Optimal. Leaf size=295 \[ -\frac {3 b (e f-d g) n x \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 e (d+e x)}+\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^3}{2 d^2 (e f-d g)}-\frac {(f+g x)^2 \left (a+b \log \left (c x^n\right )\right )^3}{2 (e f-d g) (d+e x)^2}+\frac {3 b^2 (e f-d g) n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^2 e^2}-\frac {3 b (e f+d g) n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{2 d^2 e^2}+\frac {3 b^3 (e f-d g) n^3 \text {Li}_2\left (-\frac {e x}{d}\right )}{d^2 e^2}-\frac {3 b^2 (e f+d g) n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{d^2 e^2}+\frac {3 b^3 (e f+d g) n^3 \text {Li}_3\left (-\frac {e x}{d}\right )}{d^2 e^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.33, antiderivative size = 295, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {2398, 2404,
2339, 30, 2355, 2354, 2438, 2421, 6724} \begin {gather*} -\frac {3 b^2 n^2 (d g+e f) \text {PolyLog}\left (2,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 e^2}+\frac {3 b^3 n^3 (e f-d g) \text {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^2 e^2}+\frac {3 b^3 n^3 (d g+e f) \text {PolyLog}\left (3,-\frac {e x}{d}\right )}{d^2 e^2}+\frac {3 b^2 n^2 (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 {3 b n (d g+e f) \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 e^2}+\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^3}{2 d^2 (e f-d g)}-\frac {3 b n x (e f-d g) \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 e (d+e x)}-\frac {(f+g x)^2 \left (a+b \log \left (c x^n\right )\right )^3}{2 (d+e x)^2 (e f-d g)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 2339
Rule 2354
Rule 2355
Rule 2398
Rule 2404
Rule 2421
Rule 2438
Rule 6724
Rubi steps
\begin {align*} \int \frac {(f+g x) \left (a+b \log \left (c x^n\right )\right )^3}{(d+e x)^3} \, dx &=\int \left (\frac {(e f-d g) \left (a+b \log \left (c x^n\right )\right )^3}{e (d+e x)^3}+\frac {g \left (a+b \log \left (c x^n\right )\right )^3}{e (d+e x)^2}\right ) \, dx\\ &=\frac {g \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{(d+e x)^2} \, dx}{e}+\frac {(e f-d g) \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{(d+e x)^3} \, dx}{e}\\ &=-\frac {(e f-d g) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^2 (d+e x)^2}+\frac {g x \left (a+b \log \left (c x^n\right )\right )^3}{d e (d+e x)}-\frac {(3 b g n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx}{d e}+\frac {(3 b (e f-d g) n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^2} \, dx}{2 e^2}\\ &=-\frac {(e f-d g) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^2 (d+e x)^2}+\frac {g x \left (a+b \log \left (c x^n\right )\right )^3}{d e (d+e x)}-\frac {3 b g n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{d e^2}+\frac {(3 b (e f-d g) n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)} \, dx}{2 d e^2}-\frac {(3 b (e f-d g) n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx}{2 d e}+\frac {\left (6 b^2 g n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{x} \, dx}{d e^2}\\ &=-\frac {3 b (e f-d g) n x \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 e (d+e x)}-\frac {(e f-d g) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^2 (d+e x)^2}+\frac {g x \left (a+b \log \left (c x^n\right )\right )^3}{d e (d+e x)}-\frac {3 b g n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{d e^2}-\frac {6 b^2 g n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{d e^2}+\frac {(3 b (e f-d g) n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx}{2 d^2 e^2}-\frac {(3 b (e f-d g) n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx}{2 d^2 e}+\frac {\left (3 b^2 (e f-d g) n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{d^2 e}+\frac {\left (6 b^3 g n^3\right ) \int \frac {\text {Li}_2\left (-\frac {e x}{d}\right )}{x} \, dx}{d e^2}\\ &=-\frac {3 b (e f-d g) n x \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 e (d+e x)}-\frac {(e f-d g) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^2 (d+e x)^2}+\frac {g x \left (a+b \log \left (c x^n\right )\right )^3}{d e (d+e x)}+\frac {3 b^2 (e f-d g) n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^2 