Optimal. Leaf size=821 \[ -\frac {b^2 e \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right ) n^2}{2 \left (e (-f)^{3/2}+d f \sqrt {g}\right ) \sqrt {g}}-\frac {b^2 e \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right ) n^2}{2 f \left (\sqrt {g} d+e \sqrt {-f}\right ) \sqrt {g}}-\frac {b^2 \text {Li}_3\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right ) n^2}{2 (-f)^{3/2} \sqrt {g}}+\frac {b^2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right ) n^2}{2 (-f)^{3/2} \sqrt {g}}-\frac {b e \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{\sqrt {g} d+e \sqrt {-f}}\right ) n}{2 f \left (\sqrt {g} d+e \sqrt {-f}\right ) \sqrt {g}}-\frac {b e \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {g} x+\sqrt {-f}\right )}{e \sqrt {-f}-d \sqrt {g}}\right ) n}{2 \left (e (-f)^{3/2}+d f \sqrt {g}\right ) \sqrt {g}}+\frac {b \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right ) n}{2 (-f)^{3/2} \sqrt {g}}-\frac {b \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right ) n}{2 (-f)^{3/2} \sqrt {g}}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (\sqrt {g} d+e \sqrt {-f}\right ) \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt {-f}-d \sqrt {g}\right ) \left (\sqrt {g} x+\sqrt {-f}\right )}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{\sqrt {g} d+e \sqrt {-f}}\right )}{4 (-f)^{3/2} \sqrt {g}}+\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {g} x+\sqrt {-f}\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{4 (-f)^{3/2} \sqrt {g}} \]
[Out]
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Rubi [A] time = 0.85, antiderivative size = 821, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {2409, 2397, 2394, 2393, 2391, 2396, 2433, 2374, 6589} \[ -\frac {b^2 e \text {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right ) n^2}{2 \left (e (-f)^{3/2}+d f \sqrt {g}\right ) \sqrt {g}}-\frac {b^2 e \text {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right ) n^2}{2 f \left (\sqrt {g} d+e \sqrt {-f}\right ) \sqrt {g}}-\frac {b^2 \text {PolyLog}\left (3,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right ) n^2}{2 (-f)^{3/2} \sqrt {g}}+\frac {b^2 \text {PolyLog}\left (3,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right ) n^2}{2 (-f)^{3/2} \sqrt {g}}-\frac {b e \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 \left (\sqrt {g} d+e \sqrt {-f}\right ) \sqrt {g}}-\frac {b e \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 \left (e (-f)^{3/2}+d f \sqrt {g}\right ) \sqrt {g}}+\frac {b \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)^{3/2} \sqrt {g}}-\frac {b \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)^{3/2} \sqrt {g}}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (\sqrt {g} d+e \sqrt {-f}\right ) \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt {-f}-d \sqrt {g}\right ) \left (\sqrt {g} x+\sqrt {-f}\right )}-\frac {\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)^{3/2} \sqrt {g}}+\frac {\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)^{3/2} \sqrt {g}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2374
Rule 2391
Rule 2393
Rule 2394
Rule 2396
Rule 2397
Rule 2409
Rule 2433
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx &=\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\\ &=-\frac {g \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}-\frac {g \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}-\frac {g \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{-f g-g^2 x^2} \, dx}{2 f}\\ &=-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt {-f}+d \sqrt {g}\right ) \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt {-f}-d \sqrt {g}\right ) \left (\sqrt {-f}+\sqrt {g} x\right )}-\frac {g \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}+\frac {\left (b e \sqrt {g} n\right ) \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {-f} \sqrt {g}+g x} \, dx}{2 f \left (e \sqrt {-f}-d \sqrt {g}\right )}+\frac {\left (b e \sqrt {g} n\right ) \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {-f} \sqrt {g}-g x} \, dx}{2 f \left (e \sqrt {-f}+d \sqrt {g}\right )}\\ &=-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt {-f}+d \sqrt {g}\right ) \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt {-f}-d \sqrt {g}\right ) \left (\sqrt {-f}+\sqrt {g} x\right )}-\frac {b e 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 \left (e \sqrt {-f}+d \sqrt {g}\right ) \sqrt {g}}-\frac {b e 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 \left (e (-f)^{3/2}+d f \sqrt {g}\right ) \sqrt {g}}+\frac {\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt {-f}-\sqrt {g} x} \, dx}{4 (-f)^{3/2}}+\frac {\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt {-f}+\sqrt {g} x} \, dx}{4 (-f)^{3/2}}+\frac {\left (b^2 e^2 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 \left (e \sqrt {-f}+d \sqrt {g}\right ) \sqrt {g}}+\frac {\left (b^2 e^2 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 \left (e (-f)^{3/2}+d f \sqrt {g}\right ) \sqrt {g}}\\ &=-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt {-f}+d \sqrt {g}\right ) \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt {-f}-d \sqrt {g}\right ) \left (\sqrt {-f}+\sqrt {g} x\right )}-\frac {b e 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 \left (e \sqrt {-f}+d \sqrt {g}\right ) \sqrt {g}}-\frac {\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)^{3/2} \sqrt {g}}-\frac {b e 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 \left (e (-f)^{3/2}+d f \sqrt {g}\right ) \sqrt {g}}+\frac {\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)^{3/2} \sqrt {g}}+\frac {(b e 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}{2 (-f)^{3/2} \sqrt {g}}-\frac {(b e 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}{2 (-f)^{3/2} \sqrt {g}}+\frac {\left (b^2 e 