Optimal. Leaf size=604 \[ -\frac {90 b^3 n^3 \sqrt {x}}{d f}-6 a b^2 n^2 x+12 b^3 n^3 x-6 b^3 n^3 x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )+\frac {6 b^3 n^3 \log \left (1+d f \sqrt {x}\right )}{d^2 f^2}-6 b^3 n^2 x \log \left (c x^n\right )+\frac {42 b^2 n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{d f}-3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )+6 b^2 n^2 x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {6 b^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}-\frac {9 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}+3 b n x \left (a+b \log \left (c x^n\right )\right )^2-3 b n x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {3 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2 f^2}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^3+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d^2 f^2}-\frac {12 b^3 n^3 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^2 f^2}+\frac {12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )}{d^2 f^2}-\frac {6 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^2 f^2}-\frac {24 b^3 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )}{d^2 f^2}+\frac {24 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )}{d^2 f^2}-\frac {48 b^3 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )}{d^2 f^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.38, antiderivative size = 604, normalized size of antiderivative = 1.00, number of steps
used = 24, number of rules used = 12, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {2498, 272,
45, 2417, 2333, 2332, 2342, 2341, 2438, 2421, 6724, 2430} \begin {gather*} \frac {12 b^2 n^2 \text {PolyLog}\left (2,-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}+\frac {24 b^2 n^2 \text {PolyLog}\left (3,-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}-\frac {6 b n \text {PolyLog}\left (2,-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2 f^2}-\frac {12 b^3 n^3 \text {PolyLog}\left (2,-d f \sqrt {x}\right )}{d^2 f^2}-\frac {24 b^3 n^3 \text {PolyLog}\left (3,-d f \sqrt {x}\right )}{d^2 f^2}-\frac {48 b^3 n^3 \text {PolyLog}\left (4,-d f \sqrt {x}\right )}{d^2 f^2}-\frac {6 b^2 n^2 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}+6 b^2 n^2 x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {42 b^2 n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{d f}-3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )-6 a b^2 n^2 x+\frac {3 b n \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2 f^2}-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d^2 f^2}-3 b n x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {9 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{d f}+3 b n x \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^3-6 b^3 n^2 x \log \left (c x^n\right )+\frac {6 b^3 n^3 \log \left (d f \sqrt {x}+1\right )}{d^2 f^2}-\frac {90 b^3 n^3 \sqrt {x}}{d f}-6 b^3 n^3 x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )+12 b^3 n^3 x \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 272
Rule 2332
Rule 2333
Rule 2341
Rule 2342
Rule 2417
Rule 2421
Rule 2430
Rule 2438
Rule 2498
Rule 6724
Rubi steps
\begin {align*} \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx &=\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^3+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\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}{d^2 f^2}-(3 b n) \int \left (-\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )^2+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d f \sqrt {x}}+\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2 f^2 x}\right ) \, dx\\ &=\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^3+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\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}{d^2 f^2}+\frac {1}{2} (3 b n) \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx-(3 b n) \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx+\frac {(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} \, dx}{d^2 f^2}-\frac {(3 b n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}} \, dx}{d f}\\ &=-\frac {9 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}+3 b n x \left (a+b \log \left (c x^n\right )\right )^2-3 b n x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\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}{d^2 f^2}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^3+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\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}{d^2 f^2}-\frac {6 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^2 f^2}-\left (3 b^2 n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx+\left (6 b^2 n^2\right ) \int \left (\frac {1}{2} \left (-a-b \log \left (c x^n\right )\right )+\frac {a+b \log \left (c x^n\right )}{d f \sqrt {x}}+\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2 x}\right ) \, dx+\frac {\left (12 b^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}{d^2 f^2}+\frac {\left (12 b^2 n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{\sqrt {x}} \, dx}{d f}\\ &=-\frac {48 b^3 n^3 \sqrt {x}}{d f}-3 a b^2 n^2 x+\frac {24 b^2 n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{d f}-\frac {9 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}+3 b n x \left (a+b \log \left (c x^n\right )\right )^2-3 b n x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\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}{d^2 f^2}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^3+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\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}{d^2 f^2}-\frac {6 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^2 f^2}+\frac {24 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )}{d^2 f^2}+\left (3 b^2 n^2\right ) \int \left (-a-b \log \left (c x^n\right )\right ) \, dx+\left (6 b^2 n^2\right ) \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx-\left (3 b^3 n^2\right ) \int \log \left (c x^n\right ) \, dx-\frac {\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} \, dx}{d^2 f^2}+\frac {\left (6 b^2 n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{\sqrt {x}} \, dx}{d f}-\frac {\left (24 b^3 n^3\right ) \int \frac {\text {Li}_3\left (-d f \sqrt {x}\right )}{x} \, dx}{d^2 f^2}\\ &=-\frac {72 b^3 n^3 \sqrt {x}}{d f}-6 a b^2 n^2 x+3 b^3 n^3 x-3 b^3 n^2 x \log \left (c x^n\right )+\frac {42 b^2 n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{d f}-3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )+6 b^2 n^2 x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\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 )}{d^2 f^2}-\frac {9 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}+3 b n x \left (a+b \log \left (c x^n\right )\right )^2-3 b n x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\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}{d^2 f^2}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^3+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\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}{d^2 f^2}+\frac {12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )}{d^2 f^2}-\frac {6 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^2 f^2}+\frac {24 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )}{d^2 f^2}-\frac {48 b^3 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )}{d^2 f^2}-\left (3 b^3 n^2\right ) \int \log \left (c x^n\right ) \, dx-\left (6 b^3 n^3\right ) \int \left (-\frac {1}{2}+\frac {1}{d f \sqrt {x}}+\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )-\frac {\log \left (1+d f \sqrt {x}\right )}{d^2 f^2 x}\right ) \, dx-\frac {\left (12 b^3 n^3\right ) \int \frac {\text {Li}_2\left (-d f \sqrt {x}\right )}{x} \, dx}{d^2 f^2}\\ &=-\frac {84 b^3 n^3 \sqrt {x}}{d f}-6 a b^2 n^2 x+9 b^3 n^3 x-6 b^3 n^2 x \log \left (c x^n\right )+\frac {42 b^2 n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{d f}-3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )+6 b^2 n^2 x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\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 )}{d^2 f^2}-\frac {9 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}+3 b n x \left (a+b \log \left (c x^n\right )\right )^2-3 b n x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\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}{d^2 f^2}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^3+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\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}{d^2 f^2}+\frac {12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )}{d^2 f^2}-\frac {6 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^2 f^2}-\frac {24 b^3 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )}{d^2 f^2}+\frac {24 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )}{d^2 f^2}-\frac {48 b^3 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )}{d^2 f^2}-\left (6 b^3 n^3\right ) \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \, dx+\frac {\left (6 b^3 n^3\right ) \int \frac {\log \left (1+d f \sqrt {x}\right )}{x} \, dx}{d^2 f^2}\\ &=-\frac {84 b^3 n^3 \sqrt {x}}{d f}-6 a b^2 n^2 x+9 b^3 n^3 x-6 b^3 n^3 x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )-6 b^3 n^2 x \log \left (c x^n\right )+\frac {42 b^2 n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{d f}-3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )+6 b^2 n^2 x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\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 )}{d^2 f^2}-\frac {9 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}+3 b n x \left (a+b \log \left (c x^n\right )\right )^2-3 b n x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\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}{d^2 f^2}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^3+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\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}{d^2 f^2}-\frac {12 b^3 n^3 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^2 f^2}+\frac {12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )}{d^2 f^2}-\frac {6 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^2 f^2}-\frac {24 b^3 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )}{d^2 f^2}+\frac {24 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )}{d^2 f^2}-\frac {48 b^3 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )}{d^2 f^2}+\left (3 b^3 f n^3\right ) \int \frac {\sqrt {x}}{\frac {1}{d}+f \sqrt {x}} \, dx\\ &=-\frac {84 b^3 n^3 \sqrt {x}}{d f}-6 a b^2 n^2 x+9 b^3 n^3 x-6 b^3 n^3 x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )-6 b^3 n^2 x \log \left (c x^n\right )+\frac {42 b^2 n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{d f}-3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )+6 b^2 n^2 x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\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 )}{d^2 f^2}-\frac {9 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}+3 b n x \left (a+b \log \left (c x^n\right )\right )^2-3 b n x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\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}{d^2 f^2}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^3+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\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}{d^2 f^2}-\frac {12 b^3 n^3 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^2 f^2}+\frac {12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )}{d^2 f^2}-\frac {6 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^2 f^2}-\frac {24 b^3 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )}{d^2 f^2}+\frac {24 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )}{d^2 f^2}-\frac {48 b^3 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )}{d^2 f^2}+\left (6 b^3 f n^3\right ) \text {Subst}\left (\int \frac {x^2}{\frac {1}{d}+f x} \, dx,x,\sqrt {x}\right )\\ &=-\frac {84 b^3 n^3 \sqrt {x}}{d f}-6 a b^2 n^2 x+9 b^3 n^3 x-6 b^3 n^3 x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )-6 b^3 n^2 x \log \left (c x^n\right )+\frac {42 b^2 n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{d f}-3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )+6 b^2 n^2 x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\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 )}{d^2 f^2}-\frac {9 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}+3 b n x \left (a+b \log \left (c x^n\right )\right )^2-3 b n x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\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}{d^2 f^2}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^3+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\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}{d^2 f^2}-\frac {12 b^3 n^3 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^2 f^2}+\frac {12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )}{d^2 f^2}-\frac {6 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^2 f^2}-\frac {24 b^3 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )}{d^2 f^2}+\frac {24 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )}{d^2 f^2}-\frac {48 b^3 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )}{d^2 f^2}+\left (6 b^3 f n^3\right ) \text {Subst}\left (\int \left (-\frac {1}{d f^2}+\frac {x}{f}+\frac {1}{d f^2 (1+d f x)}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {90 b^3 n^3 \sqrt {x}}{d f}-6 a b^2 n^2 x+12 b^3 n^3 x-6 b^3 n^3 x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )+\frac {6 b^3 n^3 \log \left (1+d f \sqrt {x}\right )}{d^2 f^2}-6 b^3 n^2 x \log \left (c x^n\right )+\frac {42 b^2 n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{d f}-3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )+6 b^2 n^2 x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\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 )}{d^2 f^2}-\frac {9 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}+3 b n x \left (a+b \log \left (c x^n\right )\right )^2-3 b n x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\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}{d^2 f^2}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^3+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\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}{d^2 f^2}-\frac {12 b^3 n^3 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^2 f^2}+\frac {12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )}{d^2 f^2}-\frac {6 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^2 f^2}-\frac {24 b^3 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )}{d^2 f^2}+\frac {24 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )}{d^2 f^2}-\frac {48 b^3 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )}{d^2 f^2}\\ \end {align*}
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Mathematica [A]
time = 0.31, size = 986, normalized size = 1.63 \begin {gather*} -\frac {-2 a^3 d f \sqrt {x}+18 a^2 b d f n \sqrt {x}-84 a b^2 d f n^2 \sqrt {x}+180 b^3 d f n^3 \sqrt {x}+a^3 d^2 f^2 x-6 a^2 b d^2 f^2 n x+18 a b^2 d^2 f^2 n^2 x-24 b^3 d^2 f^2 n^3 x+2 a^3 \log \left (1+d f \sqrt {x}\right )-6 a^2 b n \log \left (1+d f \sqrt {x}\right )+12 a b^2 n^2 \log \left (1+d f \sqrt {x}\right )-12 b^3 n^3 \log \left (1+d f \sqrt {x}\right )-2 a^3 d^2 f^2 x \log \left (1+d f \sqrt {x}\right )+6 a^2 b d^2 f^2 n x \log \left (1+d f \sqrt {x}\right )-12 a b^2 d^2 f^2 n^2 x \log \left (1+d f \sqrt {x}\right )+12 b^3 d^2 f^2 n^3 x \log \left (1+d f \sqrt {x}\right )-6 a^2 b d f \sqrt {x} \log \left (c x^n\right )+36 a b^2 d f n \sqrt {x} \log \left (c x^n\right )-84 b^3 d f n^2 \sqrt {x} \log \left (c x^n\right )+3 a^2 b d^2 f^2 x \log \left (c x^n\right )-12 a b^2 d^2 f^2 n x \log \left (c x^n\right )+18 b^3 d^2 f^2 n^2 x \log \left (c x^n\right )+6 a^2 b \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )-12 a b^2 n \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )+12 b^3 n^2 \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )-6 a^2 b d^2 f^2 x \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )+12 a b^2 d^2 f^2 n x \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )-12 b^3 d^2 f^2 n^2 x \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )-6 a b^2 d f \sqrt {x} \log ^2\left (c x^n\right )+18 b^3 d f n \sqrt {x} \log ^2\left (c x^n\right )+3 a b^2 d^2 f^2 x \log ^2\left (c x^n\right )-6 b^3 d^2 f^2 n x \log ^2\left (c x^n\right )+6 a b^2 \log \left (1+d f \sqrt {x}\right ) \log ^2\left (c x^n\right )-6 b^3 n \log \left (1+d f \sqrt {x}\right ) \log ^2\left (c x^n\right )-6 a b^2 d^2 f^2 x \log \left (1+d f \sqrt {x}\right ) \log ^2\left (c x^n\right )+6 b^3 d^2 f^2 n x \log \left (1+d f \sqrt {x}\right ) \log ^2\left (c x^n\right )-2 b^3 d f \sqrt {x} \log ^3\left (c x^n\right )+b^3 d^2 f^2 x \log ^3\left (c x^n\right )+2 b^3 \log \left (1+d f \sqrt {x}\right ) \log ^3\left (c x^n\right )-2 b^3 d^2 f^2 x \log \left (1+d f \sqrt {x}\right ) \log ^3\left (c x^n\right )+12 b n \left (a^2-2 a b n+2 b^2 n^2+2 b (a-b n) \log \left (c x^n\right )+b^2 \log ^2\left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )-48 b^2 n^2 \left (a-b n+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )+96 b^3 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )}{2 d^2 f^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (a +b \ln \left (c \,x^{n}\right )\right )^{3} \ln \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )\, dx\]
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(-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.
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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 \ln \left (d\,\left (f\,\sqrt {x}+\frac {1}{d}\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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