-1\right )}}{\sqrt{-d}} - \frac{2 \,{\left (b n \log \left (x e\right ) - b n + b \log \left (c\right ) + a\right )} e^{\left (-1\right )}}{\sqrt{x e + d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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