Optimal. Leaf size=849 \[ -\frac {175 b^3 d f n^3}{216 x^{3/2}}+\frac {45 b^3 d^2 f^2 n^3}{16 x}-\frac {255 b^3 d^3 f^3 n^3}{8 \sqrt {x}}+\frac {3}{8} b^3 d^4 f^4 n^3 \log \left (1+d f \sqrt {x}\right )-\frac {3 b^3 n^3 \log \left (1+d f \sqrt {x}\right )}{8 x^2}-\frac {3}{16} b^3 d^4 f^4 n^3 \log (x)+\frac {3}{16} b^3 d^4 f^4 n^3 \log ^2(x)-\frac {37 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{36 x^{3/2}}+\frac {21 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{8 x}-\frac {63 b^2 d^3 f^3 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 \sqrt {x}}+\frac {3}{4} b^2 d^4 f^4 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3 b^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac {3}{8} b^2 d^4 f^4 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {7 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{12 x^{3/2}}+\frac {9 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2}{8 x}-\frac {15 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )^2}{4 \sqrt {x}}+\frac {3}{4} b d^4 f^4 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}{4 x^2}-\frac {1}{8} d^4 f^4 \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}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^3}{2 \sqrt {x}}+\frac {1}{2} d^4 f^4 \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}{2 x^2}-\frac {d^4 f^4 \left (a+b \log \left (c x^n\right )\right )^4}{16 b n}+\frac {3}{2} b^3 d^4 f^4 n^3 \text {Li}_2\left (-d f \sqrt {x}\right )+3 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )+3 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )-6 b^3 d^4 f^4 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )-12 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )+24 b^3 d^4 f^4 n^3 \text {Li}_4\left (-d f \sqrt {x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.79, antiderivative size = 849, normalized size of antiderivative = 1.00, number of steps
used = 34, number of rules used = 16, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {2504,
2442, 46, 2424, 2342, 2341, 2423, 2438, 2338, 2421, 6724, 2413, 12, 2339, 30, 2430}
\begin {gather*} -\frac {d^4 \left (a+b \log \left (c x^n\right )\right )^4 f^4}{16 b n}-\frac {1}{8} d^4 \left (a+b \log \left (c x^n\right )\right )^3 f^4+\frac {1}{2} d^4 \log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )^3 f^4+\frac {3}{16} b^3 d^4 n^3 \log ^2(x) f^4+\frac {3}{4} b d^4 n \log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )^2 f^4+\frac {3}{8} b^3 d^4 n^3 \log \left (d \sqrt {x} f+1\right ) f^4-\frac {3}{16} b^3 d^4 n^3 \log (x) f^4+\frac {3}{4} b^2 d^4 n^2 \log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right ) f^4-\frac {3}{8} b^2 d^4 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right ) f^4+\frac {3}{2} b^3 d^4 n^3 \text {PolyLog}\left (2,-d f \sqrt {x}\right ) f^4+3 b d^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text {PolyLog}\left (2,-d f \sqrt {x}\right ) f^4+3 b^2 d^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {PolyLog}\left (2,-d f \sqrt {x}\right ) f^4-6 b^3 d^4 n^3 \text {PolyLog}\left (3,-d f \sqrt {x}\right ) f^4-12 b^2 d^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {PolyLog}\left (3,-d f \sqrt {x}\right ) f^4+24 b^3 d^4 n^3 \text {PolyLog}\left (4,-d f \sqrt {x}\right ) f^4-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )^3 f^3}{2 \sqrt {x}}-\frac {15 b d^3 n \left (a+b \log \left (c x^n\right )\right )^2 f^3}{4 \sqrt {x}}-\frac {63 b^2 d^3 n^2 \left (a+b \log \left (c x^n\right )\right ) f^3}{4 \sqrt {x}}-\frac {255 b^3 d^3 n^3 f^3}{8 \sqrt {x}}+\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^3 f^2}{4 x}+\frac {9 b d^2 n \left (a+b \log \left (c x^n\right )\right )^2 f^2}{8 x}+\frac {21 b^2 d^2 n^2 \left (a+b \log \left (c x^n\right )\right ) f^2}{8 x}+\frac {45 b^3 d^2 n^3 f^2}{16 x}-\frac {d \left (a+b \log \left (c x^n\right )\right )^3 f}{6 x^{3/2}}-\frac {7 b d n \left (a+b \log \left (c x^n\right )\right )^2 f}{12 x^{3/2}}-\frac {37 b^2 d n^2 \left (a+b \log \left (c x^n\right )\right ) f}{36 x^{3/2}}-\frac {175 b^3 d n^3 f}{216 x^{3/2}}-\frac {\log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-\frac {3 b n \log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac {3 b^3 n^3 \log \left (d \sqrt {x} f+1\right )}{8 x^2}-\frac {3 b^2 n^2 \log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 30
Rule 46
Rule 2338
Rule 2339
Rule 2341
Rule 2342
Rule 2413
Rule 2421
Rule 2423
Rule 2424
Rule 2430
Rule 2438
Rule 2442
Rule 2504
Rule 6724
