Optimal. Leaf size=694 \[ -\frac{b g^{3/2} n \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(-f)^{5/2}}+\frac{b g^{3/2} n \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(-f)^{5/2}}-\frac{2 b^2 e^3 n^2 \text{PolyLog}\left (2,\frac{d}{d+e x}\right )}{3 d^3 f}-\frac{2 b^2 e g n^2 \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{d f^2}+\frac{b^2 g^{3/2} n^2 \text{PolyLog}\left (3,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{(-f)^{5/2}}-\frac{b^2 g^{3/2} n^2 \text{PolyLog}\left (3,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right )}{(-f)^{5/2}}+\frac{2 b e^3 n \log \left (1-\frac{d}{d+e x}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d^3 f}+\frac{2 b e^2 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d^3 f x}-\frac{2 b e g n \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d f^2}+\frac{g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f^2 x}+\frac{g^{3/2} \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{d \sqrt{g}+e \sqrt{-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 (-f)^{5/2}}-\frac{g^{3/2} \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 (-f)^{5/2}}-\frac{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d f x^2}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 f x^3}-\frac{b^2 e^2 n^2}{3 d^2 f x}-\frac{b^2 e^3 n^2 \log (x)}{d^3 f}+\frac{b^2 e^3 n^2 \log (d+e x)}{3 d^3 f} \]
[Out]
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Rubi [A] time = 1.11448, antiderivative size = 717, normalized size of antiderivative = 1.03, number of steps used = 28, number of rules used = 20, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.69, Rules used = {2416, 2398, 2411, 2347, 2344, 2301, 2317, 2391, 2314, 31, 2319, 44, 2397, 2394, 2315, 2409, 2396, 2433, 2374, 6589} \[ -\frac{b g^{3/2} n \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(-f)^{5/2}}+\frac{b g^{3/2} n \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(-f)^{5/2}}+\frac{2 b^2 e^3 n^2 \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{3 d^3 f}-\frac{2 b^2 e g n^2 \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{d f^2}+\frac{b^2 g^{3/2} n^2 \text{PolyLog}\left (3,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{(-f)^{5/2}}-\frac{b^2 g^{3/2} n^2 \text{PolyLog}\left (3,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right )}{(-f)^{5/2}}-\frac{e^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 d^3 f}+\frac{2 b e^3 n \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d^3 f}+\frac{2 b e^2 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d^3 f x}-\frac{2 b e g n \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d f^2}+\frac{g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f^2 x}+\frac{g^{3/2} \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{d \sqrt{g}+e \sqrt{-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 (-f)^{5/2}}-\frac{g^{3/2} \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 (-f)^{5/2}}-\frac{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d f x^2}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 f x^3}-\frac{b^2 e^2 n^2}{3 d^2 f x}-\frac{b^2 e^3 n^2 \log (x)}{d^3 f}+\frac{b^2 e^3 n^2 \log (d+e x)}{3 d^3 f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2416
Rule 2398
Rule 2411
Rule 2347
Rule 2344
Rule 2301
Rule 2317
Rule 2391
Rule 2314
Rule 31
Rule 2319
Rule 44
Rule 2397
Rule 2394
Rule 2315
Rule 2409
Rule 2396
Rule 2433
Rule 2374
Rule 6589
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^4 \left (f+g x^2\right )} \, dx &=\int \left (\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f x^4}-\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^2 x^2}+\frac{g^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^2 \left (f+g x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^4} \, dx}{f}-\frac{g \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^2} \, dx}{f^2}+\frac{g^2 \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2} \, dx}{f^2}\\ &=-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 f x^3}+\frac{g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f^2 x}+\frac{g^2 \int \left (\frac{\sqrt{-f} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f \left (\sqrt{-f}-\sqrt{g} x\right )}+\frac{\sqrt{-f} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f \left (\sqrt{-f}+\sqrt{g} x\right )}\right ) \, dx}{f^2}+\frac{(2 b e n) \int \frac{a+b \log \left (c (d+e x)^n\right )}{x^3 (d+e x)} \, dx}{3 f}-\frac{(2 b e g n) \int \frac{a+b \log \left (c (d+e x)^n\right )}{x} \, dx}{d f^2}\\ &=-\frac{2 b e g n \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d f^2}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 f x^3}+\frac{g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f^2 