Optimal. Leaf size=907 \[ -\frac {\log \left (\frac {\sqrt {x} f}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3 e^4}{2 f^4}+\frac {3 b n \log \left (\frac {\sqrt {x} f}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2 e^4}{4 f^4}+\frac {3 b^3 n^3 \log \left (e+f \sqrt {x}\right ) e^4}{8 f^4}+\frac {3 b^3 n^3 \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right ) e^4}{2 f^4}-\frac {3 b^2 n^2 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right ) e^4}{4 f^4}+\frac {3 b^3 n^3 \text {Li}_2\left (\frac {\sqrt {x} f}{e}+1\right ) e^4}{2 f^4}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right ) e^4}{f^4}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right ) e^4}{f^4}-\frac {6 b^3 n^3 \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right ) e^4}{f^4}+\frac {12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right ) e^4}{f^4}-\frac {24 b^3 n^3 \text {Li}_4\left (-\frac {f \sqrt {x}}{e}\right ) e^4}{f^4}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3 e^3}{2 f^3}-\frac {15 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2 e^3}{4 f^3}+\frac {63 b^2 n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right ) e^3}{4 f^3}-\frac {255 b^3 n^3 \sqrt {x} e^3}{8 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^3 e^2}{4 f^2}+\frac {9 b n x \left (a+b \log \left (c x^n\right )\right )^2 e^2}{8 f^2}+\frac {45 b^3 n^3 x e^2}{16 f^2}-\frac {9 a b^2 n^2 x e^2}{4 f^2}-\frac {9 b^3 n^2 x \log \left (c x^n\right ) e^2}{4 f^2}-\frac {3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right ) e^2}{8 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3 e}{6 f}-\frac {7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2 e}{12 f}-\frac {175 b^3 n^3 x^{3/2} e}{216 f}+\frac {37 b^2 n^2 x^{3/2} \left (a+b \log \left (c x^n\right )\right ) e}{36 f}-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^3+\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 )^3+\frac {3}{8} b^3 n^3 x^2+\frac {3}{8} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {3}{4} b n x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {3}{8} b^3 n^3 x^2 \log \left (d \left (e+f \sqrt {x}\right )\right )-\frac {9}{16} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {3}{4} b^2 n^2 x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.31, antiderivative size = 907, normalized size of antiderivative = 1.00, number of steps used = 36, number of rules used = 16, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {2454, 2395, 43, 2377, 2296, 2295, 2305, 2304, 2375, 2337, 2374, 2383, 6589, 2376, 2394, 2315} \[ -\frac {\log \left (\frac {\sqrt {x} f}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3 e^4}{2 f^4}+\frac {3 b n \log \left (\frac {\sqrt {x} f}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2 e^4}{4 f^4}+\frac {3 b^3 n^3 \log \left (e+f \sqrt {x}\right ) e^4}{8 f^4}+\frac {3 b^3 n^3 \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right ) e^4}{2 f^4}-\frac {3 b^2 n^2 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right ) e^4}{4 f^4}+\frac {3 b^3 n^3 \text {PolyLog}\left (2,\frac {\sqrt {x} f}{e}+1\right ) e^4}{2 f^4}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right ) e^4}{f^4}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right ) e^4}{f^4}-\frac {6 b^3 n^3 \text {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right ) e^4}{f^4}+\frac {12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right ) e^4}{f^4}-\frac {24 b^3 n^3 \text {PolyLog}\left (4,-\frac {f \sqrt {x}}{e}\right ) e^4}{f^4}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3 e^3}{2 f^3}-\frac {15 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2 e^3}{4 f^3}+\frac {63 b^2 n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right ) e^3}{4 f^3}-\frac {255 b^3 n^3 \sqrt {x} e^3}{8 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^3 e^2}{4 f^2}+\frac {9 b n x \left (a+b \log \left (c x^n\right )\right )^2 e^2}{8 f^2}+\frac {45 b^3 n^3 x e^2}{16 f^2}-\frac {9 a b^2 n^2 x e^2}{4 f^2}-\frac {9 b^3 n^2 x \log \left (c x^n\right ) e^2}{4 f^2}-\frac {3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right ) e^2}{8 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3 e}{6 f}-\frac {7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2 e}{12 f}-\frac {175 b^3 n^3 x^{3/2} e}{216 f}+\frac {37 b^2 n^2 x^{3/2} \left (a+b \log \left (c x^n\right )\right ) e}{36 f}-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^3+\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 )^3+\frac {3}{8} b^3 n^3 x^2+\frac {3}{8} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {3}{4} b n x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {3}{8} b^3 n^3 x^2 \log \left (d \left (e+f \sqrt {x}\right )\right )-\frac {9}{16} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {3}{4} b^2 n^2 x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 2295
