Optimal. Leaf size=604 \[ \frac {12 b^2 n^2 \text {Li}_2\left (-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}+\frac {24 b^2 n^2 \text {Li}_3\left (-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}-\frac {6 b^2 n^2 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}+6 b^2 n^2 x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {42 b^2 n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{d f}-3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )-6 a b^2 n^2 x-\frac {6 b n \text {Li}_2\left (-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2 f^2}+\frac {3 b n \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2 f^2}-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d^2 f^2}-3 b n x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {9 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{d f}+3 b n x \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^3-6 b^3 n^2 x \log \left (c x^n\right )-\frac {12 b^3 n^3 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^2 f^2}-\frac {24 b^3 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )}{d^2 f^2}-\frac {48 b^3 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )}{d^2 f^2}+\frac {6 b^3 n^3 \log \left (d f \sqrt {x}+1\right )}{d^2 f^2}-\frac {90 b^3 n^3 \sqrt {x}}{d f}-6 b^3 n^3 x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )+12 b^3 n^3 x \]
[Out]
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Rubi [A] time = 0.52, 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 = {2448, 266, 43, 2370, 2296, 2295, 2305, 2304, 2391, 2374, 6589, 2383} \[ \frac {12 b^2 n^2 \text {PolyLog}\left (2,-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}+\frac {24 b^2 n^2 \text {PolyLog}\left (3,-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}-\frac {6 b n \text {PolyLog}\left (2,-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2 f^2}-\frac {12 b^3 n^3 \text {PolyLog}\left (2,-d f \sqrt {x}\right )}{d^2 f^2}-\frac {24 b^3 n^3 \text {PolyLog}\left (3,-d f \sqrt {x}\right )}{d^2 f^2}-\frac {48 b^3 n^3 \text {PolyLog}\left (4,-d f \sqrt {x}\right )}{d^2 f^2}-\frac {6 b^2 n^2 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}+6 b^2 n^2 x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {42 b^2 n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{d f}-3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )-6 a b^2 n^2 x+\frac {3 b n \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2 f^2}-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d^2 f^2}-3 b n x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {9 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{d f}+3 b n x \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^3-6 b^3 n^2 x \log \left (c x^n\right )+\frac {6 b^3 n^3 \log \left (d f \sqrt {x}+1\right )}{d^2 f^2}-\frac {90 b^3 n^3 \sqrt {x}}{d f}-6 b^3 n^3 x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )+12 b^3 n^3 x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 266
Rule 2295
Rule 2296
Rule 2304
Rule 2305
Rule 2370
Rule 2374
Rule 2383
Rule 2391
Rule 2448
Rule 6589
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 ) \operatorname {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 ) \operatorname {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 = 0.51, size = 986, normalized size = 1.63 \[ -\frac {d^2 f^2 x a^3-2 d^2 f^2 x \log \left (d \sqrt {x} f+1\right ) a^3+2 \log \left (d \sqrt {x} f+1\right ) a^3-2 d f \sqrt {x} a^3-6 b d^2 f^2 n x a^2-6 b n \log \left (d \sqrt {x} f+1\right ) a^2+6 b d^2 f^2 n x \log \left (d \sqrt {x} f+1\right ) a^2+3 b d^2 f^2 x \log \left (c x^n\right ) a^2+6 b \log \left (d \sqrt {x} f+1\right ) \log \left (c x^n\right ) a^2-6 b d^2 f^2 x \log \left (d \sqrt {x} f+1\right ) \log \left (c x^n\right ) a^2-6 b d f \sqrt {x} \log \left (c x^n\right ) a^2+18 b d f n \sqrt {x} a^2+3 b^2 d^2 f^2 x \log ^2\left (c x^n\right ) a+6 b^2 \log \left (d \sqrt {x} f+1\right ) \log ^2\left (c x^n\right ) a-6 b^2 d^2 f^2 x \log \left (d \sqrt {x} f+1\right ) \log ^2\left (c x^n\right ) a-6 b^2 d f \sqrt {x} \log ^2\left (c x^n\right ) a+18 b^2 d^2 f^2 n^2 x a+12 b^2 n^2 \log \left (d \sqrt {x} f+1\right ) a-12 b^2 d^2 f^2 n^2 x \log \left (d \sqrt {x} f+1\right ) a-12 b^2 d^2 f^2 n x \log \left (c x^n\right ) a-12 b^2 n \log \left (d \sqrt {x} f+1\right ) \log \left (c