Optimal. Leaf size=608 \[ -\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac {b n \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}+\frac {2 b f^4 n \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4}-\frac {f^4 \left (a+b \log \left (c x^n\right )\right )^3}{12 b e^4 n}+\frac {f^4 \log \left (\frac {f \sqrt {x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^4}+\frac {b f^4 n \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^4}-\frac {b f^4 n \log (x) \left (a+b \log \left (c x^n\right )\right )}{4 e^4}-\frac {f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^3 \sqrt {x}}-\frac {5 b f^3 n \left (a+b \log \left (c x^n\right )\right )}{2 e^3 \sqrt {x}}+\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 e^2 x}+\frac {3 b f^2 n \left (a+b \log \left (c x^n\right )\right )}{4 e^2 x}-\frac {f \left (a+b \log \left (c x^n\right )\right )^2}{6 e x^{3/2}}-\frac {7 b f n \left (a+b \log \left (c x^n\right )\right )}{18 e x^{3/2}}-\frac {b^2 n^2 \log \left (d \left (e+f \sqrt {x}\right )\right )}{4 x^2}-\frac {b^2 f^4 n^2 \text {Li}_2\left (\frac {\sqrt {x} f}{e}+1\right )}{e^4}-\frac {4 b^2 f^4 n^2 \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )}{e^4}+\frac {b^2 f^4 n^2 \log ^2(x)}{8 e^4}+\frac {b^2 f^4 n^2 \log \left (e+f \sqrt {x}\right )}{4 e^4}-\frac {b^2 f^4 n^2 \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{e^4}-\frac {b^2 f^4 n^2 \log (x)}{8 e^4}-\frac {21 b^2 f^3 n^2}{4 e^3 \sqrt {x}}+\frac {7 b^2 f^2 n^2}{8 e^2 x}-\frac {37 b^2 f n^2}{108 e x^{3/2}} \]
[Out]
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Rubi [A] time = 0.78, antiderivative size = 608, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 17, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.607, Rules used = {2454, 2395, 44, 2377, 2304, 2375, 2337, 2374, 6589, 2376, 2394, 2315, 2301, 2366, 12, 2302, 30} \[ \frac {2 b f^4 n \text {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4}-\frac {b^2 f^4 n^2 \text {PolyLog}\left (2,\frac {f \sqrt {x}}{e}+1\right )}{e^4}-\frac {4 b^2 f^4 n^2 \text {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right )}{e^4}-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac {b n \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {f^4 \left (a+b \log \left (c x^n\right )\right )^3}{12 b e^4 n}+\frac {f^4 \log \left (\frac {f \sqrt {x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^4}+\frac {b f^4 n \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^4}-\frac {b f^4 n \log (x) \left (a+b \log \left (c x^n\right )\right )}{4 e^4}-\frac {f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^3 \sqrt {x}}-\frac {5 b f^3 n \left (a+b \log \left (c x^n\right )\right )}{2 e^3 \sqrt {x}}+\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 e^2 x}+\frac {3 b f^2 n \left (a+b \log \left (c x^n\right )\right )}{4 e^2 x}-\frac {f \left (a+b \log \left (c x^n\right )\right )^2}{6 e x^{3/2}}-\frac {7 b f n \left (a+b \log \left (c x^n\right )\right )}{18 e x^{3/2}}-\frac {b^2 n^2 \log \left (d \left (e+f \sqrt {x}\right )\right )}{4 x^2}-\frac {21 b^2 f^3 n^2}{4 e^3 \sqrt {x}}+\frac {7 b^2 f^2 n^2}{8 e^2 x}+\frac {b^2 f^4 n^2 \log ^2(x)}{8 e^4}+\frac {b^2 f^4 n^2 \log \left (e+f \sqrt {x}\right )}{4 e^4}-\frac {b^2 f^4 n^2 \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{e^4}-\frac {b^2 f^4 n^2 \log (x)}{8 e^4}-\frac {37 b^2 f n^2}{108 e x^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 30
Rule 44
Rule 2301
Rule 2302
Rule 2304
Rule 2315
Rule 2337
Rule 2366
Rule 2374
Rule 2375
Rule 2376
Rule 2377
Rule 2394
Rule 2395
