Optimal. Leaf size=858 \[ -\frac {255 b^3 n^3 \sqrt {x}}{8 d^3 f^3}-\frac {9 a b^2 n^2 x}{4 d^2 f^2}+\frac {45 b^3 n^3 x}{16 d^2 f^2}-\frac {175 b^3 n^3 x^{3/2}}{216 d f}+\frac {3}{8} b^3 n^3 x^2+\frac {3 b^3 n^3 \log \left (1+d f \sqrt {x}\right )}{8 d^4 f^4}-\frac {3}{8} b^3 n^3 x^2 \log \left (1+d f \sqrt {x}\right )-\frac {9 b^3 n^2 x \log \left (c x^n\right )}{4 d^2 f^2}+\frac {63 b^2 n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{4 d^3 f^3}-\frac {3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{8 d^2 f^2}+\frac {37 b^2 n^2 x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{36 d f}-\frac {9}{16} b^2 n^2 x^2 \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 d^4 f^4}+\frac {3}{4} b^2 n^2 x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {15 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{4 d^3 f^3}+\frac {9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{8 d^2 f^2}-\frac {7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{12 d f}+\frac {3}{8} b n x^2 \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 d^4 f^4}-\frac {3}{4} b n x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 d f}-\frac {1}{8} x^2 \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 d^4 f^4}+\frac {1}{2} x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b^3 n^3 \text {Li}_2\left (-d f \sqrt {x}\right )}{2 d^4 f^4}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {6 b^3 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )}{d^4 f^4}+\frac {12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {24 b^3 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )}{d^4 f^4} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.63, antiderivative size = 858, normalized size of antiderivative = 1.00, number
of steps used = 30, number of rules used = 13, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules
used = {2504, 2442, 45, 2424, 2333, 2332, 2342, 2341, 2421, 2430, 6724, 2423, 2438}
\begin {gather*} \frac {3}{8} n^3 x^2 b^3-\frac {175 n^3 x^{3/2} b^3}{216 d f}+\frac {45 n^3 x b^3}{16 d^2 f^2}+\frac {3 n^3 \log \left (d \sqrt {x} f+1\right ) b^3}{8 d^4 f^4}-\frac {3}{8} n^3 x^2 \log \left (d \sqrt {x} f+1\right ) b^3-\frac {9 n^2 x \log \left (c x^n\right ) b^3}{4 d^2 f^2}-\frac {3 n^3 \text {PolyLog}\left (2,-d f \sqrt {x}\right ) b^3}{2 d^4 f^4}-\frac {6 n^3 \text {PolyLog}\left (3,-d f \sqrt {x}\right ) b^3}{d^4 f^4}-\frac {24 n^3 \text {PolyLog}\left (4,-d f \sqrt {x}\right ) b^3}{d^4 f^4}-\frac {255 n^3 \sqrt {x} b^3}{8 d^3 f^3}-\frac {9 a n^2 x b^2}{4 d^2 f^2}-\frac {9}{16} n^2 x^2 \left (a+b \log \left (c x^n\right )\right ) b^2+\frac {37 n^2 x^{3/2} \left (a+b \log \left (c x^n\right )\right ) b^2}{36 d f}-\frac {3 n^2 x \left (a+b \log \left (c x^n\right )\right ) b^2}{8 d^2 f^2}-\frac {3 n^2 \log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right ) b^2}{4 d^4 f^4}+\frac {3}{4} n^2 x^2 \log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right ) b^2+\frac {63 n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right ) b^2}{4 d^3 f^3}+\frac {3 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {PolyLog}\left (2,-d f \sqrt {x}\right ) b^2}{d^4 f^4}+\frac {12 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {PolyLog}\left (3,-d f \sqrt {x}\right ) b^2}{d^4 f^4}+\frac {3}{8} n x^2 \left (a+b \log \left (c x^n\right )\right )^2 b-\frac {7 n x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2 b}{12 d f}+\frac {9 n x \left (a+b \log \left (c x^n\right )\right )^2 b}{8 d^2 f^2}-\frac {3}{4} n x^2 \log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )^2 b+\frac {3 n \log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )^2 b}{4 d^4 f^4}-\frac {15 n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2 b}{4 d^3 f^3}-\frac {3 n \left (a+b \log \left (c x^n\right )\right )^2 \text {PolyLog}\left (2,-d f \sqrt {x}\right ) b}{d^4 f^4}-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^3+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 d f}-\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac {1}{2} x^2 \log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {\log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 d^4 f^4}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{2 d^3 f^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 2332