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 \left (e \sqrt {-f}+d \sqrt {g}\right ) \sqrt {g}}+\frac {\left (b^2 e 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 \left (e (-f)^{3/2}+d f \sqrt {g}\right ) \sqrt {g}}\\ &=-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt {-f}+d \sqrt {g}\right ) \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt {-f}-d \sqrt {g}\right ) \left (\sqrt {-f}+\sqrt {g} x\right )}-\frac {b e 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 \left (e \sqrt {-f}+d \sqrt {g}\right ) \sqrt {g}}-\frac {\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)^{3/2} \sqrt {g}}-\frac {b e 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 \left (e (-f)^{3/2}+d f \sqrt {g}\right ) \sqrt {g}}+\frac {\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)^{3/2} \sqrt {g}}-\frac {b^2 e n^2 \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \left (e (-f)^{3/2}+d f \sqrt {g}\right ) \sqrt {g}}-\frac {b^2 e n^2 \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f \left (e \sqrt {-f}+d \sqrt {g}\right ) \sqrt {g}}+\frac {(b 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 )}{2 (-f)^{3/2} \sqrt {g}}-\frac {(b 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 )}{2 (-f)^{3/2} \sqrt {g}}\\ &=-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt {-f}+d \sqrt {g}\right ) \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt {-f}-d \sqrt {g}\right ) \left (\sqrt {-f}+\sqrt {g} x\right )}-\frac {b e 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 \left (e \sqrt {-f}+d \sqrt {g}\right ) \sqrt {g}}-\frac {\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)^{3/2} \sqrt {g}}-\frac {b e 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 \left (e (-f)^{3/2}+d f \sqrt {g}\right ) \sqrt {g}}+\frac {\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)^{3/2} \sqrt {g}}-\frac {b^2 e n^2 \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \left (e (-f)^{3/2}+d f \sqrt {g}\right ) \sqrt {g}}+\frac {b 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)^{3/2} \sqrt {g}}-\frac {b^2 e n^2 \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f \left (e \sqrt {-f}+d \sqrt {g}\right ) \sqrt {g}}-\frac {b 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)^{3/2} \sqrt {g}}-\frac {\left (b^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 )}{2 (-f)^{3/2} \sqrt {g}}+\frac {\left (b^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 )}{2 (-f)^{3/2} \sqrt {g}}\\ &=-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt {-f}+d \sqrt {g}\right ) \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt {-f}-d \sqrt {g}\right ) \left (\sqrt {-f}+\sqrt {g} x\right )}-\frac {b e 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 \left (e \sqrt {-f}+d \sqrt {g}\right ) \sqrt {g}}-\frac {\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)^{3/2} \sqrt {g}}-\frac {b e 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 \left (e (-f)^{3/2}+d f \sqrt {g}\right ) \sqrt {g}}+\frac {\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)^{3/2} \sqrt {g}}-\frac {b^2 e n^2 \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \left (e (-f)^{3/2}+d f \sqrt {g}\right ) \sqrt {g}}+\frac {b 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)^{3/2} \sqrt {g}}-\frac {b^2 e n^2 \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f \left (e \sqrt {-f}+d \sqrt {g}\right ) \sqrt {g}}-\frac {b 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)^{3/2} \sqrt {g}}-\frac {b^2 n^2 \text {Li}_3\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 (-f)^{3/2} \sqrt {g}}+\frac {b^2 n^2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 (-f)^{3/2} \sqrt {g}}\\ \end {align*}
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Mathematica [C] time = 3.08, size = 1143, normalized size = 1.39 \[ \frac {\frac {b^2 \left (-\frac {\sqrt {f} \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} \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 )}+i \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 )-i \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}{\sqrt {g}}+\frac {2 b \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (\frac {\sqrt {f} \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} \left (\sqrt {g} (d+e x) \log (d+e x)+e \left (-\sqrt {g} x-i \sqrt {f}\right ) \log \left (\sqrt {g} x+i \sqrt {f}\right )\right )}{\left (i \sqrt {g} d+e \sqrt {f}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}-i \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 )+i \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}{\sqrt {g}}+\frac {2 \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}{\sqrt {g}}+\frac {2 \sqrt {f} 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^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 2 \, a b \log \left ({\left (e x + d\right )}^{n} c\right ) + a^{2}}{g^{2} x^{4} + 2 \, f g x^{2} + f^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}}{{\left (g x^{2} + f\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 46.01, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \left (e x +d \right )^{n}\right )+a \right )^{2}}{\left (g \,x^{2}+f \right )^{2}}\, dx \]
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 \[ \frac {1}{2} \, a^{2} {\left (\frac {x}{f g x^{2} + f^{2}} + \frac {\arctan \left (\frac {g x}{\sqrt {f g}}\right )}{\sqrt {f g} f}\right )} + \int \frac {b^{2} \log \left ({\left (e x + d\right )}^{n}\right )^{2} + b^{2} \log \relax (c)^{2} + 2 \, a b \log \relax (c) + 2 \, {\left (b^{2} \log \relax (c) + a b\right )} \log \left ({\left (e x + d\right )}^{n}\right )}{g^{2} x^{4} + 2 \, f g x^{2} + f^{2}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{{\left (g\,x^2+f\right )}^2} \,d x \]
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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