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^3} \, dx &=-\frac {d f \left (a+b \log \left (c x^n\right )\right )^3}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^3}{2 \sqrt {x}}+\frac {1}{2} d^4 f^4 \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}{2 x^2}-\frac {1}{4} d^4 f^4 \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}{6 x^{5/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 x^{3/2}}-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^3}+\frac {d^4 f^4 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x}-\frac {d^4 f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )^2}{4 x}\right ) \, dx\\ &=-\frac {d f \left (a+b \log \left (c x^n\right )\right )^3}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^3}{2 \sqrt {x}}+\frac {1}{2} d^4 f^4 \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}{2 x^2}-\frac {1}{4} d^4 f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )^3+\frac {1}{2} (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^3} \, dx+\frac {1}{2} (b d f n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^{5/2}} \, dx-\frac {1}{4} \left (3 b d^2 f^2 n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx+\frac {1}{2} \left (3 b d^3 f^3 n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^{3/2}} \, dx+\frac {1}{4} \left (3 b d^4 f^4 n\right ) \int \frac {\log (x) \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx-\frac {1}{2} \left (3 b d^4 f^4 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 {7 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{12 x^{3/2}}+\frac {9 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2}{8 x}-\frac {15 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )^2}{4 \sqrt {x}}+\frac {3}{4} b d^4 f^4 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}{4 x^2}-\frac {3}{8} b d^4 f^4 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}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^3}{2 \sqrt {x}}+\frac {1}{2} d^4 f^4 \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}{2 x^2}+3 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )-\frac {1}{4} \left (3 b d^4 f^4 n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{3 b n x} \, dx-\left (3 b^2 n^2\right ) \int \left (-\frac {d f \left (a+b \log \left (c x^n\right )\right )}{6 x^{5/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac {d^3 f^3 \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^3}+\frac {d^4 f^4 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x}-\frac {d^4 f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )}{4 x}\right ) \, dx+\frac {1}{3} \left (2 b^2 d f n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{x^{5/2}} \, dx-\frac {1}{2} \left (3 b^2 d^2 f^2 n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx+\left (6 b^2 d^3 f^3 n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{x^{3/2}} \, dx-\left (6 b^2 d^4 f^4 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 {8 b^3 d f n^3}{27 x^{3/2}}+\frac {3 b^3 d^2 f^2 n^3}{2 x}-\frac {24 b^3 d^3 f^3 n^3}{\sqrt {x}}-\frac {4 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{9 x^{3/2}}+\frac {3 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{2 x}-\frac {12 b^2 d^3 f^3 n^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}-\frac {7 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{12 x^{3/2}}+\frac {9 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2}{8 x}-\frac {15 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )^2}{4 \sqrt {x}}+\frac {3}{4} b d^4 f^4 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}{4 x^2}-\frac {3}{8} b d^4 f^4 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}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^3}{2 \sqrt {x}}+\frac {1}{2} d^4 f^4 \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}{2 x^2}+3 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )-12 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )-\frac {1}{4} \left (d^4 f^4\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x} \, dx+\frac {1}{2} \left (3 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^3} \, dx+\frac {1}{2} \left (b^2 d f n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{x^{5/2}} \, dx-\frac {1}{4} \left (3 b^2 d^2 f^2 n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx+\frac {1}{2} \left (3 b^2 d^3 f^3 n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{x^{3/2}} \, dx+\frac {1}{4} \left (3 b^2 d^4 f^4 n^2\right ) \int \frac {\log (x) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx-\frac {1}{2} \left (3 b^2 d^4 f^4 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 (12 b^3 d^4 f^4 n^3\right ) \int \frac {\text {Li}_3\left (-d f \sqrt {x}\right )}{x} \, dx\\ &=-\frac {14 b^3 d f n^3}{27 x^{3/2}}+\frac {9 b^3 d^2 f^2 n^3}{4 x}-\frac {30 b^3 d^3 f^3 n^3}{\sqrt {x}}-\frac {37 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{36 x^{3/2}}+\frac {21 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{8 x}-\frac {63 b^2 d^3 f^3 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 \sqrt {x}}+\frac {3}{4} b^2 d^4 f^4 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3 b^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac {3}{8} b^2 d^4 f^4 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {7 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{12 x^{3/2}}+\frac {9 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2}{8 x}-\frac {15 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )^2}{4 \sqrt {x}}+\frac {3}{4} b d^4 f^4 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}{4 x^2}-\frac {d f \left (a+b \log \left (c x^n\right )\right )^3}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^3}{2 \sqrt {x}}+\frac {1}{2} d^4 f^4 \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}{2 x^2}+3 