x}-\frac{g^2 \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt{-f}-\sqrt{g} x} \, dx}{2 (-f)^{5/2}}-\frac{g^2 \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt{-f}+\sqrt{g} x} \, dx}{2 (-f)^{5/2}}+\frac{(2 b n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^3} \, dx,x,d+e x\right )}{3 f}+\frac{\left (2 b^2 e^2 g n^2\right ) \int \frac{\log \left (-\frac{e x}{d}\right )}{d+e x} \, dx}{d f^2}\\ &=-\frac{2 b e g n \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d f^2}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 f x^3}+\frac{g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f^2 x}+\frac{g^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 (-f)^{5/2}}-\frac{g^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 (-f)^{5/2}}-\frac{2 b^2 e g n^2 \text{Li}_2\left (1+\frac{e x}{d}\right )}{d f^2}+\frac{(2 b n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{\left (-\frac{d}{e}+\frac{x}{e}\right )^3} \, dx,x,d+e x\right )}{3 d f}-\frac{(2 b e n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+e x\right )}{3 d f}-\frac{\left (b e g^{3/2} n\right ) \int \frac{\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 )}{d+e x} \, dx}{(-f)^{5/2}}+\frac{\left (b e g^{3/2} n\right ) \int \frac{\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 )}{d+e x} \, dx}{(-f)^{5/2}}\\ &=-\frac{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d f x^2}-\frac{2 b e g n \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d f^2}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 f x^3}+\frac{g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f^2 x}+\frac{g^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 (-f)^{5/2}}-\frac{g^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 (-f)^{5/2}}-\frac{2 b^2 e g n^2 \text{Li}_2\left (1+\frac{e x}{d}\right )}{d f^2}-\frac{(2 b e n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{\left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+e x\right )}{3 d^2 f}+\frac{\left (2 b e^2 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )} \, dx,x,d+e x\right )}{3 d^2 f}-\frac{\left (b g^{3/2} n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac{e \left (\frac{e \sqrt{-f}+d \sqrt{g}}{e}-\frac{\sqrt{g} x}{e}\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{x} \, dx,x,d+e x\right )}{(-f)^{5/2}}+\frac{\left (b g^{3/2} n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac{e \left (\frac{e \sqrt{-f}-d \sqrt{g}}{e}+\frac{\sqrt{g} x}{e}\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{x} \, dx,x,d+e x\right )}{(-f)^{5/2}}+\frac{\left (b^2 e n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+e x\right )}{3 d f}\\ &=-\frac{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d f x^2}+\frac{2 b e^2 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d^3 f x}-\frac{2 b e g n \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d f^2}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 f x^3}+\frac{g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f^2 x}+\frac{g^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 (-f)^{5/2}}-\frac{g^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 (-f)^{5/2}}-\frac{b g^{3/2} n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{(-f)^{5/2}}+\frac{b g^{3/2} n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{(-f)^{5/2}}-\frac{2 b^2 e g n^2 \text{Li}_2\left (1+\frac{e x}{d}\right )}{d f^2}+\frac{\left (2 b e^2 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+e x\right )}{3 d^3 f}-\frac{\left (2 b e^3 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x} \, dx,x,d+e x\right )}{3 d^3 f}+\frac{\left (b^2 e n^2\right ) \operatorname{Subst}\left (\int \left (\frac{e^2}{d (d-x)^2}+\frac{e^2}{d^2 (d-x)}+\frac{e^2}{d^2 x}\right ) \, dx,x,d+e x\right )}{3 d f}-\frac{\left (2 b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+e x\right )}{3 d^3 f}+\frac{\left (b^2 g^{3/2} n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{\sqrt{g} x}{e \sqrt{-f}-d \sqrt{g}}\right )}{x} \, dx,x,d+e x\right )}{(-f)^{5/2}}-\frac{\left (b^2 g^{3/2} n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{\sqrt{g} x}{e \sqrt{-f}+d \sqrt{g}}\right )}{x} \, dx,x,d+e x\right )}{(-f)^{5/2}}\\ &=-\frac{b^2 e^2 n^2}{3 d^2 f x}-\frac{b^2 e^3 n^2 \log (x)}{d^3 f}+\frac{b^2 e^3 n^2 \log (d+e x)}{3 d^3 f}-\frac{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d f x^2}+\frac{2 b e^2 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d^3 f x}+\frac{2 b e^3 n \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d^3 f}-\frac{2 b e g n \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d f^2}-\frac{e^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 d^3 f}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 