Optimal. Leaf size=293 \[ -\frac{3 b e^2 n \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x}}\right )}{4 \sqrt{d}}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 \sqrt{d}}-\frac{3 e \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac{(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}+\frac{3 b e^2 n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )^2}{4 \sqrt{d}}-\frac{9 b e^2 n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{8 \sqrt{d}}-\frac{3 b e^2 n \log \left (\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x}}\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{2 \sqrt{d}}-\frac{b d n \sqrt{d+e x}}{4 x^2}-\frac{11 b e n \sqrt{d+e x}}{8 x} \]
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Rubi [A] time = 0.383664, antiderivative size = 293, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 11, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.478, Rules used = {47, 63, 208, 2350, 12, 14, 51, 5984, 5918, 2402, 2315} \[ -\frac{3 b e^2 n \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x}}\right )}{4 \sqrt{d}}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 \sqrt{d}}-\frac{3 e \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac{(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}+\frac{3 b e^2 n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )^2}{4 \sqrt{d}}-\frac{9 b e^2 n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{8 \sqrt{d}}-\frac{3 b e^2 n \log \left (\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x}}\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{2 \sqrt{d}}-\frac{b d n \sqrt{d+e x}}{4 x^2}-\frac{11 b e n \sqrt{d+e x}}{8 x} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 208
Rule 2350
Rule 12
Rule 14
Rule 51
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^3} \, dx &=-\frac{3 e \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac{(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 \sqrt{d}}-(b n) \int \frac{-\sqrt{d+e x} (2 d+5 e x)-\frac{3 e^2 x^2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{\sqrt{d}}}{4 x^3} \, dx\\ &=-\frac{3 e \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac{(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 \sqrt{d}}-\frac{1}{4} (b n) \int \frac{-\sqrt{d+e x} (2 d+5 e x)-\frac{3 e^2 x^2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{\sqrt{d}}}{x^3} \, dx\\ &=-\frac{3 e \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac{(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 \sqrt{d}}-\frac{1}{4} (b n) \int \left (-\frac{2 d \sqrt{d+e x}}{x^3}-\frac{5 e \sqrt{d+e x}}{x^2}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{\sqrt{d} x}\right ) \, dx\\ &=-\frac{3 e \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac{(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 \sqrt{d}}+\frac{1}{2} (b d n) \int \frac{\sqrt{d+e x}}{x^3} \, dx+\frac{1}{4} (5 b e n) \int \frac{\sqrt{d+e x}}{x^2} \, dx+\frac{\left (3 b e^2 n\right ) \int \frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{x} \, dx}{4 \sqrt{d}}\\ &=-\frac{b d n \sqrt{d+e x}}{4 x^2}-\frac{5 b e n \sqrt{d+e x}}{4 x}-\frac{3 e \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac{(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 \sqrt{d}}+\frac{1}{8} (b d e n) \int \frac{1}{x^2 \sqrt{d+e x}} \, dx+\frac{1}{8} \left (5 b e^2 n\right ) \int \frac{1}{x \sqrt{d+e x}} \, dx+\frac{\left (3 b e^2 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 )}{2 \sqrt{d}}\\ &=-\frac{b d n \sqrt{d+e x}}{4 x^2}-\frac{11 b e n \sqrt{d+e x}}{8 x}+\frac{3 b e^2 n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )^2}{4 \sqrt{d}}-\frac{3 e \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac{(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 \sqrt{d}}+\frac{1}{4} (5 b e n) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x}\right )-\frac{1}{16} \left (b e^2 n\right ) \int \frac{1}{x \sqrt{d+e x}} \, dx-\frac{\left (3 b e^2 n\right ) \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 )}{2 d}\\ &=-\frac{b d n \sqrt{d+e x}}{4 x^2}-\frac{11 b e n \sqrt{d+e x}}{8 x}-\frac{5 b e^2 n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 \sqrt{d}}+\frac{3 b e^2 n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )^2}{4 \sqrt{d}}-\frac{3 e \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac{(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 \sqrt{d}}-\frac{3 b e^2 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 )}{2 \sqrt{d}}-\frac{1}{8} (b e n) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x}\right )+\frac{\left (3 b e^2 n\right ) \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 )}{2 d}\\ &=-\frac{b d n \sqrt{d+e x}}{4 x^2}-\frac{11 b e n \sqrt{d+e x}}{8 x}-\frac{9 b e^2 n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{8 \sqrt{d}}+\frac{3 b e^2 n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )^2}{4 \sqrt{d}}-\frac{3 e \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac{(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 \sqrt{d}}-\frac{3 b e^2 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 )}{2 \sqrt{d}}-\frac{\left (3 b e^2 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 )}{2 \sqrt{d}}\\ &=-\frac{b d n \sqrt{d+e x}}{4 x^2}-\frac{11 b e n \sqrt{d+e x}}{8 x}-\frac{9 b e^2 n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{8 \sqrt{d}}+\frac{3 b e^2 n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )^2}{4 \sqrt{d}}-\frac{3 e \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac{(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 \sqrt{d}}-\frac{3 b e^2 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 )}{2 \sqrt{d}}-\frac{3 b e^2 n \text{Li}_2\left (1-\frac{2}{1-\frac{\sqrt{d+e x}}{\sqrt{d}}}\right )}{4 \sqrt{d}}\\ \end{align*}
Mathematica [A] time = 0.584244, size = 501, normalized size = 1.71 \[ -\frac{6 b e^2 n x^2 \text{PolyLog}\left (2,\frac{1}{2}-\frac{\sqrt{d+e x}}{2 \sqrt{d}}\right )-6 b e^2 n x^2 \text{PolyLog}\left (2,\frac{1}{2} \left (\frac{\sqrt{d+e x}}{\sqrt{d}}+1\right )\right )+8 a d^{3/2} \sqrt{d+e x}-6 a e^2 x^2 \log \left (\sqrt{d}-\sqrt{d+e x}\right )+6 a e^2 x^2 \log \left (\sqrt{d+e x}+\sqrt{d}\right )+20 a \sqrt{d} e x \sqrt{d+e x}+8 b d^{3/2} \sqrt{d+e x} \log \left (c x^n\right )-6 b e^2 x^2 \log \left (c x^n\right ) \log \left (\sqrt{d}-\sqrt{d+e x}\right )+6 b e^2 x^2 \log \left (c x^n\right ) \log \left (\sqrt{d+e x}+\sqrt{d}\right )+20 b \sqrt{d} e x \sqrt{d+e x} \log \left (c x^n\right )+4 b d^{3/2} n \sqrt{d+e x}+3 b e^2 n x^2 \log ^2\left (\sqrt{d}-\sqrt{d+e x}\right )-3 b e^2 n x^2 \log ^2\left (\sqrt{d+e x}+\sqrt{d}\right )-6 b e^2 n x^2 \log \left (\sqrt{d+e x}+\sqrt{d}\right ) \log \left (\frac{1}{2}-\frac{\sqrt{d+e x}}{2 \sqrt{d}}\right )+6 b e^2 n x^2 \log \left (\sqrt{d}-\sqrt{d+e x}\right ) \log \left (\frac{1}{2} \left (\frac{\sqrt{d+e x}}{\sqrt{d}}+1\right )\right )+18 b e^2 n x^2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )+22 b \sqrt{d} e n x \sqrt{d+e x}}{16 \sqrt{d} x^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.479, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c{x}^{n} \right ) }{{x}^{3}} \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^{3}}, 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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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