Rule 2333
Rule 2341
Rule 2342
Rule 2421
Rule 2423
Rule 2424
Rule 2430
Rule 2438
Rule 2442
Rule 2504
Rule 6724
Rubi steps
\begin {align*} \int x \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}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 d f}-\frac {1}{8} x^2 \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 d^4 f^4}+\frac {1}{2} x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-(3 b n) \int \left (-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{4 d^2 f^2}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 d^3 f^3 \sqrt {x}}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{6 d f}-\frac {1}{8} x \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}{2 d^4 f^4 x}+\frac {1}{2} x \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2\right ) \, dx\\ &=\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 d f}-\frac {1}{8} x^2 \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 d^4 f^4}+\frac {1}{2} x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3+\frac {1}{8} (3 b n) \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx-\frac {1}{2} (3 b n) \int x \log \left (1+d f \sqrt {x}\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}{2 d^4 f^4}-\frac {(3 b n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}} \, dx}{2 d^3 f^3}+\frac {(3 b n) \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{4 d^2 f^2}-\frac {(b n) \int \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{2 d f}\\ &=-\frac {15 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{4 d^3 f^3}+\frac {9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{8 d^2 f^2}-\frac {7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{12 d f}+\frac {3}{8} b n x^2 \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 d^4 f^4}-\frac {3}{4} b n x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 d f}-\frac {1}{8} x^2 \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 d^4 f^4}+\frac {1}{2} x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {1}{8} \left (3 b^2 n^2\right ) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx+\left (3 b^2 n^2\right ) \int \left (-\frac {a+b \log \left (c x^n\right )}{4 d^2 f^2}+\frac {a+b \log \left (c x^n\right )}{2 d^3 f^3 \sqrt {x}}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{6 d f}-\frac {1}{8} x \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 )}{2 d^4 f^4 x}+\frac {1}{2} x \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )\right ) \, dx+\frac {\left (6 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^4 f^4}+\frac {\left (6 b^2 n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{\sqrt {x}} \, dx}{d^3 f^3}-\frac {\left (3 b^2 n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{2 d^2 f^2}+\frac {\left (2 b^2 n^2\right ) \int \sqrt {x} \left (a+b \log \left (c x^n\right )\right ) \, dx}{3 d f}\\ &=-\frac {24 b^3 n^3 \sqrt {x}}{d^3 f^3}-\frac {3 a b^2 n^2 x}{2 d^2 f^2}-\frac {8 b^3 n^3 x^{3/2}}{27 d f}+\frac {3}{32} b^3 n^3 x^2+\frac {12 b^2 n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{d^3 f^3}+\frac {4 b^2 n^2 x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 d f}-\frac {3}{16} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {15 