Optimal. Leaf size=464 \[ \frac{b d^{5/2} n \sqrt{d+e x^2} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{32 e^{3/2} \sqrt{\frac{e x^2}{d}+1}}-\frac{d^{5/2} \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{16 e^{3/2} \sqrt{\frac{e x^2}{d}+1}}+\frac{d^2 x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{16 e}+\frac{1}{6} x^3 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac{1}{8} d x^3 \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac{b d^{5/2} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{32 e^{3/2} \sqrt{\frac{e x^2}{d}+1}}-\frac{b d^{5/2} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{192 e^{3/2} \sqrt{\frac{e x^2}{d}+1}}+\frac{b d^{5/2} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{16 e^{3/2} \sqrt{\frac{e x^2}{d}+1}}-\frac{11 b d^2 n x \sqrt{d+e x^2}}{192 e}-\frac{1}{36} b e n x^5 \sqrt{d+e x^2}-\frac{23}{288} b d n x^3 \sqrt{d+e x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.59279, antiderivative size = 464, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 13, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.52, Rules used = {2341, 279, 321, 215, 2350, 12, 14, 195, 5659, 3716, 2190, 2279, 2391} \[ \frac{b d^{5/2} n \sqrt{d+e x^2} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{32 e^{3/2} \sqrt{\frac{e x^2}{d}+1}}-\frac{d^{5/2} \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{16 e^{3/2} \sqrt{\frac{e x^2}{d}+1}}+\frac{d^2 x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{16 e}+\frac{1}{6} x^3 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac{1}{8} d x^3 \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac{b d^{5/2} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{32 e^{3/2} \sqrt{\frac{e x^2}{d}+1}}-\frac{b d^{5/2} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{192 e^{3/2} \sqrt{\frac{e x^2}{d}+1}}+\frac{b d^{5/2} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{16 e^{3/2} \sqrt{\frac{e x^2}{d}+1}}-\frac{11 b d^2 n x \sqrt{d+e x^2}}{192 e}-\frac{1}{36} b e n x^5 \sqrt{d+e x^2}-\frac{23}{288} b d n x^3 \sqrt{d+e x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2341
Rule 279
Rule 321
Rule 215
Rule 2350
Rule 12
Rule 14
Rule 195
Rule 5659
Rule 3716
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int x^2 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{\left (d \sqrt{d+e x^2}\right ) \int x^2 \left (1+\frac{e x^2}{d}\right )^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx}{\sqrt{1+\frac{e x^2}{d}}}\\ &=\frac{d^2 x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{16 e}+\frac{1}{8} d x^3 \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac{1}{6} x^3 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )-\frac{d^{5/2} \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 )}{16 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}-\frac{\left (b d n \sqrt{d+e x^2}\right ) \int \frac{\sqrt{e} x \sqrt{1+\frac{e x^2}{d}} \left (3 d^2+14 d e x^2+8 e^2 x^4\right )-3 d^{5/2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{48 d e^{3/2} x} \, dx}{\sqrt{1+\frac{e x^2}{d}}}\\ &=\frac{d^2 x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{16 e}+\frac{1}{8} d x^3 \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac{1}{6} x^3 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )-\frac{d^{5/2} \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 )}{16 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}-\frac{\left (b n \sqrt{d+e x^2}\right ) \int \frac{\sqrt{e} x \sqrt{1+\frac{e x^2}{d}} \left (3 d^2+14 d e x^2+8 e^2 x^4\right )-3 d^{5/2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{x} \, dx}{48 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}\\ &=\frac{d^2 x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{16 e}+\frac{1}{8} d x^3 \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac{1}{6} x^3 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )-\frac{d^{5/2} \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 )}{16 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}-\frac{\left (b n \sqrt{d+e x^2}\right ) \int \left (3 d^2 \sqrt{e} \sqrt{1+\frac{e x^2}{d}}+14 d e^{3/2} x^2 \sqrt{1+\frac{e x^2}{d}}+8 e^{5/2} x^4 \sqrt{1+\frac{e x^2}{d}}-\frac{3 d^{5/2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{x}\right ) \, dx}{48 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}\\ &=\frac{d^2 x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{16 e}+\frac{1}{8} d x^3 \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac{1}{6} x^3 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )-\frac{d^{5/2} \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 )}{16 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}-\frac{\left (7 b d n \sqrt{d+e x^2}\right ) \int x^2 \sqrt{1+\frac{e x^2}{d}} \, dx}{24 \sqrt{1+\frac{e x^2}{d}}}+\frac{\left (b d^{5/2} n \sqrt{d+e x^2}\right ) \int \frac{\sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{x} \, dx}{16 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}-\frac{\left (b d^2 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{\left (b e n \sqrt{d+e x^2}\right ) \int x^4 \sqrt{1+\frac{e x^2}{d}} \, dx}{6 \sqrt{1+\frac{e x^2}{d}}}\\ &=-\frac{b d^2 n x \sqrt{d+e x^2}}{32 e}-\frac{7}{96} b d n x^3 \sqrt{d+e x^2}-\frac{1}{36} b e n x^5 \sqrt{d+e x^2}+\frac{d^2 x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{16 e}+\frac{1}{8} d x^3 \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac{1}{6} x^3 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )-\frac{d^{5/2} \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 )}{16 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}-\frac{\left (7 b d n \sqrt{d+e x^2}\right ) \int \frac{x^2}{\sqrt{1+\frac{e x^2}{d}}} \, dx}{96 \sqrt{1+\frac{e