Optimal. Leaf size=598 \[ -\frac{2 b e^4 n \text{PolyLog}\left (2,-\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^4}-\frac{b^2 e^4 n^2 \text{PolyLog}\left (2,\frac{f \sqrt{x}}{e}+1\right )}{f^4}+\frac{4 b^2 e^4 n^2 \text{PolyLog}\left (3,-\frac{f \sqrt{x}}{e}\right )}{f^4}+\frac{1}{2} x^2 \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{1}{2} b n x^2 \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{e^4 \log \left (\frac{f \sqrt{x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 f^4}+\frac{b e^4 n \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^4}+\frac{e^3 \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{2 f^3}-\frac{5 b e^3 n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{2 f^3}-\frac{e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{4 f^2}+\frac{b e^2 n x \left (a+b \log \left (c x^n\right )\right )}{4 f^2}+\frac{e x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{6 f}-\frac{7 b e n x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{18 f}-\frac{1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac{a b e^2 n x}{2 f^2}+\frac{b^2 e^2 n x \log \left (c x^n\right )}{2 f^2}+\frac{1}{4} b^2 n^2 x^2 \log \left (d \left (e+f \sqrt{x}\right )\right )+\frac{21 b^2 e^3 n^2 \sqrt{x}}{4 f^3}-\frac{7 b^2 e^2 n^2 x}{8 f^2}-\frac{b^2 e^4 n^2 \log \left (e+f \sqrt{x}\right )}{4 f^4}-\frac{b^2 e^4 n^2 \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{f^4}+\frac{37 b^2 e n^2 x^{3/2}}{108 f}-\frac{3}{16} b^2 n^2 x^2 \]
[Out]
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Rubi [A] time = 0.656244, antiderivative size = 598, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 13, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2454, 2395, 43, 2377, 2295, 2304, 2375, 2337, 2374, 6589, 2376, 2394, 2315} \[ -\frac{2 b e^4 n \text{PolyLog}\left (2,-\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^4}-\frac{b^2 e^4 n^2 \text{PolyLog}\left (2,\frac{f \sqrt{x}}{e}+1\right )}{f^4}+\frac{4 b^2 e^4 n^2 \text{PolyLog}\left (3,-\frac{f \sqrt{x}}{e}\right )}{f^4}+\frac{1}{2} x^2 \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{1}{2} b n x^2 \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{e^4 \log \left (\frac{f \sqrt{x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 f^4}+\frac{b e^4 n \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^4}+\frac{e^3 \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{2 f^3}-\frac{5 b e^3 n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{2 f^3}-\frac{e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{4 f^2}+\frac{b e^2 n x \left (a+b \log \left (c x^n\right )\right )}{4 f^2}+\frac{e x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{6 f}-\frac{7 b e n x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{18 f}-\frac{1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac{a b e^2 n x}{2 f^2}+\frac{b^2 e^2 n x \log \left (c x^n\right )}{2 f^2}+\frac{1}{4} b^2 n^2 x^2 \log \left (d \left (e+f \sqrt{x}\right )\right )+\frac{21 b^2 e^3 n^2 \sqrt{x}}{4 f^3}-\frac{7 b^2 e^2 n^2 x}{8 f^2}-\frac{b^2 e^4 n^2 \log \left (e+f \sqrt{x}\right )}{4 f^4}-\frac{b^2 e^4 n^2 \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{f^4}+\frac{37 b^2 e n^2 x^{3/2}}{108 f}-\frac{3}{16} b^2 n^2 x^2 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2454
Rule 2395
Rule 43
Rule 2377
Rule 2295
Rule 2304
Rule 2375
Rule 2337
Rule 2374
Rule 6589
Rule 2376
Rule 2394
Rule 2315
Rubi steps
\begin{align*} \int x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx &=\frac{e^3 \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{2 f^3}-\frac{e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{4 f^2}+\frac{e x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{6 f}-\frac{1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac{e^4 \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 f^4}+\frac{1}{2} x^2 \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2-(2 b n) \int \left (-\frac{e^2 \left (a+b \log \left (c x^n\right )\right )}{4 f^2}+\frac{e^3 \left (a+b \log \left (c x^n\right )\right )}{2 f^3 \sqrt{x}}+\frac{e \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac{1}{8} x \left (a+b \log \left (c x^n\right )\right )-\frac{e^4 \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^4 x}+\frac{1}{2} x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\\ &=\frac{e^3 \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{2 f^3}-\frac{e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{4 f^2}+\frac{e x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{6 f}-\frac{1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac{e^4 \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 f^4}+\frac{1}{2} x^2 \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{4} (b n) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx-(b n) \int x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac{\left (b e^4 n\right ) \int \frac{\log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx}{f^4}-\frac{\left (b e^3 n\right ) \int \frac{a+b \log \left (c x^n\right )}{\sqrt{x}} \, dx}{f^3}+\frac{\left (b e^2 n\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{2 f^2}-\frac{(b e n) \int \sqrt{x} \left (a+b \log \left (c x^n\right )\right ) \, dx}{3 f}\\ &=\frac{4 b^2 e^3 n^2 \sqrt{x}}{f^3}+\frac{a b e^2 n x}{2 f^2}+\frac{4 b^2 e n^2 x^{3/2}}{27 f}-\frac{1}{16} b^2 n^2 x^2-\frac{5 b e^3 n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{2 f^3}+\frac{b e^2 n x \left (a+b \log \left (c x^n\right )\right )}{4 f^2}-\frac{7 b e n x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{18 f}+\frac{1}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac{b e^4 n \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^4}-\frac{1}{2} b n x^2 \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{e^3 \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{2 f^3}-\frac{e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{4 f^2}+\frac{e x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{6 f}-\frac{1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{2} x^2 \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{e^4 \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\left (e+f \sqrt{x}\right ) \sqrt{x}} \, dx}{4 f^3}+\frac{\left (b^2 e^2 n\right ) \int \log \left (c x^n\right ) \, dx}{2 f^2}+\left (b^2 n^2\right ) \int \left (-\frac{e^2}{4 f^2}+\frac{e^3}{2 f^3 \sqrt{x}}+\frac{e \sqrt{x}}{6 f}-\frac{x}{8}-\frac{e^4 \log \left (e+f \sqrt{x}\right )}{2 f^4 x}+\frac{1}{2} x \log \left (d \left (e+f \sqrt{x}\right )\right )\right ) \, dx\\ &=\frac{5 b^2 e^3 n^2 \sqrt{x}}{f^3}+\frac{a b e^2 n x}{2 f^2}-\frac{3 b^2 e^2 n^2 x}{4 f^2}+\frac{7 b^2 e n^2 x^{3/2}}{27 f}-\frac{1}{8} b^2 n^2 x^2+\frac{b^2 e^2 n x \log \left (c x^n\right )}{2 f^2}-\frac{5 b e^3 n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{2 f^3}+\frac{b e^2 n x \left (a+b \log \left (c x^n\right )\right )}{4 f^2}-\frac{7 b e n x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{18 f}+\frac{1}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac{b e^4 n \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^4}-\frac{1}{2} b n x^2 \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{e^3 \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{2 f^3}-\frac{e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{4 f^2}+\frac{e x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{6 f}-\frac{1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{2} x^2 \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{e^4 \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 f^4}+\frac{\left (b e^4 n\right ) \int \frac{\log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx}{f^4}+\frac{1}{2} \left (b^2 n^2\right ) \int x \log \left (d \left (e+f \sqrt{x}\right )\right ) \, dx-\frac{\left (b^2 e^4 n^2\right ) \int \frac{\log \left (e+f \sqrt{x}\right )}{x} \, dx}{2 f^4}\\ &=\frac{5 b^2 e^3 n^2 \sqrt{x}}{f^3}+\frac{a b e^2 n x}{2 f^2}-\frac{3 b^2 e^2 n^2 x}{4 f^2}+\frac{7 b^2 e n^2 x^{3/2}}{27 f}-\frac{1}{8} b^2 n^2 x^2+\frac{b^2 e^2 n x \log \left (c x^n\right )}{2 f^2}-\frac{5 b e^3 n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{2 f^3}+\frac{b e^2 n x \left (a+b \log \left (c x^n\right )\right )}{4 f^2}-\frac{7 b e n x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{18 f}+\frac{1}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac{b e^4 n \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^4}-\frac{1}{2} b n x^2 \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{e^3 \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{2 f^3}-\frac{e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{4 