Optimal. Leaf size=454 \[ -\frac{B^2 i^3 n^2 (b c-a d)^4 \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right )}{2 b^4 d}+\frac{B i^3 n (b c-a d)^4 \log \left (1-\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 b^4 d}-\frac{B i^3 n (a+b x) (b c-a d)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 b^4}-\frac{B i^3 n (c+d x)^2 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{4 b^2 d}-\frac{B i^3 n (c+d x)^3 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{6 b d}+\frac{i^3 (c+d x)^4 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 d}+\frac{5 B^2 i^3 n^2 x (b c-a d)^3}{12 b^3}+\frac{B^2 i^3 n^2 (c+d x)^2 (b c-a d)^2}{12 b^2 d}+\frac{5 B^2 i^3 n^2 (b c-a d)^4 \log \left (\frac{a+b x}{c+d x}\right )}{12 b^4 d}+\frac{11 B^2 i^3 n^2 (b c-a d)^4 \log (c+d x)}{12 b^4 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.663842, antiderivative size = 544, normalized size of antiderivative = 1.2, number of steps used = 23, number of rules used = 13, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.371, Rules used = {2525, 12, 2528, 2486, 31, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 43} \[ -\frac{B^2 i^3 n^2 (b c-a d)^4 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{2 b^4 d}-\frac{B i^3 n (b c-a d)^4 \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 b^4 d}-\frac{B i^3 n (c+d x)^2 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{4 b^2 d}-\frac{A B i^3 n x (b c-a d)^3}{2 b^3}-\frac{B i^3 n (c+d x)^3 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{6 b d}+\frac{i^3 (c+d x)^4 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 d}-\frac{B^2 i^3 n (a+b x) (b c-a d)^3 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{2 b^4}+\frac{5 B^2 i^3 n^2 x (b c-a d)^3}{12 b^3}+\frac{B^2 i^3 n^2 (c+d x)^2 (b c-a d)^2}{12 b^2 d}+\frac{B^2 i^3 n^2 (b c-a d)^4 \log ^2(a+b x)}{4 b^4 d}+\frac{5 B^2 i^3 n^2 (b c-a d)^4 \log (a+b x)}{12 b^4 d}+\frac{B^2 i^3 n^2 (b c-a d)^4 \log (c+d x)}{2 b^4 d}-\frac{B^2 i^3 n^2 (b c-a d)^4 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{2 b^4 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2525
Rule 12
Rule 2528
Rule 2486
Rule 31
Rule 2524
Rule 2418
Rule 2390
Rule 2301
Rule 2394
Rule 2393
Rule 2391
Rule 43
Rubi steps
\begin{align*} \int (181 c+181 d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \, dx &=\frac{5929741 (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4 d}-\frac{(B n) \int \frac{1073283121 (b c-a d) (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{362 d}\\ &=\frac{5929741 (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4 d}-\frac{(5929741 B (b c-a d) n) \int \frac{(c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{2 d}\\ &=\frac{5929741 (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4 d}-\frac{(5929741 B (b c-a d) n) \int \left (\frac{d (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3}+\frac{(b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 (a+b x)}+\frac{d (b c-a d) (c+d x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2}+\frac{d (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b}\right ) \, dx}{2 d}\\ &=\frac{5929741 (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4 d}-\frac{(5929741 B (b c-a d) n) \int (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{2 b}-\frac{\left (5929741 B (b c-a d)^2 n\right ) \int (c+d x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{2 b^2}-\frac{\left (5929741 B (b c-a d)^3 n\right ) \int \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{2 b^3}-\frac{\left (5929741 B (b c-a d)^4 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{2 