Optimal. Leaf size=557 \[ -\frac {2 b n \text {Li}_2\left (-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4 f^4}+\frac {b n \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^4 f^4}-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 d^4 f^4}-\frac {5 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 f^3}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3 f^3}+\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}-\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{4 d^2 f^2}-\frac {7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{18 d f}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{6 d f}-\frac {1}{2} b n x^2 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {1}{2} x^2 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac {a b n x}{2 d^2 f^2}+\frac {b^2 n x \log \left (c x^n\right )}{2 d^2 f^2}+\frac {b^2 n^2 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}+\frac {4 b^2 n^2 \text {Li}_3\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {b^2 n^2 \log \left (d f \sqrt {x}+1\right )}{4 d^4 f^4}+\frac {21 b^2 n^2 \sqrt {x}}{4 d^3 f^3}-\frac {7 b^2 n^2 x}{8 d^2 f^2}+\frac {37 b^2 n^2 x^{3/2}}{108 d f}+\frac {1}{4} b^2 n^2 x^2 \log \left (d f \sqrt {x}+1\right )-\frac {3}{16} b^2 n^2 x^2 \]
[Out]
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Rubi [A] time = 0.46, antiderivative size = 557, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {2454, 2395, 43, 2377, 2295, 2304, 2374, 6589, 2376, 2391} \[ -\frac {2 b n \text {PolyLog}\left (2,-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4 f^4}+\frac {b^2 n^2 \text {PolyLog}\left (2,-d f \sqrt {x}\right )}{d^4 f^4}+\frac {4 b^2 n^2 \text {PolyLog}\left (3,-d f \sqrt {x}\right )}{d^4 f^4}+\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}+\frac {b n \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^4 f^4}-\frac {5 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{4 d^2 f^2}-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 d^4 f^4}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3 f^3}-\frac {7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{18 d f}-\frac {1}{2} b n x^2 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{6 d f}+\frac {1}{2} x^2 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac {a b n x}{2 d^2 f^2}+\frac {b^2 n x \log \left (c x^n\right )}{2 d^2 f^2}-\frac {7 b^2 n^2 x}{8 d^2 f^2}+\frac {21 b^2 n^2 \sqrt {x}}{4 d^3 f^3}-\frac {b^2 n^2 \log \left (d f \sqrt {x}+1\right )}{4 d^4 f^4}+\frac {37 b^2 n^2 x^{3/2}}{108 d f}+\frac {1}{4} b^2 n^2 x^2 \log \left (d f \sqrt {x}+1\right )-\frac {3}{16} b^2 n^2 x^2 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 2295
Rule 2304
Rule 2374
Rule 2376
Rule 2377
Rule 2391
Rule 2395
Rule 2454
Rule 6589
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 )^2 \, dx &=\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{6 d f}-\frac {1}{8} x^2 \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}+\frac {1}{2} x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-(2 b n) \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 {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{6 d f}-\frac {1}{8} x^2 \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}+\frac {1}{2} x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{4} (b n) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx-(b n) \int x \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac {(b n) \int \frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx}{d^4 f^4}-\frac {(b n) \int \frac {a+b \log \left (c x^n\right )}{\sqrt {x}} \, dx}{d^3 f^3}+\frac {(b n) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{2 d^2 f^2}-\frac {(b n) \int \sqrt {x} \left (a+b \log \left (c x^n\right )\right ) \, dx}{3 d f}\\ &=\frac {4 b^2 n^2 \sqrt {x}}{d^3 f^3}+\frac {a b n x}{2 d^2 f^2}+\frac {4 b^2 n^2 x^{3/2}}{27 d f}-\frac {1}{16} b^2 n^2 x^2-\frac {5 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 f^3}+\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}-\frac {7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{18 d f}+\frac {1}{4} b n x^2 \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 d^4 f^4}-\frac {1}{2} b n x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{6 d f}-\frac {1}{8} x^2 \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}+\frac {1}{2} x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}+\frac {\left (b^2 n\right ) \int \log \left (c x^n\right ) \, dx}{2 d^2 f^2}+\left (b^2 n^2\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 (2 b^2 n^2\right ) \int \frac {\text {Li}_2\left (-d f \sqrt {x}\right )}{x} \, dx}{d^4 f^4}\\ &=\frac {5 b^2 n^2 \sqrt {x}}{d^3 f^3}+\frac {a b n x}{2 d^2 f^2}-\frac {3 b^2 n^2 x}{4 d^2 f^2}+\frac {7 b^2 n^2 x^{3/2}}{27 d f}-\frac {1}{8} b^2 n^2 x^2+\frac {b^2 n x \log \left (c x^n\right )}{2 d^2 f^2}-\frac {5 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 f^3}+\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}-\frac {7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{18 d f}+\frac {1}{4} b n x^2 \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 d^4 f^4}-\frac {1}{2} b n x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{6 d f}-\frac {1}{8} x^2 \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}+\frac {1}{2} x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}+\frac {4 b^2 n^2 \text {Li}_3\left (-d f \sqrt {x}\right )}{d^4 f^4}+\frac {1}{2} \left (b^2 n^2\right ) \int x \log \left (1+d f \sqrt {x}\right ) \, dx-\frac {\left (b^2 n^2\right ) \int \frac {\log \left (1+d f \sqrt {x}\right )}{x} \, dx}{2 d^4 f^4}\\ &=\frac {5 b^2 n^2 \sqrt {x}}{d^3 f^3}+\frac {a b n x}{2 d^2 f^2}-\frac {3 b^2 n^2 x}{4 d^2 f^2}+\frac {7 b^2 n^2 x^{3/2}}{27 d f}-\frac {1}{8} b^2 n^2 x^2+\frac {b^2 n x \log \left (c x^n\right )}{2 d^2 f^2}-\frac {5 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 f^3}+\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}-\frac {7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{18 d f}+\frac {1}{4} b n x^2 \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 d^4 f^4}-\frac {1}{2} b n x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{6 d f}-\frac {1}{8} x^2 \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}+\frac {1}{2} x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {b^2 n^2 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}+\frac {4 b^2 n^2 \text {Li}_3\left (-d f \sqrt {x}\right )}{d^4 f^4}+\left (b^2 n^2\right ) \operatorname {Subst}\left (\int x^3 \log (1+d f x) \, dx,x,\sqrt {x}\right )\\ &=\frac {5 b^2 n^2 \sqrt {x}}{d^3 f^3}+\frac {a b n x}{2 d^2 f^2}-\frac {3 b^2 n^2 x}{4 d^2 f^2}+\frac {7 b^2 n^2 x^{3/2}}{27 d f}-\frac {1}{8} b^2 n^2 x^2+\frac {1}{4} b^2 n^2 x^2 \log \left (1+d f \sqrt {x}\right )+\frac {b^2 n x \log \left (c x^n\right )}{2 d^2 f^2}-\frac {5 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 f^3}+\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}-\frac {7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{18 d f}+\frac {1}{4} b n x^2 \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 d^4 f^4}-\frac {1}{2} b n x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{6 d f}-\frac {1}{8} x^2 \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}+\frac {1}{2} x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {b^2 n^2 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}+\frac {4 b^2 n^2 \text {Li}_3\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {1}{4} \left (b^2 d f n^2\right ) \operatorname {Subst}\left (\int \frac {x^4}{1+d f x} \, dx,x,\sqrt {x}\right )\\ &=\frac {5 b^2 n^2 \sqrt {x}}{d^3 f^3}+\frac {a b n x}{2 d^2 f^2}-\frac {3 b^2 n^2 x}{4 d^2 f^2}+\frac {7 b^2 n^2 x^{3/2}}{27 d f}-\frac {1}{8} b^2 n^2 x^2+\frac {1}{4} b^2 n^2 x^2 \log \left (1+d f \sqrt {x}\right )+\frac {b^2 n x \log \left (c x^n\right )}{2 d^2 f^2}-\frac {5 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 f^3}+\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}-\frac {7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{18 d f}+\frac {1}{4} b n x^2 \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 d^4 f^4}-\frac {1}{2} b n x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{6 