Optimal. Leaf size=555 \[ -\frac {37 b^2 d f n^2}{108 x^{3/2}}+\frac {7 b^2 d^2 f^2 n^2}{8 x}-\frac {21 b^2 d^3 f^3 n^2}{4 \sqrt {x}}+\frac {1}{4} b^2 d^4 f^4 n^2 \log \left (1+d f \sqrt {x}\right )-\frac {b^2 n^2 \log \left (1+d f \sqrt {x}\right )}{4 x^2}-\frac {1}{8} b^2 d^4 f^4 n^2 \log (x)+\frac {1}{8} b^2 d^4 f^4 n^2 \log ^2(x)-\frac {7 b d f n \left (a+b \log \left (c x^n\right )\right )}{18 x^{3/2}}+\frac {3 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac {5 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {x}}+\frac {1}{2} b d^4 f^4 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {1}{4} b d^4 f^4 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 \sqrt {x}}+\frac {1}{2} d^4 f^4 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac {d^4 f^4 \left (a+b \log \left (c x^n\right )\right )^3}{12 b n}+b^2 d^4 f^4 n^2 \text {Li}_2\left (-d f \sqrt {x}\right )+2 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )-4 b^2 d^4 f^4 n^2 \text {Li}_3\left (-d f \sqrt {x}\right ) \]
[Out]
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Rubi [A]
time = 0.38, antiderivative size = 555, normalized size of antiderivative = 1.00, number
of steps used = 19, number of rules used = 14, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules
used = {2504, 2442, 46, 2424, 2341, 2423, 2438, 2338, 2421, 6724, 2413, 12, 2339, 30}
\begin {gather*} 2 b d^4 f^4 n \text {PolyLog}\left (2,-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )+b^2 d^4 f^4 n^2 \text {PolyLog}\left (2,-d f \sqrt {x}\right )-4 b^2 d^4 f^4 n^2 \text {PolyLog}\left (3,-d f \sqrt {x}\right )-\frac {d^4 f^4 \left (a+b \log \left (c x^n\right )\right )^3}{12 b n}+\frac {1}{2} d^4 f^4 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{2} b d^4 f^4 n \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b d^4 f^4 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 \sqrt {x}}-\frac {5 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {x}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x}+\frac {3 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{6 x^{3/2}}-\frac {7 b d f n \left (a+b \log \left (c x^n\right )\right )}{18 x^{3/2}}-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac {b n \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}+\frac {1}{8} b^2 d^4 f^4 n^2 \log ^2(x)+\frac {1}{4} b^2 d^4 f^4 n^2 \log \left (d f \sqrt {x}+1\right )-\frac {1}{8} b^2 d^4 f^4 n^2 \log (x)-\frac {21 b^2 d^3 f^3 n^2}{4 \sqrt {x}}+\frac {7 b^2 d^2 f^2 n^2}{8 x}-\frac {37 b^2 d f n^2}{108 x^{3/2}}-\frac {b^2 n^2 \log \left (d f \sqrt {x}+1\right )}{4 x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 30
Rule 46
Rule 2338
Rule 2339
Rule 2341
Rule 2413
Rule 2421
Rule 2423
Rule 2424
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 )^2}{x^3} \, dx &=-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 \sqrt {x}}+\frac {1}{2} d^4 f^4 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac {1}{4} d^4 f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )^2-(2 b n) \int \left (-\frac {d f \left (a+b \log \left (c x^n\right )\right )}{6 x^{5/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )}{2 x^{3/2}}-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^3}+\frac {d^4 f^4 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x}-\frac {d^4 f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )}{4 x}\right ) \, dx\\ &=-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 \sqrt {x}}+\frac {1}{2} d^4 f^4 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac {1}{4} d^4 f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )^2+(b n) \int \frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx+\frac {1}{3} (b d f n) \int \frac {a+b \log \left (c x^n\right )}{x^{5/2}} \, dx-\frac {1}{2} \left (b d^2 f^2 n\right ) \int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx+\left (b d^3 f^3 n\right ) \int \frac {a+b \log \left (c x^n\right )}{x^{3/2}} \, dx+\frac {1}{2} \left (b d^4 f^4 n\right ) \int \frac {\log (x) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx-\left (b d^4 f^4 n\right ) \int \frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx\\ &=-\frac {4 b^2 d f n^2}{27 x^{3/2}}+\frac {b^2 d^2 f^2 n^2}{2 x}-\frac {4 b^2 d^3 