b^3 d}\\ &=-\frac{5929741 A B (b c-a d)^3 n x}{2 b^3}-\frac{5929741 B (b c-a d)^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{4 b^2 d}-\frac{5929741 B (b c-a d) n (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{6 b d}-\frac{5929741 B (b c-a d)^4 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b^4 d}+\frac{5929741 (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4 d}-\frac{\left (5929741 B^2 (b c-a d)^3 n\right ) \int \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx}{2 b^3}+\frac{\left (5929741 B^2 (b c-a d) n^2\right ) \int \frac{(b c-a d) (c+d x)^2}{a+b x} \, dx}{6 b d}+\frac{\left (5929741 B^2 (b c-a d)^2 n^2\right ) \int \frac{(b c-a d) (c+d x)}{a+b x} \, dx}{4 b^2 d}+\frac{\left (5929741 B^2 (b c-a d)^4 n^2\right ) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{2 b^4 d}\\ &=-\frac{5929741 A B (b c-a d)^3 n x}{2 b^3}-\frac{5929741 B^2 (b c-a d)^3 n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{2 b^4}-\frac{5929741 B (b c-a d)^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{4 b^2 d}-\frac{5929741 B (b c-a d) n (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{6 b d}-\frac{5929741 B (b c-a d)^4 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b^4 d}+\frac{5929741 (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4 d}+\frac{\left (5929741 B^2 (b c-a d)^2 n^2\right ) \int \frac{(c+d x)^2}{a+b x} \, dx}{6 b d}+\frac{\left (5929741 B^2 (b c-a d)^3 n^2\right ) \int \frac{c+d x}{a+b x} \, dx}{4 b^2 d}+\frac{\left (5929741 B^2 (b c-a d)^4 n^2\right ) \int \frac{1}{c+d x} \, dx}{2 b^4}+\frac{\left (5929741 B^2 (b c-a d)^4 n^2\right ) \int \left (\frac{b \log (a+b x)}{a+b x}-\frac{d \log (a+b x)}{c+d x}\right ) \, dx}{2 b^4 d}\\ &=-\frac{5929741 A B (b c-a d)^3 n x}{2 b^3}-\frac{5929741 B^2 (b c-a d)^3 n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{2 b^4}-\frac{5929741 B (b c-a d)^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{4 b^2 d}-\frac{5929741 B (b c-a d) n (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{6 b d}-\frac{5929741 B (b c-a d)^4 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b^4 d}+\frac{5929741 (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4 d}+\frac{5929741 B^2 (b c-a d)^4 n^2 \log (c+d x)}{2 b^4 d}+\frac{\left (5929741 B^2 (b c-a d)^2 n^2\right ) \int \left (\frac{d (b c-a d)}{b^2}+\frac{(b c-a d)^2}{b^2 (a+b x)}+\frac{d (c+d x)}{b}\right ) \, dx}{6 b d}+\frac{\left (5929741 B^2 (b c-a d)^3 n^2\right ) \int \left (\frac{d}{b}+\frac{b c-a d}{b (a+b x)}\right ) \, dx}{4 b^2 d}-\frac{\left (5929741 B^2 (b c-a d)^4 n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{2 b^4}+\frac{\left (5929741 B^2 (b c-a d)^4 n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{2 b^3 d}\\ &=-\frac{5929741 A B (b c-a d)^3 n x}{2 b^3}+\frac{29648705 B^2 (b c-a d)^3 n^2 x}{12 b^3}+\frac{5929741 B^2 (b c-a d)^2 n^2 (c+d x)^2}{12 b^2 d}+\frac{29648705 B^2 (b c-a d)^4 n^2 \log (a+b x)}{12 b^4 d}-\frac{5929741 B^2 (b c-a d)^3 n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{2 b^4}-\frac{5929741 B (b c-a d)^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{4 b^2 d}-\frac{5929741 B (b c-a d) n (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{6 b d}-\frac{5929741 B (b c-a d)^4 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b^4 d}+\frac{5929741 (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4 d}+\frac{5929741 B^2 (b c-a d)^4 n^2 \log (c+d x)}{2 b^4 d}-\frac{5929741 B^2 (b c-a d)^4 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{2 b^4 d}+\frac{\left (5929741 B^2 (b c-a d)^4 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{2 