Optimal. Leaf size=259 \[ -3 b \sqrt{d} e n \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x}}\right )-\frac{(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}+3 e \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )-3 \sqrt{d} e \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )-4 b e n \sqrt{d+e x}-\frac{b d n \sqrt{d+e x}}{x}+3 b \sqrt{d} e n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )^2+3 b \sqrt{d} e n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )-6 b \sqrt{d} e n \log \left (\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x}}\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \]
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Rubi [A] time = 0.324265, antiderivative size = 259, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.435, Rules used = {47, 50, 63, 208, 2350, 14, 5984, 5918, 2402, 2315} \[ -3 b \sqrt{d} e n \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x}}\right )-\frac{(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}+3 e \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )-3 \sqrt{d} e \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )-4 b e n \sqrt{d+e x}-\frac{b d n \sqrt{d+e x}}{x}+3 b \sqrt{d} e n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )^2+3 b \sqrt{d} e n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )-6 b \sqrt{d} e n \log \left (\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x}}\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 208
Rule 2350
Rule 14
Rule 5984
Rule 5918
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int \frac{(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx &=3 e \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )-\frac{(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}-3 \sqrt{d} e \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{-(d-2 e x) \sqrt{d+e x}-3 \sqrt{d} e x \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{x^2} \, dx\\ &=3 e \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )-\frac{(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}-3 \sqrt{d} e \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (-\frac{d \sqrt{d+e x}}{x^2}+\frac{2 e \sqrt{d+e x}}{x}-\frac{3 \sqrt{d} e \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{x}\right ) \, dx\\ &=3 e \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )-\frac{(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}-3 \sqrt{d} e \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )+(b d n) \int \frac{\sqrt{d+e x}}{x^2} \, dx-(2 b e n) \int \frac{\sqrt{d+e x}}{x} \, dx+\left (3 b \sqrt{d} e n\right ) \int \frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{x} \, dx\\ &=-4 b e n \sqrt{d+e x}-\frac{b d n \sqrt{d+e x}}{x}+3 e \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )-\frac{(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}-3 \sqrt{d} e \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )+\left (6 b \sqrt{d} e n\right ) \operatorname{Subst}\left (\int \frac{x \tanh ^{-1}\left (\frac{x}{\sqrt{d}}\right )}{-d+x^2} \, dx,x,\sqrt{d+e x}\right )+\frac{1}{2} (b d e n) \int \frac{1}{x \sqrt{d+e x}} \, dx-(2 b d e n) \int \frac{1}{x \sqrt{d+e x}} \, dx\\ &=-4 b e n \sqrt{d+e x}-\frac{b d n \sqrt{d+e x}}{x}+3 b \sqrt{d} e n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )^2+3 e \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )-\frac{(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}-3 \sqrt{d} e \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )+(b d n) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x}\right )-(4 b d n) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x}\right )-(6 b e 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 )\\ &=-4 b e n \sqrt{d+e x}-\frac{b d n \sqrt{d+e x}}{x}+3 b \sqrt{d} e n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )+3 b \sqrt{d} e n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )^2+3 e \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )-\frac{(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}-3 \sqrt{d} e \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )-6 b \sqrt{d} e 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 )+(6 b e 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 )\\ &=-4 b e n \sqrt{d+e x}-\frac{b d n \sqrt{d+e x}}{x}+3 b \sqrt{d} e n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )+3 b \sqrt{d} e n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )^2+3 e \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )-\frac{(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}-3 \sqrt{d} e \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )-6 b \sqrt{d} e 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 )-\left (6 b \sqrt{d} e n\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-\frac{\sqrt{d+e x}}{\sqrt{d}}}\right )\\ &=-4 b e n \sqrt{d+e x}-\frac{b d n \sqrt{d+e x}}{x}+3 b \sqrt{d} e n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )+3 b \sqrt{d} e n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )^2+3 e \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )-\frac{(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}-3 \sqrt{d} e \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )-6 b \sqrt{d} e 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 )-3 b \sqrt{d} e n \text{Li}_2\left (1-\frac{2}{1-\frac{\sqrt{d+e x}}{\sqrt{d}}}\right )\\ \end{align*}
Mathematica [A] time = 0.333317, size = 480, normalized size = 1.85 \[ \frac{-6 b \sqrt{d} e n x \text{PolyLog}\left (2,\frac{1}{2}-\frac{\sqrt{d+e x}}{2 \sqrt{d}}\right )+6 b \sqrt{d} e n x \text{PolyLog}\left (2,\frac{1}{2} \left (\frac{\sqrt{d+e x}}{\sqrt{d}}+1\right )\right )-4 a d \sqrt{d+e x}+8 a e x \sqrt{d+e x}+6 a \sqrt{d} e x \log \left (\sqrt{d}-\sqrt{d+e x}\right )-6 a \sqrt{d} e x \log \left (\sqrt{d+e x}+\sqrt{d}\right )+6 b \sqrt{d} e x \log \left (c x^n\right ) \log \left (\sqrt{d}-\sqrt{d+e x}\right )-4 b d \sqrt{d+e x} \log \left (c x^n\right )+8 b e x \sqrt{d+e x} \log \left (c x^n\right )-6 b \sqrt{d} e x \log \left (c x^n\right ) \log \left (\sqrt{d+e x}+\sqrt{d}\right )-4 b d n \sqrt{d+e x}-16 b e n x \sqrt{d+e x}-3 b \sqrt{d} e n x \log ^2\left (\sqrt{d}-\sqrt{d+e x}\right )+3 b \sqrt{d} e n x \log ^2\left (\sqrt{d+e x}+\sqrt{d}\right )-6 b \sqrt{d} e n x \log \left (\frac{1}{2} \left (\frac{\sqrt{d+e x}}{\sqrt{d}}+1\right )\right ) \log \left (\sqrt{d}-\sqrt{d+e x}\right )+6 b \sqrt{d} e n x \log \left (\sqrt{d+e x}+\sqrt{d}\right ) \log \left (\frac{1}{2}-\frac{\sqrt{d+e x}}{2 \sqrt{d}}\right )+12 b \sqrt{d} e n x \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.504, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c{x}^{n} \right ) }{{x}^{2}} \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{{\left (b e x + b d\right )} \sqrt{e x + d} \log \left (c x^{n}\right ) +{\left (a e x + a d\right )} \sqrt{e x + d}}{x^{2}}, 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{{\left (e x + d\right )}^{\frac{3}{2}}{\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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