Optimal. Leaf size=610 \[ -\frac {90 b^3 d f n^3}{\sqrt {x}}+6 b^3 d^2 f^2 n^3 \log \left (1+d f \sqrt {x}\right )-\frac {6 b^3 n^3 \log \left (1+d f \sqrt {x}\right )}{x}-3 b^3 d^2 f^2 n^3 \log (x)+\frac {3}{2} b^3 d^2 f^2 n^3 \log ^2(x)-\frac {42 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+6 b^2 d^2 f^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {6 b^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-3 b^2 d^2 f^2 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {9 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}+3 b d^2 f^2 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {3 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {1}{2} d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac {d f \left (a+b \log \left (c x^n\right )\right )^3}{\sqrt {x}}+d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}-\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^4}{8 b n}+12 b^3 d^2 f^2 n^3 \text {Li}_2\left (-d f \sqrt {x}\right )+12 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )+6 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )-24 b^3 d^2 f^2 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )-24 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )+48 b^3 d^2 f^2 n^3 \text {Li}_4\left (-d f \sqrt {x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.55, antiderivative size = 610, normalized size of antiderivative = 1.00, number of steps
used = 28, number of rules used = 16, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {2504,
2442, 46, 2424, 2342, 2341, 2423, 2438, 2338, 2421, 6724, 2413, 12, 2339, 30, 2430}
\begin {gather*} 12 b^2 d^2 f^2 n^2 \text {PolyLog}\left (2,-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-24 b^2 d^2 f^2 n^2 \text {PolyLog}\left (3,-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )+6 b d^2 f^2 n \text {PolyLog}\left (2,-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+12 b^3 d^2 f^2 n^3 \text {PolyLog}\left (2,-d f \sqrt {x}\right )-24 b^3 d^2 f^2 n^3 \text {PolyLog}\left (3,-d f \sqrt {x}\right )+48 b^3 d^2 f^2 n^3 \text {PolyLog}\left (4,-d f \sqrt {x}\right )+6 b^2 d^2 f^2 n^2 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-3 b^2 d^2 f^2 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {6 b^2 n^2 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {42 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}-\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^4}{8 b n}-\frac {1}{2} d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3+d^2 f^2 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3+3 b d^2 f^2 n \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}-\frac {d f \left (a+b \log \left (c x^n\right )\right )^3}{\sqrt {x}}-\frac {3 b n \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {9 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}+\frac {3}{2} b^3 d^2 f^2 n^3 \log ^2(x)+6 b^3 d^2 f^2 n^3 \log \left (d f \sqrt {x}+1\right )-3 b^3 d^2 f^2 n^3 \log (x)-\frac {90 b^3 d f n^3}{\sqrt {x}}-\frac {6 b^3 n^3 \log \left (d f \sqrt {x}+1\right )}{x} \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^2} \, dx &=-\frac {d f \left (a+b \log \left (c x^n\right )\right )^3}{\sqrt {x}}+d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}-\frac {1}{2} d^2 f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )^3-(3 b n) \int \left (-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{x^{3/2}}-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^2}+\frac {d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {d^2 f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )^2}{2 x}\right ) \, dx\\ &=-\frac {d f \left (a+b \log \left (c x^n\right )\right )^3}{\sqrt {x}}+d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}-\frac {1}{2} d^2 f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )^3+(3 b n) \int \frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx+(3 b d f n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^{3/2}} \, dx+\frac {1}{2} \left (3 b d^2 f^2 n\right ) \int \frac {\log (x) \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx-\left (3 b d^2 f^2 n\right ) \int \frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx\\ &=-\frac {9 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}+3 b d^2 f^2 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {3 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {3}{2} b d^2 f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )^2-\frac {d f \left (a+b \log \left (c x^n\right )\right )^3}{\sqrt {x}}+d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}+6 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )-\frac {1}{2} \left (3 b d^2 f^2 n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{3 b n x} \, dx-\left (6 b^2 n^2\right ) \int \left (-\frac {d f \left (a+b \log \left (c x^n\right )\right )}{x^{3/2}}-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x^2}+\frac {d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {d^2 f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 x}\right ) \, dx+\left (12 b^2 d f n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{x^{3/2}} \, dx-\left (12 b^2 d^2 f^2 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )}{x} \, dx\\ &=-\frac {48 b^3 d f n^3}{\sqrt {x}}-\frac {24 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}-\frac {9 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}+3 b d^2 f^2 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {3 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {3}{2} b d^2 f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )^2-\frac {d f \left (a+b \log \left (c x^n\right )\right )^3}{\sqrt {x}}+d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}+6 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )-24 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )-\frac {1}{2} \left (d^2 f^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x} \, dx+\left (6 b^2 n^2\right ) \int \frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx+\left (6 b^2 d f n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{x^{3/2}} \, dx+\left (3 b^2 d^2 f^2 n^2\right ) \int \frac {\log (x) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx-\left (6 b^2 d^2 f^2 n^2\right ) \int \frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx+\left (24 b^3 d^2 f^2 n^3\right ) \int \frac {\text {Li}_3\left (-d f \sqrt {x}\right )}{x} \, dx\\ &=-\frac {72 b^3 d f n^3}{\sqrt {x}}-\frac {42 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+6 b^2 d^2 f^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {6 b^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-3 b^2 d^2 f^2 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {9 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}+3 b d^2 f^2 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {3 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {d f \left (a+b \log \left (c x^n\right )\right )^3}{\sqrt {x}}+d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}+12 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )+6 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )-24 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )+48 b^3 d^2 f^2 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )-\frac {\left (d^2 f^2\right ) \text {Subst}\left (\int x^3 \, dx,x,a+b \log \left (c x^n\right )\right )}{2 b n}-\left (3 b^2 d^2 f^2 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 b n x} \, dx-\left (6 b^3 n^3\right ) \int \left (-\frac {d f}{x^{3/2}}-\frac {\log \left (1+d f \sqrt {x}\right )}{x^2}+\frac {d^2 f^2 \log \left (1+d f \sqrt {x}\right )}{x}-\frac {d^2 f^2 \log (x)}{2 x}\right ) \, dx-\left (12 b^3 d^2 f^2 n^3\right ) \int \frac {\text {Li}_2\left (-d f \sqrt {x}\right )}{x} \, dx\\ &=-\frac {84 b^3 d f n^3}{\sqrt {x}}-\frac {42 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+6 b^2 d^2 f^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {6 b^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-3 b^2 d^2 f^2 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {9 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}+3 b d^2 f^2 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {3 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {d f \left (a+b \log \left (c x^n\right )\right )^3}{\sqrt {x}}+d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}-\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^4}{8 b n}+12 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )+6 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )-24 b^3 d^2 f^2 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )-24 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )+48 b^3 d^2 f^2 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )-\frac {1}{2} \left (3 b d^2 f^2 n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx+\left (6 b^3 n^3\right ) \int \frac {\log \left (1+d f \sqrt {x}\right )}{x^2} \, dx+\left (3 b^3 d^2 f^2 n^3\right ) \int \frac {\log (x)}{x} \, dx-\left (6 b^3 d^2 f^2 n^3\right ) \int \frac {\log \left (1+d f \sqrt {x}\right )}{x} \, dx\\ &=-\frac {84 b^3 d f n^3}{\sqrt {x}}+\frac {3}{2} b^3 d^2 f^2 n^3 \log ^2(x)-\frac {42 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+6 b^2 d^2 f^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {6 b^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-3 b^2 d^2 f^2 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {9 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}+3 b d^2 f^2 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {3 