Optimal. Leaf size=919 \[ \frac {b^2 e \sqrt {g} \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right ) n^2}{2 f \left (e (-f)^{3/2}+d f \sqrt {g}\right )}+\frac {b^2 e \sqrt {g} \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right ) n^2}{2 f^2 \left (\sqrt {g} d+e \sqrt {-f}\right )}+\frac {2 b^2 e \text {Li}_2\left (\frac {e x}{d}+1\right ) n^2}{d f^2}-\frac {3 b^2 \sqrt {g} \text {Li}_3\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right ) n^2}{2 (-f)^{5/2}}+\frac {3 b^2 \sqrt {g} \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right ) n^2}{2 (-f)^{5/2}}+\frac {2 b e \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right ) n}{d f^2}+\frac {b e \sqrt {g} \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 ) n}{2 f^2 \left (\sqrt {g} d+e \sqrt {-f}\right )}+\frac {b e \sqrt {g} \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 ) n}{2 f \left (e (-f)^{3/2}+d f \sqrt {g}\right )}+\frac {3 b \sqrt {g} \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 ) n}{2 (-f)^{5/2}}-\frac {3 b \sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right ) n}{2 (-f)^{5/2}}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f^2 x}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f^2 \left (\sqrt {g} d+e \sqrt {-f}\right ) \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f^2 \left (e \sqrt {-f}-d \sqrt {g}\right ) \left (\sqrt {g} x+\sqrt {-f}\right )}-\frac {3 \sqrt {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 )}{4 (-f)^{5/2}}+\frac {3 \sqrt {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 )}{4 (-f)^{5/2}} \]
[Out]
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Rubi [A] time = 1.61, antiderivative size = 919, normalized size of antiderivative = 1.00, number of steps used = 35, number of rules used = 11, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {2416, 2397, 2394, 2315, 2409, 2393, 2391, 2396, 2433, 2374, 6589} \[ \frac {b^2 e \sqrt {g} \text {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right ) n^2}{2 f \left (e (-f)^{3/2}+d f \sqrt {g}\right )}+\frac {b^2 e \sqrt {g} \text {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right ) n^2}{2 f^2 \left (\sqrt {g} d+e \sqrt {-f}\right )}+\frac {2 b^2 e \text {PolyLog}\left (2,\frac {e x}{d}+1\right ) n^2}{d f^2}-\frac {3 b^2 \sqrt {g} \text {PolyLog}\left (3,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right ) n^2}{2 (-f)^{5/2}}+\frac {3 b^2 \sqrt {g} \text {PolyLog}\left (3,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right ) n^2}{2 (-f)^{5/2}}+\frac {2 b e \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right ) n}{d f^2}+\frac {b e \sqrt {g} \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 ) n}{2 f^2 \left (\sqrt {g} d+e \sqrt {-f}\right )}+\frac {b e \sqrt {g} \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 ) n}{2 f \left (e (-f)^{3/2}+d f \sqrt {g}\right )}+\frac {3 b \sqrt {g} \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 ) n}{2 (-f)^{5/2}}-\frac {3 b \sqrt {g} \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 ) n}{2 (-f)^{5/2}}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f^2 x}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f^2 \left (\sqrt {g} d+e \sqrt {-f}\right ) \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f^2 \left (e \sqrt {-f}-d \sqrt {g}\right ) \left (\sqrt {g} x+\sqrt {-f}\right )}-\frac {3 \sqrt {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 )}{4 (-f)^{5/2}}+\frac {3 \sqrt {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 )}{4 (-f)^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2315
Rule 2374
Rule 2391
Rule 2393
Rule 2394
Rule 2396
Rule 2397
Rule 2409
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^2 \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^2}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f \left (f+g x^2\right )^2}-\frac {g \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^2} \, dx}{f^2}-\frac {g \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2} \, dx}{f^2}-\frac {g \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx}{f}\\ &=-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f^2 x}-\frac {g \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 {g \int \left (-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (\sqrt {-f} \sqrt {g}-g x\right )^2}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (\sqrt {-f} \sqrt {g}+g x\right )^2}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f \left (-f g-g^2 x^2\right )}\right ) \, dx}{f}+\frac {(2 b e