Optimal. Leaf size=117 \[ \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{2 b n}-2 k \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {k \log \left (\frac {f \sqrt {x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}+4 b k n \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.15, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2375, 2337, 2374, 6589} \[ -2 k \text {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )+4 b k n \text {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right )+\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{2 b n}-\frac {k \log \left (\frac {f \sqrt {x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2337
Rule 2374
Rule 2375
Rule 6589
Rubi steps
\begin {align*} \int \frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}-\frac {(f k) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\left (e+f \sqrt {x}\right ) \sqrt {x}} \, dx}{4 b n}\\ &=\frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}-\frac {k \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}+k \int \frac {\log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx\\ &=\frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}-\frac {k \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}-2 k \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )+(2 b k n) \int \frac {\text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )}{x} \, dx\\ &=\frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}-\frac {k \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}-2 k \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )+4 b k n \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.17, size = 186, normalized size = 1.59 \[ \frac {1}{2} \left (4 a \log \left (-\frac {f \sqrt {x}}{e}\right ) \log \left (d \left (e+f \sqrt {x}\right )^k\right )+4 a k \text {Li}_2\left (\frac {\sqrt {x} f}{e}+1\right )+2 b \log (x) \log \left (c x^n\right ) \log \left (d \left (e+f \sqrt {x}\right )^k\right )-4 b k \log \left (c x^n\right ) \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )-2 b k \log (x) \log \left (c x^n\right ) \log \left (\frac {f \sqrt {x}}{e}+1\right )-b n \log ^2(x) \log \left (d \left (e+f \sqrt {x}\right )^k\right )+8 b k n \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )+b k n \log ^2(x) \log \left (\frac {f \sqrt {x}}{e}+1\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f \sqrt {x} + e\right )}^{k} d\right )}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f \sqrt {x} + e\right )}^{k} d\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right ) \ln \left (d \left (f \sqrt {x}+e \right )^{k}\right )}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {b e n \log \relax (d) \log \relax (x)^{2} - 2 \, b e \log \relax (d) \log \relax (x) \log \left (x^{n}\right ) + {\left (b e n \log \relax (x)^{2} - 2 \, b e \log \relax (x) \log \left (x^{n}\right ) - 2 \, {\left (b e \log \relax (c) + a e\right )} \log \relax (x)\right )} k \log \left (f \sqrt {x} + e\right ) - 2 \, {\left (b e \log \relax (c) \log \relax (d) + a e \log \relax (d)\right )} \log \relax (x) - \frac {b f k n x \log \relax (x)^{2} - 2 \, {\left (b f k \log \relax (c) + a f k\right )} x \log \relax (x) + 4 \, {\left (a f k - {\left (2 \, f k n - f k \log \relax (c)\right )} b\right )} x - 2 \, {\left (b f k x \log \relax (x) - 2 \, b f k x\right )} \log \left (x^{n}\right )}{\sqrt {x}}}{2 \, e} + \int -\frac {b f^{2} k n \log \relax (x)^{2} - 2 \, b f^{2} k \log \relax (x) \log \left (x^{n}\right ) - 2 \, {\left (b f^{2} k \log \relax (c) + a f^{2} k\right )} \log \relax (x)}{4 \, {\left (e f \sqrt {x} + e^{2}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\ln \left (d\,{\left (e+f\,\sqrt {x}\right )}^k\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________