Optimal. Leaf size=176 \[ \frac{8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \sqrt{d+e x^2}}+\frac{4 e \left (a+b \log \left (c x^n\right )\right )}{3 d^2 x \sqrt{d+e x^2}}-\frac{a+b \log \left (c x^n\right )}{3 d x^3 \sqrt{d+e x^2}}-\frac{8 b e^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{3 d^3}+\frac{14 b e n \sqrt{d+e x^2}}{9 d^3 x}-\frac{b n \sqrt{d+e x^2}}{9 d^2 x^3} \]
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Rubi [A] time = 0.166127, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {271, 191, 2350, 12, 1265, 451, 217, 206} \[ \frac{8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \sqrt{d+e x^2}}+\frac{4 e \left (a+b \log \left (c x^n\right )\right )}{3 d^2 x \sqrt{d+e x^2}}-\frac{a+b \log \left (c x^n\right )}{3 d x^3 \sqrt{d+e x^2}}-\frac{8 b e^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{3 d^3}+\frac{14 b e n \sqrt{d+e x^2}}{9 d^3 x}-\frac{b n \sqrt{d+e x^2}}{9 d^2 x^3} \]
Antiderivative was successfully verified.
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Rule 271
Rule 191
Rule 2350
Rule 12
Rule 1265
Rule 451
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^{3/2}} \, dx &=-\frac{a+b \log \left (c x^n\right )}{3 d x^3 \sqrt{d+e x^2}}+\frac{4 e \left (a+b \log \left (c x^n\right )\right )}{3 d^2 x \sqrt{d+e x^2}}+\frac{8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \sqrt{d+e x^2}}-(b n) \int \frac{-d^2+4 d e x^2+8 e^2 x^4}{3 d^3 x^4 \sqrt{d+e x^2}} \, dx\\ &=-\frac{a+b \log \left (c x^n\right )}{3 d x^3 \sqrt{d+e x^2}}+\frac{4 e \left (a+b \log \left (c x^n\right )\right )}{3 d^2 x \sqrt{d+e x^2}}+\frac{8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \sqrt{d+e x^2}}-\frac{(b n) \int \frac{-d^2+4 d e x^2+8 e^2 x^4}{x^4 \sqrt{d+e x^2}} \, dx}{3 d^3}\\ &=-\frac{b n \sqrt{d+e x^2}}{9 d^2 x^3}-\frac{a+b \log \left (c x^n\right )}{3 d x^3 \sqrt{d+e x^2}}+\frac{4 e \left (a+b \log \left (c x^n\right )\right )}{3 d^2 x \sqrt{d+e x^2}}+\frac{8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \sqrt{d+e x^2}}+\frac{(b n) \int \frac{-14 d^2 e-24 d e^2 x^2}{x^2 \sqrt{d+e x^2}} \, dx}{9 d^4}\\ &=-\frac{b n \sqrt{d+e x^2}}{9 d^2 x^3}+\frac{14 b e n \sqrt{d+e x^2}}{9 d^3 x}-\frac{a+b \log \left (c x^n\right )}{3 d x^3 \sqrt{d+e x^2}}+\frac{4 e \left (a+b \log \left (c x^n\right )\right )}{3 d^2 x \sqrt{d+e x^2}}+\frac{8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \sqrt{d+e x^2}}-\frac{\left (8 b e^2 n\right ) \int \frac{1}{\sqrt{d+e x^2}} \, dx}{3 d^3}\\ &=-\frac{b n \sqrt{d+e x^2}}{9 d^2 x^3}+\frac{14 b e n \sqrt{d+e x^2}}{9 d^3 x}-\frac{a+b \log \left (c x^n\right )}{3 d x^3 \sqrt{d+e x^2}}+\frac{4 e \left (a+b \log \left (c x^n\right )\right )}{3 d^2 x \sqrt{d+e x^2}}+\frac{8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \sqrt{d+e x^2}}-\frac{\left (8 b e^2 n\right ) \operatorname{Subst}\left (\int \frac{1}{1-e x^2} \, dx,x,\frac{x}{\sqrt{d+e x^2}}\right )}{3 d^3}\\ &=-\frac{b n \sqrt{d+e x^2}}{9 d^2 x^3}+\frac{14 b e n \sqrt{d+e x^2}}{9 d^3 x}-\frac{8 b e^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{3 d^3}-\frac{a+b \log \left (c x^n\right )}{3 d x^3 \sqrt{d+e x^2}}+\frac{4 e \left (a+b \log \left (c x^n\right )\right )}{3 d^2 x \sqrt{d+e x^2}}+\frac{8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \sqrt{d+e x^2}}\\ \end{align*}
Mathematica [A] time = 0.158234, size = 144, normalized size = 0.82 \[ \frac{-3 a d^2+12 a d e x^2+24 a e^2 x^4-3 b \left (d^2-4 d e x^2-8 e^2 x^4\right ) \log \left (c x^n\right )-b d^2 n-24 b e^{3/2} n x^3 \sqrt{d+e x^2} \log \left (\sqrt{e} \sqrt{d+e x^2}+e x\right )+13 b d e n x^2+14 b e^2 n x^4}{9 d^3 x^3 \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.418, 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 [A] time = 1.65842, size = 840, normalized size = 4.77 \begin{align*} \left [\frac{12 \,{\left (b e^{2} n x^{5} + b d e n x^{3}\right )} \sqrt{e} \log \left (-2 \, e x^{2} + 2 \, \sqrt{e x^{2} + d} \sqrt{e} x - d\right ) +{\left (2 \,{\left (7 \, b e^{2} n + 12 \, a e^{2}\right )} x^{4} - b d^{2} n - 3 \, a d^{2} +{\left (13 \, b d e n + 12 \, a d e\right )} x^{2} + 3 \,{\left (8 \, b e^{2} x^{4} + 4 \, b d e x^{2} - b d^{2}\right )} \log \left (c\right ) + 3 \,{\left (8 \, b e^{2} n x^{4} + 4 \, b d e n x^{2} - b d^{2} n\right )} \log \left (x\right )\right )} \sqrt{e x^{2} + d}}{9 \,{\left (d^{3} e x^{5} + d^{4} x^{3}\right )}}, \frac{24 \,{\left (b e^{2} n x^{5} + b d e n x^{3}\right )} \sqrt{-e} \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right ) +{\left (2 \,{\left (7 \, b e^{2} n + 12 \, a e^{2}\right )} x^{4} - b d^{2} n - 3 \, a d^{2} +{\left (13 \, b d e n + 12 \, a d e\right )} x^{2} + 3 \,{\left (8 \, b e^{2} x^{4} + 4 \, b d e x^{2} - b d^{2}\right )} \log \left (c\right ) + 3 \,{\left (8 \, b e^{2} n x^{4} + 4 \, b d e n x^{2} - b d^{2} n\right )} \log \left (x\right )\right )} \sqrt{e x^{2} + d}}{9 \,{\left (d^{3} e x^{5} + d^{4} x^{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{b \log \left (c x^{n}\right ) + a}{{\left (e x^{2} + d\right )}^{\frac{3}{2}} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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