Optimal. Leaf size=255 \[ -2 b d^{3/2} n \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x}}\right )-2 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{2}{3} (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )+2 d \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )+2 b d^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )^2+\frac{16}{3} b d^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )-4 b d^{3/2} n \log \left (\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x}}\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )-\frac{4}{9} b n (d+e x)^{3/2}-\frac{16}{3} b d n \sqrt{d+e x} \]
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Rubi [A] time = 0.456245, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 11, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.478, Rules used = {2346, 63, 208, 2348, 12, 5984, 5918, 2402, 2315, 2319, 50} \[ -2 b d^{3/2} n \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x}}\right )-2 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{2}{3} (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )+2 d \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )+2 b d^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )^2+\frac{16}{3} b d^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )-4 b d^{3/2} n \log \left (\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x}}\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )-\frac{4}{9} b n (d+e x)^{3/2}-\frac{16}{3} b d n \sqrt{d+e x} \]
Antiderivative was successfully verified.
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Rule 2346
Rule 63
Rule 208
Rule 2348
Rule 12
Rule 5984
Rule 5918
Rule 2402
Rule 2315
Rule 2319
Rule 50
Rubi steps
\begin{align*} \int \frac{(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=d \int \frac{\sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx+e \int \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right ) \, dx\\ &=\frac{2}{3} (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )+d^2 \int \frac{a+b \log \left (c x^n\right )}{x \sqrt{d+e x}} \, dx+(d e) \int \frac{a+b \log \left (c x^n\right )}{\sqrt{d+e x}} \, dx-\frac{1}{3} (2 b n) \int \frac{(d+e x)^{3/2}}{x} \, dx\\ &=-\frac{4}{9} b n (d+e x)^{3/2}+2 d \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )+\frac{2}{3} (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )-2 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{3} (2 b d n) \int \frac{\sqrt{d+e x}}{x} \, dx-(2 b d n) \int \frac{\sqrt{d+e x}}{x} \, dx-\left (b d^2 n\right ) \int -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{\sqrt{d} x} \, dx\\ &=-\frac{16}{3} b d n \sqrt{d+e x}-\frac{4}{9} b n (d+e x)^{3/2}+2 d \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )+\frac{2}{3} (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )-2 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )+\left (2 b d^{3/2} n\right ) \int \frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{x} \, dx-\frac{1}{3} \left (2 b d^2 n\right ) \int \frac{1}{x \sqrt{d+e x}} \, dx-\left (2 b d^2 n\right ) \int \frac{1}{x \sqrt{d+e x}} \, dx\\ &=-\frac{16}{3} b d n \sqrt{d+e x}-\frac{4}{9} b n (d+e x)^{3/2}+2 d \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )+\frac{2}{3} (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )-2 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )+\left (4 b d^{3/2} 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 )-\frac{\left (4 b d^2 n\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{3 e}-\frac{\left (4 b d^2 n\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{e}\\ &=-\frac{16}{3} b d n \sqrt{d+e x}-\frac{4}{9} b n (d+e x)^{3/2}+\frac{16}{3} b d^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )+2 b d^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )^2+2 d \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )+\frac{2}{3} (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )-2 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )-(4 b d 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 )\\ &=-\frac{16}{3} b d n \sqrt{d+e x}-\frac{4}{9} b n (d+e x)^{3/2}+\frac{16}{3} b d^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )+2 b d^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )^2+2 d \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )+\frac{2}{3} (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )-2 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )-4 b d^{3/2} 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 )+(4 b d 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 )\\ &=-\frac{16}{3} b d n \sqrt{d+e x}-\frac{4}{9} b n (d+e x)^{3/2}+\frac{16}{3} b d^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )+2 b d^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )^2+2 d \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )+\frac{2}{3} (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )-2 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )-4 b d^{3/2} 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 )-\left (4 b d^{3/2} 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 )\\ &=-\frac{16}{3} b d n \sqrt{d+e x}-\frac{4}{9} b n (d+e x)^{3/2}+\frac{16}{3} b d^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )+2 b d^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )^2+2 d \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )+\frac{2}{3} (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )-2 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )-4 b d^{3/2} 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 b d^{3/2} n \text{Li}_2\left (1-\frac{2}{1-\frac{\sqrt{d+e x}}{\sqrt{d}}}\right )\\ \end{align*}
Mathematica [A] time = 0.253343, size = 375, normalized size = 1.47 \[ -\frac{1}{2} b d^{3/2} n \left (2 \text{PolyLog}\left (2,\frac{1}{2}-\frac{\sqrt{d+e x}}{2 \sqrt{d}}\right )+\log \left (\sqrt{d}-\sqrt{d+e x}\right ) \left (\log \left (\sqrt{d}-\sqrt{d+e x}\right )+2 \log \left (\frac{1}{2} \left (\frac{\sqrt{d+e x}}{\sqrt{d}}+1\right )\right )\right )\right )+\frac{1}{2} b d^{3/2} n \left (2 \text{PolyLog}\left (2,\frac{1}{2} \left (\frac{\sqrt{d+e x}}{\sqrt{d}}+1\right )\right )+\log \left (\sqrt{d+e x}+\sqrt{d}\right ) \left (\log \left (\sqrt{d+e x}+\sqrt{d}\right )+2 \log \left (\frac{1}{2}-\frac{\sqrt{d+e x}}{2 \sqrt{d}}\right )\right )\right )+d^{3/2} \log \left (\sqrt{d}-\sqrt{d+e x}\right ) \left (a+b \log \left (c x^n\right )\right )-d^{3/2} \log \left (\sqrt{d+e x}+\sqrt{d}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{2}{3} (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )+2 a d \sqrt{d+e x}+2 b d \sqrt{d+e x} \log \left (c x^n\right )-\frac{4}{9} b n (d+e x)^{3/2}+\frac{16}{3} b d n \left (\sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )-\sqrt{d+e x}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.485, 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{{\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}, 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}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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