Optimal. Leaf size=907 \[ -\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} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.88, 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 = {2504,
2442, 45, 2424, 2333, 2332, 2342, 2341, 2422, 2375, 2421, 2430, 6724, 2423, 2441, 2352}
\begin {gather*} -\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 ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 2332
Rule 2333
Rule 2341
Rule 2342
Rule 2352
Rule 2375
Rule 2421
Rule 2422
Rule 2423
Rule 2424
Rule 2430
Rule 2441
Rule 2442
Rule 2504
Rule 6724
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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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] Leaf count is larger than twice the leaf count of optimal. \(1968\) vs. \(2(907)=1814\).
time = 0.47, size = 1968, normalized size = 2.17 \begin {gather*} \frac {216 a^3 e^3 f \sqrt {x}-1620 a^2 b e^3 f n \sqrt {x}+6804 a b^2 e^3 f n^2 \sqrt {x}-13770 b^3 e^3 f n^3 \sqrt {x}-108 a^3 e^2 f^2 x+486 a^2 b e^2 f^2 n x-1134 a b^2 e^2 f^2 n^2 x+1215 b^3 e^2 f^2 n^3 x+72 a^3 e f^3 x^{3/2}-252 a^2 b e f^3 n x^{3/2}+444 a b^2 e f^3 n^2 x^{3/2}-350 b^3 e f^3 n^3 x^{3/2}-54 a^3 f^4 x^2+162 a^2 b f^4 n x^2-243 a b^2 f^4 n^2 x^2+162 b^3 f^4 n^3 x^2-216 a^3 e^4 \log \left (e+f \sqrt {x}\right )+324 a^2 b e^4 n \log \left (e+f \sqrt {x}\right )-324 a b^2 e^4 n^2 \log \left (e+f \sqrt {x}\right )+162 b^3 e^4 n^3 \log \left (e+f \sqrt {x}\right )+216 a^3 f^4 x^2 \log \left (d \left (e+f \sqrt {x}\right )\right )-324 a^2 b f^4 n x^2 \log \left (d \left (e+f \sqrt {x}\right )\right )+324 a b^2 f^4 n^2 x^2 \log \left (d \left (e+f \sqrt {x}\right )\right )-162 b^3 f^4 n^3 x^2 \log \left (d \left (e+f \sqrt {x}\right )\right )+648 a^2 b e^4 n \log \left (e+f \sqrt {x}\right ) \log (x)-648 a b^2 e^4 n^2 \log \left (e+f \sqrt {x}\right ) \log (x)+324 b^3 e^4 n^3 \log \left (e+f \sqrt {x}\right ) \log (x)-648 a^2 b e^4 n \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x)+648 a b^2 e^4 n^2 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x)-324 b^3 e^4 n^3 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x)-648 a b^2 e^4 n^2 \log \left (e+f \sqrt {x}\right ) \log ^2(x)+324 b^3 e^4 n^3 \log \left (e+f \sqrt {x}\right ) \log ^2(x)+648 a b^2 e^4 n^2 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log ^2(x)-324 b^3 e^4 n^3 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log ^2(x)+216 b^3 e^4 n^3 \log \left (e+f \sqrt {x}\right ) \log ^3(x)-216 b^3 e^4 n^3 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log ^3(x)+648 a^2 b e^3 f \sqrt {x} \log \left (c x^n\right )-3240 a b^2 e^3 f n \sqrt {x} \log \left (c x^n\right )+6804 b^3 e^3 f n^2 \sqrt {x} \log \left (c x^n\right )-324 a^2 b e^2 f^2 x \log \left (c x^n\right )+972 a b^2 e^2 f^2 n x \log \left (c x^n\right )-1134 b^3 e^2 f^2 n^2 x \log \left (c x^n\right )+216 a^2 b e f^3 x^{3/2} \log \left (c x^n\right )-504 a b^2 e f^3 n x^{3/2} \log \left (c x^n\right )+444 b^3 e f^3 n^2 x^{3/2} \log \left (c x^n\right )-162 a^2 b f^4 x^2 \log \left (c x^n\right )+324 a b^2 f^4 n x^2 \log \left (c x^n\right )-243 b^3 f^4 n^2 x^2 \log \left (c x^n\right )-648 a^2 b e^4 \log \left (e+f \sqrt {x}\right ) \log \left (c x^n\right )+648 a b^2 e^4 n \log \left (e+f \sqrt {x}\right ) \log \left (c x^n\right )-324 b^3 e^4 n^2 \log \left (e+f \sqrt {x}\right ) \log \left (c x^n\right )+648 a^2 b f^4 x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \log \left (c x^n\right )-648 a b^2 f^4 n x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \log \left (c x^n\right )+324 b^3 f^4 n^2 x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \log \left (c x^n\right )+1296 a b^2 e^4 n \log \left (e+f \sqrt {x}\right ) \log (x) \log \left (c x^n\right )-648 b^3 e^4 n^2 \log \left (e+f \sqrt {x}\right ) \log (x) \log \left (c x^n\right )-1296 a b^2 e^4 n \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x) \log \left (c x^n\right )+648 b^3 e^4 n^2 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x) \log \left (c x^n\right )-648 b^3 e^4 n^2 \log \left (e+f \sqrt {x}\right ) \log ^2(x) \log \left (c x^n\right )+648 b^3 e^4 n^2 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log ^2(x) \log \left (c x^n\right )+648 a b^2 e^3 f \sqrt {x} \log ^2\left (c x^n\right )-1620 b^3 e^3 f n \sqrt {x} \log ^2\left (c x^n\right )-324 a b^2 e^2 f^2 x \log ^2\left (c x^n\right )+486 b^3 e^2 f^2 n x \log ^2\left (c x^n\right )+216 a b^2 e f^3 x^{3/2} \log ^2\left (c x^n\right )-252 b^3 e f^3 n x^{3/2} \log ^2\left (c x^n\right )-162 a b^2 f^4 x^2 \log ^2\left (c x^n\right )+162 b^3 f^4 n x^2 \log ^2\left (c x^n\right )-648 a b^2 e^4 \log \left (e+f \sqrt {x}\right ) \log ^2\left (c x^n\right )+324 b^3 e^4 n \log \left (e+f \sqrt {x}\right ) \log ^2\left (c x^n\right )+648 a b^2 f^4 x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \log ^2\left (c x^n\right )-324 b^3 f^4 n x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \log ^2\left (c x^n\right )+648 b^3 e^4 n \log \left (e+f \sqrt {x}\right ) \log (x) \log ^2\left (c x^n\right )-648 b^3 e^4 n \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x) \log ^2\left (c x^n\right )+216 b^3 e^3 f \sqrt {x} \log ^3\left (c x^n\right )-108 b^3 e^2 f^2 x \log ^3\left (c x^n\right )+72 b^3 e f^3 x^{3/2} \log ^3\left (c x^n\right )-54 b^3 f^4 x^2 \log ^3\left (c x^n\right )-216 b^3 e^4 \log \left (e+f \sqrt {x}\right ) \log ^3\left (c x^n\right )+216 b^3 f^4 x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \log ^3\left (c x^n\right )-648 b e^4 n \left (2 a^2-2 a b n+b^2 n^2-2 b (-2 a+b n) \log \left (c x^n\right )+2 b^2 \log ^2\left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )+2592 b^2 e^4 n^2 \left (2 a-b n+2 b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )-10368 b^3 e^4 n^3 \text {Li}_4\left (-\frac {f \sqrt {x}}{e}\right )}{432 f^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int x \left (a +b \ln \left (c \,x^{n}\right )\right )^{3} \ln \left (d \left (e +f \sqrt {x}\right )\right )\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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