x^2}{d}}}+\frac{\left (b d^{5/2} 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 )}{16 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}-\frac{\left (b d^2 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{\left (b e n \sqrt{d+e x^2}\right ) \int \frac{x^4}{\sqrt{1+\frac{e x^2}{d}}} \, dx}{36 \sqrt{1+\frac{e x^2}{d}}}\\ &=-\frac{13 b d^2 n x \sqrt{d+e x^2}}{192 e}-\frac{23}{288} b d n x^3 \sqrt{d+e x^2}-\frac{1}{36} b e n x^5 \sqrt{d+e x^2}-\frac{b d^{5/2} 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^{5/2} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{32 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}+\frac{d^2 x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{16 e}+\frac{1}{8} d x^3 \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac{1}{6} x^3 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )-\frac{d^{5/2} \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 )}{16 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}+\frac{\left (b d n \sqrt{d+e x^2}\right ) \int \frac{x^2}{\sqrt{1+\frac{e x^2}{d}}} \, dx}{48 \sqrt{1+\frac{e x^2}{d}}}-\frac{\left (b d^{5/2} 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 )}{8 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}+\frac{\left (7 b d^2 n \sqrt{d+e x^2}\right ) \int \frac{1}{\sqrt{1+\frac{e x^2}{d}}} \, dx}{192 e \sqrt{1+\frac{e x^2}{d}}}\\ &=-\frac{11 b d^2 n x \sqrt{d+e x^2}}{192 e}-\frac{23}{288} b d n x^3 \sqrt{d+e x^2}-\frac{1}{36} b e n x^5 \sqrt{d+e x^2}+\frac{b d^{5/2} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{192 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}-\frac{b d^{5/2} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{32 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}+\frac{b d^{5/2} 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 )}{16 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}+\frac{d^2 x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{16 e}+\frac{1}{8} d x^3 \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac{1}{6} x^3 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )-\frac{d^{5/2} \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 )}{16 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}-\frac{\left (b d^{5/2} 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 )}{16 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}-\frac{\left (b d^2 n \sqrt{d+e x^2}\right ) \int \frac{1}{\sqrt{1+\frac{e x^2}{d}}} \, dx}{96 e \sqrt{1+\frac{e x^2}{d}}}\\ &=-\frac{11 b d^2 n x \sqrt{d+e x^2}}{192 e}-\frac{23}{288} b d n x^3 \sqrt{d+e x^2}-\frac{1}{36} b e n x^5 \sqrt{d+e x^2}-\frac{b d^{5/2} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{192 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}-\frac{b d^{5/2} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{32 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}+\frac{b d^{5/2} 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 )}{16 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}+\frac{d^2 x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{16 e}+\frac{1}{8} d x^3 \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac{1}{6} x^3 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )-\frac{d^{5/2} \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 )}{16 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}-\frac{\left (b d^{5/2} 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 )}{32 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}\\ &=-\frac{11 b d^2 n x \sqrt{d+e x^2}}{192 e}-\frac{23}{288} b d n x^3 \sqrt{d+e x^2}-\frac{1}{36} b e n x^5 \sqrt{d+e x^2}-\frac{b d^{5/2} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{192 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}-\frac{b d^{5/2} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{32 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}+\frac{b d^{5/2} 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 )}{16 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}+\frac{d^2 x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{16 e}+\frac{1}{8} d x^3 \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac{1}{6} x^3 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )-\frac{d^{5/2} \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 )}{16 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}+\frac{b d^{5/2} n \sqrt{d+e x^2} \text{Li}_2\left (e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{32 e^{3/2} \sqrt{1+\frac{e x^2}{d}}}\\ \end{align*}
Mathematica [C] time = 1.04213, size = 331, normalized size = 0.71 \[ \frac{-144 b e^{5/2} n x^5 \sqrt{d+e x^2} \, _3F_2\left (-\frac{1}{2},\frac{5}{2},\frac{5}{2};\frac{7}{2},\frac{7}{2};-\frac{e x^2}{d}\right )-400 b d e^{3/2} n x^3 \sqrt{d+e x^2} \, _3F_2\left (-\frac{1}{2},\frac{3}{2},\frac{3}{2};\frac{5}{2},\frac{5}{2};-\frac{e x^2}{d}\right )-75 \left (\sqrt{\frac{e x^2}{d}+1} \left (3 d^3 \log \left (\sqrt{e} \sqrt{d+e x^2}+e x\right ) (a-b n \log (x))-a \sqrt{e} x \sqrt{d+e x^2} \left (3 d^2+14 d e x^2+8 e^2 x^4\right )-b \log \left (c x^n\right ) \left (\sqrt{e} x \sqrt{d+e x^2} \left (3 d^2+14 d e x^2+8 e^2 x^4\right )-3 d^3 \log \left (\sqrt{e} \sqrt{d+e x^2}+e x\right )\right )\right )+3 b d^{5/2} n \log (x) \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )\right )}{3600 e^{3/2} \sqrt{\frac{e x^2}{d}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.414, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b e x^{4} + b d x^{2}\right )} \sqrt{e x^{2} + d} \log \left (c x^{n}\right ) +{\left (a e x^{4} + a d x^{2}\right )} \sqrt{e x^{2} + d}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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 )} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]