Optimal. Leaf size=970 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 1.65316, antiderivative size = 994, normalized size of antiderivative = 1.02, number of steps used = 38, number of rules used = 19, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.655, Rules used = {2416, 2398, 2411, 2347, 2344, 2301, 2317, 2391, 2314, 31, 2396, 2433, 2374, 6589, 2413, 2418, 2390, 2394, 2393} \[ \frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 e^2}{2 f^2 \left (g d^2+e^2 f\right )}+\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2 e^2}{2 d^2 f^2}+\frac{b^2 n^2 \log (x) e^2}{d^2 f^2}-\frac{b n \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right ) e^2}{d^2 f^2}-\frac{b^2 n^2 \text{PolyLog}\left (2,\frac{e x}{d}+1\right ) e^2}{d^2 f^2}-\frac{b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right ) e}{d^2 f^2 x}-\frac{b \left (\sqrt{-f} \sqrt{g} d+e f\right ) g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{\sqrt{g} d+e \sqrt{-f}}\right ) e}{2 f^3 \left (g d^2+e^2 f\right )}-\frac{b \left (e f-d \sqrt{-f} \sqrt{g}\right ) g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e \left (\sqrt{g} x+\sqrt{-f}\right )}{e \sqrt{-f}-d \sqrt{g}}\right ) e}{2 f^3 \left (g d^2+e^2 f\right )}-\frac{b^2 \left (\sqrt{g} d+e \sqrt{-f}\right ) g n^2 \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right ) e}{2 (-f)^{5/2} \left (g d^2+e^2 f\right )}-\frac{b^2 \left (\sqrt{-f} \sqrt{g} d+e f\right ) g n^2 \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{\sqrt{g} d+e \sqrt{-f}}\right ) e}{2 f^3 \left (g d^2+e^2 f\right )}-\frac{2 g \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^3}-\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 \left (g x^2+f\right )}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 x^2}+\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{\sqrt{g} d+e \sqrt{-f}}\right )}{f^3}+\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{g} x+\sqrt{-f}\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{f^3}+\frac{2 b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{f^3}+\frac{2 b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{\sqrt{g} d+e \sqrt{-f}}\right )}{f^3}-\frac{4 b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{f^3}-\frac{2 b^2 g n^2 \text{PolyLog}\left (3,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{f^3}-\frac{2 b^2 g n^2 \text{PolyLog}\left (3,\frac{\sqrt{g} (d+e x)}{\sqrt{g} d+e \sqrt{-f}}\right )}{f^3}+\frac{4 b^2 g n^2 \text{PolyLog}\left (3,\frac{e x}{d}+1\right )}{f^3} \]
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 2396
Rule 2433
Rule 2374
Rule 6589
Rule 2413
Rule 2418
Rule 2390
Rule 2394
Rule 2393
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3 \left (f+g x^2\right )^2} \, dx &=\int \left (\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^2 x^3}-\frac{2 g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^3 x}+\frac{g^2 x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^2 \left (f+g x^2\right )^2}+\frac{2 g^2 x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^3 \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^3} \, dx}{f^2}-\frac{(2 g) \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x} \, dx}{f^3}+\frac{\left (2 g^2\right ) \int \frac{x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2} \, dx}{f^3}+\frac{g^2 \int \frac{x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx}{f^2}\\ &=-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 x^2}-\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 \left (f+g x^2\right )}-\frac{2 g \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^3}+\frac{\left (2 g^2\right ) \int \left (-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 \sqrt{g} \left (\sqrt{-f}-\sqrt{g} x\right )}+\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 \sqrt{g} \left (\sqrt{-f}+\sqrt{g} x\right )}\right ) \, dx}{f^3}+\frac{(b e n) \int \frac{a+b \log \left (c (d+e x)^n\right )}{x^2 (d+e x)} \, dx}{f^2}+\frac{(4 b e g n) \int \frac{\log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx}{f^3}+\frac{(b e g n) \int \frac{a+b \log \left (c (d+e x)^n\right )}{(d+e x) \left (f+g x^2\right )} \, dx}{f^2}\\ &=-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 x^2}-\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 \left (f+g x^2\right )}-\frac{2 g \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^3}-\frac{g^{3/2} \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt{-f}-\sqrt{g} x} \, dx}{f^3}+\frac{g^{3/2} \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt{-f}+\sqrt{g} x} \, dx}{f^3}+\frac{(b 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 )}{f^2}+\frac{(4 b g n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (-\frac{e \left (-\frac{d}{e}+\frac{x}{e}\right )}{d}\right )}{x} \, dx,x,d+e x\right )}{f^3}+\frac{(b e g n) \int \left (\frac{e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (e^2 f+d^2 g\right ) (d+e x)}-\frac{g (-d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (e^2 f+d^2 g\right ) \left (f+g x^2\right )}\right ) \, dx}{f^2}\\ &=-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 x^2}-\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 \left (f+g x^2\right )}-\frac{2 g \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^3}+\frac{g \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 )}{f^3}+\frac{g \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 )}{f^3}-\frac{4 b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (1+\frac{e x}{d}\right )}{f^3}+\frac{(b 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 )}{d f^2}-\frac{(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 )} \, dx,x,d+e x\right )}{d f^2}-\frac{(2 b e g n) \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^3}-\frac{(2 b e g n) \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^3}+\frac{\left (b e^3 g n\right ) \int \frac{a+b \log \left (c (d+e x)^n\right )}{d+e x} \, dx}{f^2 \left (e^2 f+d^2 g\right )}-\frac{\left (b e g^2 n\right ) \int \frac{(-d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x^2} \, dx}{f^2 \left (e^2 f+d^2 g\right )}+\frac{\left (4 b^2 g n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{d}\right )}{x} \, dx,x,d+e x\right )}{f^3}\\ &=-\frac{b e n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d^2 f^2 x}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 x^2}-\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 \left (f+g x^2\right )}-\frac{2 g \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^3}+\frac{g \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 )}{f^3}+\frac{g \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 )}{f^3}-\frac{4 b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (1+\frac{e x}{d}\right )}{f^3}+\frac{4 b^2 g n^2 \text{Li}_3\left (1+\frac{e x}{d}\right )}{f^3}-\frac{(b e n) \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 )}{d^2 f^2}+\frac{\left (b e^2 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x} \, dx,x,d+e x\right )}{d^2 f^2}-\frac{(2 b g n) \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^3}-\frac{(2 b g n) \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^3}+\frac{\left (b e^2 g n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x} \, dx,x,d+e x\right )}{f^2 \left (e^2 f+d^2 g\right )}-\frac{\left (b e g^2 n\right ) \int \left (\frac{\left (-d \sqrt{-f}-\frac{e f}{\sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f \left (\sqrt{-f}-\sqrt{g} x\right )}+\frac{\left (-d \sqrt{-f}+\frac{e f}{\sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f \left (\sqrt{-f}+\sqrt{g} x\right )}\right ) \, dx}{f^2 \left (e^2 f+d^2 g\right )}+\frac{\left (b^2 e n^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+e x\right )}{d^2 f^2}\\ &=\frac{b^2 e^2 n^2 \log (x)}{d^2 f^2}-\frac{b e n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d^2 f^2 x}-\frac{b e^2 n \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d^2 f^2}+\frac{e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 d^2 f^2}+\frac{e^2 g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 \left (e^2 f+d^2 g\right )}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 x^2}-\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 \left (f+g x^2\right )}-\frac{2 g \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^3}+\frac{g \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 )}{f^3}+\frac{g \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 )}{f^3}+\frac{2 b g 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^3}+\frac{2 b g 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^3}-\frac{4 b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (1+\frac{e x}{d}\right )}{f^3}+\frac{4 b^2 g n^2 \text{Li}_3\left (1+\frac{e x}{d}\right )}{f^3}-\frac{\left (b e \left (\frac{d}{\sqrt{-f}}+\frac{e}{\sqrt{g}}\right ) g^2 n\right ) \int \frac{a+b \log \left (c (d+e x)^n\right )}{\sqrt{-f}+\sqrt{g} x} \, dx}{2 f^2 \left (e^2 f+d^2 g\right )}+\frac{\left (b e \left (\frac{d f}{(-f)^{3/2}}+\frac{e}{\sqrt{g}}\right ) g^2 n\right ) \int \frac{a+b \log \left (c (d+e x)^n\right )}{\sqrt{-f}-\sqrt{g} x} \, dx}{2 f^2 \left (e^2 f+d^2 g\right )}+\frac{\left (b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{d}\right )}{x} \, dx,x,d+e x\right )}{d^2 f^2}-\frac{\left (2 b^2 g 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^3}-\frac{\left (2 b^2 g 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^3}\\ &=\frac{b^2 e^2 n^2 \log (x)}{d^2 f^2}-\frac{b e n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d^2 f^2 x}-\frac{b e^2 n \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d^2 f^2}+\frac{e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 d^2 f^2}+\frac{e^2 g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 \left (e^2 f+d^2 g\right )}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 x^2}-\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 \left (f+g x^2\right )}-\frac{2 g \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^3}-\frac{b e \left (e f+d \sqrt{-f} \sqrt{g}\right ) g n \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 f^3 \left (e^2 f+d^2 g\right )}+\frac{g \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 )}{f^3}-\frac{b e \left (e \sqrt{-f}+d \sqrt{g}\right ) g n \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 (-f)^{5/2} \left (e^2 f+d^2 g\right )}+\frac{g \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 )}{f^3}+\frac{2 b g 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^3}+\frac{2 b g 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^3}-\frac{b^2 e^2 n^2 \text{Li}_2\left (1+\frac{e x}{d}\right )}{d^2 f^2}-\frac{4 b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (1+\frac{e x}{d}\right )}{f^3}-\frac{2 b^2 g n^2 \text{Li}_3\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{f^3}-\frac{2 b^2 g n^2 \text{Li}_3\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{f^3}+\frac{4 b^2 g n^2 \text{Li}_3\left (1+\frac{e x}{d}\right )}{f^3}+\frac{\left (b^2 e^2 \left (e \sqrt{-f}+d \sqrt{g}\right ) g n^2\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 (-f)^{5/2} \left (e^2 f+d^2 g\right )}+\frac{\left (b^2 e^2 \left (e f+d \sqrt{-f} \sqrt{g}\right ) g n^2\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 f^3 \left (e^2 f+d^2 g\right )}\\ &=\frac{b^2 e^2 n^2 \log (x)}{d^2 f^2}-\frac{b e n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d^2 f^2 x}-\frac{b e^2 n \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d^2 f^2}+\frac{e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 d^2 f^2}+\frac{e^2 g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 \left (e^2 f+d^2 g\right )}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 x^2}-\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 \left (f+g x^2\right )}-\frac{2 g \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^3}-\frac{b e \left (e f+d \sqrt{-f} \sqrt{g}\right ) g n \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 f^3 \left (e^2 f+d^2 g\right )}+\frac{g \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 )}{f^3}-\frac{b e \left (e \sqrt{-f}+d \sqrt{g}\right ) g n \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 (-f)^{5/2} \left (e^2 f+d^2 g\right )}+\frac{g \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 )}{f^3}+\frac{2 b g 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^3}+\frac{2 b g 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^3}-\frac{b^2 e^2 n^2 \text{Li}_2\left (1+\frac{e x}{d}\right )}{d^2 f^2}-\frac{4 b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (1+\frac{e x}{d}\right )}{f^3}-\frac{2 b^2 g n^2 \text{Li}_3\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{f^3}-\frac{2 b^2 g n^2 \text{Li}_3\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{f^3}+\frac{4 b^2 g n^2 \text{Li}_3\left (1+\frac{e x}{d}\right )}{f^3}+\frac{\left (b^2 e \left (e \sqrt{-f}+d \sqrt{g}\right ) g n^2\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 (-f)^{5/2} \left (e^2 f+d^2 g\right )}+\frac{\left (b^2 e \left (e f+d \sqrt{-f} \sqrt{g}\right ) g n^2\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 f^3 \left (e^2 f+d^2 g\right )}\\ &=\frac{b^2 e^2 n^2 \log (x)}{d^2 f^2}-\frac{b e n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d^2 f^2 x}-\frac{b e^2 n \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d^2 f^2}+\frac{e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 d^2 f^2}+\frac{e^2 g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 \left (e^2 f+d^2 g\right )}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 x^2}-\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 \left (f+g x^2\right )}-\frac{2 g \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^3}-\frac{b e \left (e f+d \sqrt{-f} \sqrt{g}\right ) g n \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 f^3 \left (e^2 f+d^2 g\right )}+\frac{g \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 )}{f^3}-\frac{b e \left (e \sqrt{-f}+d \sqrt{g}\right ) g n \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 (-f)^{5/2} \left (e^2 f+d^2 g\right )}+\frac{g \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 )}{f^3}-\frac{b^2 e \left (e \sqrt{-f}+d \sqrt{g}\right ) g n^2 \text{Li}_2\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 (-f)^{5/2} \left (e^2 f+d^2 g\right )}+\frac{2 b g 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^3}-\frac{b^2 e \left (e f+d \sqrt{-f} \sqrt{g}\right ) g n^2 \text{Li}_2\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 f^3 \left (e^2 f+d^2 g\right )}+\frac{2 b g 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^3}-\frac{b^2 e^2 n^2 \text{Li}_2\left (1+\frac{e x}{d}\right )}{d^2 f^2}-\frac{4 b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (1+\frac{e x}{d}\right )}{f^3}-\frac{2 b^2 g n^2 \text{Li}_3\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{f^3}-\frac{2 b^2 g n^2 \text{Li}_3\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{f^3}+\frac{4 b^2 g n^2 \text{Li}_3\left (1+\frac{e x}{d}\right )}{f^3}\\ \end{align*}
Mathematica [C] time = 2.93581, size = 1391, normalized size = 1.43 \[ \text{result too large to display} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.444, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{2}}{{x}^{3} \left ( g{x}^{2}+f \right ) ^{2}}}\, dx \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}{2} \, a^{2}{\left (\frac{2 \, g x^{2} + f}{f^{2} g x^{4} + f^{3} x^{2}} - \frac{2 \, g \log \left (g x^{2} + f\right )}{f^{3}} + \frac{4 \, g \log \left (x\right )}{f^{3}}\right )} + \int \frac{b^{2} \log \left ({\left (e x + d\right )}^{n}\right )^{2} + b^{2} \log \left (c\right )^{2} + 2 \, a b \log \left (c\right ) + 2 \,{\left (b^{2} \log \left (c\right ) + a b\right )} \log \left ({\left (e x + d\right )}^{n}\right )}{g^{2} x^{7} + 2 \, f g x^{5} + f^{2} x^{3}}\,{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^{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^{2} x^{7} + 2 \, f g x^{5} + f^{2} x^{3}}, 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 )}^{2} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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