Optimal. Leaf size=226 \[ \frac{b e n \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x}}\right )}{d^{3/2}}+\frac{e \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{3/2}}-\frac{\sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{d x}-\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )^2}{d^{3/2}}-\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{d^{3/2}}+\frac{2 b e n \log \left (\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x}}\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{d^{3/2}}-\frac{b n \sqrt{d+e x}}{d x} \]
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Rubi [A] time = 0.270374, antiderivative size = 226, 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 = {51, 63, 208, 2350, 14, 47, 5984, 5918, 2402, 2315} \[ \frac{b e n \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x}}\right )}{d^{3/2}}+\frac{e \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{3/2}}-\frac{\sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{d x}-\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )^2}{d^{3/2}}-\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{d^{3/2}}+\frac{2 b e n \log \left (\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x}}\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{d^{3/2}}-\frac{b n \sqrt{d+e x}}{d x} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 2350
Rule 14
Rule 47
Rule 5984
Rule 5918
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c x^n\right )}{x^2 \sqrt{d+e x}} \, dx &=-\frac{\sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{d x}+\frac{e \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{3/2}}-(b n) \int \frac{-\frac{\sqrt{d+e x}}{d}+\frac{e x \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{d^{3/2}}}{x^2} \, dx\\ &=-\frac{\sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{d x}+\frac{e \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{3/2}}-(b n) \int \left (-\frac{\sqrt{d+e x}}{d x^2}+\frac{e \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{d^{3/2} x}\right ) \, dx\\ &=-\frac{\sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{d x}+\frac{e \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{\sqrt{d+e x}}{x^2} \, dx}{d}-\frac{(b e n) \int \frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{x} \, dx}{d^{3/2}}\\ &=-\frac{b n \sqrt{d+e x}}{d x}-\frac{\sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{d x}+\frac{e \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 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^{3/2}}+\frac{(b e n) \int \frac{1}{x \sqrt{d+e x}} \, dx}{2 d}\\ &=-\frac{b n \sqrt{d+e x}}{d x}-\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )^2}{d^{3/2}}-\frac{\sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{d x}+\frac{e \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) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{d}+\frac{(2 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^2}\\ &=-\frac{b n \sqrt{d+e x}}{d x}-\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{d^{3/2}}-\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )^2}{d^{3/2}}-\frac{\sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{d x}+\frac{e \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 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^{3/2}}-\frac{(2 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^2}\\ &=-\frac{b n \sqrt{d+e x}}{d x}-\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{d^{3/2}}-\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )^2}{d^{3/2}}-\frac{\sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{d x}+\frac{e \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 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^{3/2}}+\frac{(2 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^{3/2}}\\ &=-\frac{b n \sqrt{d+e x}}{d x}-\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{d^{3/2}}-\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )^2}{d^{3/2}}-\frac{\sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{d x}+\frac{e \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 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^{3/2}}+\frac{b e 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 = 0.245857, size = 392, normalized size = 1.73 \[ -\frac{-2 b e n x \text{PolyLog}\left (2,\frac{1}{2}-\frac{\sqrt{d+e x}}{2 \sqrt{d}}\right )+2 b e n x \text{PolyLog}\left (2,\frac{1}{2} \left (\frac{\sqrt{d+e x}}{\sqrt{d}}+1\right )\right )+4 a \sqrt{d} \sqrt{d+e x}+2 a e x \log \left (\sqrt{d}-\sqrt{d+e x}\right )-2 a e x \log \left (\sqrt{d+e x}+\sqrt{d}\right )+2 b e x \log \left (c x^n\right ) \log \left (\sqrt{d}-\sqrt{d+e x}\right )+4 b \sqrt{d} \sqrt{d+e x} \log \left (c x^n\right )-2 b e x \log \left (c x^n\right ) \log \left (\sqrt{d+e x}+\sqrt{d}\right )+4 b \sqrt{d} n \sqrt{d+e x}-b e n x \log ^2\left (\sqrt{d}-\sqrt{d+e x}\right )+b e n x \log ^2\left (\sqrt{d+e x}+\sqrt{d}\right )-2 b e n x \log \left (\frac{1}{2} \left (\frac{\sqrt{d+e x}}{\sqrt{d}}+1\right )\right ) \log \left (\sqrt{d}-\sqrt{d+e x}\right )+2 b e n x \log \left (\sqrt{d+e x}+\sqrt{d}\right ) \log \left (\frac{1}{2}-\frac{\sqrt{d+e x}}{2 \sqrt{d}}\right )+4 b e n x \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 d^{3/2} x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.535, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c{x}^{n} \right ) }{{x}^{2}}{\frac{1}{\sqrt{ex+d}}}}\, 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 x^{3} + d 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} \sqrt{d + e x}}\, 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}{\sqrt{e x + d} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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