Optimal. Leaf size=610 \[ 12 b^2 d^2 f^2 n^2 \text{PolyLog}\left (2,-d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-24 b^2 d^2 f^2 n^2 \text{PolyLog}\left (3,-d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )+6 b d^2 f^2 n \text{PolyLog}\left (2,-d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+12 b^3 d^2 f^2 n^3 \text{PolyLog}\left (2,-d f \sqrt{x}\right )-24 b^3 d^2 f^2 n^3 \text{PolyLog}\left (3,-d f \sqrt{x}\right )+48 b^3 d^2 f^2 n^3 \text{PolyLog}\left (4,-d f \sqrt{x}\right )+6 b^2 d^2 f^2 n^2 \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-3 b^2 d^2 f^2 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{6 b^2 n^2 \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{42 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt{x}}-\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^4}{8 b n}-\frac{1}{2} d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3+d^2 f^2 \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3+3 b d^2 f^2 n \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{\log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}-\frac{d f \left (a+b \log \left (c x^n\right )\right )^3}{\sqrt{x}}-\frac{3 b n \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac{9 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{x}}+\frac{3}{2} b^3 d^2 f^2 n^3 \log ^2(x)+6 b^3 d^2 f^2 n^3 \log \left (d f \sqrt{x}+1\right )-3 b^3 d^2 f^2 n^3 \log (x)-\frac{90 b^3 d f n^3}{\sqrt{x}}-\frac{6 b^3 n^3 \log \left (d f \sqrt{x}+1\right )}{x} \]
[Out]
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Rubi [A] time = 0.778395, antiderivative size = 610, normalized size of antiderivative = 1., number of steps used = 28, number of rules used = 16, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.533, Rules used = {2454, 2395, 44, 2377, 2305, 2304, 2376, 2391, 2301, 2374, 6589, 2366, 12, 2302, 30, 2383} \[ 12 b^2 d^2 f^2 n^2 \text{PolyLog}\left (2,-d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-24 b^2 d^2 f^2 n^2 \text{PolyLog}\left (3,-d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )+6 b d^2 f^2 n \text{PolyLog}\left (2,-d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+12 b^3 d^2 f^2 n^3 \text{PolyLog}\left (2,-d f \sqrt{x}\right )-24 b^3 d^2 f^2 n^3 \text{PolyLog}\left (3,-d f \sqrt{x}\right )+48 b^3 d^2 f^2 n^3 \text{PolyLog}\left (4,-d f \sqrt{x}\right )+6 b^2 d^2 f^2 n^2 \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-3 b^2 d^2 f^2 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{6 b^2 n^2 \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{42 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt{x}}-\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^4}{8 b n}-\frac{1}{2} d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3+d^2 f^2 \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3+3 b d^2 f^2 n \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{\log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}-\frac{d f \left (a+b \log \left (c x^n\right )\right )^3}{\sqrt{x}}-\frac{3 b n \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac{9 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{x}}+\frac{3}{2} b^3 d^2 f^2 n^3 \log ^2(x)+6 b^3 d^2 f^2 n^3 \log \left (d f \sqrt{x}+1\right )-3 b^3 d^2 f^2 n^3 \log (x)-\frac{90 b^3 d f n^3}{\sqrt{x}}-\frac{6 b^3 n^3 \log \left (d f \sqrt{x}+1\right )}{x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2454
Rule 2395
Rule 44
Rule 2377
Rule 2305
Rule 2304
Rule 2376
Rule 2391
Rule 2301
Rule 2374
Rule 6589
Rule 2366
Rule 12
Rule 2302
Rule 30
Rule 2383
Rubi steps
\begin{align*} \int \frac{\log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^2} \, dx &=-\frac{d f \left (a+b \log \left (c x^n\right )\right )^3}{\sqrt{x}}+d^2 f^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}-\frac{1}{2} d^2 f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )^3-(3 b n) \int \left (-\frac{d f \left (a+b \log \left (c x^n\right )\right )^2}{x^{3/2}}-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^2}+\frac{d^2 f^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac{d^2 f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )^2}{2 x}\right ) \, dx\\ &=-\frac{d f \left (a+b \log \left (c x^n\right )\right )^3}{\sqrt{x}}+d^2 f^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}-\frac{1}{2} d^2 f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )^3+(3 b n) \int \frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx+(3 b d f n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^{3/2}} \, dx+\frac{1}{2} \left (3 b d^2 f^2 n\right ) \int \frac{\log (x) \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx-\left (3 b d^2 f^2 n\right ) \int \frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx\\ &=-\frac{9 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{x}}+3 b d^2 f^2 n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{3 b n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac{3}{2} b d^2 f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )^2-\frac{d f \left (a+b \log \left (c x^n\right )\right )^3}{\sqrt{x}}+d^2 f^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}+6 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-d f \sqrt{x}\right )-\frac{1}{2} \left (3 b d^2 f^2 n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{3 b n x} \, dx-\left (6 b^2 n^2\right ) \int \left (-\frac{d f \left (a+b \log \left (c x^n\right )\right )}{x^{3/2}}-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x^2}+\frac{d^2 f^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{d^2 f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 x}\right ) \, dx+\left (12 b^2 d f n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{x^{3/2}} \, dx-\left (12 b^2 d^2 f^2 n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )}{x} \, dx\\ &=-\frac{48 b^3 d f n^3}{\sqrt{x}}-\frac{24 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt{x}}-\frac{9 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{x}}+3 b d^2 f^2 n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{3 b n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac{3}{2} b d^2 f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )^2-\frac{d f \left (a+b \log \left (c x^n\right )\right )^3}{\sqrt{x}}+d^2 f^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}+6 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-d f \sqrt{x}\right )-24 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-d f \sqrt{x}\right )-\frac{1}{2} \left (d^2 f^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{x} \, dx+\left (6 b^2 n^2\right ) \int \frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx+\left (6 b^2 d f n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{x^{3/2}} \, dx+\left (3 b^2 d^2 f^2 n^2\right ) \int \frac{\log (x) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx-\left (6 b^2 d^2 f^2 n^2\right ) \int \frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx+\left (24 b^3 d^2 f^2 n^3\right ) \int \frac{\text{Li}_3\left (-d f \sqrt{x}\right )}{x} \, dx\\ &=-\frac{72 b^3 d f n^3}{\sqrt{x}}-\frac{42 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt{x}}+6 b^2 d^2 f^2 n^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{6 b^2 n^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-3 b^2 d^2 f^2 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{9 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{x}}+3 b d^2 f^2 n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{3 b n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac{d f \left (a+b \log \left (c x^n\right )\right )^3}{\sqrt{x}}+d^2 f^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}+12 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )+6 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-d f \sqrt{x}\right )-24 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-d f \sqrt{x}\right )+48 b^3 d^2 f^2 n^3 \text{Li}_4\left (-d f \sqrt{x}\right )-\frac{\left (d^2 f^2\right ) \operatorname{Subst}\left (\int x^3 \, dx,x,a+b \log \left (c x^n\right )\right )}{2 b n}-\left (3 b^2 d^2 f^2 n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 b n x} \, dx-\left (6 b^3 n^3\right ) \int \left (-\frac{d f}{x^{3/2}}-\frac{\log \left (1+d f \sqrt{x}\right )}{x^2}+\frac{d^2 f^2 \log \left (1+d f \sqrt{x}\right )}{x}-\frac{d^2 f^2 \log (x)}{2 x}\right ) \, dx-\left (12 b^3 d^2 f^2 n^3\right ) \int \frac{\text{Li}_2\left (-d f \sqrt{x}\right )}{x} \, dx\\ &=-\frac{84 b^3 d f n^3}{\sqrt{x}}-\frac{42 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt{x}}+6 b^2 d^2 f^2 n^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{6 b^2 n^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-3 b^2 d^2 f^2 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{9 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{x}}+3 b d^2 f^2 n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{3 b n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac{d f \left (a+b \log \left (c x^n\right )\right )^3}{\sqrt{x}}+d^2 f^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}-\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^4}{8 b n}+12 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )+6 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-d f \sqrt{x}\right )-24 b^3 d^2 f^2 n^3 \text{Li}_3\left (-d f \sqrt{x}\right )-24 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-d f \sqrt{x}\right )+48 b^3 d^2 f^2 n^3 \text{Li}_4\left (-d f \sqrt{x}\right )-\frac{1}{2} \left (3 b d^2 f^2 n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx+\left (6 b^3 n^3\right ) \int \frac{\log \left (1+d f \sqrt{x}\right )}{x^2} \, dx+\left (3 b^3 d^2 f^2 n^3\right ) \int \frac{\log (x)}{x} \, dx-\left (6 b^3 d^2 f^2 n^3\right ) \int \frac{\log \left (1+d f \sqrt{x}\right )}{x} \, dx\\ &=-\frac{84 b^3 d f n^3}{\sqrt{x}}+\frac{3}{2} b^3 d^2 f^2 n^3 \log ^2(x)-\frac{42 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt{x}}+6 b^2 d^2 f^2 n^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{6 b^2 n^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-3 b^2 d^2 f^2 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{9 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{x}}+3 b d^2 f^2 n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{3 b n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac{d f \left (a+b \log \left (c x^n\right )\right )^3}{\sqrt{x}}+d^2 f^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}-\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^4}{8 b n}+12 b^3 d^2 f^2 n^3 \text{Li}_2\left (-d f \sqrt{x}\right )+12 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )+6 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-d f \sqrt{x}\right )-24 b^3 d^2 f^2 n^3 \text{Li}_3\left (-d f \sqrt{x}\right )-24 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-d f \sqrt{x}\right )+48 b^3 d^2 f^2 n^3 \text{Li}_4\left (-d f \sqrt{x}\right )-\frac{1}{2} \left (3 d^2 f^2\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )+\left (12 b^3 n^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+d f x)}{x^3} \, dx,x,\sqrt{x}\right )\\ &=-\frac{84 b^3 d f n^3}{\sqrt{x}}-\frac{6 b^3 n^3 \log \left (1+d f \sqrt{x}\right )}{x}+\frac{3}{2} b^3 d^2 f^2 n^3 \log ^2(x)-\frac{42 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt{x}}+6 b^2 d^2 f^2 n^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{6 b^2 n^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-3 b^2 d^2 f^2 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{9 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{x}}+3 b d^2 f^2 n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{3 b n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac{1}{2} d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac{d f \left (a+b \log \left (c x^n\right )\right )^3}{\sqrt{x}}+d^2 f^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}-\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^4}{8 b n}+12 b^3 d^2 f^2 n^3 \text{Li}_2\left (-d f \sqrt{x}\right )+12 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )+6 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-d f \sqrt{x}\right )-24 b^3 d^2 f^2 n^3 \text{Li}_3\left (-d f \sqrt{x}\right )-24 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-d f \sqrt{x}\right )+48 b^3 d^2 f^2 n^3 \text{Li}_4\left (-d f \sqrt{x}\right )+\left (6 b^3 d f n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 (1+d f x)} \, dx,x,\sqrt{x}\right )\\ &=-\frac{84 b^3 d f n^3}{\sqrt{x}}-\frac{6 b^3 n^3 \log \left (1+d f \sqrt{x}\right )}{x}+\frac{3}{2} b^3 d^2 f^2 n^3 \log ^2(x)-\frac{42 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt{x}}+6 b^2 d^2 f^2 n^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{6 b^2 n^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-3 b^2 d^2 f^2 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{9 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{x}}+3 b d^2 f^2 n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{3 b n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac{1}{2} d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac{d f \left (a+b \log \left (c x^n\right )\right )^3}{\sqrt{x}}+d^2 f^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}-\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^4}{8 b n}+12 b^3 d^2 f^2 n^3 \text{Li}_2\left (-d f \sqrt{x}\right )+12 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )+6 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-d f \sqrt{x}\right )-24 b^3 d^2 f^2 n^3 \text{Li}_3\left (-d f \sqrt{x}\right )-24 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-d f \sqrt{x}\right )+48 b^3 d^2 f^2 n^3 \text{Li}_4\left (-d f \sqrt{x}\right )+\left (6 b^3 d f n^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{x^2}-\frac{d f}{x}+\frac{d^2 f^2}{1+d f x}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{90 b^3 d f n^3}{\sqrt{x}}+6 b^3 d^2 f^2 n^3 \log \left (1+d f \sqrt{x}\right )-\frac{6 b^3 n^3 \log \left (1+d f \sqrt{x}\right )}{x}-3 b^3 d^2 f^2 n^3 \log (x)+\frac{3}{2} b^3 d^2 f^2 n^3 \log ^2(x)-\frac{42 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt{x}}+6 b^2 d^2 f^2 n^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{6 b^2 n^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-3 b^2 d^2 f^2 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{9 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{x}}+3 b d^2 f^2 n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{3 b n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac{1}{2} d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac{d f \left (a+b \log \left (c x^n\right )\right )^3}{\sqrt{x}}+d^2 f^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}-\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^4}{8 b n}+12 b^3 d^2 f^2 n^3 \text{Li}_2\left (-d f \sqrt{x}\right )+12 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )+6 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-d f \sqrt{x}\right )-24 b^3 d^2 f^2 n^3 \text{Li}_3\left (-d f \sqrt{x}\right )-24 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-d f \sqrt{x}\right )+48 b^3 d^2 f^2 n^3 \text{Li}_4\left (-d f \sqrt{x}\right )\\ \end{align*}
Mathematica [B] time = 0.858118, size = 1455, normalized size = 2.39 \[ \text{result too large to display} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.063, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{3}}{{x}^{2}}\ln \left ( d \left ({d}^{-1}+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 \frac{{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f \sqrt{x} + \frac{1}{d}\right )} d\right )}{x^{2}}\,{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 (\frac{{\left (b^{3} \log \left (c x^{n}\right )^{3} + 3 \, a b^{2} \log \left (c x^{n}\right )^{2} + 3 \, a^{2} b \log \left (c x^{n}\right ) + a^{3}\right )} \log \left (d f \sqrt{x} + 1\right )}{x^{2}}, 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 \frac{{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f \sqrt{x} + \frac{1}{d}\right )} d\right )}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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