f x^3}+\frac{g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f^2 x}+\frac{g^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 (-f)^{5/2}}-\frac{g^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 (-f)^{5/2}}-\frac{b g^{3/2} n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{(-f)^{5/2}}+\frac{b g^{3/2} n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{(-f)^{5/2}}-\frac{2 b^2 e g n^2 \text{Li}_2\left (1+\frac{e x}{d}\right )}{d f^2}+\frac{b^2 g^{3/2} n^2 \text{Li}_3\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{(-f)^{5/2}}-\frac{b^2 g^{3/2} n^2 \text{Li}_3\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{(-f)^{5/2}}-\frac{\left (2 b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{d}\right )}{x} \, dx,x,d+e x\right )}{3 d^3 f}\\ &=-\frac{b^2 e^2 n^2}{3 d^2 f x}-\frac{b^2 e^3 n^2 \log (x)}{d^3 f}+\frac{b^2 e^3 n^2 \log (d+e x)}{3 d^3 f}-\frac{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d f x^2}+\frac{2 b e^2 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d^3 f x}+\frac{2 b e^3 n \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d^3 f}-\frac{2 b e g n \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d f^2}-\frac{e^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 d^3 f}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 f x^3}+\frac{g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f^2 x}+\frac{g^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 (-f)^{5/2}}-\frac{g^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 (-f)^{5/2}}-\frac{b g^{3/2} n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{(-f)^{5/2}}+\frac{b g^{3/2} n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{(-f)^{5/2}}+\frac{2 b^2 e^3 n^2 \text{Li}_2\left (1+\frac{e x}{d}\right )}{3 d^3 f}-\frac{2 b^2 e g n^2 \text{Li}_2\left (1+\frac{e x}{d}\right )}{d f^2}+\frac{b^2 g^{3/2} n^2 \text{Li}_3\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{(-f)^{5/2}}-\frac{b^2 g^{3/2} n^2 \text{Li}_3\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{(-f)^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.791029, size = 930, normalized size = 1.34 \[ \frac{6 \sqrt{f} g x^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 d^3-2 f^{3/2} \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 d^3+6 g^{3/2} x^3 \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 d^3+2 i b n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (-3 d^3 g^{3/2} \left (\log (d+e x) \log \left (\frac{e \left (i \sqrt{g} x+\sqrt{f}\right )}{e \sqrt{f}-i d \sqrt{g}}\right )+\text{PolyLog}\left (2,-\frac{i \sqrt{g} (d+e x)}{e \sqrt{f}-i d \sqrt{g}}\right )\right ) x^3+3 d^3 g^{3/2} \left (\log (d+e x) \log \left (\frac{e \left (\sqrt{f}-i \sqrt{g} x\right )}{i \sqrt{g} d+e \sqrt{f}}\right )+\text{PolyLog}\left (2,\frac{i \sqrt{g} (d+e x)}{i \sqrt{g} d+e \sqrt{f}}\right )\right ) x^3+6 i d^2 \sqrt{f} g (e x \log (x)-(d+e x) \log (d+e x)) x^2+i f^{3/2} \left (-2 e^3 \log (x) x^3+d e (d-2 e x) x+2 \left (d^3+e^3 x^3\right ) \log (d+e x)\right )\right )+i b^2 n^2 \left (3 d^3 g^{3/2} \left (\log \left (1-\frac{\sqrt{g} (d+e x)}{d \sqrt{g}-i e \sqrt{f}}\right ) \log ^2(d+e x)+2 \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}-i e \sqrt{f}}\right ) \log (d+e x)-2 \text{PolyLog}\left (3,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}-i e \sqrt{f}}\right )\right ) x^3-3 d^3 g^{3/2} \left (\log \left (1-\frac{\sqrt{g} (d+e x)}{\sqrt{g} d+i e \sqrt{f}}\right ) \log ^2(d+e x)+2 \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{\sqrt{g} d+i e \sqrt{f}}\right ) \log (d+e x)-2 \text{PolyLog}\left (3,\frac{\sqrt{g} (d+e x)}{\sqrt{g} d+i e \sqrt{f}}\right )\right ) x^3+6 i d^2 \sqrt{f} g \left (-(d+e x) \log ^2(d+e x)+2 e x \log \left (-\frac{e x}{d}\right ) \log (d+e x)+2 e x \text{PolyLog}\left (2,\frac{e x}{d}+1\right )\right ) x^2+2 i f^{3/2} \left (\log ^2(d+e x) d^3+e x \log (d+e x) d^2+e^2 x^2 d-2 e^2 x^2 \log (d+e x) d+e^3 x^3 \log ^2(d+e x)+3 e^3 x^3 \log (x)-3 e^3 x^3 \log (d+e x)-2 e^3 x^3 \log \left (-\frac{e x}{d}\right ) \log (d+e x)-2 e^3 x^3 \text{PolyLog}\left (2,\frac{e x}{d}+1\right )\right )\right )}{6 d^3 f^{5/2} x^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 9.815, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{2}}{{x}^{4} \left ( g{x}^{2}+f \right ) }}\, 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 (\frac{b^{2} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 2 \, a b \log \left ({\left (e x + d\right )}^{n} c\right ) + a^{2}}{g x^{6} + f x^{4}}, 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 )}^{2}}{{\left (g x^{2} + f\right )} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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