Optimal. Leaf size=253 \[ \frac{3 b e n \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x}}\right )}{d^{5/2}}-\frac{3 e \left (a+b \log \left (c x^n\right )\right )}{d^2 \sqrt{d+e x}}+\frac{3 e \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{5/2}}-\frac{a+b \log \left (c x^n\right )}{d x \sqrt{d+e x}}-\frac{b n \sqrt{d+e x}}{d^2 x}-\frac{3 b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )^2}{d^{5/2}}-\frac{5 b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{d^{5/2}}+\frac{6 b e n \log \left (\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x}}\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{d^{5/2}} \]
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Rubi [A] time = 0.517039, antiderivative size = 255, normalized size of antiderivative = 1.01, number of steps used = 15, number of rules used = 12, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.522, Rules used = {51, 63, 208, 2350, 12, 14, 47, 50, 5984, 5918, 2402, 2315} \[ \frac{3 b e n \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x}}\right )}{d^{5/2}}-\frac{3 \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{d^2 x}+\frac{3 e \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{5/2}}+\frac{2 \left (a+b \log \left (c x^n\right )\right )}{d x \sqrt{d+e x}}-\frac{b n \sqrt{d+e x}}{d^2 x}-\frac{3 b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )^2}{d^{5/2}}-\frac{5 b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{d^{5/2}}+\frac{6 b e n \log \left (\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x}}\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{d^{5/2}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 2350
Rule 12
Rule 14
Rule 47
Rule 50
Rule 5984
Rule 5918
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c x^n\right )}{x^2 (d+e x)^{3/2}} \, dx &=\frac{2 \left (a+b \log \left (c x^n\right )\right )}{d x \sqrt{d+e x}}-\frac{3 \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{d^2 x}+\frac{3 e \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{5/2}}-(b n) \int \frac{-\frac{\sqrt{d} (d+3 e x)}{\sqrt{d+e x}}+3 e x \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{d^{5/2} x^2} \, dx\\ &=\frac{2 \left (a+b \log \left (c x^n\right )\right )}{d x \sqrt{d+e x}}-\frac{3 \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{d^2 x}+\frac{3 e \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{5/2}}-\frac{(b n) \int \frac{-\frac{\sqrt{d} (d+3 e x)}{\sqrt{d+e x}}+3 e x \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{x^2} \, dx}{d^{5/2}}\\ &=\frac{2 \left (a+b \log \left (c x^n\right )\right )}{d x \sqrt{d+e x}}-\frac{3 \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{d^2 x}+\frac{3 e \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{5/2}}-\frac{(b n) \int \left (\frac{2 e^2}{\sqrt{d} \sqrt{d+e x}}-\frac{\sqrt{d} \sqrt{d+e x}}{x^2}-\frac{2 e \sqrt{d+e x}}{\sqrt{d} x}+\frac{3 e \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{x}\right ) \, dx}{d^{5/2}}\\ &=-\frac{4 b e n \sqrt{d+e x}}{d^3}+\frac{2 \left (a+b \log \left (c x^n\right )\right )}{d x \sqrt{d+e x}}-\frac{3 \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{d^2 x}+\frac{3 e \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{5/2}}+\frac{(b n) \int \frac{\sqrt{d+e x}}{x^2} \, dx}{d^2}+\frac{(2 b e n) \int \frac{\sqrt{d+e x}}{x} \, dx}{d^3}-\frac{(3 b e n) \int \frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{x} \, dx}{d^{5/2}}\\ &=-\frac{b n \sqrt{d+e x}}{d^2 x}+\frac{2 \left (a+b \log \left (c x^n\right )\right )}{d x \sqrt{d+e x}}-\frac{3 \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{d^2 x}+\frac{3 e \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{5/2}}-\frac{(6 b e 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^{5/2}}+\frac{(b e n) \int \frac{1}{x \sqrt{d+e x}} \, dx}{2 d^2}+\frac{(2 b e n) \int \frac{1}{x \sqrt{d+e x}} \, dx}{d^2}\\ &=-\frac{b n \sqrt{d+e x}}{d^2 x}-\frac{3 b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )^2}{d^{5/2}}+\frac{2 \left (a+b \log \left (c x^n\right )\right )}{d x \sqrt{d+e x}}-\frac{3 \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{d^2 x}+\frac{3 e \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{5/2}}+\frac{(b n) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{d^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^2}+\frac{(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 )}{d^3}\\ &=-\frac{b n \sqrt{d+e x}}{d^2 x}-\frac{5 b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{d^{5/2}}-\frac{3 b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )^2}{d^{5/2}}+\frac{2 \left (a+b \log \left (c x^n\right )\right )}{d x \sqrt{d+e x}}-\frac{3 \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{d^2 x}+\frac{3 e \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{5/2}}+\frac{6 b 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 )}{d^{5/2}}-\frac{(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 )}{d^3}\\ &=-\frac{b n \sqrt{d+e x}}{d^2 x}-\frac{5 b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{d^{5/2}}-\frac{3 b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )^2}{d^{5/2}}+\frac{2 \left (a+b \log \left (c x^n\right )\right )}{d x \sqrt{d+e x}}-\frac{3 \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{d^2 x}+\frac{3 e \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{5/2}}+\frac{6 b 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 )}{d^{5/2}}+\frac{(6 b e 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^{5/2}}\\ &=-\frac{b n \sqrt{d+e x}}{d^2 x}-\frac{5 b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{d^{5/2}}-\frac{3 b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )^2}{d^{5/2}}+\frac{2 \left (a+b \log \left (c x^n\right )\right )}{d x \sqrt{d+e x}}-\frac{3 \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{d^2 x}+\frac{3 e \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{5/2}}+\frac{6 b 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 )}{d^{5/2}}+\frac{3 b e n \text{Li}_2\left (1-\frac{2}{1-\frac{\sqrt{d+e x}}{\sqrt{d}}}\right )}{d^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.47813, size = 506, normalized size = 2. \[ 2 e \left (\frac{3 b n \left (2 \text{PolyLog}\left (2,\frac{\sqrt{d}-\sqrt{d+e x}}{2 \sqrt{d}}\right )+\log ^2\left (\sqrt{d}-\sqrt{d+e x}\right )+2 \log \left (\frac{\sqrt{d+e x}+\sqrt{d}}{2 \sqrt{d}}\right ) \log \left (\sqrt{d}-\sqrt{d+e x}\right )\right )}{8 d^{5/2}}-\frac{3 b n \left (2 \text{PolyLog}\left (2,\frac{\sqrt{d+e x}+\sqrt{d}}{2 \sqrt{d}}\right )+\log ^2\left (\sqrt{d+e x}+\sqrt{d}\right )+2 \log \left (\frac{\sqrt{d}-\sqrt{d+e x}}{2 \sqrt{d}}\right ) \log \left (\sqrt{d+e x}+\sqrt{d}\right )\right )}{8 d^{5/2}}-\frac{a+b \log \left (c x^n\right )}{d^2 \sqrt{d+e x}}+\frac{a+b \log \left (c x^n\right )}{4 d^2 \left (\sqrt{d}-\sqrt{d+e x}\right )}-\frac{a+b \log \left (c x^n\right )}{4 d^2 \left (\sqrt{d+e x}+\sqrt{d}\right )}-\frac{3 \log \left (\sqrt{d}-\sqrt{d+e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d^{5/2}}+\frac{3 \log \left (\sqrt{d+e x}+\sqrt{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d^{5/2}}-\frac{2 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{d^{5/2}}+\frac{b n \left (\frac{1}{\sqrt{d}-\sqrt{d+e x}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{\sqrt{d}}\right )}{4 d^2}-\frac{b n \left (\frac{1}{\sqrt{d+e x}+\sqrt{d}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{\sqrt{d}}\right )}{4 d^2}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.573, 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{\sqrt{e x + d} b \log \left (c x^{n}\right ) + \sqrt{e x + d} a}{e^{2} x^{4} + 2 \, d e x^{3} + d^{2} x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \log{\left (c x^{n} \right )}}{x^{2} \left (d + e x\right )^{\frac{3}{2}}}\, dx \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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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