Rule 2454
Rule 6589
Rubi steps
\begin {align*} \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx &=-\frac {f \left (a+b \log \left (c x^n\right )\right )^2}{6 e x^{3/2}}+\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 e^2 x}-\frac {f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^3 \sqrt {x}}+\frac {f^4 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^4}-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac {f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )^2}{4 e^4}-(2 b n) \int \left (-\frac {f \left (a+b \log \left (c x^n\right )\right )}{6 e x^{5/2}}+\frac {f^2 \left (a+b \log \left (c x^n\right )\right )}{4 e^2 x^2}-\frac {f^3 \left (a+b \log \left (c x^n\right )\right )}{2 e^3 x^{3/2}}+\frac {f^4 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^4 x}-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^3}-\frac {f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )}{4 e^4 x}\right ) \, dx\\ &=-\frac {f \left (a+b \log \left (c x^n\right )\right )^2}{6 e x^{3/2}}+\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 e^2 x}-\frac {f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^3 \sqrt {x}}+\frac {f^4 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^4}-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac {f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )^2}{4 e^4}+(b n) \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx+\frac {(b f n) \int \frac {a+b \log \left (c x^n\right )}{x^{5/2}} \, dx}{3 e}-\frac {\left (b f^2 n\right ) \int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx}{2 e^2}+\frac {\left (b f^3 n\right ) \int \frac {a+b \log \left (c x^n\right )}{x^{3/2}} \, dx}{e^3}+\frac {\left (b f^4 n\right ) \int \frac {\log (x) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx}{2 e^4}-\frac {\left (b f^4 n\right ) \int \frac {\log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx}{e^4}\\ &=-\frac {4 b^2 f n^2}{27 e x^{3/2}}+\frac {b^2 f^2 n^2}{2 e^2 x}-\frac {4 b^2 f^3 n^2}{e^3 \sqrt {x}}-\frac {7 b f n \left (a+b \log \left (c x^n\right )\right )}{18 e x^{3/2}}+\frac {3 b f^2 n \left (a+b \log \left (c x^n\right )\right )}{4 e^2 x}-\frac {5 b f^3 n \left (a+b \log \left (c x^n\right )\right )}{2 e^3 \sqrt {x}}+\frac {b f^4 n \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^4}-\frac {b n \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {b f^4 n \log (x) \left (a+b \log \left (c x^n\right )\right )}{4 e^4}-\frac {f \left (a+b \log \left (c x^n\right )\right )^2}{6 e x^{3/2}}+\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 e^2 x}-\frac {f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^3 \sqrt {x}}-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}+\frac {f^5 \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\left (e+f \sqrt {x}\right ) \sqrt {x}} \, dx}{4 e^4}-\frac {\left (b f^4 n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 b n x} \, dx}{2 e^4}-\left (b^2 n^2\right ) \int \left (-\frac {f}{6 e x^{5/2}}+\frac {f^2}{4 e^2 x^2}-\frac {f^3}{2 e^3 x^{3/2}}+\frac {f^4 \log \left (e+f \sqrt {x}\right )}{2 e^4 x}-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right )}{2 x^3}-\frac {f^4 \log (x)}{4 e^4 x}\right ) \, dx\\ &=-\frac {7 b^2 f n^2}{27 e x^{3/2}}+\frac {3 b^2 f^2 n^2}{4 e^2 x}-\frac {5 b^2 f^3 n^2}{e^3 \sqrt {x}}-\frac {7 b f n \left (a+b \log \left (c x^n\right )\right )}{18 e x^{3/2}}+\frac {3 b f^2 n \left (a+b \log \left (c x^n\right )\right )}{4 e^2 x}-\frac {5 b f^3 n \left (a+b \log \left (c x^n\right )\right )}{2 e^3 \sqrt {x}}+\frac {b f^4 n \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^4}-\frac {b n \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {b f^4 n \log (x) \left (a+b \log \left (c x^n\right )\right )}{4 e^4}-\frac {f \left (a+b \log \left (c x^n\right )\right )^2}{6 e x^{3/2}}+\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 e^2 x}-\frac {f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^3 \sqrt {x}}-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}+\frac {f^4 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^4}-\frac {f^4 \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx}{4 e^4}-\frac {\left (b f^4 n\right ) \int \frac {\log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx}{e^4}+\frac {1}{2} \left (b^2 n^2\right ) \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right )}{x^3} \, dx+\frac {\left (b^2 f^4 n^2\right ) \int \frac {\log (x)}{x} \, dx}{4 e^4}-\frac {\left (b^2 f^4 n^2\right ) \int \frac {\log \left (e+f \sqrt {x}\right )}{x} \, dx}{2 e^4}\\ &=-\frac {7 b^2 f n^2}{27 e x^{3/2}}+\frac {3 b^2 f^2 n^2}{4 e^2 x}-\frac {5 b^2 f^3 n^2}{e^3 \sqrt {x}}+\frac {b^2 f^4 n^2 \log ^2(x)}{8 e^4}-\frac {7 b f n \left (a+b \log \left (c x^n\right )\right )}{18 e x^{3/2}}+\frac {3 b f^2 n \left (a+b \log \left (c x^n\right )\right )}{4 e^2 x}-\frac {5 b f^3 n \left (a+b \log \left (c x^n\right )\right )}{2 e^3 \sqrt {x}}+\frac {b f^4 n \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^4}-\frac {b n \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {b f^4 n \log (x) \left (a+b \log \left (c x^n\right )\right )}{4 e^4}-\frac {f \left (a+b \log \left (c x^n\right )\right )^2}{6 e x^{3/2}}+\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 e^2 x}-\frac {f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^3 \sqrt {x}}-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}+\frac {f^4 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^4}+\frac {2 b f^4 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )}{e^4}-\frac {f^4 \operatorname {Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )}{4 b e^4 n}+\left (b^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (d (e+f x))}{x^5} \, dx,x,\sqrt {x}\right )-\frac {\left (b^2 f^4 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (e+f x)}{x} \, dx,x,\sqrt {x}\right )}{e^4}-\frac {\left (2 b^2 f^4 n^2\right ) \int \frac {\text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )}{x} \, dx}{e^4}\\ &=-\frac {7 b^2 f n^2}{27 e x^{3/2}}+\frac {3 b^2 f^2 n^2}{4 e^2 x}-\frac {5 b^2 f^3 n^2}{e^3 \sqrt {x}}-\frac {b^2 n^2 \log \left (d \left (e+f \sqrt {x}\right )\right )}{4 x^2}-\frac {b^2 f^4 n^2 \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{e^4}+\frac {b^2 f^4 n^2 \log ^2(x)}{8 e^4}-\frac {7 b f n \left (a+b \log \left (c x^n\right )\right )}{18 e x^{3/2}}+\frac {3 b f^2 n \left (a+b \log \left (c x^n\right )\right )}{4 e^2 x}-\frac {5 b f^3 n \left (a+b \log \left (c x^n\right )\right )}{2 e^3 \sqrt {x}}+\frac {b f^4 n \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^4}-\frac {b n \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {b f^4 n \log (x) \left (a+b \log \left (c x^n\right )\right )}{4 e^4}-\frac {f \left (a+b \log \left (c x^n\right )\right )^2}{6 e x^{3/2}}+\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 e^2 x}-\frac {f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^3 \sqrt {x}}-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}+\frac {f^4 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^4}-\frac {f^4 \left (a+b \log \left (c x^n\right )\right )^3}{12 b e^4 n}+\frac {2 b f^4 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )}{e^4}-\frac {4 b^2 f^4 n^2 \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )}{e^4}+\frac {1}{4} \left (b^2 f n^2\right ) \operatorname {Subst}\left (\int \frac {1}{x^4 (e+f x)} \, dx,x,\sqrt {x}\right )+\frac {\left (b^2 f^5 n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {f x}{e}\right )}{e+f x} \, dx,x,\sqrt {x}\right )}{e^4}\\ &=-\frac {7 b^2 f n^2}{27 e x^{3/2}}+\frac {3 b^2 f^2 n^2}{4 e^2 x}-\frac {5 b^2 f^3 n^2}{e^3 \sqrt {x}}-\frac {b^2 n^2 \log \left (d \left (e+f \sqrt {x}\right )\right )}{4 x^2}-\frac {b^2 f^4 n^2 \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{e^4}+\frac {b^2 f^4 n^2 \log ^2(x)}{8 e^4}-\frac {7 b f n \left (a+b \log \left (c x^n\right )\right )}{18 e x^{3/2}}+\frac {3 b f^2 n \left (a+b \log \left (c x^n\right )\right )}{4 e^2 x}-\frac {5 b f^3 n \left (a+b \log \left (c x^n\right )\right )}{2 e^3 \sqrt {x}}+\frac {b f^4 n \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^4}-\frac {b n \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {b f^4 n \log (x) \left (a+b \log \left (c x^n\right )\right )}{4 e^4}-\frac {f \left (a+b \log \left (c x^n\right )\right )^2}{6 e x^{3/2}}+\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 e^2 x}-\frac {f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^3 \sqrt {x}}-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}+\frac {f^4 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^4}-\frac {f^4 \left (a+b \log \left (c x^n\right )\right )^3}{12 b e^4 n}-\frac {b^2 f^4 n^2 \text {Li}_2\left (1+\frac {f \sqrt {x}}{e}\right )}{e^4}+\frac {2 b f^4 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )}{e^4}-\frac {4 b^2 f^4 n^2 \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )}{e^4}+\frac {1}{4} \left (b^2 f n^2\right ) \operatorname {Subst}\left (\int \left (\frac {1}{e x^4}-\frac {f}{e^2 x^3}+\frac {f^2}{e^3 x^2}-\frac {f^3}{e^4 x}+\frac {f^4}{e^4 (e+f x)}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {37 b^2 f n^2}{108 e x^{3/2}}+\frac {7 b^2 f^2 n^2}{8 e^2 x}-\frac {21 b^2 f^3 n^2}{4 e^3 \sqrt {x}}+\frac {b^2 f^4 n^2 \log \left (e+f \sqrt {x}\right )}{4 e^4}-\frac {b^2 n^2 \log \left (d \left (e+f \sqrt {x}\right )\right )}{4 x^2}-\frac {b^2 f^4 n^2 \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{e^4}-\frac {b^2 f^4 n^2 \log (x)}{8 e^4}+\frac {b^2 f^4 n^2 \log ^2(x)}{8 e^4}-\frac {7 b f n \left (a+b \log \left (c x^n\right )\right )}{18 e x^{3/2}}+\frac {3 b f^2 n \left (a+b \log \left (c x^n\right )\right )}{4 e^2 x}-\frac {5 b f^3 n \left (a+b \log \left (c x^n\right )\right )}{2 e^3 \sqrt {x}}+\frac {b f^4 n \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^4}-\frac {b n \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {b f^4 n \log (x) \left (a+b \log \left (c x^n\right )\right )}{4 e^4}-\frac {f \left (a+b \log \left (c x^n\right )\right )^2}{6 e x^{3/2}}+\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 e^2 x}-\frac {f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^3 \sqrt {x}}-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}+\frac {f^4 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^4}-\frac {f^4 \left (a+b \log \left (c x^n\right )\right )^3}{12 b e^4 n}-\frac {b^2 f^4 n^2 \text {Li}_2\left (1+\frac {f \sqrt {x}}{e}\right )}{e^4}+\frac {2 b f^4 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )}{e^4}-\frac {4 b^2 f^4 n^2 \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )}{e^4}\\ \end {align*}