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {d f \left (a+b \log \left (c x^n\right )\right )^3}{\sqrt {x}}+d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}-\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^4}{8 b n}+12 b^3 d^2 f^2 n^3 \text {Li}_2\left (-d f \sqrt {x}\right )+12 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )+6 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )-24 b^3 d^2 f^2 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )-24 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )+48 b^3 d^2 f^2 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )-\frac {1}{2} \left (3 d^2 f^2\right ) \text {Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )+\left (12 b^3 n^3\right ) \text {Subst}\left (\int \frac {\log (1+d f x)}{x^3} \, dx,x,\sqrt {x}\right )\\ &=-\frac {84 b^3 d f n^3}{\sqrt {x}}-\frac {6 b^3 n^3 \log \left (1+d f \sqrt {x}\right )}{x}+\frac {3}{2} b^3 d^2 f^2 n^3 \log ^2(x)-\frac {42 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+6 b^2 d^2 f^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {6 b^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-3 b^2 d^2 f^2 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {9 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}+3 b d^2 f^2 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {3 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {1}{2} d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac {d f \left (a+b \log \left (c x^n\right )\right )^3}{\sqrt {x}}+d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}-\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^4}{8 b n}+12 b^3 d^2 f^2 n^3 \text {Li}_2\left (-d f \sqrt {x}\right )+12 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )+6 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )-24 b^3 d^2 f^2 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )-24 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )+48 b^3 d^2 f^2 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )+\left (6 b^3 d f n^3\right ) \text {Subst}\left (\int \frac {1}{x^2 (1+d f x)} \, dx,x,\sqrt {x}\right )\\ &=-\frac {84 b^3 d f n^3}{\sqrt {x}}-\frac {6 b^3 n^3 \log \left (1+d f \sqrt {x}\right )}{x}+\frac {3}{2} b^3 d^2 f^2 n^3 \log ^2(x)-\frac {42 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+6 b^2 d^2 f^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {6 b^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-3 b^2 d^2 f^2 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {9 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}+3 b d^2 f^2 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {3 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {1}{2} d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac {d f \left (a+b \log \left (c x^n\right )\right )^3}{\sqrt {x}}+d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}-\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^4}{8 b n}+12 b^3 d^2 f^2 n^3 \text {Li}_2\left (-d f \sqrt {x}\right )+12 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )+6 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )-24 b^3 d^2 f^2 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )-24 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )+48 b^3 d^2 f^2 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )+\left (6 b^3 d f n^3\right ) \text {Subst}\left (\int \left (\frac {1}{x^2}-\frac {d f}{x}+\frac {d^2 f^2}{1+d f x}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {90 b^3 d f n^3}{\sqrt {x}}+6 b^3 d^2 f^2 n^3 \log \left (1+d f \sqrt {x}\right )-\frac {6 b^3 n^3 \log \left (1+d f \sqrt {x}\right )}{x}-3 b^3 d^2 f^2 n^3 \log (x)+\frac {3}{2} b^3 d^2 f^2 n^3 \log ^2(x)-\frac {42 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+6 b^2 d^2 f^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {6 b^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-3 b^2 d^2 f^2 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {9 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}+3 b d^2 f^2 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {3 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {1}{2} d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac {d f \left (a+b \log \left (c x^n\right )\right )^3}{\sqrt {x}}+d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}-\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^4}{8 b n}+12 b^3 d^2 f^2 n^3 \text {Li}_2\left (-d f \sqrt {x}\right )+12 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )+6 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )-24 b^3 d^2 f^2 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )-24 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )+48 b^3 d^2 f^2 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1455\) vs. \(2(610)=1220\).