Rule 2296
Rule 2304
Rule 2305
Rule 2315
Rule 2337
Rule 2374
Rule 2375
Rule 2376
Rule 2377
Rule 2383
Rule 2394
Rule 2395
Rule 2454
Rule 6589
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 )^3 \, dx &=\frac {e^3 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{2 f^3}-\frac {e^2 x \left (a+b \log \left (c x^n\right )\right )^3}{4 f^2}+\frac {e x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 f}-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac {e^4 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{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 )^3-(3 b n) \int \left (-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 f^2}+\frac {e^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 f^3 \sqrt {x}}+\frac {e \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{6 f}-\frac {1}{8} x \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 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 )^2\right ) \, dx\\ &=\frac {e^3 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{2 f^3}-\frac {e^2 x \left (a+b \log \left (c x^n\right )\right )^3}{4 f^2}+\frac {e x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 f}-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac {e^4 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{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 )^3+\frac {1}{8} (3 b n) \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx-\frac {1}{2} (3 b n) \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 {\left (3 b e^4 n\right ) \int \frac {\log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx}{2 f^4}-\frac {\left (3 b e^3 n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}} \, dx}{2 f^3}+\frac {\left (3 b e^2 n\right ) \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{4 f^2}-\frac {(b e n) \int \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{2 f}\\ &=-\frac {15 b e^3 n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{4 f^3}+\frac {9 b e^2 n x \left (a+b \log \left (c x^n\right )\right )^2}{8 f^2}-\frac {7 b e n x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{12 f}+\frac {3}{8} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac {3 b e^4 n \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 f^4}-\frac {3}{4} b n 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^3 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{2 f^3}-\frac {e^2 x \left (a+b \log \left (c x^n\right )\right )^3}{4 f^2}+\frac {e x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 f}-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^3+\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 )^3-\frac {e^4 \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{\left (e+f \sqrt {x}\right ) \sqrt {x}} \, dx}{4 f^3}-\frac {1}{8} \left (3 b^2 n^2\right ) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx+\left (3 b^2 n^2\right ) \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 {\left (6 b^2 e^3 n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{\sqrt {x}} \, dx}{f^3}-\frac {\left (3 b^2 e^2 n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{2 f^2}+\frac {\left (2 b^2 e n^2\right ) \int \sqrt {x} \left (a+b \log \left (c x^n\right )\right ) \, dx}{3 f}\\ &=-\frac {24 b^3 e^3 n^3 \sqrt {x}}{f^3}-\frac {3 a b^2 e^2 n^2 x}{2 f^2}-\frac {8 b^3 e n^3 x^{3/2}}{27 f}+\frac {3}{32} b^3 n^3 x^2+\frac {12 b^2 e^3 n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{f^3}+\frac {4 b^2 e n^2 x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 f}-\frac {3}{16} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {15 b e^3 n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{4 f^3}+\frac {9 b e^2 n x \left (a+b \log \left (c x^n\right )\right )^2}{8 f^2}-\frac {7 b e n x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{12 f}+\frac {3}{8} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac {3 b e^4 n \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 f^4}-\frac {3}{4} b n 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^3 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{2 f^3}-\frac {e^2 x \left (a+b \log \left (c x^n\right )\right )^3}{4 f^2}+\frac {e x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 f}-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^3+\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 )^3-\frac {e^4 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 f^4}+\frac {\left (3 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 )^2}{x} \, dx}{2 f^4}-\frac {1}{8} \left (3 b^2 n^2\right ) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac {1}{2} \left (3 b^2 n^2\right ) \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 (3 b^2 e^4 n^2\right ) \int \frac {\log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx}{2 f^4}+\frac {\left (3 b^2 e^3 n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{\sqrt {x}} \, dx}{2 f^3}-\frac {\left (3 b^2 e^2 n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{4 f^2}-\frac {\left (3 b^3 e^2 n^2\right ) \int \log \left (c x^n\right ) \, dx}{2 f^2}+\frac {\left (b^2 e n^2\right ) \int \sqrt {x} \left (a+b \log \left (c x^n\right )\right ) \, dx}{2 f}\\ &=-\frac {30 b^3 e^3 n^3 \sqrt {x}}{f^3}-\frac {9 a b^2 e^2 n^2 x}{4 f^2}+\frac {3 b^3 e^2 n^3 x}{2 f^2}-\frac {14 b^3 e n^3 x^{3/2}}{27 f}+\frac {3}{16} b^3 n^3 x^2-\frac {3 b^3 e^2 n^2 x \log \left (c x^n\right )}{2 f^2}+\frac {63 b^2 e^3 n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{4 f^3}-\frac {3 b^2 e^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{8 f^2}+\frac {37 b^2 e n^2 x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{36 f}-\frac {9}{16} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {3 b^2 e^4 n^2 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 f^4}+\frac {3}{4} b^2 n^2 x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {15 b e^3 n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{4 f^3}+\frac {9 b e^2 n x \left (a+b \log \left (c x^n\right )\right )^2}{8 f^2}-\frac {7 b e n x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{12 f}+\frac {3}{8} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {3}{4} b n 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^3 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{2 f^3}-\frac {e^2 x \left (a+b \log \left (c x^n\right )\right )^3}{4 f^2}+\frac {e x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 f}-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^3+\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 )^3-\frac {e^4 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 f^4}-\frac {3 b e^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )}{f^4}+\frac {\left (3 b e^4 n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\left (e+f \sqrt {x}\right ) \sqrt {x}} \, dx}{8 f^3}+\frac {\left (6 b^2 e^4 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )}{x} \, dx}{f^4}-\frac {\left (3 b^3 e^2 n^2\right ) \int \log \left (c x^n\right ) \, dx}{4 f^2}-\frac {1}{2} \left (3 b^3 n^3\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 {63 b^3 e^3 n^3 \sqrt {x}}{2 f^3}-\frac {9 a b^2 e^2 n^2 x}{4 f^2}+\frac {21 b^3 e^2 n^3 x}{8 f^2}-\frac {37 b^3 e n^3 x^{3/2}}{54 f}+\frac {9}{32} b^3 n^3 x^2-\frac {9 b^3 e^2 n^2 x \log \left (c x^n\right )}{4 f^2}+\frac {63 b^2 e^3 n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{4 f^3}-\frac {3 b^2 e^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{8 f^2}+\frac {37 b^2 e n^2 x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{36 f}-\frac {9}{16} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {3 b^2 e^4 n^2 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 f^4}+\frac {3}{4} b^2 n^2 x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {15 b e^3 n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{4 f^3}+\frac {9 b e^2 n x \left (a+b \log \left (c x^n\right )\right )^2}{8 f^2}-\frac {7 b e n x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{12 f}+\frac {3}{8} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {3}{4} b n x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {3 b e^4 n \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 f^4}+\frac {e^3 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{2 f^3}-\frac {e^2 x \left (a+b \log \left (c x^n\right )\right )^3}{4 f^2}+\frac {e x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 f}-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^3+\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 )^3-\frac {e^4 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 f^4}-\frac {3 b e^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )}{f^4}+\frac {12 b^2 e^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )}{f^4}-\frac {\left (3 b^2 e^4 n^2\right ) \int \frac {\log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx}{2 f^4}-\frac {1}{4} \left (3 b^3 n^3\right ) \int x \log \left (d \left (e+f \sqrt {x}\right )\right ) \, dx+\frac {\left (3 b^3 e^4 n^3\right ) \int \frac {\log \left (e+f \sqrt {x}\right )}{x} \, dx}{4 f^4}-\frac {\left (12 b^3 e^4 n^3\right ) \int \frac {\text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )}{x} \, dx}{f^4}\\ &=-\frac {63 b^3 e^3 n^3 \sqrt {x}}{2 f^3}-\frac {9 a b^2 e^2 n^2 x}{4 f^2}+\frac {21 b^3 e^2 n^3 x}{8 f^2}-\frac {37 b^3 e n^3 x^{3/2}}{54 f}+\frac {9}{32} b^3 n^3 x^2-\frac {9 b^3 e^2 n^2 x \log \left (c x^n\right )}{4 f^2}+\frac {63 b^2 e^3 n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{4 f^3}-\frac {3 b^2 e^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{8 f^2}+\frac {37 b^2 e n^2 x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{36 f}-\frac {9}{16} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {3 b^2 e^4 n^2 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 f^4}+\frac {3}{4} b^2 n^2 x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {15 b e^3 n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{4 f^3}+\frac {9 b e^2 n x \left (a+b \log \left (c x^n\right )\right )^2}{8 f^2}-\frac {7 b e n x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{12 f}+\frac {3}{8} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {3}{4} b n x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {3 b e^4 n \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 f^4}+\frac {e^3 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{2 f^3}-\frac {e^2 x \left (a+b \log \left (c x^n\right )\right )^3}{4 f^2}+\frac {e x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 f}-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^3+\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 )^3-\frac {e^4 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 f^4}+\frac {3 b^2 e^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )}{f^4}-\frac {3 b e^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )}{f^4}+\frac {12 b^2 e^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )}{f^4}-\frac {24 b^3 e^4 n^3 \text {Li}_4\left (-\frac {f \sqrt {x}}{e}\right )}{f^4}-\frac {1}{2} \left (3 b^3 n^3\right ) \operatorname {Subst}\left (\int x^3 \log (d (e+f x)) \, dx,x,\sqrt {x}\right )+\frac {\left (3 b^3 e^4 n^3\right ) \operatorname {Subst}\left (\int \frac {\log (e+f x)}{x} \, dx,x,\sqrt {x}\right )}{2 f^4}-\frac {\left (3 b^3 e^4 n^3\right ) \int \frac {\text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )}{x} \, dx}{f^4}\\ &=-\frac {63 b^3 e^3 n^3 \sqrt {x}}{2 f^3}-\frac {9 a b^2 e^2 n^2 x}{4 f^2}+\frac {21 b^3 e^2 n^3 x}{8 f^2}-\frac {37 b^3 e n^3 x^{3/2}}{54 f}+\frac {9}{32} b^3 n^3 x^2-\frac {3}{8} b^3 n^3 x^2 \log \left (d \left (e+f \sqrt {x}\right )\right )+\frac {3 b^3 e^4 n^3 \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{2 f^4}-\frac {9 b^3 e^2 n^2 x \log \left (c x^n\right )}{4 f^2}+\frac {63 b^2 e^3 n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{4 f^3}-\frac {3 b^2 e^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{8 f^2}+\frac {37 b^2 e n^2 x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{36 f}-\frac {9}{16} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {3 b^2 e^4 n^2 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 f^4}+\frac {3}{4} b^2 n^2 x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {15 b e^3 n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{4 f^3}+\frac {9 b e^2 n x \left (a+b \log \left (c x^n\right )\right )^2}{8 f^2}-\frac {7 b e n x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{12 f}+\frac {3}{8} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {3}{4} b n x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {3 b e^4 n \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 f^4}+\frac {e^3 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{2 f^3}-\frac {e^2 x \left (a+b \log \left (c x^n\right )\right )^3}{4 f^2}+\frac {e x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 f}-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^3+\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 )^3-\frac {e^4 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 f^4}+\frac {3 b^2 e^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )}{f^4}-\frac {3 b e^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )}{f^4}-\frac {6 b^3 e^4 n^3 \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )}{f^4}+\frac {12 b^2 e^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )}{f^4}-\frac {24 b^3 e^4 n^3 \text {Li}_4\left (-\frac {f \sqrt {x}}{e}\right )}{f^4}-\frac {\left (3 b^3 e^4 n^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {f x}{e}\right )}{e+f x} \, dx,x,\sqrt {x}\right )}{2 f^3}+\frac {1}{8} \left (3 b^3 f n^3\right ) \operatorname {Subst}\left (\int \frac {x^4}{e+f x} \, dx,x,\sqrt {x}\right )\\ &=-\frac {63 b^3 e^3 n^3 \sqrt {x}}{2 f^3}-\frac {9 a b^2 e^2 n^2 x}{4 f^2}+\frac {21 b^3 e^2 n^3 x}{8 f^2}-\frac {37 b^3 e n^3 x^{3/2}}{54 f}+\frac {9}{32} b^3 n^3 x^2-\frac {3}{8} b^3 n^3 x^2 \log \left (d \left (e+f \sqrt {x}\right )\right )+\frac {3 b^3 e^4 n^3 \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{2 f^4}-\frac {9 b^3 e^2 n^2 x \log \left (c x^n\right )}{4 f^2}+\frac {63 b^2 e^3 n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{4 f^3}-\frac {3 b^2 e^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{8 f^2}+\frac {37 b^2 e n^2 x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{36 f}-\frac {9}{16} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {3 b^2 e^4 n^2 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 f^4}+\frac {3}{4} b^2 n^2 x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {15 