Optimal. Leaf size=158 \[ -\frac{d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt{d+e x^2}}-\frac{2 d \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac{8 b d^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{3 e^3}+\frac{5 b d n \sqrt{d+e x^2}}{3 e^3}-\frac{b n \left (d+e x^2\right )^{3/2}}{9 e^3} \]
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Rubi [A] time = 0.218035, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {266, 43, 2350, 12, 1251, 897, 1153, 208} \[ -\frac{d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt{d+e x^2}}-\frac{2 d \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac{8 b d^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{3 e^3}+\frac{5 b d n \sqrt{d+e x^2}}{3 e^3}-\frac{b n \left (d+e x^2\right )^{3/2}}{9 e^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rule 2350
Rule 12
Rule 1251
Rule 897
Rule 1153
Rule 208
Rubi steps
\begin{align*} \int \frac{x^5 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{3/2}} \, dx &=-\frac{d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt{d+e x^2}}-\frac{2 d \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-(b n) \int \frac{-8 d^2-4 d e x^2+e^2 x^4}{3 e^3 x \sqrt{d+e x^2}} \, dx\\ &=-\frac{d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt{d+e x^2}}-\frac{2 d \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac{(b n) \int \frac{-8 d^2-4 d e x^2+e^2 x^4}{x \sqrt{d+e x^2}} \, dx}{3 e^3}\\ &=-\frac{d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt{d+e x^2}}-\frac{2 d \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac{(b n) \operatorname{Subst}\left (\int \frac{-8 d^2-4 d e x+e^2 x^2}{x \sqrt{d+e x}} \, dx,x,x^2\right )}{6 e^3}\\ &=-\frac{d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt{d+e x^2}}-\frac{2 d \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac{(b n) \operatorname{Subst}\left (\int \frac{-3 d^2-6 d x^2+x^4}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x^2}\right )}{3 e^4}\\ &=-\frac{d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt{d+e x^2}}-\frac{2 d \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac{(b n) \operatorname{Subst}\left (\int \left (-5 d e+e x^2-\frac{8 d^2}{-\frac{d}{e}+\frac{x^2}{e}}\right ) \, dx,x,\sqrt{d+e x^2}\right )}{3 e^4}\\ &=\frac{5 b d n \sqrt{d+e x^2}}{3 e^3}-\frac{b n \left (d+e x^2\right )^{3/2}}{9 e^3}-\frac{d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt{d+e x^2}}-\frac{2 d \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac{\left (8 b d^2 n\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x^2}\right )}{3 e^4}\\ &=\frac{5 b d n \sqrt{d+e x^2}}{3 e^3}-\frac{b n \left (d+e x^2\right )^{3/2}}{9 e^3}-\frac{8 b d^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{3 e^3}-\frac{d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt{d+e x^2}}-\frac{2 d \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}\\ \end{align*}
Mathematica [A] time = 0.178271, size = 160, normalized size = 1.01 \[ \frac{-24 a d^2-12 a d e x^2+3 a e^2 x^4-3 b \left (8 d^2+4 d e x^2-e^2 x^4\right ) \log \left (c x^n\right )+24 b d^{3/2} n \log (x) \sqrt{d+e x^2}-24 b d^{3/2} n \sqrt{d+e x^2} \log \left (\sqrt{d} \sqrt{d+e x^2}+d\right )+14 b d^2 n+13 b d e n x^2-b e^2 n x^4}{9 e^3 \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.408, size = 0, normalized size = 0. \begin{align*} \int{{x}^{5} \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 [A] time = 1.60782, size = 818, normalized size = 5.18 \begin{align*} \left [\frac{12 \,{\left (b d e n x^{2} + b d^{2} n\right )} \sqrt{d} \log \left (-\frac{e x^{2} - 2 \, \sqrt{e x^{2} + d} \sqrt{d} + 2 \, d}{x^{2}}\right ) -{\left ({\left (b e^{2} n - 3 \, a e^{2}\right )} x^{4} - 14 \, b d^{2} n + 24 \, a d^{2} -{\left (13 \, b d e n - 12 \, a d e\right )} x^{2} - 3 \,{\left (b e^{2} x^{4} - 4 \, b d e x^{2} - 8 \, b d^{2}\right )} \log \left (c\right ) - 3 \,{\left (b e^{2} n x^{4} - 4 \, b d e n x^{2} - 8 \, b d^{2} n\right )} \log \left (x\right )\right )} \sqrt{e x^{2} + d}}{9 \,{\left (e^{4} x^{2} + d e^{3}\right )}}, \frac{24 \,{\left (b d e n x^{2} + b d^{2} n\right )} \sqrt{-d} \arctan \left (\frac{\sqrt{-d}}{\sqrt{e x^{2} + d}}\right ) -{\left ({\left (b e^{2} n - 3 \, a e^{2}\right )} x^{4} - 14 \, b d^{2} n + 24 \, a d^{2} -{\left (13 \, b d e n - 12 \, a d e\right )} x^{2} - 3 \,{\left (b e^{2} x^{4} - 4 \, b d e x^{2} - 8 \, b d^{2}\right )} \log \left (c\right ) - 3 \,{\left (b e^{2} n x^{4} - 4 \, b d e n x^{2} - 8 \, b d^{2} n\right )} \log \left (x\right )\right )} \sqrt{e x^{2} + d}}{9 \,{\left (e^{4} x^{2} + d e^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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^{5}}{{\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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