f^2}+\frac{e x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{6 f}-\frac{1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{2} x^2 \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{e^4 \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 f^4}-\frac{2 b e^4 n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{f^4}+\left (b^2 n^2\right ) \operatorname{Subst}\left (\int x^3 \log (d (e+f x)) \, dx,x,\sqrt{x}\right )-\frac{\left (b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (e+f x)}{x} \, dx,x,\sqrt{x}\right )}{f^4}+\frac{\left (2 b^2 e^4 n^2\right ) \int \frac{\text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{x} \, dx}{f^4}\\ &=\frac{5 b^2 e^3 n^2 \sqrt{x}}{f^3}+\frac{a b e^2 n x}{2 f^2}-\frac{3 b^2 e^2 n^2 x}{4 f^2}+\frac{7 b^2 e n^2 x^{3/2}}{27 f}-\frac{1}{8} b^2 n^2 x^2+\frac{1}{4} b^2 n^2 x^2 \log \left (d \left (e+f \sqrt{x}\right )\right )-\frac{b^2 e^4 n^2 \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{f^4}+\frac{b^2 e^2 n x \log \left (c x^n\right )}{2 f^2}-\frac{5 b e^3 n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{2 f^3}+\frac{b e^2 n x \left (a+b \log \left (c x^n\right )\right )}{4 f^2}-\frac{7 b e n x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{18 f}+\frac{1}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac{b e^4 n \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^4}-\frac{1}{2} b n x^2 \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{e^3 \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{2 f^3}-\frac{e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{4 f^2}+\frac{e x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{6 f}-\frac{1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{2} x^2 \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{e^4 \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 f^4}-\frac{2 b e^4 n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{f^4}+\frac{4 b^2 e^4 n^2 \text{Li}_3\left (-\frac{f \sqrt{x}}{e}\right )}{f^4}+\frac{\left (b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{f x}{e}\right )}{e+f x} \, dx,x,\sqrt{x}\right )}{f^3}-\frac{1}{4} \left (b^2 f n^2\right ) \operatorname{Subst}\left (\int \frac{x^4}{e+f x} \, dx,x,\sqrt{x}\right )\\ &=\frac{5 b^2 e^3 n^2 \sqrt{x}}{f^3}+\frac{a b e^2 n x}{2 f^2}-\frac{3 b^2 e^2 n^2 x}{4 f^2}+\frac{7 b^2 e n^2 x^{3/2}}{27 f}-\frac{1}{8} b^2 n^2 x^2+\frac{1}{4} b^2 n^2 x^2 \log \left (d \left (e+f \sqrt{x}\right )\right )-\frac{b^2 e^4 n^2 \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{f^4}+\frac{b^2 e^2 n x \log \left (c x^n\right )}{2 f^2}-\frac{5 b e^3 n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{2 f^3}+\frac{b e^2 n x \left (a+b \log \left (c x^n\right )\right )}{4 f^2}-\frac{7 b e n x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{18 f}+\frac{1}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac{b e^4 n \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^4}-\frac{1}{2} b n x^2 \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{e^3 \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{2 f^3}-\frac{e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{4 f^2}+\frac{e x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{6 f}-\frac{1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{2} x^2 \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{e^4 \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 f^4}-\frac{b^2 e^4 n^2 \text{Li}_2\left (1+\frac{f \sqrt{x}}{e}\right )}{f^4}-\frac{2 b e^4 n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{f^4}+\frac{4 b^2 e^4 n^2 \text{Li}_3\left (-\frac{f \sqrt{x}}{e}\right )}{f^4}-\frac{1}{4} \left (b^2 f n^2\right ) \operatorname{Subst}\left (\int \left (-\frac{e^3}{f^4}+\frac{e^2 x}{f^3}-\frac{e x^2}{f^2}+\frac{x^3}{f}+\frac{e^4}{f^4 (e+f x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{21 b^2 e^3 n^2 \sqrt{x}}{4 f^3}+\frac{a b e^2 n x}{2 f^2}-\frac{7 b^2 e^2 n^2 x}{8 f^2}+\frac{37 b^2 e n^2 x^{3/2}}{108 f}-\frac{3}{16} b^2 n^2 x^2-\frac{b^2 e^4 n^2 \log \left (e+f \sqrt{x}\right )}{4 f^4}+\frac{1}{4} b^2 n^2 x^2 \log \left (d \left (e+f \sqrt{x}\right )\right )-\frac{b^2 e^4 n^2 \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{f^4}+\frac{b^2 e^2 n x \log \left (c x^n\right )}{2 f^2}-\frac{5 b e^3 n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{2 