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )+3 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )-12 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )+24 b^3 d^4 f^4 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )-\frac {\left (d^4 f^4\right ) \text {Subst}\left (\int x^3 \, dx,x,a+b \log \left (c x^n\right )\right )}{4 b n}-\frac {1}{4} \left (3 b^2 d^4 f^4 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 b n x} \, dx-\frac {1}{2} \left (3 b^3 n^3\right ) \int \left (-\frac {d f}{6 x^{5/2}}+\frac {d^2 f^2}{4 x^2}-\frac {d^3 f^3}{2 x^{3/2}}-\frac {\log \left (1+d f \sqrt {x}\right )}{2 x^3}+\frac {d^4 f^4 \log \left (1+d f \sqrt {x}\right )}{2 x}-\frac {d^4 f^4 \log (x)}{4 x}\right ) \, dx-\left (3 b^3 d^4 f^4 n^3\right ) \int \frac {\text {Li}_2\left (-d f \sqrt {x}\right )}{x} \, dx\\ &=-\frac {37 b^3 d f n^3}{54 x^{3/2}}+\frac {21 b^3 d^2 f^2 n^3}{8 x}-\frac {63 b^3 d^3 f^3 n^3}{2 \sqrt {x}}-\frac {37 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{36 x^{3/2}}+\frac {21 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{8 x}-\frac {63 b^2 d^3 f^3 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 \sqrt {x}}+\frac {3}{4} b^2 d^4 f^4 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3 b^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac {3}{8} b^2 d^4 f^4 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {7 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{12 x^{3/2}}+\frac {9 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2}{8 x}-\frac {15 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )^2}{4 \sqrt {x}}+\frac {3}{4} b d^4 f^4 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}{4 x^2}-\frac {d f \left (a+b \log \left (c x^n\right )\right )^3}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^3}{2 \sqrt {x}}+\frac {1}{2} d^4 f^4 \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}{2 x^2}-\frac {d^4 f^4 \left (a+b \log \left (c x^n\right )\right )^4}{16 b n}+3 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )+3 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )-6 b^3 d^4 f^4 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )-12 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )+24 b^3 d^4 f^4 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )-\frac {1}{8} \left (3 b d^4 f^4 n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx+\frac {1}{4} \left (3 b^3 n^3\right ) \int \frac {\log \left (1+d f \sqrt {x}\right )}{x^3} \, dx+\frac {1}{8} \left (3 b^3 d^4 f^4 n^3\right ) \int \frac {\log (x)}{x} \, dx-\frac {1}{4} \left (3 b^3 d^4 f^4 n^3\right ) \int \frac {\log \left (1+d f \sqrt {x}\right )}{x} \, dx\\ &=-\frac {37 b^3 d f n^3}{54 x^{3/2}}+\frac {21 b^3 d^2 f^2 n^3}{8 x}-\frac {63 b^3 d^3 f^3 n^3}{2 \sqrt {x}}+\frac {3}{16} b^3 d^4 f^4 n^3 \log ^2(x)-\frac {37 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{36 x^{3/2}}+\frac {21 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{8 x}-\frac {63 b^2 d^3 f^3 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 \sqrt {x}}+\frac {3}{4} b^2 d^4 f^4 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3 b^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac {3}{8} b^2 d^4 f^4 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {7 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{12 x^{3/2}}+\frac {9 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2}{8 x}-\frac {15 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )^2}{4 \sqrt {x}}+\frac {3}{4} b d^4 f^4 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}{4 x^2}-\frac {d f \left (a+b \log \left (c x^n\right )\right )^3}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^3}{2 \sqrt {x}}+\frac {1}{2} d^4 f^4 \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}{2 x^2}-\frac {d^4 f^4 \left (a+b \log \left (c x^n\right )\right )^4}{16 b n}+\frac {3}{2} b^3 d^4 f^4 n^3 \text {Li}_2\left (-d f \sqrt {x}\right )+3 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )+3 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )-6 b^3 d^4 f^4 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )-12 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )+24 b^3 d^4 f^4 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )-\frac {1}{8} \left (3 d^4 f^4\right ) \text {Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )+\frac {1}{2} \left (3 b^3 n^3\right ) \text {Subst}\left (\int \frac {\log (1+d f x)}{x^5} \, dx,x,\sqrt {x}\right )\\ &=-\frac {37 b^3 d f n^3}{54 x^{3/2}}+\frac {21 b^3 d^2 f^2 n^3}{8 x}-\frac {63 b^3 d^3 f^3 n^3}{2 \sqrt {x}}-\frac {3 b^3 n^3 \log \left (1+d f \sqrt {x}\right )}{8 x^2}+\frac {3}{16} b^3 d^4 f^4 n^3 \log ^2(x)-\frac {37 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{36 x^{3/2}}+\frac {21 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{8 x}-\frac {63 b^2 d^3 f^3 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 \sqrt {x}}+\frac {3}{4} b^2 d^4 f^4 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3 b^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac {3}{8} b^2 d^4 f^4 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {7 