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{4 d^3 f^3}+\frac {9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{8 d^2 f^2}-\frac {7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{12 d f}+\frac {3}{8} b n x^2 \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 d^4 f^4}-\frac {3}{4} b n x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 d f}-\frac {1}{8} x^2 \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 d^4 f^4}+\frac {1}{2} x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}+\frac {12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {1}{8} \left (3 b^2 n^2\right ) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac {1}{2} \left (3 b^2 n^2\right ) \int x \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx-\frac {\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} \, dx}{2 d^4 f^4}+\frac {\left (3 b^2 n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{\sqrt {x}} \, dx}{2 d^3 f^3}-\frac {\left (3 b^2 n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{4 d^2 f^2}-\frac {\left (3 b^3 n^2\right ) \int \log \left (c x^n\right ) \, dx}{2 d^2 f^2}+\frac {\left (b^2 n^2\right ) \int \sqrt {x} \left (a+b \log \left (c x^n\right )\right ) \, dx}{2 d f}-\frac {\left (12 b^3 n^3\right ) \int \frac {\text {Li}_3\left (-d f \sqrt {x}\right )}{x} \, dx}{d^4 f^4}\\ &=-\frac {30 b^3 n^3 \sqrt {x}}{d^3 f^3}-\frac {9 a b^2 n^2 x}{4 d^2 f^2}+\frac {3 b^3 n^3 x}{2 d^2 f^2}-\frac {14 b^3 n^3 x^{3/2}}{27 d f}+\frac {3}{16} b^3 n^3 x^2-\frac {3 b^3 n^2 x \log \left (c x^n\right )}{2 d^2 f^2}+\frac {63 b^2 n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{4 d^3 f^3}-\frac {3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{8 d^2 f^2}+\frac {37 b^2 n^2 x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{36 d f}-\frac {9}{16} b^2 n^2 x^2 \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 d^4 f^4}+\frac {3}{4} b^2 n^2 x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {15 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{4 d^3 f^3}+\frac {9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{8 d^2 f^2}-\frac {7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{12 d f}+\frac {3}{8} b n x^2 \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 d^4 f^4}-\frac {3}{4} b n x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 d f}-\frac {1}{8} x^2 \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 d^4 f^4}+\frac {1}{2} x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}+\frac {12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {24 b^3 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {\left (3 b^3 n^2\right ) \int \log \left (c x^n\right ) \, dx}{4 d^2 f^2}-\frac {1}{2} \left (3 b^3 n^3\right ) \int \left (-\frac {1}{4 d^2 f^2}+\frac {1}{2 d^3 f^3 \sqrt {x}}+\frac {\sqrt {x}}{6 d f}-\frac {x}{8}-\frac {\log \left (1+d f \sqrt {x}\right )}{2 d^4 f^4 x}+\frac {1}{2} x \log \left (1+d f \sqrt {x}\right )\right ) \, dx-\frac {\left (3 b^3 n^3\right ) \int \frac {\text {Li}_2\left (-d f \sqrt {x}\right )}{x} \, dx}{d^4 f^4}\\ &=-\frac {63 b^3 n^3 \sqrt {x}}{2 d^3 f^3}-\frac {9 a b^2 n^2 x}{4 d^2 f^2}+\frac {21 b^3 n^3 x}{8 d^2 f^2}-\frac {37 b^3 n^3 x^{3/2}}{54 d f}+\frac {9}{32} b^3 n^3 x^2-\frac {9 b^3 n^2 x \log \left (c x^n\right )}{4 d^2 f^2}+\frac {63 b^2 n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{4 d^3 f^3}-\frac {3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{8 d^2 f^2}+\frac {37 b^2 n^2 x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{36 d f}-\frac {9}{16} b^2 n^2 x^2 \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 d^4 f^4}+\frac {3}{4} b^2 n^2 x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {15 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{4 