Optimal. Leaf size=400 \[ -\frac{b e^{3/2} n \sqrt{d+e x^2} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{2 \sqrt{d} \sqrt{\frac{e x^2}{d}+1}}+\frac{e^{3/2} \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{d} \sqrt{\frac{e x^2}{d}+1}}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac{e \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x}+\frac{b e^{3/2} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{2 \sqrt{d} \sqrt{\frac{e x^2}{d}+1}}+\frac{4 b e^{3/2} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{3 \sqrt{d} \sqrt{\frac{e x^2}{d}+1}}-\frac{b e^{3/2} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{\sqrt{d} \sqrt{\frac{e x^2}{d}+1}}-\frac{b n \left (d+e x^2\right )^{3/2}}{9 x^3}-\frac{4 b e n \sqrt{d+e x^2}}{3 x} \]
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Rubi [A] time = 0.444425, antiderivative size = 400, 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, 451, 5659, 3716, 2190, 2279, 2391} \[ -\frac{b e^{3/2} n \sqrt{d+e x^2} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{2 \sqrt{d} \sqrt{\frac{e x^2}{d}+1}}+\frac{e^{3/2} \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{d} \sqrt{\frac{e x^2}{d}+1}}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac{e \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x}+\frac{b e^{3/2} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{2 \sqrt{d} \sqrt{\frac{e x^2}{d}+1}}+\frac{4 b e^{3/2} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{3 \sqrt{d} \sqrt{\frac{e x^2}{d}+1}}-\frac{b e^{3/2} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{\sqrt{d} \sqrt{\frac{e x^2}{d}+1}}-\frac{b n \left (d+e x^2\right )^{3/2}}{9 x^3}-\frac{4 b e n \sqrt{d+e x^2}}{3 x} \]
Antiderivative was successfully verified.
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Rule 2341
Rule 277
Rule 215
Rule 2350
Rule 451
Rule 5659
Rule 3716
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx &=\frac{\left (d \sqrt{d+e x^2}\right ) \int \frac{\left (1+\frac{e x^2}{d}\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx}{\sqrt{1+\frac{e x^2}{d}}}\\ &=-\frac{e \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 x^3}+\frac{e^{3/2} \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 d n \sqrt{d+e x^2}\right ) \int \left (-\frac{\left (d+4 e x^2\right ) \sqrt{1+\frac{e x^2}{d}}}{3 d x^4}+\frac{e^{3/2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d^{3/2} x}\right ) \, dx}{\sqrt{1+\frac{e x^2}{d}}}\\ &=-\frac{e \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 x^3}+\frac{e^{3/2} \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{\left (d+4 e x^2\right ) \sqrt{1+\frac{e x^2}{d}}}{x^4} \, dx}{3 \sqrt{1+\frac{e x^2}{d}}}-\frac{\left (b e^{3/2} 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 \left (d+e x^2\right )^{3/2}}{9 x^3}-\frac{e \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 x^3}+\frac{e^{3/2} \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 (4 b e n \sqrt{d+e x^2}\right ) \int \frac{\sqrt{1+\frac{e x^2}{d}}}{x^2} \, dx}{3 \sqrt{1+\frac{e x^2}{d}}}-\frac{\left (b e^{3/2} 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{4 b e n \sqrt{d+e x^2}}{3 x}-\frac{b n \left (d+e x^2\right )^{3/2}}{9 x^3}+\frac{b e^{3/2} 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{e \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 x^3}+\frac{e^{3/2} \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 e^{3/2} 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{\left (4 b e^2 n \sqrt{d+e x^2}\right ) \int \frac{1}{\sqrt{1+\frac{e x^2}{d}}} \, dx}{3 d \sqrt{1+\frac{e x^2}{d}}}\\ &=-\frac{4 b e n \sqrt{d+e x^2}}{3 x}-\frac{b n \left (d+e x^2\right )^{3/2}}{9 x^3}+\frac{4 b e^{3/2} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{3 \sqrt{d} \sqrt{1+\frac{e x^2}{d}}}+\frac{b e^{3/2} 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 e^{3/2} 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{e \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 x^3}+\frac{e^{3/2} \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 e^{3/2} 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{4 b e n \sqrt{d+e x^2}}{3 x}-\frac{b n \left (d+e x^2\right )^{3/2}}{9 x^3}+\frac{4 b e^{3/2} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{3 \sqrt{d} \sqrt{1+\frac{e x^2}{d}}}+\frac{b e^{3/2} 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 e^{3/2} 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{e \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 x^3}+\frac{e^{3/2} \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 e^{3/2} 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{4 b e n \sqrt{d+e x^2}}{3 x}-\frac{b n \left (d+e x^2\right )^{3/2}}{9 x^3}+\frac{4 b e^{3/2} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{3 \sqrt{d} \sqrt{1+\frac{e x^2}{d}}}+\frac{b e^{3/2} 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 e^{3/2} 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{e \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 x^3}+\frac{e^{3/2} \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 e^{3/2} 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 = 0.756763, size = 269, normalized size = 0.67 \[ \frac{b e n \sqrt{d+e x^2} \left (-\, _3F_2\left (-\frac{1}{2},-\frac{1}{2},-\frac{1}{2};\frac{1}{2},\frac{1}{2};-\frac{e x^2}{d}\right )-\log (x) \sqrt{\frac{e x^2}{d}+1}+\frac{\sqrt{e} x \log (x) \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d}}\right )}{x \sqrt{\frac{e x^2}{d}+1}}+e^{3/2} \log \left (\sqrt{e} \sqrt{d+e x^2}+e x\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )-\frac{\sqrt{d+e x^2} \left (d+4 e x^2\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{3 x^3}+\frac{b d n \sqrt{d+e x^2} \left (-\, _2F_1\left (-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};-\frac{e x^2}{d}\right )-3 \log (x) \left (\frac{e x^2}{d}+1\right )^{3/2}\right )}{9 x^3 \sqrt{\frac{e x^2}{d}+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.423, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c{x}^{n} \right ) }{{x}^{4}} \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{{\left (b e x^{2} + b d\right )} \sqrt{e x^{2} + d} \log \left (c x^{n}\right ) +{\left (a e x^{2} + a d\right )} \sqrt{e x^{2} + d}}{x^{4}}, 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 ) \left (d + e x^{2}\right )^{\frac{3}{2}}}{x^{4}}\, 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 (e x^{2} + d\right )}^{\frac{3}{2}}{\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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