Optimal. Leaf size=378 \[ -\frac{3 b d^{5/2} n \sqrt{\frac{e x^2}{d}+1} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{16 \sqrt{e} \sqrt{d+e x^2}}+\frac{3 d^{5/2} \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 \sqrt{e} \sqrt{d+e x^2}}+\frac{3}{8} d x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac{1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac{3 b d^{5/2} n \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{16 \sqrt{e} \sqrt{d+e x^2}}-\frac{9 b d^2 n \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{32 \sqrt{e}}-\frac{3 b d^{5/2} n \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{8 \sqrt{e} \sqrt{d+e x^2}}-\frac{9}{32} b d n x \sqrt{d+e x^2}-\frac{1}{16} b n x \left (d+e x^2\right )^{3/2} \]
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Rubi [A] time = 0.269006, antiderivative size = 378, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 11, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2321, 195, 217, 206, 2327, 2325, 5659, 3716, 2190, 2279, 2391} \[ -\frac{3 b d^{5/2} n \sqrt{\frac{e x^2}{d}+1} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{16 \sqrt{e} \sqrt{d+e x^2}}+\frac{3 d^{5/2} \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 \sqrt{e} \sqrt{d+e x^2}}+\frac{3}{8} d x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac{1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac{3 b d^{5/2} n \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{16 \sqrt{e} \sqrt{d+e x^2}}-\frac{9 b d^2 n \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{32 \sqrt{e}}-\frac{3 b d^{5/2} n \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{8 \sqrt{e} \sqrt{d+e x^2}}-\frac{9}{32} b d n x \sqrt{d+e x^2}-\frac{1}{16} b n x \left (d+e x^2\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 2321
Rule 195
Rule 217
Rule 206
Rule 2327
Rule 2325
Rule 5659
Rule 3716
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac{1}{4} (3 d) \int \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right ) \, dx-\frac{1}{4} (b n) \int \left (d+e x^2\right )^{3/2} \, dx\\ &=-\frac{1}{16} b n x \left (d+e x^2\right )^{3/2}+\frac{3}{8} d x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac{1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac{1}{8} \left (3 d^2\right ) \int \frac{a+b \log \left (c x^n\right )}{\sqrt{d+e x^2}} \, dx-\frac{1}{16} (3 b d n) \int \sqrt{d+e x^2} \, dx-\frac{1}{8} (3 b d n) \int \sqrt{d+e x^2} \, dx\\ &=-\frac{9}{32} b d n x \sqrt{d+e x^2}-\frac{1}{16} b n x \left (d+e x^2\right )^{3/2}+\frac{3}{8} d x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac{1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )-\frac{1}{32} \left (3 b d^2 n\right ) \int \frac{1}{\sqrt{d+e x^2}} \, dx-\frac{1}{16} \left (3 b d^2 n\right ) \int \frac{1}{\sqrt{d+e x^2}} \, dx+\frac{\left (3 d^2 \sqrt{1+\frac{e x^2}{d}}\right ) \int \frac{a+b \log \left (c x^n\right )}{\sqrt{1+\frac{e x^2}{d}}} \, dx}{8 \sqrt{d+e x^2}}\\ &=-\frac{9}{32} b d n x \sqrt{d+e x^2}-\frac{1}{16} b n x \left (d+e x^2\right )^{3/2}+\frac{3}{8} d x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac{1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac{3 d^{5/2} \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 )}{8 \sqrt{e} \sqrt{d+e x^2}}-\frac{1}{32} \left (3 b d^2 n\right ) \operatorname{Subst}\left (\int \frac{1}{1-e x^2} \, dx,x,\frac{x}{\sqrt{d+e x^2}}\right )-\frac{1}{16} \left (3 b d^2 n\right ) \operatorname{Subst}\left (\int \frac{1}{1-e x^2} \, dx,x,\frac{x}{\sqrt{d+e x^2}}\right )-\frac{\left (3 b d^{5/2} n \sqrt{1+\frac{e x^2}{d}}\right ) \int \frac{\sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{x} \, dx}{8 \sqrt{e} \sqrt{d+e x^2}}\\ &=-\frac{9}{32} b d n x \sqrt{d+e x^2}-\frac{1}{16} b n x \left (d+e x^2\right )^{3/2}-\frac{9 b d^2 n \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{32 \sqrt{e}}+\frac{3}{8} d x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac{1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac{3 d^{5/2} \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 )}{8 \sqrt{e} \sqrt{d+e x^2}}-\frac{\left (3 b d^{5/2} 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 )}{8 \sqrt{e} \sqrt{d+e x^2}}\\ &=-\frac{9}{32} b d n x \sqrt{d+e x^2}-\frac{1}{16} b n x \left (d+e x^2\right )^{3/2}+\frac{3 b d^{5/2} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{16 \sqrt{e} \sqrt{d+e x^2}}-\frac{9 b d^2 n \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{32 \sqrt{e}}+\frac{3}{8} d x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac{1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac{3 d^{5/2} \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 )}{8 \sqrt{e} \sqrt{d+e x^2}}+\frac{\left (3 b d^{5/2} 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 )}{4 \sqrt{e} \sqrt{d+e x^2}}\\ &=-\frac{9}{32} b d n x \sqrt{d+e x^2}-\frac{1}{16} b n x \left (d+e x^2\right )^{3/2}+\frac{3 b d^{5/2} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{16 \sqrt{e} \sqrt{d+e x^2}}-\frac{9 b d^2 n \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{32 \sqrt{e}}-\frac{3 b d^{5/2} 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 )}{8 \sqrt{e} \sqrt{d+e x^2}}+\frac{3}{8} d x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac{1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac{3 d^{5/2} \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 )}{8 \sqrt{e} \sqrt{d+e x^2}}+\frac{\left (3 b d^{5/2} 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 )}{8 \sqrt{e} \sqrt{d+e x^2}}\\ &=-\frac{9}{32} b d n x \sqrt{d+e x^2}-\frac{1}{16} b n x \left (d+e x^2\right )^{3/2}+\frac{3 b d^{5/2} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{16 \sqrt{e} \sqrt{d+e x^2}}-\frac{9 b d^2 n \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{32 \sqrt{e}}-\frac{3 b d^{5/2} 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 )}{8 \sqrt{e} \sqrt{d+e x^2}}+\frac{3}{8} d x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac{1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac{3 d^{5/2} \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 )}{8 \sqrt{e} \sqrt{d+e x^2}}+\frac{\left (3 b d^{5/2} 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 )}{16 \sqrt{e} \sqrt{d+e x^2}}\\ &=-\frac{9}{32} b d n x \sqrt{d+e x^2}-\frac{1}{16} b n x \left (d+e x^2\right )^{3/2}+\frac{3 b d^{5/2} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{16 \sqrt{e} \sqrt{d+e x^2}}-\frac{9 b d^2 n \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{32 \sqrt{e}}-\frac{3 b d^{5/2} 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 )}{8 \sqrt{e} \sqrt{d+e x^2}}+\frac{3}{8} d x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac{1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac{3 d^{5/2} \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 )}{8 \sqrt{e} \sqrt{d+e x^2}}-\frac{3 b d^{5/2} 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 )}{16 \sqrt{e} \sqrt{d+e x^2}}\\ \end{align*}
Mathematica [C] time = 0.899494, size = 314, normalized size = 0.83 \[ \frac{9 \left (-4 b d \sqrt{e} n x \sqrt{d+e x^2} \, _3F_2\left (\frac{1}{2},\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2};-\frac{e x^2}{d}\right )+\sqrt{\frac{e x^2}{d}+1} \left (3 d^2 \log \left (\sqrt{e} \sqrt{d+e x^2}+e x\right ) (a-b n \log (x))+\sqrt{e} x \sqrt{d+e x^2} \left (5 a d+2 a e x^2-2 b d n\right )+b \log \left (c x^n\right ) \left (3 d^2 \log \left (\sqrt{e} \sqrt{d+e x^2}+e x\right )+\sqrt{e} x \sqrt{d+e x^2} \left (5 d+2 e x^2\right )\right )\right )+b d^{3/2} n (3 \log (x)-2) \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )\right )-8 b e^{3/2} n x^3 \sqrt{d+e x^2} \, _3F_2\left (-\frac{1}{2},\frac{3}{2},\frac{3}{2};\frac{5}{2},\frac{5}{2};-\frac{e x^2}{d}\right )}{72 \sqrt{e} \sqrt{\frac{e x^2}{d}+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.478, size = 0, normalized size = 0. \begin{align*} \int \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \, 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 ({\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\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 \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{\left (e x^{2} + d\right )}^{\frac{3}{2}}{\left (b \log \left (c x^{n}\right ) + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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