f^3 n^2}{\sqrt {x}}-\frac {7 b d f n \left (a+b \log \left (c x^n\right )\right )}{18 x^{3/2}}+\frac {3 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac {5 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {x}}+\frac {1}{2} b d^4 f^4 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {1}{4} b d^4 f^4 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 \sqrt {x}}+\frac {1}{2} d^4 f^4 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}+2 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )-\frac {1}{2} \left (b d^4 f^4 n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 b n x} \, dx-\left (b^2 n^2\right ) \int \left (-\frac {d f}{6 x^{5/2}}+\frac {d^2 f^2}{4 x^2}-\frac {d^3 f^3}{2 x^{3/2}}-\frac {\log \left (1+d f \sqrt {x}\right )}{2 x^3}+\frac {d^4 f^4 \log \left (1+d f \sqrt {x}\right )}{2 x}-\frac {d^4 f^4 \log (x)}{4 x}\right ) \, dx-\left (2 b^2 d^4 f^4 n^2\right ) \int \frac {\text {Li}_2\left (-d f \sqrt {x}\right )}{x} \, dx\\ &=-\frac {7 b^2 d f n^2}{27 x^{3/2}}+\frac {3 b^2 d^2 f^2 n^2}{4 x}-\frac {5 b^2 d^3 f^3 n^2}{\sqrt {x}}-\frac {7 b d f n \left (a+b \log \left (c x^n\right )\right )}{18 x^{3/2}}+\frac {3 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac {5 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {x}}+\frac {1}{2} b d^4 f^4 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {1}{4} b d^4 f^4 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 \sqrt {x}}+\frac {1}{2} d^4 f^4 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}+2 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )-4 b^2 d^4 f^4 n^2 \text {Li}_3\left (-d f \sqrt {x}\right )-\frac {1}{4} \left (d^4 f^4\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx+\frac {1}{2} \left (b^2 n^2\right ) \int \frac {\log \left (1+d f \sqrt {x}\right )}{x^3} \, dx+\frac {1}{4} \left (b^2 d^4 f^4 n^2\right ) \int \frac {\log (x)}{x} \, dx-\frac {1}{2} \left (b^2 d^4 f^4 n^2\right ) \int \frac {\log \left (1+d f \sqrt {x}\right )}{x} \, dx\\ &=-\frac {7 b^2 d f n^2}{27 x^{3/2}}+\frac {3 b^2 d^2 f^2 n^2}{4 x}-\frac {5 b^2 d^3 f^3 n^2}{\sqrt {x}}+\frac {1}{8} b^2 d^4 f^4 n^2 \log ^2(x)-\frac {7 b d f n \left (a+b \log \left (c x^n\right )\right )}{18 x^{3/2}}+\frac {3 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac {5 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {x}}+\frac {1}{2} b d^4 f^4 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {1}{4} b d^4 f^4 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 \sqrt {x}}+\frac {1}{2} d^4 f^4 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}+b^2 d^4 f^4 n^2 \text {Li}_2\left (-d f \sqrt {x}\right )+2 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )-4 b^2 d^4 f^4 n^2 \text {Li}_3\left (-d f \sqrt {x}\right )-\frac {\left (d^4 f^4\right ) \text {Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )}{4 b n}+\left (b^2 n^2\right ) \text {Subst}\left (\int \frac {\log (1+d f x)}{x^5} \, dx,x,\sqrt {x}\right )\\ &=-\frac {7 b^2 d f n^2}{27 x^{3/2}}+\frac {3 b^2 d^2 f^2 n^2}{4 x}-\frac {5 b^2 d^3 f^3 n^2}{\sqrt {x}}-\frac {b^2 n^2 \log \left (1+d f \sqrt {x}\right )}{4 x^2}+\frac {1}{8} b^2 d^4 f^4 n^2 \log ^2(x)-\frac {7 b d f n \left (a+b \log \left (c x^n\right )\right )}{18 x^{3/2}}+\frac {3 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac {5 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {x}}+\frac {1}{2} b d^4 f^4 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {1}{4} b d^4 f^4 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 \sqrt {x}}+\frac {1}{2} d^4 f^4 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac {d^4 f^4 \left (a+b \log \left (c x^n\right )\right )^3}{12 b n}+b^2 d^4 f^4 n^2 \text {Li}_2\left (-d f \sqrt {x}\right )+2 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )-4 b^2 d^4 f^4 n^2 \text {Li}_3\left (-d f \sqrt {x}\right )+\frac {1}{4} \left (b^2 d f n^2\right ) \text {Subst}\left (\int \frac {1}{x^4 (1+d f x)} \, dx,x,\sqrt {x}\right )\\ &=-\frac {7 b^2 d f n^2}{27 x^{3/2}}+\frac {3 b^2 d^2 f^2 n^2}{4 x}-\frac {5 b^2 d^3 f^3 n^2}{\sqrt {x}}-\frac {b^2 n^2 \log \left (1+d f \sqrt {x}\right )}{4 x^2}+\frac {1}{8} b^2 d^4 f^4 n^2 \log ^2(x)-\frac {7 b d f n \left (a+b \log \left (c x^n\right )\right )}{18 