n) \int \frac {a+b \log \left (c (d+e x)^n\right )}{x} \, dx}{d f^2}\\ &=\frac {2 b e n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d f^2}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f^2 x}+\frac {g \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 \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}{\left (\sqrt {-f} \sqrt {g}-g x\right )^2} \, dx}{4 f^2}+\frac {g^2 \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (\sqrt {-f} \sqrt {g}+g x\right )^2} \, dx}{4 f^2}+\frac {g^2 \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{-f g-g^2 x^2} \, dx}{2 f^2}-\frac {\left (2 b^2 e^2 n^2\right ) \int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx}{d f^2}\\ &=\frac {2 b e n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d f^2}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f^2 x}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f^2 \left (e \sqrt {-f}+d \sqrt {g}\right ) \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f^2 \left (e \sqrt {-f}-d \sqrt {g}\right ) \left (\sqrt {-f}+\sqrt {g} x\right )}-\frac {\sqrt {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 )}{2 (-f)^{5/2}}+\frac {\sqrt {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 )}{2 (-f)^{5/2}}+\frac {2 b^2 e n^2 \text {Li}_2\left (1+\frac {e x}{d}\right )}{d f^2}+\frac {g^2 \int \left (-\frac {\sqrt {-f} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f g \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 g \left (\sqrt {-f}+\sqrt {g} x\right )}\right ) \, dx}{2 f^2}+\frac {\left (b e \sqrt {g} 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 \sqrt {g} 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 {a+b \log \left (c (d+e x)^n\right )}{\sqrt {-f} \sqrt {g}+g x} \, dx}{2 f^2 \left (e \sqrt {-f}-d \sqrt {g}\right )}-\frac {\left (b e g^{3/2} n\right ) \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {-f} \sqrt {g}-g x} \, dx}{2 f^2 \left (e \sqrt {-f}+d \sqrt {g}\right )}\\ &=\frac {2 b e n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d f^2}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f^2 x}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f^2 \left (e \sqrt {-f}+d \sqrt {g}\right ) \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f^2 \left (e \sqrt {-f}-d \sqrt {g}\right ) \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {b e \sqrt {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^2 \left (e \sqrt {-f}+d \sqrt {g}\right )}-\frac {\sqrt {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 )}{2 (-f)^{5/2}}-\frac {b e \sqrt {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^2 \left (e \sqrt {-f}-d \sqrt {g}\right )}+\frac {\sqrt {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 )}{2 (-f)^{5/2}}+\frac {2 b^2 e n^2 \text {Li}_2\left (1+\frac {e x}{d}\right )}{d f^2}+\frac {g \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt {-f}-\sqrt {g} x} \, dx}{4 (-f)^{5/2}}+\frac {g \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt {-f}+\sqrt {g} x} \, dx}{4 (-f)^{5/2}}+\frac {\left (b \sqrt {g} 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 \sqrt {g} 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^2 \sqrt {g} n^2\right ) \int \frac {\log \left (\frac {e \left (\sqrt {-f} \sqrt {g}+g x\right )}{e \sqrt {-f} \sqrt {g}-d g}\right )}{d+e x} \, dx}{2 f^2 \left (e \sqrt {-f}-d \sqrt {g}\right )}-\frac {\left (b^2 e^2 \sqrt {g} n^2\right ) \int \frac {\log \left (\frac {e \left (\sqrt {-f} \sqrt {g}-g x\right )}{e \sqrt {-f} \sqrt {g}+d g}\right )}{d+e x} \, dx}{2 f^2 \left (e \sqrt {-f}+d \sqrt {g}\right )}\\ &=\frac {2 b e n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d f^2}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f^2 x}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f^2 \left (e \sqrt {-f}+d \sqrt {g}\right ) \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f^2 \left (e \sqrt {-f}-d \sqrt {g}\right ) \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {b e \sqrt {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^2 \left (e \sqrt {-f}+d \sqrt {g}\right )}-\frac {3 \sqrt {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 )}{4 (-f)^{5/2}}-\frac {b e \sqrt {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^2 \left (e \sqrt {-f}-d \sqrt {g}\right )}+\frac {3 \sqrt {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 )}{4 (-f)^{5/2}}+\frac {b \sqrt {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)^{5/2}}-\frac {b \sqrt {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)^{5/2}}+\frac {2 b^2 e n^2 \text {Li}_2\left (1+\frac {e x}{d}\right )}{d f^2}+\frac {\left (b e \sqrt {g} 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}{2 (-f)^{5/2}}-\frac {\left (b e \sqrt {g} 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}{2 (-f)^{5/2}}-\frac {\left (b^2 \sqrt {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)^{5/2}}+\frac {\left (b^2 \sqrt {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)^{5/2}}+\frac {\left (b^2 e \sqrt {g} n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {g x}{e \sqrt {-f} \sqrt {g}-d g}\right )}{x} \, dx,x,d+e x\right )}{2 f^2 \left (e \sqrt {-f}-d \sqrt {g}\right )}-\frac {\left (b^2 e \sqrt {g} n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {g x}{e \sqrt {-f} \sqrt {g}+d g}\right )}{x} \, dx,x,d+e x\right )}{2 f^2 \left (e \sqrt {-f}+d \sqrt {g}\right )}\\ &=\frac {2 b e n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d f^2}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f^2 x}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f^2 \left (e \sqrt {-f}+d \sqrt {g}\right ) \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f^2 \left (e \sqrt {-f}-d \sqrt {g}\right ) \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {b e \sqrt {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^2 \left (e \sqrt {-f}+d \sqrt {g}\right )}-\frac {3 \sqrt {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 )}{4 (-f)^{5/2}}-\frac {b e \sqrt {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^2 \left (e \sqrt {-f}-d \sqrt {g}\right )}+\frac {3 \sqrt {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 )}{4 (-f)^{5/2}}-\frac {b^2 e \sqrt {g} n^2 \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f^2 \left (e \sqrt {-f}-d \sqrt {g}\right )}+\frac {b \sqrt {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)^{5/2}}+\frac {b^2 e \sqrt {g} n^2 \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f^2 \left (e \sqrt {-f}+d \sqrt {g}\right )}-\frac {b \sqrt {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)^{5/2}}+\frac {2 b^2 e n^2 \text {Li}_2\left (1+\frac {e x}{d}\right )}{d f^2}-\frac {b^2 \sqrt {g} 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 \sqrt {g} n^2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{(-f)^{5/2}}+\frac {\left (b \sqrt {g} 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 )}{2 (-f)^{5/2}}-\frac {\left (b \sqrt {g} 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 )}{2 (-f)^{5/2}}\\ &=\frac {2 b e n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d f^2}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f^2 x}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f^2 \left (e \sqrt {-f}+d \sqrt {g}\right ) \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f^2 \left (e \sqrt {-f}-d \sqrt {g}\right ) \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {b e \sqrt {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^2 \left (e \sqrt {-f}+d \sqrt {g}\right )}-\frac {3 \sqrt {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 )}{4 (-f)^{5/2}}-\frac {b e \sqrt {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^2 \left (e \sqrt {-f}-d \sqrt {g}\right )}+\frac {3 \sqrt {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 )}{4 (-f)^{5/2}}-\frac {b^2 e \sqrt {g} n^2 \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f^2 \left (e \sqrt {-f}-d \sqrt {g}\right )}+\frac {3 b \sqrt {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 )}{2 (-f)^{5/2}}+\frac {b^2 e \sqrt {g} n^2 \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f^2 \left (e \sqrt {-f}+d \sqrt {g}\right )}-\frac {3 b \sqrt {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 )}{2 (-f)^{5/2}}+\frac {2 b^2 e n^2 \text {Li}_2\left (1+\frac {e x}{d}\right )}{d f^2}-\frac {b^2 \sqrt {g} 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 \sqrt {g} n^2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{(-f)^{5/2}}-\frac {\left (b^2 \sqrt {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 )}{2 (-f)^{5/2}}+\frac {\left (b^2 \sqrt {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 )}{2 (-f)^{5/2}}\\ &=\frac {2 b e n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d f^2}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f^2 x}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f^2 \left (e \sqrt {-f}+d \sqrt {g}\right ) \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f^2 \left (e \sqrt {-f}-d \sqrt {g}\right ) \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {b e \sqrt {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^2 \left (e \sqrt {-f}+d \sqrt {g}\right )}-\frac {3 \sqrt {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 )}{4 (-f)^{5/2}}-\frac {b e \sqrt {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^2 \left (e \sqrt {-f}-d \sqrt {g}\right )}+\frac {3 \sqrt {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 )}{4 (-f)^{5/2}}-\frac {b^2 e \sqrt {g} n^2 \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f^2 \left (e \sqrt {-f}-d \sqrt {g}\right )}+\frac {3 b \sqrt {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 )}{2 (-f)^{5/2}}+\frac {b^2 e \sqrt {g} n^2 \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f^2 \left (e \sqrt {-f}+d \sqrt {g}\right )}-\frac {3 b \sqrt {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 )}{2 (-f)^{5/2}}+\frac {2 b^2 e n^2 \text {Li}_2\left (1+\frac {e x}{d}\right )}{d f^2}-\frac {3 b^2 \sqrt {g} n^2 \text {Li}_3\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 (-f)^{5/2}}+\frac {3 b^2 \sqrt {g} n^2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 (-f)^{5/2}}\\ \end {align*}
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Mathematica [C] time = 3.69, size = 1304, normalized size = 1.42 \[ \frac {b^2 \left (\frac {\sqrt {f} \sqrt {g} \left (-\sqrt {g} (d+e x) \log ^2(d+e x)+2 e \left (\sqrt {g} x+i \sqrt {f}\right ) \log \left (\frac {e \left (\sqrt {f}-i \sqrt {g} x\right )}{i \sqrt {g} d+e \sqrt {f}}\right ) \log (d+e x)+2 e \left (\sqrt {g} x+i \sqrt {f}\right ) \text {Li}_2\left (\frac {i \sqrt {g} (d+e x)}{i \sqrt {g} d+e \sqrt {f}}\right )\right )}{\left (i \sqrt {g} d+e \sqrt {f}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}-\frac {\sqrt {f} \sqrt {g} \left (\log (d+e x) \left (\sqrt {g} (d+e x) \log (d+e x)+2 i e \left (i \sqrt {g} x+\sqrt {f}\right ) \log \left (\frac {e \left (i \sqrt {g} x+\sqrt {f}\right )}{e \sqrt {f}-i d \sqrt {g}}\right )\right )+2 i e \left (i \sqrt {g} x+\sqrt {f}\right ) \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+i e \sqrt {f}}\right )\right )}{\left (e \sqrt {f}-i d \sqrt {g}\right ) \left (i \sqrt {g} x+\sqrt {f}\right )}+\frac {4 \sqrt {f} \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 )}{d x}-3 i \sqrt {g} \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 )+3 i \sqrt {g} \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 )\right ) n^2+2 b \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (\frac {4 \sqrt {f} (e x \log (x)-(d+e x) \log (d+e x))}{d x}-\frac {\sqrt {f} \sqrt {g} \left (\sqrt {g} (d+e x) \log (d+e x)+i e \left (i \sqrt {g} x+\sqrt {f}\right ) \log \left (i \sqrt {f}-\sqrt {g} x\right )\right )}{\left (e \sqrt {f}-i d \sqrt {g}\right ) \left (i \sqrt {g} x+\sqrt {f}\right )}+\frac {\sqrt {f} \sqrt {g} \left (e \left (\sqrt {g} x+i \sqrt {f}\right ) \log \left (\sqrt {g} x+i \sqrt {f}\right )-\sqrt {g} (d+e x) \log (d+e x)\right )}{\left (i \sqrt {g} d+e \sqrt {f}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}+3 i \sqrt {g} \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 )-3 i \sqrt {g} \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 )\right ) n-6 \sqrt {g} \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-\frac {4 \sqrt {f} \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2}{x}-\frac {2 \sqrt {f} g x \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2}{g x^2+f}}{4 f^{5/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 2 \, a b \log \left ({\left (e x + d\right )}^{n} c\right ) + a^{2}}{g^{2} x^{6} + 2 \, f g x^{4} + f^{2} x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}}{{\left (g x^{2} + f\right )}^{2} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 17.50, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \left (e x +d \right )^{n}\right )+a \right )^{2}}{\left (g \,x^{2}+f \right )^{2} x^{2}}\, dx \]
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 \[ -\frac {1}{2} \, a^{2} {\left (\frac {3 \, g x^{2} + 2 \, f}{f^{2} g x^{3} + f^{3} x} + \frac {3 \, g \arctan \left (\frac {g x}{\sqrt {f g}}\right )}{\sqrt {f g} f^{2}}\right )} + \int \frac {b^{2} \log \left ({\left (e x + d\right )}^{n}\right )^{2} + b^{2} \log \relax (c)^{2} + 2 \, a b \log \relax (c) + 2 \, {\left (b^{2} \log \relax (c) + a b\right )} \log \left ({\left (e x + d\right )}^{n}\right )}{g^{2} x^{6} + 2 \, f g x^{4} + f^{2} x^{2}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{x^2\,{\left (g\,x^2+f\right )}^2} \,d x \]
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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