time = 0.56, size = 1455, normalized size = 2.39 \begin {gather*} d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a^3+3 a^2 b n+6 a b^2 n^2+6 b^3 n^3+3 a^2 b \left (-n \log (x)+\log \left (c x^n\right )\right )+6 a b^2 n \left (-n \log (x)+\log \left (c x^n\right )\right )+6 b^3 n^2 \left (-n \log (x)+\log \left (c x^n\right )\right )+3 a b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2+3 b^3 n \left (-n \log (x)+\log \left (c x^n\right )\right )^2+b^3 \left (-n \log (x)+\log \left (c x^n\right )\right )^3\right )-d^2 f^2 \log \left (\sqrt {x}\right ) \left (a^3+3 a^2 b n+6 a b^2 n^2+6 b^3 n^3+3 a^2 b \left (-n \log (x)+\log \left (c x^n\right )\right )+6 a b^2 n \left (-n \log (x)+\log \left (c x^n\right )\right )+6 b^3 n^2 \left (-n \log (x)+\log \left (c x^n\right )\right )+3 a b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2+3 b^3 n \left (-n \log (x)+\log \left (c x^n\right )\right )^2+b^3 \left (-n \log (x)+\log \left (c x^n\right )\right )^3\right )-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a^3+3 a^2 b n+6 a b^2 n^2+6 b^3 n^3+3 a^2 b n \log (x)+6 a b^2 n^2 \log (x)+6 b^3 n^3 \log (x)+3 a b^2 n^2 \log ^2(x)+3 b^3 n^3 \log ^2(x)+b^3 n^3 \log ^3(x)+3 a^2 b \left (-n \log (x)+\log \left (c x^n\right )\right )+6 a b^2 n \left (-n \log (x)+\log \left (c x^n\right )\right )+6 b^3 n^2 \left (-n \log (x)+\log \left (c x^n\right )\right )+6 a b^2 n \log (x) \left (-n \log (x)+\log \left (c x^n\right )\right )+6 b^3 n^2 \log (x) \left (-n \log (x)+\log \left (c x^n\right )\right )+3 b^3 n^2 \log ^2(x) \left (-n \log (x)+\log \left (c x^n\right )\right )+3 a b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2+3 b^3 n \left (-n \log (x)+\log \left (c x^n\right )\right )^2+3 b^3 n \log (x) \left (-n \log (x)+\log \left (c x^n\right )\right )^2+b^3 \left (-n \log (x)+\log \left (c x^n\right )\right )^3\right )}{x}+\frac {-a^3 d f-3 a^2 b d f n-6 a b^2 d f n^2-6 b^3 d f n^3-3 a^2 b d f \left (-n \log (x)+\log \left (c x^n\right )\right )-6 a b^2 d f n \left (-n \log (x)+\log \left (c x^n\right )\right )-6 b^3 d f n^2 \left (-n \log (x)+\log \left (c x^n\right )\right )-3 a b^2 d f \left (-n \log (x)+\log \left (c x^n\right )\right )^2-3 b^3 d f n \left (-n \log (x)+\log \left (c x^n\right )\right )^2-b^3 d f \left (-n \log (x)+\log \left (c x^n\right )\right )^3}{\sqrt {x}}+3 b d f n \left (a^2+2 a b n+2 b^2 n^2+2 a b \left (-n \log (x)+\log \left (c x^n\right )\right )+2 b^2 n \left (-n \log (x)+\log \left (c x^n\right )\right )+b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2\right ) \left (\left (-\frac {1}{\sqrt {x}}+d f \log \left (1+d f \sqrt {x}\right )-d f \log \left (\sqrt {x}\right )\right ) \left (-2 \log \left (\sqrt {x}\right )+\log (x)\right )+2 \left (-\frac {1}{\sqrt {x}}-\frac {\log \left (\sqrt {x}\right )}{\sqrt {x}}+d f \log \left (1+d f \sqrt {x}\right ) \log \left (\sqrt {x}\right )-\frac {1}{2} d f \log ^2\left (\sqrt {x}\right )+d f \text {Li}_2\left (-d f \sqrt {x}\right )\right )\right )+3 b^2 d f n^2 \left (a+b n+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right ) \left (\left (-\frac {1}{\sqrt {x}}+d f \log \left (1+d f \sqrt {x}\right )-d f \log \left (\sqrt {x}\right )\right ) \left (-2 \log \left (\sqrt {x}\right )+\log (x)\right )^2+4 \left (-2 \log \left (\sqrt {x}\right )+\log (x)\right ) \left (-\frac {1}{\sqrt {x}}-\frac {\log \left (\sqrt {x}\right )}{\sqrt {x}}+d f \log \left (1+d f \sqrt {x}\right ) \log \left (\sqrt {x}\right )-\frac {1}{2} d f \log ^2\left (\sqrt {x}\right )+d f \text {Li}_2\left (-d f \sqrt {x}\right )\right )+4 \left (-\frac {2}{\sqrt {x}}-\frac {2 \log \left (\sqrt {x}\right )}{\sqrt {x}}-\frac {\log ^2\left (\sqrt {x}\right )}{\sqrt {x}}+d f \log \left (1+d f \sqrt {x}\right ) \log ^2\left (\sqrt {x}\right )-\frac {1}{3} d f \log ^3\left (\sqrt {x}\right )+2 d f \log \left (\sqrt {x}\right ) \text {Li}_2\left (-d f \sqrt {x}\right )-2 d f \text {Li}_3\left (-d f \sqrt {x}\right )\right )\right )+\frac {b^3 d f n^3 \left (1+\frac {1}{d f \sqrt {x}}\right ) \left (2 \left (-d f \sqrt {x}+d^2 f^2 x \log \left (1+\frac {1}{d f \sqrt {x}}\right )\right ) \log ^3(x)-12 d f \sqrt {x} \log ^2(x) \left (1+d f \sqrt {x} \text {Li}_2\left (-\frac {1}{d f \sqrt {x}}\right )\right )-48 d f \sqrt {x} \log (x) \left (1+d f \sqrt {x} \text {Li}_3\left (-\frac {1}{d f \sqrt {x}}\right )\right )-96 d f \sqrt {x} \left (1+d f \sqrt {x} \text {Li}_4\left (-\frac {1}{d f \sqrt {x}}\right )\right )\right )}{2 \left (1+d f \sqrt {x}\right ) \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right )^{3} \ln \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )}{x^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________