x^{3/2}}+\frac {3 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac {5 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {x}}+\frac {1}{2} b d^4 f^4 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {1}{4} b d^4 f^4 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 \sqrt {x}}+\frac {1}{2} d^4 f^4 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac {d^4 f^4 \left (a+b \log \left (c x^n\right )\right )^3}{12 b n}+b^2 d^4 f^4 n^2 \text {Li}_2\left (-d f \sqrt {x}\right )+2 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )-4 b^2 d^4 f^4 n^2 \text {Li}_3\left (-d f \sqrt {x}\right )+\frac {1}{4} \left (b^2 d f n^2\right ) \text {Subst}\left (\int \left (\frac {1}{x^4}-\frac {d f}{x^3}+\frac {d^2 f^2}{x^2}-\frac {d^3 f^3}{x}+\frac {d^4 f^4}{1+d f x}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {37 b^2 d f n^2}{108 x^{3/2}}+\frac {7 b^2 d^2 f^2 n^2}{8 x}-\frac {21 b^2 d^3 f^3 n^2}{4 \sqrt {x}}+\frac {1}{4} b^2 d^4 f^4 n^2 \log \left (1+d f \sqrt {x}\right )-\frac {b^2 n^2 \log \left (1+d f \sqrt {x}\right )}{4 x^2}-\frac {1}{8} b^2 d^4 f^4 n^2 \log (x)+\frac {1}{8} b^2 d^4 f^4 n^2 \log ^2(x)-\frac {7 b d f n \left (a+b \log \left (c x^n\right )\right )}{18 x^{3/2}}+\frac {3 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac {5 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {x}}+\frac {1}{2} b d^4 f^4 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {1}{4} b d^4 f^4 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 \sqrt {x}}+\frac {1}{2} d^4 f^4 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac {d^4 f^4 \left (a+b \log \left (c x^n\right )\right )^3}{12 b n}+b^2 d^4 f^4 n^2 \text {Li}_2\left (-d f \sqrt {x}\right )+2 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )-4 b^2 d^4 f^4 n^2 \text {Li}_3\left (-d f \sqrt {x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.34, size = 881, normalized size = 1.59 \begin {gather*} -\frac {36 a^2 d f \sqrt {x}+84 a b d f n \sqrt {x}+74 b^2 d f n^2 \sqrt {x}-54 a^2 d^2 f^2 x-162 a b d^2 f^2 n x-189 b^2 d^2 f^2 n^2 x+108 a^2 d^3 f^3 x^{3/2}+540 a b d^3 f^3 n x^{3/2}+1134 b^2 d^3 f^3 n^2 x^{3/2}+108 a^2 \log \left (1+d f \sqrt {x}\right )+108 a b n \log \left (1+d f \sqrt {x}\right )+54 b^2 n^2 \log \left (1+d f \sqrt {x}\right )-108 a^2 d^4 f^4 x^2 \log \left (1+d f \sqrt {x}\right )-108 a b d^4 f^4 n x^2 \log \left (1+d f \sqrt {x}\right )-54 b^2 d^4 f^4 n^2 x^2 \log \left (1+d f \sqrt {x}\right )+54 a^2 d^4 f^4 x^2 \log (x)+54 a b d^4 f^4 n x^2 \log (x)+27 b^2 d^4 f^4 n^2 x^2 \log (x)-54 a b d^4 f^4 n x^2 \log ^2(x)-27 b^2 d^4 f^4 n^2 x^2 \log ^2(x)+18 b^2 d^4 f^4 n^2 x^2 \log ^3(x)+72 a b d f \sqrt {x} \log \left (c x^n\right )+84 b^2 d f n \sqrt {x} \log \left (c x^n\right )-108 a b d^2 f^2 x \log \left (c x^n\right )-162 b^2 d^2 f^2 n x \log \left (c x^n\right )+216 a b d^3 f^3 x^{3/2} \log \left (c x^n\right )+540 b^2 d^3 f^3 n x^{3/2} \log \left (c x^n\right )+216 a b \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )+108 b^2 n \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )-216 a b d^4 f^4 x^2 \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )-108 b^2 d^4 f^4 n x^2 \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )+108 a b d^4 f^4 x^2 \log (x) \log \left (c x^n\right )+54 b^2 d^4 f^4 n x^2 \log (x) \log \left (c x^n\right )-54 b^2 d^4 f^4 n x^2 \log ^2(x) \log \left (c x^n\right )+36 b^2 d f \sqrt {x} \log ^2\left (c x^n\right )-54 b^2 d^2 f^2 x \log ^2\left (c x^n\right )+108 b^2 d^3 f^3 x^{3/2} \log ^2\left (c x^n\right )+108 b^2 \log \left (1+d f \sqrt {x}\right ) \log ^2\left (c x^n\right )-108 b^2 d^4 f^4 x^2 \log \left (1+d f \sqrt {x}\right ) \log ^2\left (c x^n\right )+54 b^2 d^4 f^4 x^2 \log (x) \log ^2\left (c x^n\right )-216 b d^4 f^4 n x^2 \left (2 a+b n+2 b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )+864 b^2 d^4 f^4 n^2 x^2 \text {Li}_3\left (-d f \sqrt {x}\right )}{216 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right )^{2} \ln \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )}{x^{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\ln \left (d\,\left (f\,\sqrt {x}+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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