Optimal. Leaf size=202 \[ -\frac {b n (d g+e f) \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 e^2}+\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 (e f-d g)}-\frac {b n x (e f-d g) \left (a+b \log \left (c x^n\right )\right )}{d^2 e (d+e x)}-\frac {(f+g x)^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 (d+e x)^2 (e f-d g)}-\frac {b^2 n^2 (d g+e f) \text {Li}_2\left (-\frac {e x}{d}\right )}{d^2 e^2}+\frac {b^2 n^2 (e f-d g) \log (d+e x)}{d^2 e^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.41, antiderivative size = 278, normalized size of antiderivative = 1.38, number of steps used = 13, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2357, 2319, 2347, 2344, 2301, 2317, 2391, 2314, 31, 2318} \[ -\frac {b^2 n^2 (e f-d g) \text {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^2 e^2}-\frac {2 b^2 g n^2 \text {PolyLog}\left (2,-\frac {e x}{d}\right )}{d e^2}-\frac {b n (e f-d g) \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 e^2}+\frac {(e f-d g) \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 e^2}-\frac {b n x (e f-d g) \left (a+b \log \left (c x^n\right )\right )}{d^2 e (d+e x)}-\frac {(e f-d g) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2 (d+e x)^2}-\frac {2 b g n \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d e^2}+\frac {g x \left (a+b \log \left (c x^n\right )\right )^2}{d e (d+e x)}+\frac {b^2 n^2 (e f-d g) \log (d+e x)}{d^2 e^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 2301
Rule 2314
Rule 2317
Rule 2318
Rule 2319
Rule 2344
Rule 2347
Rule 2357
Rule 2391
Rubi steps
\begin {align*} \int \frac {(f+g x) \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx &=\int \left (\frac {(e f-d g) \left (a+b \log \left (c x^n\right )\right )^2}{e (d+e x)^3}+\frac {g \left (a+b \log \left (c x^n\right )\right )^2}{e (d+e x)^2}\right ) \, dx\\ &=\frac {g \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx}{e}+\frac {(e f-d g) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx}{e}\\ &=-\frac {(e f-d g) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2 (d+e x)^2}+\frac {g x \left (a+b \log \left (c x^n\right )\right )^2}{d e (d+e x)}-\frac {(2 b g n) \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{d e}+\frac {(b (e f-d g) n) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx}{e^2}\\ &=-\frac {(e f-d g) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2 (d+e x)^2}+\frac {g x \left (a+b \log \left (c x^n\right )\right )^2}{d e (d+e x)}-\frac {2 b g n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d e^2}+\frac {(b (e f-d g) n) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{d e^2}-\frac {(b (e f-d g) n) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{d e}+\frac {\left (2 b^2 g n^2\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{d e^2}\\ &=-\frac {b (e f-d g) n x \left (a+b \log \left (c x^n\right )\right )}{d^2 e (d+e x)}-\frac {(e f-d g) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2 (d+e x)^2}+\frac {g x \left (a+b \log \left (c x^n\right )\right )^2}{d e (d+e x)}-\frac {2 b g n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d e^2}-\frac {2 b^2 g n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{d e^2}+\frac {(b (e f-d g) n) \int \frac {a+b \log \left (c x^n\right )}{x} \, dx}{d^2 e^2}-\frac {(b (e f-d g) n) \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{d^2 e}+\frac {\left (b^2 (e f-d g) n^2\right ) \int \frac {1}{d+e x} \, dx}{d^2 e}\\ &=-\frac {b (e f-d g) n x \left (a+b \log \left (c x^n\right )\right )}{d^2 e (d+e x)}+\frac {(e f-d g) \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 e^2}-\frac {(e f-d g) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2 (d+e x)^2}+\frac {g x \left (a+b \log \left (c x^n\right )\right )^2}{d e (d+e x)}+\frac {b^2 (e f-d g) n^2 \log (d+e x)}{d^2 e^2}-\frac {2 b g n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d e^2}-\frac {b (e f-d g) n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^2 e^2}-\frac {2 b^2 g n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{d e^2}+\frac {\left (b^2 (e f-d g) n^2\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{d^2 e^2}\\ &=-\frac {b (e f-d g) n x \left (a+b \log \left (c x^n\right )\right )}{d^2 e (d+e x)}+\frac {(e f-d g) \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 e^2}-\frac {(e f-d g) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2 (d+e x)^2}+\frac {g x \left (a+b \log \left (c x^n\right )\right )^2}{d e (d+e x)}+\frac {b^2 (e f-d g) n^2 \log (d+e x)}{d^2 e^2}-\frac {2 b g n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d e^2}-\frac {b (e f-d g) n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^2 e^2}-\frac {2 b^2 g n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{d e^2}-\frac {b^2 (e f-d g) n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{d^2 e^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.25, size = 244, normalized size = 1.21 \[ \frac {\frac {(e f-d g) \left (-2 b n (d+e x) \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )+(d+e x) \left (a+b \log \left (c x^n\right )\right )^2+2 b d n \left (a+b \log \left (c x^n\right )\right )-2 b^2 n^2 (d+e x) \text {Li}_2\left (-\frac {e x}{d}\right )-2 b^2 n^2 (d+e x) (\log (x)-\log (d+e x))\right )}{d^2 (d+e x)}+\frac {2 g \left (\left (a+b \log \left (c x^n\right )\right ) \left (a+b \log \left (c x^n\right )-2 b n \log \left (\frac {e x}{d}+1\right )\right )-2 b^2 n^2 \text {Li}_2\left (-\frac {e x}{d}\right )\right )}{d}-\frac {(e f-d g) \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2}-\frac {2 g \left (a+b \log \left (c x^n\right )\right )^2}{d+e x}}{2 e^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{2} g x + a^{2} f + {\left (b^{2} g x + b^{2} f\right )} \log \left (c x^{n}\right )^{2} + 2 \, {\left (a b g x + a b f\right )} \log \left (c x^{n}\right )}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (g x + f\right )} {\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.39, size = 2163, normalized size = 10.71 \[ \text {Expression too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ a b f n {\left (\frac {1}{d e^{2} x + d^{2} e} - \frac {\log \left (e x + d\right )}{d^{2} e} + \frac {\log \relax (x)}{d^{2} e}\right )} - a b g n {\left (\frac {1}{e^{3} x + d e^{2}} + \frac {\log \left (e x + d\right )}{d e^{2}} - \frac {\log \relax (x)}{d e^{2}}\right )} - \frac {{\left (2 \, e x + d\right )} a b g \log \left (c x^{n}\right )}{e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}} - \frac {{\left (2 \, e x + d\right )} a^{2} g}{2 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}} - \frac {a b f \log \left (c x^{n}\right )}{e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e} - \frac {a^{2} f}{2 \, {\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} - \frac {{\left (2 \, b^{2} e g x + {\left (e f + d g\right )} b^{2}\right )} \log \left (x^{n}\right )^{2}}{2 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}} + \int \frac {b^{2} e^{2} g x^{2} \log \relax (c)^{2} + b^{2} e^{2} f x \log \relax (c)^{2} + {\left (2 \, {\left (e^{2} g n + e^{2} g \log \relax (c)\right )} b^{2} x^{2} + {\left (e^{2} f n + 3 \, d e g n + 2 \, e^{2} f \log \relax (c)\right )} b^{2} x + {\left (d e f n + d^{2} g n\right )} b^{2}\right )} \log \left (x^{n}\right )}{e^{5} x^{4} + 3 \, d e^{4} x^{3} + 3 \, d^{2} e^{3} x^{2} + d^{3} e^{2} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (f+g\,x\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{{\left (d+e\,x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{2} \left (f + g x\right )}{\left (d + e x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________