e^2}-\frac {3 b g n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{d e^2}-\frac {3 b (e f-d g) n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{2 d^2 e^2}-\frac {6 b^2 g n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{d e^2}+\frac {6 b^3 g n^3 \text {Li}_3\left (-\frac {e x}{d}\right )}{d e^2}+\frac {(3 (e f-d g)) \text {Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )}{2 d^2 e^2}+\frac {\left (3 b^2 (e f-d g) n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{x} \, dx}{d^2 e^2}-\frac {\left (3 b^3 (e f-d g) n^3\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{d^2 e^2}\\ &=-\frac {3 b (e f-d g) n x \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 e (d+e x)}+\frac {(e f-d g) \left (a+b \log \left (c x^n\right )\right )^3}{2 d^2 e^2}-\frac {(e f-d g) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^2 (d+e x)^2}+\frac {g x \left (a+b \log \left (c x^n\right )\right )^3}{d e (d+e x)}+\frac {3 b^2 (e f-d g) n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^2 e^2}-\frac {3 b g n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{d e^2}-\frac {3 b (e f-d g) n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{2 d^2 e^2}+\frac {3 b^3 (e f-d g) n^3 \text {Li}_2\left (-\frac {e x}{d}\right )}{d^2 e^2}-\frac {6 b^2 g n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{d e^2}-\frac {3 b^2 (e f-d g) n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{d^2 e^2}+\frac {6 b^3 g n^3 \text {Li}_3\left (-\frac {e x}{d}\right )}{d e^2}+\frac {\left (3 b^3 (e f-d g) n^3\right ) \int \frac {\text {Li}_2\left (-\frac {e x}{d}\right )}{x} \, dx}{d^2 e^2}\\ &=-\frac {3 b (e f-d g) n x \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 e (d+e x)}+\frac {(e f-d g) \left (a+b \log \left (c x^n\right )\right )^3}{2 d^2 e^2}-\frac {(e f-d g) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^2 (d+e x)^2}+\frac {g x \left (a+b \log \left (c x^n\right )\right )^3}{d e (d+e x)}+\frac {3 b^2 (e f-d g) n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^2 e^2}-\frac {3 b g n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{d e^2}-\frac {3 b (e f-d g) n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{2 d^2 e^2}+\frac {3 b^3 (e f-d g) n^3 \text {Li}_2\left (-\frac {e x}{d}\right )}{d^2 e^2}-\frac {6 b^2 g n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{d e^2}-\frac {3 b^2 (e f-d g) n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{d^2 e^2}+\frac {6 b^3 g n^3 \text {Li}_3\left (-\frac {e x}{d}\right )}{d e^2}+\frac {3 b^3 (e f-d g) n^3 \text {Li}_3\left (-\frac {e x}{d}\right )}{d^2 e^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.26, size = 339, normalized size = 1.15 \begin {gather*} \frac {-\frac {(e f-d g) \left (a+b \log \left (c x^n\right )\right )^3}{(d+e x)^2}-\frac {2 g \left (a+b \log \left (c x^n\right )\right )^3}{d+e x}+\frac {2 g \left (\left (a+b \log \left (c x^n\right )\right )^2 \left (a+b \log \left (c x^n\right )-3 b n \log \left (1+\frac {e x}{d}\right )\right )-6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )+6 b^3 n^3 \text {Li}_3\left (-\frac {e x}{d}\right )\right )}{d}+\frac {(e f-d g) \left (3 b d n \left (a+b \log \left (c x^n\right )\right )^2+(d+e x) \left (a+b \log \left (c x^n\right )\right )^3-3 b n (d+e x) \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )-3 b n (d+e x) \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 (1+\frac {e x}{d}\right )\right )-2 b^2 n^2 \text {Li}_2\left (-\frac {e x}{d}\right )\right )-6 b^2 n^2 (d+e x) \left (\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )-b n \text {Li}_3\left (-\frac {e x}{d}\right )\right )\right )}{d^2 (d+e x)}}{2 e^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.37, size = 11535, normalized size = 39.10
method | result | size |
risch | \(\text {Expression too large to display}\) | \(11535\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{3} \left (f + g x\right )}{\left (d + e x\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (f+g\,x\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{{\left (d+e\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________