Optimal. Leaf size=345 \[ -\frac{b \sqrt{e} n \sqrt{d+e x^2} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{2 \sqrt{d} \sqrt{\frac{e x^2}{d}+1}}-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x}+\frac{\sqrt{e} \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{d} \sqrt{\frac{e x^2}{d}+1}}-\frac{b n \sqrt{d+e x^2}}{x}+\frac{b \sqrt{e} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{2 \sqrt{d} \sqrt{\frac{e x^2}{d}+1}}+\frac{b \sqrt{e} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{\frac{e x^2}{d}+1}}-\frac{b \sqrt{e} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{\sqrt{d} \sqrt{\frac{e x^2}{d}+1}} \]
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Rubi [A] time = 0.394141, antiderivative size = 345, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {2341, 277, 215, 2350, 14, 5659, 3716, 2190, 2279, 2391} \[ -\frac{b \sqrt{e} n \sqrt{d+e x^2} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{2 \sqrt{d} \sqrt{\frac{e x^2}{d}+1}}-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x}+\frac{\sqrt{e} \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{d} \sqrt{\frac{e x^2}{d}+1}}-\frac{b n \sqrt{d+e x^2}}{x}+\frac{b \sqrt{e} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{2 \sqrt{d} \sqrt{\frac{e x^2}{d}+1}}+\frac{b \sqrt{e} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{\frac{e x^2}{d}+1}}-\frac{b \sqrt{e} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{\sqrt{d} \sqrt{\frac{e x^2}{d}+1}} \]
Antiderivative was successfully verified.
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Rule 2341
Rule 277
Rule 215
Rule 2350
Rule 14
Rule 5659
Rule 3716
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx &=\frac{\sqrt{d+e x^2} \int \frac{\sqrt{1+\frac{e x^2}{d}} \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx}{\sqrt{1+\frac{e x^2}{d}}}\\ &=-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x}+\frac{\sqrt{e} \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{d} \sqrt{1+\frac{e x^2}{d}}}-\frac{\left (b n \sqrt{d+e x^2}\right ) \int \frac{-\sqrt{1+\frac{e x^2}{d}}+\frac{\sqrt{e} x \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d}}}{x^2} \, dx}{\sqrt{1+\frac{e x^2}{d}}}\\ &=-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x}+\frac{\sqrt{e} \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{d} \sqrt{1+\frac{e x^2}{d}}}-\frac{\left (b n \sqrt{d+e x^2}\right ) \int \left (-\frac{\sqrt{1+\frac{e x^2}{d}}}{x^2}+\frac{\sqrt{e} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} x}\right ) \, dx}{\sqrt{1+\frac{e x^2}{d}}}\\ &=-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x}+\frac{\sqrt{e} \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{d} \sqrt{1+\frac{e x^2}{d}}}+\frac{\left (b n \sqrt{d+e x^2}\right ) \int \frac{\sqrt{1+\frac{e x^2}{d}}}{x^2} \, dx}{\sqrt{1+\frac{e x^2}{d}}}-\frac{\left (b \sqrt{e} n \sqrt{d+e x^2}\right ) \int \frac{\sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{x} \, dx}{\sqrt{d} \sqrt{1+\frac{e x^2}{d}}}\\ &=-\frac{b n \sqrt{d+e x^2}}{x}-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x}+\frac{\sqrt{e} \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{d} \sqrt{1+\frac{e x^2}{d}}}-\frac{\left (b \sqrt{e} n \sqrt{d+e x^2}\right ) \operatorname{Subst}\left (\int x \coth (x) \, dx,x,\sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )\right )}{\sqrt{d} \sqrt{1+\frac{e x^2}{d}}}+\frac{\left (b e n \sqrt{d+e x^2}\right ) \int \frac{1}{\sqrt{1+\frac{e x^2}{d}}} \, dx}{d \sqrt{1+\frac{e x^2}{d}}}\\ &=-\frac{b n \sqrt{d+e x^2}}{x}+\frac{b \sqrt{e} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{1+\frac{e x^2}{d}}}+\frac{b \sqrt{e} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{2 \sqrt{d} \sqrt{1+\frac{e x^2}{d}}}-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x}+\frac{\sqrt{e} \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{d} \sqrt{1+\frac{e x^2}{d}}}+\frac{\left (2 b \sqrt{e} n \sqrt{d+e x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} x}{1-e^{2 x}} \, dx,x,\sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )\right )}{\sqrt{d} \sqrt{1+\frac{e x^2}{d}}}\\ &=-\frac{b n \sqrt{d+e x^2}}{x}+\frac{b \sqrt{e} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{1+\frac{e x^2}{d}}}+\frac{b \sqrt{e} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{2 \sqrt{d} \sqrt{1+\frac{e x^2}{d}}}-\frac{b \sqrt{e} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{\sqrt{d} \sqrt{1+\frac{e x^2}{d}}}-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x}+\frac{\sqrt{e} \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{d} \sqrt{1+\frac{e x^2}{d}}}+\frac{\left (b \sqrt{e} n \sqrt{d+e x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )\right )}{\sqrt{d} \sqrt{1+\frac{e x^2}{d}}}\\ &=-\frac{b n \sqrt{d+e x^2}}{x}+\frac{b \sqrt{e} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{1+\frac{e x^2}{d}}}+\frac{b \sqrt{e} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{2 \sqrt{d} \sqrt{1+\frac{e x^2}{d}}}-\frac{b \sqrt{e} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{\sqrt{d} \sqrt{1+\frac{e x^2}{d}}}-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x}+\frac{\sqrt{e} \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{d} \sqrt{1+\frac{e x^2}{d}}}+\frac{\left (b \sqrt{e} n \sqrt{d+e x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{2 \sqrt{d} \sqrt{1+\frac{e x^2}{d}}}\\ &=-\frac{b n \sqrt{d+e x^2}}{x}+\frac{b \sqrt{e} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{1+\frac{e x^2}{d}}}+\frac{b \sqrt{e} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{2 \sqrt{d} \sqrt{1+\frac{e x^2}{d}}}-\frac{b \sqrt{e} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{\sqrt{d} \sqrt{1+\frac{e x^2}{d}}}-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x}+\frac{\sqrt{e} \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{d} \sqrt{1+\frac{e x^2}{d}}}-\frac{b \sqrt{e} n \sqrt{d+e x^2} \text{Li}_2\left (e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{2 \sqrt{d} \sqrt{1+\frac{e x^2}{d}}}\\ \end{align*}
Mathematica [C] time = 0.559439, size = 183, normalized size = 0.53 \[ \frac{b n \sqrt{d+e x^2} \left (-\, _3F_2\left (-\frac{1}{2},-\frac{1}{2},-\frac{1}{2};\frac{1}{2},\frac{1}{2};-\frac{e x^2}{d}\right )-\log (x) \sqrt{\frac{e x^2}{d}+1}+\frac{\sqrt{e} x \log (x) \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d}}\right )}{x \sqrt{\frac{e x^2}{d}+1}}-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{x}+\sqrt{e} \log \left (\sqrt{e} \sqrt{d+e x^2}+e x\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.443, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c{x}^{n} \right ) }{{x}^{2}}\sqrt{e{x}^{2}+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^{2} + d} b \log \left (c x^{n}\right ) + \sqrt{e x^{2} + d} a}{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{\left (a + b \log{\left (c x^{n} \right )}\right ) \sqrt{d + e x^{2}}}{x^{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{\sqrt{e x^{2} + d}{\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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