d^3 f^3}+\frac {9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{8 d^2 f^2}-\frac {7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{12 d f}+\frac {3}{8} b n x^2 \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 d^4 f^4}-\frac {3}{4} b n x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 d f}-\frac {1}{8} x^2 \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 d^4 f^4}+\frac {1}{2} x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {6 b^3 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )}{d^4 f^4}+\frac {12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {24 b^3 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {1}{4} \left (3 b^3 n^3\right ) \int x \log \left (1+d f \sqrt {x}\right ) \, dx+\frac {\left (3 b^3 n^3\right ) \int \frac {\log \left (1+d f \sqrt {x}\right )}{x} \, dx}{4 d^4 f^4}\\ &=-\frac {63 b^3 n^3 \sqrt {x}}{2 d^3 f^3}-\frac {9 a b^2 n^2 x}{4 d^2 f^2}+\frac {21 b^3 n^3 x}{8 d^2 f^2}-\frac {37 b^3 n^3 x^{3/2}}{54 d f}+\frac {9}{32} b^3 n^3 x^2-\frac {9 b^3 n^2 x \log \left (c x^n\right )}{4 d^2 f^2}+\frac {63 b^2 n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{4 d^3 f^3}-\frac {3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{8 d^2 f^2}+\frac {37 b^2 n^2 x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{36 d f}-\frac {9}{16} b^2 n^2 x^2 \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 d^4 f^4}+\frac {3}{4} b^2 n^2 x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {15 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{4 d^3 f^3}+\frac {9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{8 d^2 f^2}-\frac {7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{12 d f}+\frac {3}{8} b n x^2 \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 d^4 f^4}-\frac {3}{4} b n x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 d f}-\frac {1}{8} x^2 \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 d^4 f^4}+\frac {1}{2} x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b^3 n^3 \text {Li}_2\left (-d f \sqrt {x}\right )}{2 d^4 f^4}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {6 b^3 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )}{d^4 f^4}+\frac {12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {24 b^3 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {1}{2} \left (3 b^3 n^3\right ) \text {Subst}\left (\int x^3 \log (1+d f x) \, dx,x,\sqrt {x}\right )\\ &=-\frac {63 b^3 n^3 \sqrt {x}}{2 d^3 f^3}-\frac {9 a b^2 n^2 x}{4 d^2 f^2}+\frac {21 b^3 n^3 x}{8 d^2 f^2}-\frac {37 b^3 n^3 x^{3/2}}{54 d f}+\frac {9}{32} b^3 n^3 x^2-\frac {3}{8} b^3 n^3 x^2 \log \left (1+d f \sqrt {x}\right )-\frac {9 b^3 n^2 x \log \left (c x^n\right )}{4 d^2 f^2}+\frac {63 b^2 n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{4 d^3 f^3}-\frac {3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{8 d^2 f^2}+\frac {37 b^2 n^2 x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{36 d f}-\frac {9}{16} b^2 n^2 x^2 \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 d^4 f^4}+\frac {3}{4} b^2 n^2 x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {15 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{4 d^3 f^3}+\frac {9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{8 d^2 f^2}-\frac {7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{12 d f}+\frac {3}{8} b n x^2 \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 d^4 f^4}-\frac {3}{4} b n x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 d f}-\frac {1}{8} x^2 \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 d^4 f^4}+\frac {1}{2} x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b^3 n^3 \text {Li}_2\left (-d f \sqrt {x}\right )}{2 d^4 f^4}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {6 