Optimal. Leaf size=328 \[ -\frac{b \sqrt{d} n \sqrt{\frac{e x^2}{d}+1} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{2 e^{3/2} \sqrt{d+e x^2}}+\frac{\sqrt{d} \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{3/2} \sqrt{d+e x^2}}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{d+e x^2}}+\frac{b \sqrt{d} n \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{2 e^{3/2} \sqrt{d+e x^2}}+\frac{b \sqrt{d} n \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{e^{3/2} \sqrt{d+e x^2}}-\frac{b \sqrt{d} n \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{e^{3/2} \sqrt{d+e x^2}} \]
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Rubi [A] time = 0.474172, antiderivative size = 328, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 11, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.44, Rules used = {2341, 288, 215, 2350, 14, 21, 5659, 3716, 2190, 2279, 2391} \[ -\frac{b \sqrt{d} n \sqrt{\frac{e x^2}{d}+1} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{2 e^{3/2} \sqrt{d+e x^2}}+\frac{\sqrt{d} \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{3/2} \sqrt{d+e x^2}}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{d+e x^2}}+\frac{b \sqrt{d} n \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{2 e^{3/2} \sqrt{d+e x^2}}+\frac{b \sqrt{d} n \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{e^{3/2} \sqrt{d+e x^2}}-\frac{b \sqrt{d} n \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{e^{3/2} \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
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Rule 2341
Rule 288
Rule 215
Rule 2350
Rule 14
Rule 21
Rule 5659
Rule 3716
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{3/2}} \, dx &=\frac{\sqrt{1+\frac{e x^2}{d}} \int \frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (1+\frac{e x^2}{d}\right )^{3/2}} \, dx}{d \sqrt{d+e x^2}}\\ &=-\frac{x \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{d+e x^2}}+\frac{\sqrt{d} \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{3/2} \sqrt{d+e x^2}}-\frac{\left (b n \sqrt{1+\frac{e x^2}{d}}\right ) \int \frac{-\frac{d x}{e \sqrt{1+\frac{e x^2}{d}}}+\frac{d^{3/2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{e^{3/2}}}{x} \, dx}{d \sqrt{d+e x^2}}\\ &=-\frac{x \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{d+e x^2}}+\frac{\sqrt{d} \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{3/2} \sqrt{d+e x^2}}-\frac{\left (b n \sqrt{1+\frac{e x^2}{d}}\right ) \int \left (\frac{d^2 \sqrt{1+\frac{e x^2}{d}}}{e \left (-d-e x^2\right )}+\frac{d^{3/2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{e^{3/2} x}\right ) \, dx}{d \sqrt{d+e x^2}}\\ &=-\frac{x \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{d+e x^2}}+\frac{\sqrt{d} \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{3/2} \sqrt{d+e x^2}}-\frac{\left (b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}}\right ) \int \frac{\sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{x} \, dx}{e^{3/2} \sqrt{d+e x^2}}-\frac{\left (b d n \sqrt{1+\frac{e x^2}{d}}\right ) \int \frac{\sqrt{1+\frac{e x^2}{d}}}{-d-e x^2} \, dx}{e \sqrt{d+e x^2}}\\ &=-\frac{x \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{d+e x^2}}+\frac{\sqrt{d} \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{3/2} \sqrt{d+e x^2}}-\frac{\left (b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}}\right ) \operatorname{Subst}\left (\int x \coth (x) \, dx,x,\sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )\right )}{e^{3/2} \sqrt{d+e x^2}}+\frac{\left (b n \sqrt{1+\frac{e x^2}{d}}\right ) \int \frac{1}{\sqrt{1+\frac{e x^2}{d}}} \, dx}{e \sqrt{d+e x^2}}\\ &=\frac{b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{e^{3/2} \sqrt{d+e x^2}}+\frac{b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{2 e^{3/2} \sqrt{d+e x^2}}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{d+e x^2}}+\frac{\sqrt{d} \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{3/2} \sqrt{d+e x^2}}+\frac{\left (2 b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}}\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 )}{e^{3/2} \sqrt{d+e x^2}}\\ &=\frac{b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{e^{3/2} \sqrt{d+e x^2}}+\frac{b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{2 e^{3/2} \sqrt{d+e x^2}}-\frac{b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}} \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 )}{e^{3/2} \sqrt{d+e x^2}}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{d+e x^2}}+\frac{\sqrt{d} \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{3/2} \sqrt{d+e x^2}}+\frac{\left (b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}}\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 )}{e^{3/2} \sqrt{d+e x^2}}\\ &=\frac{b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{e^{3/2} \sqrt{d+e x^2}}+\frac{b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{2 e^{3/2} \sqrt{d+e x^2}}-\frac{b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}} \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 )}{e^{3/2} \sqrt{d+e x^2}}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{d+e x^2}}+\frac{\sqrt{d} \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{3/2} \sqrt{d+e x^2}}+\frac{\left (b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}}\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 e^{3/2} \sqrt{d+e x^2}}\\ &=\frac{b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{e^{3/2} \sqrt{d+e x^2}}+\frac{b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{2 e^{3/2} \sqrt{d+e x^2}}-\frac{b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}} \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 )}{e^{3/2} \sqrt{d+e x^2}}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{d+e x^2}}+\frac{\sqrt{d} \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{3/2} \sqrt{d+e x^2}}-\frac{b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}} \text{Li}_2\left (e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{2 e^{3/2} \sqrt{d+e x^2}}\\ \end{align*}
Mathematica [C] time = 0.4827, size = 217, normalized size = 0.66 \[ -\frac{b n \sqrt{\frac{e x^2}{d}+1} \left (e^{3/2} x^3 \left (d+e x^2\right ) \, _3F_2\left (\frac{3}{2},\frac{3}{2},\frac{3}{2};\frac{5}{2},\frac{5}{2};-\frac{e x^2}{d}\right )+9 d^2 \sqrt{e} x \log (x) \sqrt{\frac{e x^2}{d}+1}-9 d^{3/2} \log (x) \left (d+e x^2\right ) \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )\right )}{9 d e^{3/2} \left (d+e x^2\right )^{3/2}}+\frac{\log \left (\sqrt{e} \sqrt{d+e x^2}+e x\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{e^{3/2}}-\frac{x \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{e \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.405, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \left ( e{x}^{2}+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^{2} + d} b x^{2} \log \left (c x^{n}\right ) + \sqrt{e x^{2} + d} a x^{2}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{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{x^{2} \left (a + b \log{\left (c x^{n} \right )}\right )}{\left (d + e x^{2}\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{{\left (b \log \left (c x^{n}\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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