d f}-\frac {1}{8} x^2 \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}+\frac {1}{2} x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {b^2 n^2 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}+\frac {4 b^2 n^2 \text {Li}_3\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {1}{4} \left (b^2 d f n^2\right ) \operatorname {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 {21 b^2 n^2 \sqrt {x}}{4 d^3 f^3}+\frac {a b n x}{2 d^2 f^2}-\frac {7 b^2 n^2 x}{8 d^2 f^2}+\frac {37 b^2 n^2 x^{3/2}}{108 d f}-\frac {3}{16} b^2 n^2 x^2-\frac {b^2 n^2 \log \left (1+d f \sqrt {x}\right )}{4 d^4 f^4}+\frac {1}{4} b^2 n^2 x^2 \log \left (1+d f \sqrt {x}\right )+\frac {b^2 n x \log \left (c x^n\right )}{2 d^2 f^2}-\frac {5 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 f^3}+\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}-\frac {7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{18 d f}+\frac {1}{4} b n x^2 \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 d^4 f^4}-\frac {1}{2} b n x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{6 d f}-\frac {1}{8} x^2 \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}+\frac {1}{2} x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {b^2 n^2 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}+\frac {4 b^2 n^2 \text {Li}_3\left (-d f \sqrt {x}\right )}{d^4 f^4}\\ \end {align*}
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Mathematica [A] time = 0.41, size = 769, normalized size = 1.38 \[ \frac {-54 a^2 d^4 f^4 x^2+216 a^2 d^4 f^4 x^2 \log \left (d f \sqrt {x}+1\right )+72 a^2 d^3 f^3 x^{3/2}-108 a^2 d^2 f^2 x+216 a^2 d f \sqrt {x}-216 a^2 \log \left (d f \sqrt {x}+1\right )-108 a b d^4 f^4 x^2 \log \left (c x^n\right )+432 a b d^4 f^4 x^2 \log \left (c x^n\right ) \log \left (d f \sqrt {x}+1\right )+144 a b d^3 f^3 x^{3/2} \log \left (c x^n\right )-216 a b d^2 f^2 x \log \left (c x^n\right )+432 b n \text {Li}_2\left (-d f \sqrt {x}\right ) \left (-2 a-2 b \log \left (c x^n\right )+b n\right )+432 a b d f \sqrt {x} \log \left (c x^n\right )-432 a b \log \left (c x^n\right ) \log \left (d f \sqrt {x}+1\right )+108 a b d^4 f^4 n x^2-216 a b d^4 f^4 n x^2 \log \left (d f \sqrt {x}+1\right )-168 a b d^3 f^3 n x^{3/2}+324 a b d^2 f^2 n x-1080 a b d f n \sqrt {x}+216 a b n \log \left (d f \sqrt {x}+1\right )-54 b^2 d^4 f^4 x^2 \log ^2\left (c x^n\right )+216 b^2 d^4 f^4 x^2 \log ^2\left (c x^n\right ) \log \left (d f \sqrt {x}+1\right )+108 b^2 d^4 f^4 n x^2 \log \left (c x^n\right )-216 b^2 d^4 f^4 n x^2 \log \left (c x^n\right ) \log \left (d f \sqrt {x}+1\right )+72 b^2 d^3 f^3 x^{3/2} \log ^2\left (c x^n\right )-168 b^2 d^3 f^3 n x^{3/2} \log \left (c x^n\right )-108 b^2 d^2 f^2 x \log ^2\left (c x^n\right )+324 b^2 d^2 f^2 n x \log \left (c x^n\right )+216 b^2 d f \sqrt {x} \log ^2\left (c x^n\right )-216 b^2 \log ^2\left (c x^n\right ) \log \left (d f \sqrt {x}+1\right )-1080 b^2 d f n \sqrt {x} \log \left (c x^n\right )+216 b^2 n \log \left (c x^n\right ) \log \left (d f \sqrt {x}+1\right )-81 b^2 d^4 f^4 n^2 x^2+108 b^2 d^4 f^4 n^2 x^2 \log \left (d f \sqrt {x}+1\right )+148 b^2 d^3 f^3 n^2 x^{3/2}-378 b^2 d^2 f^2 n^2 x+1728 b^2 n^2 \text {Li}_3\left (-d f \sqrt {x}\right )+2268 b^2 d f n^2 \sqrt {x}-108 b^2 n^2 \log \left (d f \sqrt {x}+1\right )}{432 d^4 f^4} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.04, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} x \log \left (c x^{n}\right )^{2} + 2 \, a b x \log \left (c x^{n}\right ) + a^{2} x\right )} \log \left (d f \sqrt {x} + 1\right ), x\right ) \]
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 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \,x^{n}\right )+a \right )^{2} x \ln \left (\left (f \sqrt {x}+\frac {1}{d}\right ) d \right )\, dx \]
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 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x\,\ln \left (d\,\left (f\,\sqrt {x}+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \]
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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