b^3 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )}{d^4 f^4}+\frac {12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {24 b^3 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )}{d^4 f^4}+\frac {1}{8} \left (3 b^3 d f n^3\right ) \text {Subst}\left (\int \frac {x^4}{1+d f x} \, dx,x,\sqrt {x}\right )\\ &=-\frac {63 b^3 n^3 \sqrt {x}}{2 d^3 f^3}-\frac {9 a b^2 n^2 x}{4 d^2 f^2}+\frac {21 b^3 n^3 x}{8 d^2 f^2}-\frac {37 b^3 n^3 x^{3/2}}{54 d f}+\frac {9}{32} b^3 n^3 x^2-\frac {3}{8} b^3 n^3 x^2 \log \left (1+d f \sqrt {x}\right )-\frac {9 b^3 n^2 x \log \left (c x^n\right )}{4 d^2 f^2}+\frac {63 b^2 n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{4 d^3 f^3}-\frac {3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{8 d^2 f^2}+\frac {37 b^2 n^2 x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{36 d f}-\frac {9}{16} b^2 n^2 x^2 \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 d^4 f^4}+\frac {3}{4} b^2 n^2 x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {15 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{4 d^3 f^3}+\frac {9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{8 d^2 f^2}-\frac {7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{12 d f}+\frac {3}{8} b n x^2 \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 d^4 f^4}-\frac {3}{4} b n x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 d f}-\frac {1}{8} x^2 \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 d^4 f^4}+\frac {1}{2} x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b^3 n^3 \text {Li}_2\left (-d f \sqrt {x}\right )}{2 d^4 f^4}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {6 b^3 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )}{d^4 f^4}+\frac {12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {24 b^3 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )}{d^4 f^4}+\frac {1}{8} \left (3 b^3 d f n^3\right ) \text {Subst}\left (\int \left (-\frac {1}{d^4 f^4}+\frac {x}{d^3 f^3}-\frac {x^2}{d^2 f^2}+\frac {x^3}{d f}+\frac {1}{d^4 f^4 (1+d f x)}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {255 b^3 n^3 \sqrt {x}}{8 d^3 f^3}-\frac {9 a b^2 n^2 x}{4 d^2 f^2}+\frac {45 b^3 n^3 x}{16 d^2 f^2}-\frac {175 b^3 n^3 x^{3/2}}{216 d f}+\frac {3}{8} b^3 n^3 x^2+\frac {3 b^3 n^3 \log \left (1+d f \sqrt {x}\right )}{8 d^4 f^4}-\frac {3}{8} b^3 n^3 x^2 \log \left (1+d f \sqrt {x}\right )-\frac {9 b^3 n^2 x \log \left (c x^n\right )}{4 d^2 f^2}+\frac {63 b^2 n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{4 d^3 f^3}-\frac {3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{8 d^2 f^2}+\frac {37 b^2 n^2 x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{36 d f}-\frac {9}{16} b^2 n^2 x^2 \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 d^4 f^4}+\frac {3}{4} b^2 n^2 x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {15 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{4 d^3 f^3}+\frac {9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{8 d^2 f^2}-\frac {7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{12 d f}+\frac {3}{8} b n x^2 \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 d^4 f^4}-\frac {3}{4} b n x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 d f}-\frac {1}{8} x^2 \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 d^4 f^4}+\frac {1}{2} x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b^3 n^3 \text {Li}_2\left (-d f \sqrt {x}\right )}{2 d^4 f^4}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {6 b^3 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )}{d^4 f^4}+\frac {12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {24 b^3 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )}{d^4 f^4}\\ \end {align*}
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Mathematica [A]
