Optimal. Leaf size=936 \[ \text{result too large to display} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.52967, antiderivative size = 936, normalized size of antiderivative = 1., number of steps used = 34, number of rules used = 18, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.621, Rules used = {2416, 2401, 2389, 2296, 2295, 2390, 2305, 2304, 2413, 2418, 2301, 2394, 2393, 2391, 2396, 2433, 2374, 6589} \[ \frac{n^2 (d+e x)^2 b^2}{4 e^2 g^2}-\frac{2 d n^2 x b^2}{e g^2}+\frac{2 d n (d+e x) \log \left (c (d+e x)^n\right ) b^2}{e^2 g^2}-\frac{e (-f)^{3/2} \left (\sqrt{g} d+e \sqrt{-f}\right ) n^2 \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right ) b^2}{2 g^3 \left (g d^2+e^2 f\right )}-\frac{e (-f)^{3/2} \left (e \sqrt{-f}-d \sqrt{g}\right ) n^2 \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{\sqrt{g} d+e \sqrt{-f}}\right ) b^2}{2 g^3 \left (g d^2+e^2 f\right )}+\frac{2 f n^2 \text{PolyLog}\left (3,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right ) b^2}{g^3}+\frac{2 f n^2 \text{PolyLog}\left (3,\frac{\sqrt{g} (d+e x)}{\sqrt{g} d+e \sqrt{-f}}\right ) b^2}{g^3}+\frac{2 a d n x b}{e g^2}-\frac{n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) b}{2 e^2 g^2}-\frac{e f \left (\sqrt{-f} \sqrt{g} d+e f\right ) 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 ) b}{2 g^3 \left (g d^2+e^2 f\right )}-\frac{e (-f)^{3/2} \left (\sqrt{g} d+e \sqrt{-f}\right ) 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 ) b}{2 g^3 \left (g d^2+e^2 f\right )}-\frac{2 f 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 ) b}{g^3}-\frac{2 f 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 ) b}{g^3}+\frac{(d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2 g^2}-\frac{d (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 g^2}+\frac{e^2 f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g^3 \left (g d^2+e^2 f\right )}-\frac{f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g^3 \left (g x^2+f\right )}-\frac{f \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 )}{g^3}-\frac{f \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 )}{g^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2416
Rule 2401
Rule 2389
Rule 2296
Rule 2295
Rule 2390
Rule 2305
Rule 2304
Rule 2413
Rule 2418
Rule 2301
Rule 2394
Rule 2393
Rule 2391
Rule 2396
Rule 2433
Rule 2374
Rule 6589
Rubi steps
\begin{align*} \int \frac{x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx &=\int \left (\frac{x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g^2}+\frac{f^2 x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g^2 \left (f+g x^2\right )^2}-\frac{2 f x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g^2 \left (f+g x^2\right )}\right ) \, dx\\ &=\frac{\int x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx}{g^2}-\frac{(2 f) \int \frac{x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2} \, dx}{g^2}+\frac{f^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}{g^2}\\ &=-\frac{f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g^3 \left (f+g x^2\right )}+\frac{\int \left (-\frac{d \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}\right ) \, dx}{g^2}-\frac{(2 f) \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}{g^2}+\frac{\left (b e f^2 n\right ) \int \frac{a+b \log \left (c (d+e x)^n\right )}{(d+e x) \left (f+g x^2\right )} \, dx}{g^3}\\ &=-\frac{f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g^3 \left (f+g x^2\right )}+\frac{f \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt{-f}-\sqrt{g} x} \, dx}{g^{5/2}}-\frac{f \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt{-f}+\sqrt{g} x} \, dx}{g^{5/2}}+\frac{\int (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx}{e g^2}-\frac{d \int \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx}{e g^2}+\frac{\left (b e f^2 n\right ) \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}{g^3}\\ &=-\frac{f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g^3 \left (f+g x^2\right )}-\frac{f \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 )}{g^3}-\frac{f \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 )}{g^3}+\frac{\operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e^2 g^2}-\frac{d \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e^2 g^2}+\frac{(2 b e f 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}{g^3}+\frac{(2 b e f 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}{g^3}+\frac{\left (b e^3 f^2 n\right ) \int \frac{a+b \log \left (c (d+e x)^n\right )}{d+e x} \, dx}{g^3 \left (e^2 f+d^2 g\right )}-\frac{\left (b e f^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}{g^2 \left (e^2 f+d^2 g\right )}\\ &=-\frac{d (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 g^2}+\frac{(d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2 g^2}-\frac{f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g^3 \left (f+g x^2\right )}-\frac{f \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 )}{g^3}-\frac{f \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 )}{g^3}+\frac{(2 b f 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 )}{g^3}+\frac{(2 b f 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 )}{g^3}-\frac{(b n) \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e^2 g^2}+\frac{(2 b d n) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e^2 g^2}+\frac{\left (b e^2 f^2 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x} \, dx,x,d+e x\right )}{g^3 \left (e^2 f+d^2 g\right )}-\frac{\left (b e f^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}{g^2 \left (e^2 f+d^2 g\right )}\\ &=\frac{2 a b d n x}{e g^2}+\frac{b^2 n^2 (d+e x)^2}{4 e^2 g^2}-\frac{b n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2 g^2}+\frac{e^2 f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g^3 \left (e^2 f+d^2 g\right )}-\frac{d (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 g^2}+\frac{(d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2 g^2}-\frac{f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g^3 \left (f+g x^2\right )}-\frac{f \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 )}{g^3}-\frac{f \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 )}{g^3}-\frac{2 b f 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 )}{g^3}-\frac{2 b f 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 )}{g^3}+\frac{\left (2 b^2 d n\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e^2 g^2}-\frac{\left (b e f^2 \left (\frac{d}{\sqrt{-f}}+\frac{e}{\sqrt{g}}\right ) n\right ) \int \frac{a+b \log \left (c (d+e x)^n\right )}{\sqrt{-f}+\sqrt{g} x} \, dx}{2 g^2 \left (e^2 f+d^2 g\right )}+\frac{\left (b e f^2 \left (\frac{d f}{(-f)^{3/2}}+\frac{e}{\sqrt{g}}\right ) n\right ) \int \frac{a+b \log \left (c (d+e x)^n\right )}{\sqrt{-f}-\sqrt{g} x} \, dx}{2 g^2 \left (e^2 f+d^2 g\right )}+\frac{\left (2 b^2 f 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 )}{g^3}+\frac{\left (2 b^2 f 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 )}{g^3}\\ &=\frac{2 a b d n x}{e g^2}-\frac{2 b^2 d n^2 x}{e g^2}+\frac{b^2 n^2 (d+e x)^2}{4 e^2 g^2}+\frac{2 b^2 d n (d+e x) \log \left (c (d+e x)^n\right )}{e^2 g^2}-\frac{b n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2 g^2}+\frac{e^2 f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g^3 \left (e^2 f+d^2 g\right )}-\frac{d (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 g^2}+\frac{(d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2 g^2}-\frac{f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g^3 \left (f+g x^2\right )}-\frac{b e f \left (e f+d \sqrt{-f} \sqrt{g}\right ) 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 g^3 \left (e^2 f+d^2 g\right )}-\frac{f \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 )}{g^3}-\frac{b e f \left (e f-d \sqrt{-f} \sqrt{g}\right ) 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 g^3 \left (e^2 f+d^2 g\right )}-\frac{f \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 )}{g^3}-\frac{2 b f 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 )}{g^3}-\frac{2 b f 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 )}{g^3}+\frac{2 b^2 f n^2 \text{Li}_3\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{g^3}+\frac{2 b^2 f n^2 \text{Li}_3\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{g^3}+\frac{\left (b^2 e^2 f^2 \left (\frac{d}{\sqrt{-f}}+\frac{e}{\sqrt{g}}\right ) 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 g^{5/2} \left (e^2 f+d^2 g\right )}+\frac{\left (b^2 e^2 f^2 \left (\frac{d f}{(-f)^{3/2}}+\frac{e}{\sqrt{g}}\right ) 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 g^{5/2} \left (e^2 f+d^2 g\right )}\\ &=\frac{2 a b d n x}{e g^2}-\frac{2 b^2 d n^2 x}{e g^2}+\frac{b^2 n^2 (d+e x)^2}{4 e^2 g^2}+\frac{2 b^2 d n (d+e x) \log \left (c (d+e x)^n\right )}{e^2 g^2}-\frac{b n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2 g^2}+\frac{e^2 f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g^3 \left (e^2 f+d^2 g\right )}-\frac{d (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 g^2}+\frac{(d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2 g^2}-\frac{f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g^3 \left (f+g x^2\right )}-\frac{b e f \left (e f+d \sqrt{-f} \sqrt{g}\right ) 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 g^3 \left (e^2 f+d^2 g\right )}-\frac{f \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 )}{g^3}-\frac{b e f \left (e f-d \sqrt{-f} \sqrt{g}\right ) 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 g^3 \left (e^2 f+d^2 g\right )}-\frac{f \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 )}{g^3}-\frac{2 b f 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 )}{g^3}-\frac{2 b f 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 )}{g^3}+\frac{2 b^2 f n^2 \text{Li}_3\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{g^3}+\frac{2 b^2 f n^2 \text{Li}_3\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{g^3}+\frac{\left (b^2 e f^2 \left (\frac{d}{\sqrt{-f}}+\frac{e}{\sqrt{g}}\right ) 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 g^{5/2} \left (e^2 f+d^2 g\right )}+\frac{\left (b^2 e f^2 \left (\frac{d f}{(-f)^{3/2}}+\frac{e}{\sqrt{g}}\right ) 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 g^{5/2} \left (e^2 f+d^2 g\right )}\\ &=\frac{2 a b d n x}{e g^2}-\frac{2 b^2 d n^2 x}{e g^2}+\frac{b^2 n^2 (d+e x)^2}{4 e^2 g^2}+\frac{2 b^2 d n (d+e x) \log \left (c (d+e x)^n\right )}{e^2 g^2}-\frac{b n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2 g^2}+\frac{e^2 f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g^3 \left (e^2 f+d^2 g\right )}-\frac{d (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 g^2}+\frac{(d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2 g^2}-\frac{f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g^3 \left (f+g x^2\right )}-\frac{b e f \left (e f+d \sqrt{-f} \sqrt{g}\right ) 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 g^3 \left (e^2 f+d^2 g\right )}-\frac{f \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 )}{g^3}-\frac{b e f \left (e f-d \sqrt{-f} \sqrt{g}\right ) 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 g^3 \left (e^2 f+d^2 g\right )}-\frac{f \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 )}{g^3}-\frac{b^2 e f \left (e f-d \sqrt{-f} \sqrt{g}\right ) n^2 \text{Li}_2\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 g^3 \left (e^2 f+d^2 g\right )}-\frac{2 b f 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 )}{g^3}-\frac{b^2 e f \left (e f+d \sqrt{-f} \sqrt{g}\right ) n^2 \text{Li}_2\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 g^3 \left (e^2 f+d^2 g\right )}-\frac{2 b f 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 )}{g^3}+\frac{2 b^2 f n^2 \text{Li}_3\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{g^3}+\frac{2 b^2 f n^2 \text{Li}_3\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{g^3}\\ \end{align*}
Mathematica [C] time = 2.7655, size = 1254, normalized size = 1.34 \[ \frac{b^2 \left (\frac{i \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{PolyLog}\left (2,\frac{i \sqrt{g} (d+e x)}{i \sqrt{g} d+e \sqrt{f}}\right )\right ) f^{3/2}}{\left (i \sqrt{g} d+e \sqrt{f}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}-\frac{\left (\log (d+e x) \left (2 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 )-i \sqrt{g} (d+e x) \log (d+e x)\right )+2 e \left (i \sqrt{g} x+\sqrt{f}\right ) \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{\sqrt{g} d+i e \sqrt{f}}\right )\right ) f^{3/2}}{\left (e \sqrt{f}-i d \sqrt{g}\right ) \left (i \sqrt{g} x+\sqrt{f}\right )}-4 \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 ) f-4 \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 ) f+\frac{g \left (-2 \left (d^2-e^2 x^2\right ) \log ^2(d+e x)+\left (6 d^2+4 e x d-2 e^2 x^2\right ) \log (d+e x)+e x (e x-6 d)\right )}{e^2}\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{\left (i \sqrt{g} (d+e x) \log (d+e x)-e \left (i \sqrt{g} x+\sqrt{f}\right ) \log \left (i \sqrt{f}-\sqrt{g} x\right )\right ) f^{3/2}}{\left (e \sqrt{f}-i d \sqrt{g}\right ) \left (i \sqrt{g} x+\sqrt{f}\right )}+\frac{i \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 ) f^{3/2}}{\left (i \sqrt{g} d+e \sqrt{f}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}-4 \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 ) f-4 \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 ) f+\frac{g \left (e x (2 d-e x)-2 \left (d^2-e^2 x^2\right ) \log (d+e x)\right )}{e^2}\right ) n+2 g x^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2-\frac{2 f^2 \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 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (g x^2+f\right )}{4 g^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 1.707, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{5} \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{2}}{ \left ( g{x}^{2}+f \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{2} \, a^{2}{\left (\frac{f^{2}}{g^{4} x^{2} + f g^{3}} - \frac{x^{2}}{g^{2}} + \frac{2 \, f \log \left (g x^{2} + f\right )}{g^{3}}\right )} + \int \frac{b^{2} x^{5} \log \left ({\left (e x + d\right )}^{n}\right )^{2} + 2 \,{\left (b^{2} \log \left (c\right ) + a b\right )} x^{5} \log \left ({\left (e x + d\right )}^{n}\right ) +{\left (b^{2} \log \left (c\right )^{2} + 2 \, a b \log \left (c\right )\right )} x^{5}}{g^{2} x^{4} + 2 \, f g x^{2} + f^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} x^{5} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 2 \, a b x^{5} \log \left ({\left (e x + d\right )}^{n} c\right ) + a^{2} x^{5}}{g^{2} x^{4} + 2 \, f g x^{2} + f^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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} x^{5}}{{\left (g x^{2} + f\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]