b^4 d}+\frac{\left (5929741 B^2 (b c-a d)^4 n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{2 b^3 d}\\ &=-\frac{5929741 A B (b c-a d)^3 n x}{2 b^3}+\frac{29648705 B^2 (b c-a d)^3 n^2 x}{12 b^3}+\frac{5929741 B^2 (b c-a d)^2 n^2 (c+d x)^2}{12 b^2 d}+\frac{29648705 B^2 (b c-a d)^4 n^2 \log (a+b x)}{12 b^4 d}+\frac{5929741 B^2 (b c-a d)^4 n^2 \log ^2(a+b x)}{4 b^4 d}-\frac{5929741 B^2 (b c-a d)^3 n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{2 b^4}-\frac{5929741 B (b c-a d)^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{4 b^2 d}-\frac{5929741 B (b c-a d) n (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{6 b d}-\frac{5929741 B (b c-a d)^4 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b^4 d}+\frac{5929741 (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4 d}+\frac{5929741 B^2 (b c-a d)^4 n^2 \log (c+d x)}{2 b^4 d}-\frac{5929741 B^2 (b c-a d)^4 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{2 b^4 d}+\frac{\left (5929741 B^2 (b c-a d)^4 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{2 b^4 d}\\ &=-\frac{5929741 A B (b c-a d)^3 n x}{2 b^3}+\frac{29648705 B^2 (b c-a d)^3 n^2 x}{12 b^3}+\frac{5929741 B^2 (b c-a d)^2 n^2 (c+d x)^2}{12 b^2 d}+\frac{29648705 B^2 (b c-a d)^4 n^2 \log (a+b x)}{12 b^4 d}+\frac{5929741 B^2 (b c-a d)^4 n^2 \log ^2(a+b x)}{4 b^4 d}-\frac{5929741 B^2 (b c-a d)^3 n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{2 b^4}-\frac{5929741 B (b c-a d)^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{4 b^2 d}-\frac{5929741 B (b c-a d) n (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{6 b d}-\frac{5929741 B (b c-a d)^4 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b^4 d}+\frac{5929741 (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4 d}+\frac{5929741 B^2 (b c-a d)^4 n^2 \log (c+d x)}{2 b^4 d}-\frac{5929741 B^2 (b c-a d)^4 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{2 b^4 d}-\frac{5929741 B^2 (b c-a d)^4 n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{2 b^4 d}\\ \end{align*}
Mathematica [A] time = 0.321483, size = 409, normalized size = 0.9 \[ \frac{i^3 \left ((c+d x)^4 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2-\frac{B n (b c-a d) \left (-3 B n (b c-a d)^3 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+3 b^2 (c+d x)^2 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+2 b^3 (c+d x)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+6 (b c-a d)^3 \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+6 A b d x (b c-a d)^2-B n (b c-a d) \left (2 b d x (b c-a d)+2 (b c-a d)^2 \log (a+b x)+b^2 (c+d x)^2\right )+6 B d (a+b x) (b c-a d)^2 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-6 B n (b c-a d)^3 \log (c+d x)-3 B n (b c-a d)^2 ((b c-a d) \log (a+b x)+b d x)\right )}{3 b^4}\right )}{4 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.51, size = 0, normalized size = 0. \begin{align*} \int \left ( dix+ci \right ) ^{3} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 3.78814, size = 2874, normalized size = 6.33 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (A^{2} d^{3} i^{3} x^{3} + 3 \, A^{2} c d^{2} i^{3} x^{2} + 3 \, A^{2} c^{2} d i^{3} x + A^{2} c^{3} i^{3} +{\left (B^{2} d^{3} i^{3} x^{3} + 3 \, B^{2} c d^{2} i^{3} x^{2} + 3 \, B^{2} c^{2} d i^{3} x + B^{2} c^{3} i^{3}\right )} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \,{\left (A B d^{3} i^{3} x^{3} + 3 \, A B c d^{2} i^{3} x^{2} + 3 \, A B c^{2} d i^{3} x + A B c^{3} i^{3}\right )} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d i x + c i\right )}^{3}{\left (B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]