b e^3 n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{4 f^3}+\frac {9 b e^2 n x \left (a+b \log \left (c x^n\right )\right )^2}{8 f^2}-\frac {7 b e n x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{12 f}+\frac {3}{8} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {3}{4} b n x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {3 b e^4 n \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 f^4}+\frac {e^3 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{2 f^3}-\frac {e^2 x \left (a+b \log \left (c x^n\right )\right )^3}{4 f^2}+\frac {e x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 f}-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^3+\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 )^3-\frac {e^4 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 f^4}+\frac {3 b^3 e^4 n^3 \text {Li}_2\left (1+\frac {f \sqrt {x}}{e}\right )}{2 f^4}+\frac {3 b^2 e^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )}{f^4}-\frac {3 b e^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )}{f^4}-\frac {6 b^3 e^4 n^3 \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )}{f^4}+\frac {12 b^2 e^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )}{f^4}-\frac {24 b^3 e^4 n^3 \text {Li}_4\left (-\frac {f \sqrt {x}}{e}\right )}{f^4}+\frac {1}{8} \left (3 b^3 f n^3\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 {255 b^3 e^3 n^3 \sqrt {x}}{8 f^3}-\frac {9 a b^2 e^2 n^2 x}{4 f^2}+\frac {45 b^3 e^2 n^3 x}{16 f^2}-\frac {175 b^3 e n^3 x^{3/2}}{216 f}+\frac {3}{8} b^3 n^3 x^2+\frac {3 b^3 e^4 n^3 \log \left (e+f \sqrt {x}\right )}{8 f^4}-\frac {3}{8} b^3 n^3 x^2 \log \left (d \left (e+f \sqrt {x}\right )\right )+\frac {3 b^3 e^4 n^3 \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{2 f^4}-\frac {9 b^3 e^2 n^2 x \log \left (c x^n\right )}{4 f^2}+\frac {63 b^2 e^3 n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{4 f^3}-\frac {3 b^2 e^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{8 f^2}+\frac {37 b^2 e n^2 x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{36 f}-\frac {9}{16} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {3 b^2 e^4 n^2 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 f^4}+\frac {3}{4} b^2 n^2 x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {15 b e^3 n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{4 f^3}+\frac {9 b e^2 n x \left (a+b \log \left (c x^n\right )\right )^2}{8 f^2}-\frac {7 b e n x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{12 f}+\frac {3}{8} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {3}{4} b n x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {3 b e^4 n \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 f^4}+\frac {e^3 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{2 f^3}-\frac {e^2 x \left (a+b \log \left (c x^n\right )\right )^3}{4 f^2}+\frac {e x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 f}-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^3+\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 )^3-\frac {e^4 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 f^4}+\frac {3 b^3 e^4 n^3 \text {Li}_2\left (1+\frac {f \sqrt {x}}{e}\right )}{2 f^4}+\frac {3 b^2 e^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )}{f^4}-\frac {3 b e^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )}{f^4}-\frac {6 b^3 e^4 n^3 \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )}{f^4}+\frac {12 b^2 e^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )}{f^4}-\frac {24 b^3 e^4 n^3 \text {Li}_4\left (-\frac {f \sqrt {x}}{e}\right )}{f^4}\\ \end {align*}
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Mathematica [B] time = 0.88, size = 1968, normalized size = 2.17 \[ \text {result too large to display} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{3} x \log \left (c x^{n}\right )^{3} + 3 \, a b^{2} x \log \left (c x^{n}\right )^{2} + 3 \, a^{2} b x \log \left (c x^{n}\right ) + a^{3} x\right )} \log \left (d f \sqrt {x} + d e\right ), x\right ) \]
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 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x \log \left ({\left (f \sqrt {x} + e\right )} d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.17, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \,x^{n}\right )+a \right )^{3} x \ln \left (\left (f \sqrt {x}+e \right ) d \right )\, dx \]
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 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x \log \left ({\left (f \sqrt {x} + e\right )} d\right )\,{d x} \]
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 \[ \int x\,\ln \left (d\,\left (e+f\,\sqrt {x}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3 \,d x \]
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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