Optimal. Leaf size=409 \[ \frac{b d^{3/2} n \sqrt{d+e x^2} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{16 e^{3/2} \sqrt{\frac{e x^2}{d}+1}}-\frac{d^{3/2} \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 e^{3/2} \sqrt{\frac{e x^2}{d}+1}}+\frac{1}{4} x^3 \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac{d x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{8 e}-\frac{b d^{3/2} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{16 e^{3/2} \sqrt{\frac{e x^2}{d}+1}}-\frac{b d^{3/2} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{32 e^{3/2} \sqrt{\frac{e x^2}{d}+1}}+\frac{b d^{3/2} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{8 e^{3/2} \sqrt{\frac{e x^2}{d}+1}}-\frac{1}{16} b n x^3 \sqrt{d+e x^2}-\frac{3 b d n x \sqrt{d+e x^2}}{32 e} \]
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Rubi [A] time = 0.41311, antiderivative size = 410, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 12, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.48, Rules used = {2341, 279, 321, 215, 2350, 388, 195, 5659, 3716, 2190, 2279, 2391} \[ \frac{b d^{3/2} n \sqrt{d+e x^2} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{16 e^{3/2} \sqrt{\frac{e x^2}{d}+1}}-\frac{d^{3/2} \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 e^{3/2} \sqrt{\frac{e x^2}{d}+1}}+\frac{1}{4} x^3 \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac{d x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{8 e}-\frac{b d^{3/2} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{16 e^{3/2} \sqrt{\frac{e x^2}{d}+1}}-\frac{b d^{3/2} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{32 e^{3/2} \sqrt{\frac{e x^2}{d}+1}}+\frac{b d^{3/2} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{8 e^{3/2} \sqrt{\frac{e x^2}{d}+1}}-\frac{b n x \left (d+e x^2\right )^{3/2}}{16 e}-\frac{b d n x \sqrt{d+e x^2}}{32 e} \]
Antiderivative was successfully verified.
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Rule 2341
Rule 279
Rule 321
Rule 215
Rule 2350
Rule 388
Rule 195
Rule 5659
Rule 3716
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int x^2 \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{\sqrt{d+e x^2} \int x^2 \sqrt{1+\frac{e x^2}{d}} \left (a+b \log \left (c x^n\right )\right ) \, dx}{\sqrt{1+\frac{e x^2}{d}}}\\ &=\frac{d x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{8 e}+\frac{1}{4} x^3 \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac{d^{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 )}{8 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}-\frac{\left (b n \sqrt{d+e x^2}\right ) \int \left (\frac{\left (d+2 e x^2\right ) \sqrt{1+\frac{e x^2}{d}}}{8 e}-\frac{d^{3/2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 e^{3/2} x}\right ) \, dx}{\sqrt{1+\frac{e x^2}{d}}}\\ &=\frac{d x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{8 e}+\frac{1}{4} x^3 \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac{d^{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 )}{8 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}+\frac{\left (b d^{3/2} n \sqrt{d+e x^2}\right ) \int \frac{\sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{x} \, dx}{8 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}-\frac{\left (b n \sqrt{d+e x^2}\right ) \int \left (d+2 e x^2\right ) \sqrt{1+\frac{e x^2}{d}} \, dx}{8 e \sqrt{1+\frac{e x^2}{d}}}\\ &=-\frac{b n x \left (d+e x^2\right )^{3/2}}{16 e}+\frac{d x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{8 e}+\frac{1}{4} x^3 \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac{d^{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 )}{8 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}+\frac{\left (b d^{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 )}{8 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}-\frac{\left (b d n \sqrt{d+e x^2}\right ) \int \sqrt{1+\frac{e x^2}{d}} \, dx}{16 e \sqrt{1+\frac{e x^2}{d}}}\\ &=-\frac{b d n x \sqrt{d+e x^2}}{32 e}-\frac{b n x \left (d+e x^2\right )^{3/2}}{16 e}-\frac{b d^{3/2} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{16 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}+\frac{d x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{8 e}+\frac{1}{4} x^3 \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac{d^{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 )}{8 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}-\frac{\left (b d^{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 )}{4 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}-\frac{\left (b d n \sqrt{d+e x^2}\right ) \int \frac{1}{\sqrt{1+\frac{e x^2}{d}}} \, dx}{32 e \sqrt{1+\frac{e x^2}{d}}}\\ &=-\frac{b d n x \sqrt{d+e x^2}}{32 e}-\frac{b n x \left (d+e x^2\right )^{3/2}}{16 e}-\frac{b d^{3/2} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{32 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}-\frac{b d^{3/2} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{16 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}+\frac{b d^{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 )}{8 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}+\frac{d x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{8 e}+\frac{1}{4} x^3 \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac{d^{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 )}{8 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}-\frac{\left (b d^{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 )}{8 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}\\ &=-\frac{b d n x \sqrt{d+e x^2}}{32 e}-\frac{b n x \left (d+e x^2\right )^{3/2}}{16 e}-\frac{b d^{3/2} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{32 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}-\frac{b d^{3/2} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{16 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}+\frac{b d^{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 )}{8 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}+\frac{d x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{8 e}+\frac{1}{4} x^3 \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac{d^{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 )}{8 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}-\frac{\left (b d^{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 )}{16 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}\\ &=-\frac{b d n x \sqrt{d+e x^2}}{32 e}-\frac{b n x \left (d+e x^2\right )^{3/2}}{16 e}-\frac{b d^{3/2} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{32 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}-\frac{b d^{3/2} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{16 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}+\frac{b d^{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 )}{8 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}+\frac{d x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{8 e}+\frac{1}{4} x^3 \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac{d^{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 )}{8 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}+\frac{b d^{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 )}{16 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}\\ \end{align*}
Mathematica [C] time = 0.418249, size = 250, normalized size = 0.61 \[ \frac{-8 b e^{3/2} n x^3 \sqrt{d+e x^2} \, _3F_2\left (-\frac{1}{2},\frac{3}{2},\frac{3}{2};\frac{5}{2},\frac{5}{2};-\frac{e x^2}{d}\right )+9 \sqrt{\frac{e x^2}{d}+1} \left (d^2 \log \left (\sqrt{e} \sqrt{d+e x^2}+e x\right ) (b n \log (x)-a)+a \sqrt{e} x \sqrt{d+e x^2} \left (d+2 e x^2\right )+b \log \left (c x^n\right ) \left (\sqrt{e} x \sqrt{d+e x^2} \left (d+2 e x^2\right )-d^2 \log \left (\sqrt{e} \sqrt{d+e x^2}+e x\right )\right )\right )-9 b d^{3/2} n \log (x) \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{72 e^{3/2} \sqrt{\frac{e x^2}{d}+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.482, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \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 (\sqrt{e x^{2} + d} b x^{2} \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 x^{2} \left (a + b \log{\left (c x^{n} \right )}\right ) \sqrt{d + e 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 \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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