f^3}+\frac{b e^2 n x \left (a+b \log \left (c x^n\right )\right )}{4 f^2}-\frac{7 b e n x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{18 f}+\frac{1}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac{b e^4 n \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^4}-\frac{1}{2} b n x^2 \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{e^3 \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{2 f^3}-\frac{e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{4 f^2}+\frac{e x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{6 f}-\frac{1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{2} x^2 \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{e^4 \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 f^4}-\frac{b^2 e^4 n^2 \text{Li}_2\left (1+\frac{f \sqrt{x}}{e}\right )}{f^4}-\frac{2 b e^4 n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{f^4}+\frac{4 b^2 e^4 n^2 \text{Li}_3\left (-\frac{f \sqrt{x}}{e}\right )}{f^4}\\ \end{align*}
Mathematica [A] time = 0.48844, size = 960, normalized size = 1.61 \[ \frac{-216 b^2 n^2 \log \left (e+f \sqrt{x}\right ) \log ^2(x) e^4+216 b^2 n^2 \log \left (\frac{\sqrt{x} f}{e}+1\right ) \log ^2(x) e^4-216 b^2 \log \left (e+f \sqrt{x}\right ) \log ^2\left (c x^n\right ) e^4-216 a^2 \log \left (e+f \sqrt{x}\right ) e^4-108 b^2 n^2 \log \left (e+f \sqrt{x}\right ) e^4+216 a b n \log \left (e+f \sqrt{x}\right ) e^4-216 b^2 n^2 \log \left (e+f \sqrt{x}\right ) \log (x) e^4+432 a b n \log \left (e+f \sqrt{x}\right ) \log (x) e^4+216 b^2 n^2 \log \left (\frac{\sqrt{x} f}{e}+1\right ) \log (x) e^4-432 a b n \log \left (\frac{\sqrt{x} f}{e}+1\right ) \log (x) e^4-432 a b \log \left (e+f \sqrt{x}\right ) \log \left (c x^n\right ) e^4+216 b^2 n \log \left (e+f \sqrt{x}\right ) \log \left (c x^n\right ) e^4+432 b^2 n \log \left (e+f \sqrt{x}\right ) \log (x) \log \left (c x^n\right ) e^4-432 b^2 n \log \left (\frac{\sqrt{x} f}{e}+1\right ) \log (x) \log \left (c x^n\right ) e^4+432 b n \left (-2 a+b n-2 b \log \left (c x^n\right )\right ) \text{PolyLog}\left (2,-\frac{f \sqrt{x}}{e}\right ) e^4+1728 b^2 n^2 \text{PolyLog}\left (3,-\frac{f \sqrt{x}}{e}\right ) e^4+216 b^2 f \sqrt{x} \log ^2\left (c x^n\right ) e^3+432 a b f \sqrt{x} \log \left (c x^n\right ) e^3-1080 b^2 f n \sqrt{x} \log \left (c x^n\right ) e^3+2268 b^2 f n^2 \sqrt{x} e^3+216 a^2 f \sqrt{x} e^3-1080 a b f n \sqrt{x} e^3-108 b^2 f^2 x \log ^2\left (c x^n\right ) e^2-108 a^2 f^2 x e^2-378 b^2 f^2 n^2 x e^2+324 a b f^2 n x e^2-216 a b f^2 x \log \left (c x^n\right ) e^2+324 b^2 f^2 n x \log \left (c x^n\right ) e^2+72 b^2 f^3 x^{3/2} \log ^2\left (c x^n\right ) e+72 a^2 f^3 x^{3/2} e+148 b^2 f^3 n^2 x^{3/2} e-168 a b f^3 n x^{3/2} e+144 a b f^3 x^{3/2} \log \left (c x^n\right ) e-168 b^2 f^3 n x^{3/2} \log \left (c x^n\right ) e-54 a^2 f^4 x^2-81 b^2 f^4 n^2 x^2+108 a b f^4 n x^2-54 b^2 f^4 x^2 \log ^2\left (c x^n\right )+216 b^2 f^4 x^2 \log \left (d \left (e+f \sqrt{x}\right )\right ) \log ^2\left (c x^n\right )+216 a^2 f^4 x^2 \log \left (d \left (e+f \sqrt{x}\right )\right )+108 b^2 f^4 n^2 x^2 \log \left (d \left (e+f \sqrt{x}\right )\right )-216 a b f^4 n x^2 \log \left (d \left (e+f \sqrt{x}\right )\right )-108 a b f^4 x^2 \log \left (c x^n\right )+108 b^2 f^4 n x^2 \log \left (c x^n\right )+432 a b f^4 x^2 \log \left (d \left (e+f \sqrt{x}\right )\right ) \log \left (c x^n\right )-216 b^2 f^4 n x^2 \log \left (d \left (e+f \sqrt{x}\right )\right ) \log \left (c x^n\right )}{432 f^4} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.025, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}\ln \left ( d \left ( e+f\sqrt{x} \right ) \right ) \, 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*} \int{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x \log \left ({\left (f \sqrt{x} + e\right )} d\right )\,{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 ({\left (b^{2} x \log \left (c x^{n}\right )^{2} + 2 \, a b x \log \left (c x^{n}\right ) + a^{2} x\right )} \log \left (d f \sqrt{x} + d e\right ), 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{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x \log \left ({\left (f \sqrt{x} + e\right )} d\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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