Optimal. Leaf size=694 \[ \frac {2 b e^3 n \log \left (1-\frac {d}{d+e x}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d^3 f}+\frac {2 b e^2 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d^3 f x}-\frac {2 b e g n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d f^2}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f^2 x}-\frac {b g^{3/2} n \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(-f)^{5/2}}+\frac {b g^{3/2} n \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(-f)^{5/2}}+\frac {g^{3/2} \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{d \sqrt {g}+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 (-f)^{5/2}}-\frac {g^{3/2} \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 (-f)^{5/2}}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 f x^3}-\frac {b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d f x^2}-\frac {2 b^2 e^3 n^2 \text {Li}_2\left (\frac {d}{d+e x}\right )}{3 d^3 f}-\frac {b^2 e^3 n^2 \log (x)}{d^3 f}+\frac {b^2 e^3 n^2 \log (d+e x)}{3 d^3 f}-\frac {b^2 e^2 n^2}{3 d^2 f x}-\frac {2 b^2 e g n^2 \text {Li}_2\left (\frac {e x}{d}+1\right )}{d f^2}+\frac {b^2 g^{3/2} n^2 \text {Li}_3\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{(-f)^{5/2}}-\frac {b^2 g^{3/2} n^2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right )}{(-f)^{5/2}} \]
[Out]
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Rubi [A] time = 1.11, antiderivative size = 717, normalized size of antiderivative = 1.03, number of steps used = 28, number of rules used = 20, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.690, 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 31
Rule 44
Rule 2301
Rule 2314
Rule 2315
Rule 2317
Rule 2319
Rule 2344
Rule 2347
Rule 2374
Rule 2391
Rule 2394
Rule 2396
Rule 2397
Rule 2398
Rule 2409
Rule 2411
Rule 2416
Rule 2433
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*}
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Mathematica [C] time = 0.85, 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 {Li}_2\left (-\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 {Li}_2\left (\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 {Li}_2\left (\frac {\sqrt {g} (d+e x)}{d \sqrt {g}-i e \sqrt {f}}\right ) \log (d+e x)-2 \text {Li}_3\left (\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 {Li}_2\left (\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+i e \sqrt {f}}\right ) \log (d+e x)-2 \text {Li}_3\left (\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 (-\left ((d+e x) \log ^2(d+e x)\right )+2 e x \log \left (-\frac {e x}{d}\right ) \log (d+e x)+2 e x \text {Li}_2\left (\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 {Li}_2\left (\frac {e x}{d}+1\right )\right )\right )}{6 d^3 f^{5/2} x^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 2 \, a b \log \left ({\left (e x + d\right )}^{n} c\right ) + a^{2}}{g x^{6} + f x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}}{{\left (g x^{2} + f\right )} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 34.98, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \left (e x +d \right )^{n}\right )+a \right )^{2}}{\left (g \,x^{2}+f \right ) x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{3} \, a^{2} {\left (\frac {3 \, g^{2} \arctan \left (\frac {g x}{\sqrt {f g}}\right )}{\sqrt {f g} f^{2}} + \frac {3 \, g x^{2} - f}{f^{2} x^{3}}\right )} + \int \frac {b^{2} \log \left ({\left (e x + d\right )}^{n}\right )^{2} + b^{2} \log \relax (c)^{2} + 2 \, a b \log \relax (c) + 2 \, {\left (b^{2} \log \relax (c) + a b\right )} \log \left ({\left (e x + d\right )}^{n}\right )}{g x^{6} + f x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{x^4\,\left (g\,x^2+f\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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