Optimal. Leaf size=397 \[ \frac{b f^2 n \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 g^3}+\frac{b f^2 n \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right )}{2 g^3}+\frac{f^2 \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{d \sqrt{g}+e \sqrt{-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^3}+\frac{f^2 \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^3}-\frac{f x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}+\frac{x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 g}+\frac{b d^2 f n \log (d+e x)}{2 e^2 g^2}-\frac{b d^2 n x^2}{8 e^2 g}+\frac{b d^3 n x}{4 e^3 g}-\frac{b d^4 n \log (d+e x)}{4 e^4 g}-\frac{b d f n x}{2 e g^2}+\frac{b d n x^3}{12 e g}+\frac{b f n x^2}{4 g^2}-\frac{b n x^4}{16 g} \]
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Rubi [A] time = 0.510295, antiderivative size = 397, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {266, 43, 2416, 2395, 260, 2394, 2393, 2391} \[ \frac{b f^2 n \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 g^3}+\frac{b f^2 n \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right )}{2 g^3}+\frac{f^2 \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{d \sqrt{g}+e \sqrt{-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^3}+\frac{f^2 \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^3}-\frac{f x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}+\frac{x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 g}+\frac{b d^2 f n \log (d+e x)}{2 e^2 g^2}-\frac{b d^2 n x^2}{8 e^2 g}+\frac{b d^3 n x}{4 e^3 g}-\frac{b d^4 n \log (d+e x)}{4 e^4 g}-\frac{b d f n x}{2 e g^2}+\frac{b d n x^3}{12 e g}+\frac{b f n x^2}{4 g^2}-\frac{b n x^4}{16 g} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rule 2416
Rule 2395
Rule 260
Rule 2394
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x^2} \, dx &=\int \left (-\frac{f x \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2}+\frac{x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}+\frac{f^2 x \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2 \left (f+g x^2\right )}\right ) \, dx\\ &=-\frac{f \int x \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{g^2}+\frac{f^2 \int \frac{x \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x^2} \, dx}{g^2}+\frac{\int x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{g}\\ &=-\frac{f x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}+\frac{x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 g}+\frac{f^2 \int \left (-\frac{a+b \log \left (c (d+e x)^n\right )}{2 \sqrt{g} \left (\sqrt{-f}-\sqrt{g} x\right )}+\frac{a+b \log \left (c (d+e x)^n\right )}{2 \sqrt{g} \left (\sqrt{-f}+\sqrt{g} x\right )}\right ) \, dx}{g^2}+\frac{(b e f n) \int \frac{x^2}{d+e x} \, dx}{2 g^2}-\frac{(b e n) \int \frac{x^4}{d+e x} \, dx}{4 g}\\ &=-\frac{f x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}+\frac{x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 g}-\frac{f^2 \int \frac{a+b \log \left (c (d+e x)^n\right )}{\sqrt{-f}-\sqrt{g} x} \, dx}{2 g^{5/2}}+\frac{f^2 \int \frac{a+b \log \left (c (d+e x)^n\right )}{\sqrt{-f}+\sqrt{g} x} \, dx}{2 g^{5/2}}+\frac{(b e f n) \int \left (-\frac{d}{e^2}+\frac{x}{e}+\frac{d^2}{e^2 (d+e x)}\right ) \, dx}{2 g^2}-\frac{(b e n) \int \left (-\frac{d^3}{e^4}+\frac{d^2 x}{e^3}-\frac{d x^2}{e^2}+\frac{x^3}{e}+\frac{d^4}{e^4 (d+e x)}\right ) \, dx}{4 g}\\ &=-\frac{b d f n x}{2 e g^2}+\frac{b d^3 n x}{4 e^3 g}+\frac{b f n x^2}{4 g^2}-\frac{b d^2 n x^2}{8 e^2 g}+\frac{b d n x^3}{12 e g}-\frac{b n x^4}{16 g}+\frac{b d^2 f n \log (d+e x)}{2 e^2 g^2}-\frac{b d^4 n \log (d+e x)}{4 e^4 g}-\frac{f x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}+\frac{x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 g}+\frac{f^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 g^3}+\frac{f^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 g^3}-\frac{\left (b e f^2 n\right ) \int \frac{\log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{d+e x} \, dx}{2 g^3}-\frac{\left (b e f^2 n\right ) \int \frac{\log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{d+e x} \, dx}{2 g^3}\\ &=-\frac{b d f n x}{2 e g^2}+\frac{b d^3 n x}{4 e^3 g}+\frac{b f n x^2}{4 g^2}-\frac{b d^2 n x^2}{8 e^2 g}+\frac{b d n x^3}{12 e g}-\frac{b n x^4}{16 g}+\frac{b d^2 f n \log (d+e x)}{2 e^2 g^2}-\frac{b d^4 n \log (d+e x)}{4 e^4 g}-\frac{f x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}+\frac{x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 g}+\frac{f^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 g^3}+\frac{f^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 g^3}-\frac{\left (b f^2 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{g} x}{e \sqrt{-f}-d \sqrt{g}}\right )}{x} \, dx,x,d+e x\right )}{2 g^3}-\frac{\left (b f^2 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{g} x}{e \sqrt{-f}+d \sqrt{g}}\right )}{x} \, dx,x,d+e x\right )}{2 g^3}\\ &=-\frac{b d f n x}{2 e g^2}+\frac{b d^3 n x}{4 e^3 g}+\frac{b f n x^2}{4 g^2}-\frac{b d^2 n x^2}{8 e^2 g}+\frac{b d n x^3}{12 e g}-\frac{b n x^4}{16 g}+\frac{b d^2 f n \log (d+e x)}{2 e^2 g^2}-\frac{b d^4 n \log (d+e x)}{4 e^4 g}-\frac{f x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}+\frac{x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 g}+\frac{f^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 g^3}+\frac{f^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 g^3}+\frac{b f^2 n \text{Li}_2\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 g^3}+\frac{b f^2 n \text{Li}_2\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 g^3}\\ \end{align*}
Mathematica [A] time = 0.287433, size = 331, normalized size = 0.83 \[ \frac{24 b f^2 n \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )+24 b f^2 n \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right )+24 f^2 \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{d \sqrt{g}+e \sqrt{-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+24 f^2 \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-24 f g x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )+12 g^2 x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac{12 b f g n \left (2 d^2 \log (d+e x)+e x (e x-2 d)\right )}{e^2}-\frac{b g^2 n \left (e x \left (6 d^2 e x-12 d^3-4 d e^2 x^2+3 e^3 x^3\right )+12 d^4 \log (d+e x)\right )}{e^4}}{48 g^3} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.518, size = 905, normalized size = 2.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{4} \, a{\left (\frac{2 \, f^{2} \log \left (g x^{2} + f\right )}{g^{3}} + \frac{g x^{4} - 2 \, f x^{2}}{g^{2}}\right )} + b \int \frac{x^{5} \log \left ({\left (e x + d\right )}^{n}\right ) + x^{5} \log \left (c\right )}{g x^{2} + f}\,{d x} \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 (\frac{b x^{5} \log \left ({\left (e x + d\right )}^{n} c\right ) + a x^{5}}{g x^{2} + f}, x\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 ({\left (e x + d\right )}^{n} c\right ) + a\right )} x^{5}}{g x^{2} + f}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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