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Mathematica [A] time = 0.60, size = 1078, normalized size = 1.77 \[ -\frac {108 b^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \log ^2\left (c x^n\right ) e^4+108 a^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) e^4+54 b^2 n^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) e^4+108 a b n \log \left (d \left (e+f \sqrt {x}\right )\right ) e^4+216 a b \log \left (d \left (e+f \sqrt {x}\right )\right ) \log \left (c x^n\right ) e^4+108 b^2 n \log \left (d \left (e+f \sqrt {x}\right )\right ) \log \left (c x^n\right ) e^4+36 b^2 f \sqrt {x} \log ^2\left (c x^n\right ) e^3+72 a b f \sqrt {x} \log \left (c x^n\right ) e^3+84 b^2 f n \sqrt {x} \log \left (c x^n\right ) e^3+74 b^2 f n^2 \sqrt {x} e^3+36 a^2 f \sqrt {x} e^3+84 a b f n \sqrt {x} e^3-54 b^2 f^2 x \log ^2\left (c x^n\right ) e^2-54 a^2 f^2 x e^2-189 b^2 f^2 n^2 x e^2-162 a b f^2 n x e^2-108 a b f^2 x \log \left (c x^n\right ) e^2-162 b^2 f^2 n x \log \left (c x^n\right ) e^2+108 b^2 f^3 x^{3/2} \log ^2\left (c x^n\right ) e+108 a^2 f^3 x^{3/2} e+1134 b^2 f^3 n^2 x^{3/2} e+540 a b f^3 n x^{3/2} e+216 a b f^3 x^{3/2} \log \left (c x^n\right ) e+540 b^2 f^3 n x^{3/2} \log \left (c x^n\right ) e+18 b^2 f^4 n^2 x^2 \log ^3(x)-27 b^2 f^4 n^2 x^2 \log ^2(x)-54 a b f^4 n x^2 \log ^2(x)-108 b^2 f^4 n^2 x^2 \log \left (e+f \sqrt {x}\right ) \log ^2(x)+108 b^2 f^4 n^2 x^2 \log \left (\frac {\sqrt {x} f}{e}+1\right ) \log ^2(x)-108 b^2 f^4 x^2 \log \left (e+f \sqrt {x}\right ) \log ^2\left (c x^n\right )+54 b^2 f^4 x^2 \log (x) \log ^2\left (c x^n\right )-108 a^2 f^4 x^2 \log \left (e+f \sqrt {x}\right )-54 b^2 f^4 n^2 x^2 \log \left (e+f \sqrt {x}\right )-108 a b f^4 n x^2 \log \left (e+f \sqrt {x}\right )+54 a^2 f^4 x^2 \log (x)+27 b^2 f^4 n^2 x^2 \log (x)+54 a b f^4 n x^2 \log (x)+108 b^2 f^4 n^2 x^2 \log \left (e+f \sqrt {x}\right ) \log (x)+216 a b f^4 n x^2 \log \left (e+f \sqrt {x}\right ) \log (x)-108 b^2 f^4 n^2 x^2 \log \left (\frac {\sqrt {x} f}{e}+1\right ) \log (x)-216 a b f^4 n x^2 \log \left (\frac {\sqrt {x} f}{e}+1\right ) \log (x)-54 b^2 f^4 n x^2 \log ^2(x) \log \left (c x^n\right )-216 a b f^4 x^2 \log \left (e+f \sqrt {x}\right ) \log \left (c x^n\right )-108 b^2 f^4 n x^2 \log \left (e+f \sqrt {x}\right ) \log \left (c x^n\right )+108 a b f^4 x^2 \log (x) \log \left (c x^n\right )+54 b^2 f^4 n x^2 \log (x) \log \left (c x^n\right )+216 b^2 f^4 n x^2 \log \left (e+f \sqrt {x}\right ) \log (x) \log \left (c x^n\right )-216 b^2 f^4 n x^2 \log \left (\frac {\sqrt {x} f}{e}+1\right ) \log (x) \log \left (c x^n\right )-216 b f^4 n x^2 \left (2 a+b n+2 b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )+864 b^2 f^4 n^2 x^2 \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )}{216 e^4 x^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} \log \left (c x^{n}\right )^{2} + 2 \, a b \log \left (c x^{n}\right ) + a^{2}\right )} \log \left (d f \sqrt {x} + d e\right )}{x^{3}}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f \sqrt {x} + e\right )} d\right )}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.16, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{2} \ln \left (\left (f \sqrt {x}+e \right ) d \right )}{x^{3}}\, dx \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f \sqrt {x} + e\right )} d\right )}{x^{3}}\,{d x} \]
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 \[ \int \frac {\ln \left (d\,\left (e+f\,\sqrt {x}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^3} \,d x \]
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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