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{12 x^{3/2}}+\frac {9 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2}{8 x}-\frac {15 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )^2}{4 \sqrt {x}}+\frac {3}{4} b d^4 f^4 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}{4 x^2}-\frac {1}{8} d^4 f^4 \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}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^3}{2 \sqrt {x}}+\frac {1}{2} d^4 f^4 \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}{2 x^2}-\frac {d^4 f^4 \left (a+b \log \left (c x^n\right )\right )^4}{16 b n}+\frac {3}{2} b^3 d^4 f^4 n^3 \text {Li}_2\left (-d f \sqrt {x}\right )+3 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )+3 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )-6 b^3 d^4 f^4 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )-12 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )+24 b^3 d^4 f^4 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )+\frac {1}{8} \left (3 b^3 d f n^3\right ) \text {Subst}\left (\int \frac {1}{x^4 (1+d f x)} \, dx,x,\sqrt {x}\right )\\ &=-\frac {37 b^3 d f n^3}{54 x^{3/2}}+\frac {21 b^3 d^2 f^2 n^3}{8 x}-\frac {63 b^3 d^3 f^3 n^3}{2 \sqrt {x}}-\frac {3 b^3 n^3 \log \left (1+d f \sqrt {x}\right )}{8 x^2}+\frac {3}{16} b^3 d^4 f^4 n^3 \log ^2(x)-\frac {37 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{36 x^{3/2}}+\frac {21 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{8 x}-\frac {63 b^2 d^3 f^3 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 \sqrt {x}}+\frac {3}{4} b^2 d^4 f^4 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3 b^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac {3}{8} b^2 d^4 f^4 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {7 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{12 x^{3/2}}+\frac {9 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2}{8 x}-\frac {15 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )^2}{4 \sqrt {x}}+\frac {3}{4} b d^4 f^4 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}{4 x^2}-\frac {1}{8} d^4 f^4 \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}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^3}{2 \sqrt {x}}+\frac {1}{2} d^4 f^4 \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}{2 x^2}-\frac {d^4 f^4 \left (a+b \log \left (c x^n\right )\right )^4}{16 b n}+\frac {3}{2} b^3 d^4 f^4 n^3 \text {Li}_2\left (-d f \sqrt {x}\right )+3 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )+3 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )-6 b^3 d^4 f^4 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )-12 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )+24 b^3 d^4 f^4 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )+\frac {1}{8} \left (3 b^3 d f n^3\right ) \text {Subst}\left (\int \left (\frac {1}{x^4}-\frac {d f}{x^3}+\frac {d^2 f^2}{x^2}-\frac {d^3 f^3}{x}+\frac {d^4 f^4}{1+d f x}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {175 b^3 d f n^3}{216 x^{3/2}}+\frac {45 b^3 d^2 f^2 n^3}{16 x}-\frac {255 b^3 d^3 f^3 n^3}{8 \sqrt {x}}+\frac {3}{8} b^3 d^4 f^4 n^3 \log \left (1+d f \sqrt {x}\right )-\frac {3 b^3 n^3 \log \left (1+d f \sqrt {x}\right )}{8 x^2}-\frac {3}{16} b^3 d^4 f^4 n^3 \log (x)+\frac {3}{16} b^3 d^4 f^4 n^3 \log ^2(x)-\frac {37 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{36 x^{3/2}}+\frac {21 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{8 x}-\frac {63 b^2 d^3 f^3 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 \sqrt {x}}+\frac {3}{4} b^2 d^4 f^4 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3 b^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac {3}{8} b^2 d^4 f^4 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {7 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{12 x^{3/2}}+\frac {9 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2}{8 x}-\frac {15 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )^2}{4 \sqrt {x}}+\frac {3}{4} b d^4 f^4 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}{4 x^2}-\frac {1}{8} d^4 f^4 \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}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^3}{2 \sqrt {x}}+\frac {1}{2} d^4 f^4 \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}{2 x^2}-\frac {d^4 f^4 \left (a+b \log \left (c x^n\right )\right )^4}{16 b n}+\frac {3}{2} b^3 d^4 f^4 n^3 \text {Li}_2\left (-d f \sqrt {x}\right )+3 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )+3 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )-6 b^3 d^4 f^4 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )-12 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )+24 b^3 d^4 f^4 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(2009\) vs. \(2(849)=1698\).
time = 0.67, size = 2009, normalized size = 2.37 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right )^{3} \ln \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )}{x^{3}}\, dx\]
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 \frac {\ln \left (d\,\left (f\,\sqrt {x}+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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