time = 0.37, size = 1432, normalized size = 1.67 \begin {gather*} \frac {216 a^3 d f \sqrt {x}-1620 a^2 b d f n \sqrt {x}+6804 a b^2 d f n^2 \sqrt {x}-13770 b^3 d f n^3 \sqrt {x}-108 a^3 d^2 f^2 x+486 a^2 b d^2 f^2 n x-1134 a b^2 d^2 f^2 n^2 x+1215 b^3 d^2 f^2 n^3 x+72 a^3 d^3 f^3 x^{3/2}-252 a^2 b d^3 f^3 n x^{3/2}+444 a b^2 d^3 f^3 n^2 x^{3/2}-350 b^3 d^3 f^3 n^3 x^{3/2}-54 a^3 d^4 f^4 x^2+162 a^2 b d^4 f^4 n x^2-243 a b^2 d^4 f^4 n^2 x^2+162 b^3 d^4 f^4 n^3 x^2-216 a^3 \log \left (1+d f \sqrt {x}\right )+324 a^2 b n \log \left (1+d f \sqrt {x}\right )-324 a b^2 n^2 \log \left (1+d f \sqrt {x}\right )+162 b^3 n^3 \log \left (1+d f \sqrt {x}\right )+216 a^3 d^4 f^4 x^2 \log \left (1+d f \sqrt {x}\right )-324 a^2 b d^4 f^4 n x^2 \log \left (1+d f \sqrt {x}\right )+324 a b^2 d^4 f^4 n^2 x^2 \log \left (1+d f \sqrt {x}\right )-162 b^3 d^4 f^4 n^3 x^2 \log \left (1+d f \sqrt {x}\right )+648 a^2 b d f \sqrt {x} \log \left (c x^n\right )-3240 a b^2 d f n \sqrt {x} \log \left (c x^n\right )+6804 b^3 d f n^2 \sqrt {x} \log \left (c x^n\right )-324 a^2 b d^2 f^2 x \log \left (c x^n\right )+972 a b^2 d^2 f^2 n x \log \left (c x^n\right )-1134 b^3 d^2 f^2 n^2 x \log \left (c x^n\right )+216 a^2 b d^3 f^3 x^{3/2} \log \left (c x^n\right )-504 a b^2 d^3 f^3 n x^{3/2} \log \left (c x^n\right )+444 b^3 d^3 f^3 n^2 x^{3/2} \log \left (c x^n\right )-162 a^2 b d^4 f^4 x^2 \log \left (c x^n\right )+324 a b^2 d^4 f^4 n x^2 \log \left (c x^n\right )-243 b^3 d^4 f^4 n^2 x^2 \log \left (c x^n\right )-648 a^2 b \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )+648 a b^2 n \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )-324 b^3 n^2 \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )+648 a^2 b d^4 f^4 x^2 \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )-648 a b^2 d^4 f^4 n x^2 \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )+324 b^3 d^4 f^4 n^2 x^2 \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )+648 a b^2 d f \sqrt {x} \log ^2\left (c x^n\right )-1620 b^3 d f n \sqrt {x} \log ^2\left (c x^n\right )-324 a b^2 d^2 f^2 x \log ^2\left (c x^n\right )+486 b^3 d^2 f^2 n x \log ^2\left (c x^n\right )+216 a b^2 d^3 f^3 x^{3/2} \log ^2\left (c x^n\right )-252 b^3 d^3 f^3 n x^{3/2} \log ^2\left (c x^n\right )-162 a b^2 d^4 f^4 x^2 \log ^2\left (c x^n\right )+162 b^3 d^4 f^4 n x^2 \log ^2\left (c x^n\right )-648 a b^2 \log \left (1+d f \sqrt {x}\right ) \log ^2\left (c x^n\right )+324 b^3 n \log \left (1+d f \sqrt {x}\right ) \log ^2\left (c x^n\right )+648 a b^2 d^4 f^4 x^2 \log \left (1+d f \sqrt {x}\right ) \log ^2\left (c x^n\right )-324 b^3 d^4 f^4 n x^2 \log \left (1+d f \sqrt {x}\right ) \log ^2\left (c x^n\right )+216 b^3 d f \sqrt {x} \log ^3\left (c x^n\right )-108 b^3 d^2 f^2 x \log ^3\left (c x^n\right )+72 b^3 d^3 f^3 x^{3/2} \log ^3\left (c x^n\right )-54 b^3 d^4 f^4 x^2 \log ^3\left (c x^n\right )-216 b^3 \log \left (1+d f \sqrt {x}\right ) \log ^3\left (c x^n\right )+216 b^3 d^4 f^4 x^2 \log \left (1+d f \sqrt {x}\right ) \log ^3\left (c x^n\right )-648 b n \left (2 a^2-2 a b n+b^2 n^2-2 b (-2 a+b n) \log \left (c x^n\right )+2 b^2 \log ^2\left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )+2592 b^2 n^2 \left (2 a-b n+2 b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )-10368 b^3 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )}{432 d^4 f^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int x \left (a +b \ln \left (c \,x^{n}\right )\right )^{3} \ln \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )\, dx\]
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x\,\ln \left (d\,\left (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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