x^n\right ) a+12 b^2 d^2 f^2 n x \log \left (d \sqrt {x} f+1\right ) \log \left (c x^n\right ) a+36 b^2 d f n \sqrt {x} \log \left (c x^n\right ) a-84 b^2 d f n^2 \sqrt {x} a+b^3 d^2 f^2 x \log ^3\left (c x^n\right )+2 b^3 \log \left (d \sqrt {x} f+1\right ) \log ^3\left (c x^n\right )-2 b^3 d^2 f^2 x \log \left (d \sqrt {x} f+1\right ) \log ^3\left (c x^n\right )-2 b^3 d f \sqrt {x} \log ^3\left (c x^n\right )-6 b^3 d^2 f^2 n x \log ^2\left (c x^n\right )-6 b^3 n \log \left (d \sqrt {x} f+1\right ) \log ^2\left (c x^n\right )+6 b^3 d^2 f^2 n x \log \left (d \sqrt {x} f+1\right ) \log ^2\left (c x^n\right )+18 b^3 d f n \sqrt {x} \log ^2\left (c x^n\right )-24 b^3 d^2 f^2 n^3 x-12 b^3 n^3 \log \left (d \sqrt {x} f+1\right )+12 b^3 d^2 f^2 n^3 x \log \left (d \sqrt {x} f+1\right )+18 b^3 d^2 f^2 n^2 x \log \left (c x^n\right )+12 b^3 n^2 \log \left (d \sqrt {x} f+1\right ) \log \left (c x^n\right )-12 b^3 d^2 f^2 n^2 x \log \left (d \sqrt {x} f+1\right ) \log \left (c x^n\right )-84 b^3 d f n^2 \sqrt {x} \log \left (c x^n\right )+12 b n \left (a^2-2 b n a+2 b^2 n^2+b^2 \log ^2\left (c x^n\right )+2 b (a-b n) \log \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 )+180 b^3 d f n^3 \sqrt {x}}{2 d^2 f^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{3} \log \left (c x^{n}\right )^{3} + 3 \, a b^{2} \log \left (c x^{n}\right )^{2} + 3 \, a^{2} b \log \left (c x^{n}\right ) + a^{3}\right )} \log \left (d f \sqrt {x} + 1\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.12, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \,x^{n}\right )+a \right )^{3} \ln \left (\left (f \sqrt {x}+\frac {1}{d}\right ) d \right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ {\left (b^{3} x \log \left (x^{n}\right )^{3} - 3 \, {\left (b^{3} {\left (n - \log \relax (c)\right )} - a b^{2}\right )} x \log \left (x^{n}\right )^{2} + 3 \, {\left ({\left (2 \, n^{2} - 2 \, n \log \relax (c) + \log \relax (c)^{2}\right )} b^{3} - 2 \, a b^{2} {\left (n - \log \relax (c)\right )} + a^{2} b\right )} x \log \left (x^{n}\right ) + {\left (3 \, {\left (2 \, n^{2} - 2 \, n \log \relax (c) + \log \relax (c)^{2}\right )} a b^{2} - {\left (6 \, n^{3} - 6 \, n^{2} \log \relax (c) + 3 \, n \log \relax (c)^{2} - \log \relax (c)^{3}\right )} b^{3} - 3 \, a^{2} b {\left (n - \log \relax (c)\right )} + a^{3}\right )} x\right )} \log \left (d f \sqrt {x} + 1\right ) - \frac {9 \, b^{3} d f x^{2} \log \left (x^{n}\right )^{3} + 9 \, {\left (3 \, a b^{2} d f - {\left (5 \, d f n - 3 \, d f \log \relax (c)\right )} b^{3}\right )} x^{2} \log \left (x^{n}\right )^{2} + 3 \, {\left (9 \, a^{2} b d f - 6 \, {\left (5 \, d f n - 3 \, d f \log \relax (c)\right )} a b^{2} + {\left (38 \, d f n^{2} - 30 \, d f n \log \relax (c) + 9 \, d f \log \relax (c)^{2}\right )} b^{3}\right )} x^{2} \log \left (x^{n}\right ) + {\left (9 \, a^{3} d f - 9 \, {\left (5 \, d f n - 3 \, d f \log \relax (c)\right )} a^{2} b + 3 \, {\left (38 \, d f n^{2} - 30 \, d f n \log \relax (c) + 9 \, d f \log \relax (c)^{2}\right )} a b^{2} - {\left (130 \, d f n^{3} - 114 \, d f n^{2} \log \relax (c) + 45 \, d f n \log \relax (c)^{2} - 9 \, d f \log \relax (c)^{3}\right )} b^{3}\right )} x^{2}}{27 \, \sqrt {x}} + \int \frac {b^{3} d^{2} f^{2} x \log \left (x^{n}\right )^{3} + 3 \, {\left (a b^{2} d^{2} f^{2} - {\left (d^{2} f^{2} n - d^{2} f^{2} \log \relax (c)\right )} b^{3}\right )} x \log \left (x^{n}\right )^{2} + 3 \, {\left (a^{2} b d^{2} f^{2} - 2 \, {\left (d^{2} f^{2} n - d^{2} f^{2} \log \relax (c)\right )} a b^{2} + {\left (2 \, d^{2} f^{2} n^{2} - 2 \, d^{2} f^{2} n \log \relax (c) + d^{2} f^{2} \log \relax (c)^{2}\right )} b^{3}\right )} x \log \left (x^{n}\right ) + {\left (a^{3} d^{2} f^{2} - 3 \, {\left (d^{2} f^{2} n - d^{2} f^{2} \log \relax (c)\right )} a^{2} b + 3 \, {\left (2 \, d^{2} f^{2} n^{2} - 2 \, d^{2} f^{2} n \log \relax (c) + d^{2} f^{2} \log \relax (c)^{2}\right )} a b^{2} - {\left (6 \, d^{2} f^{2} n^{3} - 6 \, d^{2} f^{2} n^{2} \log \relax (c) + 3 \, d^{2} f^{2} n \log \relax (c)^{2} - d^{2} f^{2} \log \relax (c)^{3}\right )} b^{3}\right )} x}{2 \, {\left (d f \sqrt {x} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \ln \left (d\,\left (f\,\sqrt {x}+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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