Optimal. Leaf size=130 \[ -\frac {2 b d n \text {Li}_2\left (-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {d \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {2 a b n x}{e}-\frac {2 b^2 n x \log \left (c x^n\right )}{e}+\frac {2 b^2 d n^2 \text {Li}_3\left (-\frac {e x}{d}\right )}{e^2}+\frac {2 b^2 n^2 x}{e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.16, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2353, 2296, 2295, 2317, 2374, 6589} \[ -\frac {2 b d n \text {PolyLog}\left (2,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {2 b^2 d n^2 \text {PolyLog}\left (3,-\frac {e x}{d}\right )}{e^2}-\frac {d \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {2 a b n x}{e}-\frac {2 b^2 n x \log \left (c x^n\right )}{e}+\frac {2 b^2 n^2 x}{e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2295
Rule 2296
Rule 2317
Rule 2353
Rule 2374
Rule 6589
Rubi steps
\begin {align*} \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx &=\int \left (\frac {\left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{e (d+e x)}\right ) \, dx\\ &=\frac {\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{e}-\frac {d \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx}{e}\\ &=\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^2}+\frac {(2 b d n) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{x} \, dx}{e^2}-\frac {(2 b n) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{e}\\ &=-\frac {2 a b n x}{e}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^2}-\frac {2 b d n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{e^2}-\frac {\left (2 b^2 n\right ) \int \log \left (c x^n\right ) \, dx}{e}+\frac {\left (2 b^2 d n^2\right ) \int \frac {\text {Li}_2\left (-\frac {e x}{d}\right )}{x} \, dx}{e^2}\\ &=-\frac {2 a b n x}{e}+\frac {2 b^2 n^2 x}{e}-\frac {2 b^2 n x \log \left (c x^n\right )}{e}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^2}-\frac {2 b d n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{e^2}+\frac {2 b^2 d n^2 \text {Li}_3\left (-\frac {e x}{d}\right )}{e^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 103, normalized size = 0.79 \[ \frac {-2 b d n \left (\text {Li}_2\left (-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \text {Li}_3\left (-\frac {e x}{d}\right )\right )-d \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+e x \left (a+b \log \left (c x^n\right )\right )^2-2 b e n x \left (a+b \log \left (c x^n\right )-b n\right )}{e^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} x \log \left (c x^{n}\right )^{2} + 2 \, a b x \log \left (c x^{n}\right ) + a^{2} x}{e x + d}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.68, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{2} x}{e x +d}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ a^{2} {\left (\frac {x}{e} - \frac {d \log \left (e x + d\right )}{e^{2}}\right )} + \int \frac {b^{2} x \log \left (x^{n}\right )^{2} + 2 \, {\left (b^{2} \log \relax (c) + a b\right )} x \log \left (x^{n}\right ) + {\left (b^{2} \log \relax (c)^{2} + 2 \, a b \log \relax (c)\right )} x}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{d+e\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________