Optimal. Leaf size=201 \[ -\frac{2 b n \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x}}\right )}{d^{3/2}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{3/2}}+\frac{2 \left (a+b \log \left (c x^n\right )\right )}{d \sqrt{d+e x}}+\frac{2 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )^2}{d^{3/2}}+\frac{4 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{d^{3/2}}-\frac{4 b n \log \left (\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x}}\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{d^{3/2}} \]
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Rubi [A] time = 0.31977, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.435, Rules used = {2347, 63, 208, 2348, 12, 5984, 5918, 2402, 2315, 2319} \[ -\frac{2 b n \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x}}\right )}{d^{3/2}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{3/2}}+\frac{2 \left (a+b \log \left (c x^n\right )\right )}{d \sqrt{d+e x}}+\frac{2 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )^2}{d^{3/2}}+\frac{4 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{d^{3/2}}-\frac{4 b n \log \left (\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x}}\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{d^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2347
Rule 63
Rule 208
Rule 2348
Rule 12
Rule 5984
Rule 5918
Rule 2402
Rule 2315
Rule 2319
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c x^n\right )}{x (d+e x)^{3/2}} \, dx &=\frac{\int \frac{a+b \log \left (c x^n\right )}{x \sqrt{d+e x}} \, dx}{d}-\frac{e \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^{3/2}} \, dx}{d}\\ &=\frac{2 \left (a+b \log \left (c x^n\right )\right )}{d \sqrt{d+e x}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{3/2}}-\frac{(b n) \int -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{\sqrt{d} x} \, dx}{d}-\frac{(2 b n) \int \frac{1}{x \sqrt{d+e x}} \, dx}{d}\\ &=\frac{2 \left (a+b \log \left (c x^n\right )\right )}{d \sqrt{d+e x}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{3/2}}+\frac{(2 b n) \int \frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{x} \, dx}{d^{3/2}}-\frac{(4 b n) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{d e}\\ &=\frac{4 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{d^{3/2}}+\frac{2 \left (a+b \log \left (c x^n\right )\right )}{d \sqrt{d+e x}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{3/2}}+\frac{(4 b n) \operatorname{Subst}\left (\int \frac{x \tanh ^{-1}\left (\frac{x}{\sqrt{d}}\right )}{-d+x^2} \, dx,x,\sqrt{d+e x}\right )}{d^{3/2}}\\ &=\frac{4 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{d^{3/2}}+\frac{2 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )^2}{d^{3/2}}+\frac{2 \left (a+b \log \left (c x^n\right )\right )}{d \sqrt{d+e x}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{3/2}}-\frac{(4 b n) \operatorname{Subst}\left (\int \frac{\tanh ^{-1}\left (\frac{x}{\sqrt{d}}\right )}{1-\frac{x}{\sqrt{d}}} \, dx,x,\sqrt{d+e x}\right )}{d^2}\\ &=\frac{4 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{d^{3/2}}+\frac{2 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )^2}{d^{3/2}}+\frac{2 \left (a+b \log \left (c x^n\right )\right )}{d \sqrt{d+e x}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{3/2}}-\frac{4 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \log \left (\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x}}\right )}{d^{3/2}}+\frac{(4 b n) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1-\frac{x}{\sqrt{d}}}\right )}{1-\frac{x^2}{d}} \, dx,x,\sqrt{d+e x}\right )}{d^2}\\ &=\frac{4 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{d^{3/2}}+\frac{2 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )^2}{d^{3/2}}+\frac{2 \left (a+b \log \left (c x^n\right )\right )}{d \sqrt{d+e x}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{3/2}}-\frac{4 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \log \left (\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x}}\right )}{d^{3/2}}-\frac{(4 b n) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-\frac{\sqrt{d+e x}}{\sqrt{d}}}\right )}{d^{3/2}}\\ &=\frac{4 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{d^{3/2}}+\frac{2 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )^2}{d^{3/2}}+\frac{2 \left (a+b \log \left (c x^n\right )\right )}{d \sqrt{d+e x}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{3/2}}-\frac{4 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \log \left (\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x}}\right )}{d^{3/2}}-\frac{2 b n \text{Li}_2\left (1-\frac{2}{1-\frac{\sqrt{d+e x}}{\sqrt{d}}}\right )}{d^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.267341, size = 295, normalized size = 1.47 \[ \frac{-b n \left (2 \text{PolyLog}\left (2,\frac{1}{2}-\frac{\sqrt{d+e x}}{2 \sqrt{d}}\right )+\log \left (\sqrt{d}-\sqrt{d+e x}\right ) \left (\log \left (\sqrt{d}-\sqrt{d+e x}\right )+2 \log \left (\frac{1}{2} \left (\frac{\sqrt{d+e x}}{\sqrt{d}}+1\right )\right )\right )\right )+b n \left (2 \text{PolyLog}\left (2,\frac{1}{2} \left (\frac{\sqrt{d+e x}}{\sqrt{d}}+1\right )\right )+\log \left (\sqrt{d+e x}+\sqrt{d}\right ) \left (\log \left (\sqrt{d+e x}+\sqrt{d}\right )+2 \log \left (\frac{1}{2}-\frac{\sqrt{d+e x}}{2 \sqrt{d}}\right )\right )\right )+\frac{4 \sqrt{d} \left (a+b \log \left (c x^n\right )\right )}{\sqrt{d+e x}}+2 \log \left (\sqrt{d}-\sqrt{d+e x}\right ) \left (a+b \log \left (c x^n\right )\right )-2 \log \left (\sqrt{d+e x}+\sqrt{d}\right ) \left (a+b \log \left (c x^n\right )\right )+8 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{2 d^{3/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.525, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c{x}^{n} \right ) }{x} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{e x + d} b \log \left (c x^{n}\right ) + \sqrt{e x + d} a}{e^{2} x^{3} + 2 \, d e x^{2} + d^{2} x}, x\right ) \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{b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{\frac{3}{2}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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