Optimal. Leaf size=242 \[ \frac{B d^2 i^2 n \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right )}{b^3 g^3}-\frac{d^2 i^2 \log \left (1-\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 g^3}-\frac{d i^2 (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 g^3 (a+b x)}-\frac{i^2 (c+d x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 b g^3 (a+b x)^2}-\frac{B d i^2 n (c+d x)}{b^2 g^3 (a+b x)}-\frac{B i^2 n (c+d x)^2}{4 b g^3 (a+b x)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.560004, antiderivative size = 354, normalized size of antiderivative = 1.46, number of steps used = 18, number of rules used = 11, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.256, Rules used = {2528, 2525, 12, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ \frac{B d^2 i^2 n \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g^3}+\frac{d^2 i^2 \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 g^3}-\frac{2 d i^2 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 g^3 (a+b x)}-\frac{i^2 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 b^3 g^3 (a+b x)^2}+\frac{B d^2 i^2 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g^3}-\frac{3 B d i^2 n (b c-a d)}{2 b^3 g^3 (a+b x)}-\frac{B i^2 n (b c-a d)^2}{4 b^3 g^3 (a+b x)^2}-\frac{B d^2 i^2 n \log ^2(a+b x)}{2 b^3 g^3}-\frac{3 B d^2 i^2 n \log (a+b x)}{2 b^3 g^3}+\frac{3 B d^2 i^2 n \log (c+d x)}{2 b^3 g^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2528
Rule 2525
Rule 12
Rule 44
Rule 2524
Rule 2418
Rule 2390
Rule 2301
Rule 2394
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{(123 c+123 d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^3} \, dx &=\int \left (\frac{15129 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^3 (a+b x)^3}+\frac{30258 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^3 (a+b x)^2}+\frac{15129 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^3 (a+b x)}\right ) \, dx\\ &=\frac{\left (15129 d^2\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{b^2 g^3}+\frac{(30258 d (b c-a d)) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{b^2 g^3}+\frac{\left (15129 (b c-a d)^2\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{b^2 g^3}\\ &=-\frac{15129 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b^3 g^3 (a+b x)^2}-\frac{30258 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^3 (a+b x)}+\frac{15129 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^3}-\frac{\left (15129 B d^2 n\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}{b^3 g^3}+\frac{(30258 B d (b c-a d) n) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^3}+\frac{\left (15129 B (b c-a d)^2 n\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{2 b^3 g^3}\\ &=-\frac{15129 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b^3 g^3 (a+b x)^2}-\frac{30258 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^3 (a+b x)}+\frac{15129 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^3}-\frac{\left (15129 B d^2 n\right ) \int \left (\frac{b \log (a+b x)}{a+b x}-\frac{d \log (a+b x)}{c+d x}\right ) \, dx}{b^3 g^3}+\frac{\left (30258 B d (b c-a d)^2 n\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^3}+\frac{\left (15129 B (b c-a d)^3 n\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{2 b^3 g^3}\\ &=-\frac{15129 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b^3 g^3 (a+b x)^2}-\frac{30258 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^3 (a+b x)}+\frac{15129 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^3}-\frac{\left (15129 B d^2 n\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b^2 g^3}+\frac{\left (15129 B d^3 n\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^3 g^3}+\frac{\left (30258 B d (b c-a d)^2 n\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^2}-\frac{b d}{(b c-a d)^2 (a+b x)}+\frac{d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^3 g^3}+\frac{\left (15129 B (b c-a d)^3 n\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^3}-\frac{b d}{(b c-a d)^2 (a+b x)^2}+\frac{b d^2}{(b c-a d)^3 (a+b x)}-\frac{d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{2 b^3 g^3}\\ &=-\frac{15129 B (b c-a d)^2 n}{4 b^3 g^3 (a+b x)^2}-\frac{45387 B d (b c-a d) n}{2 b^3 g^3 (a+b x)}-\frac{45387 B d^2 n \log (a+b x)}{2 b^3 g^3}-\frac{15129 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b^3 g^3 (a+b x)^2}-\frac{30258 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^3 (a+b x)}+\frac{15129 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^3}+\frac{45387 B d^2 n \log (c+d x)}{2 b^3 g^3}+\frac{15129 B d^2 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g^3}-\frac{\left (15129 B d^2 n\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^3 g^3}-\frac{\left (15129 B d^2 n\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 g^3}\\ &=-\frac{15129 B (b c-a d)^2 n}{4 b^3 g^3 (a+b x)^2}-\frac{45387 B d (b c-a d) n}{2 b^3 g^3 (a+b x)}-\frac{45387 B d^2 n \log (a+b x)}{2 b^3 g^3}-\frac{15129 B d^2 n \log ^2(a+b x)}{2 b^3 g^3}-\frac{15129 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b^3 g^3 (a+b x)^2}-\frac{30258 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^3 (a+b x)}+\frac{15129 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^3}+\frac{45387 B d^2 n \log (c+d x)}{2 b^3 g^3}+\frac{15129 B d^2 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g^3}-\frac{\left (15129 B d^2 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^3 g^3}\\ &=-\frac{15129 B (b c-a d)^2 n}{4 b^3 g^3 (a+b x)^2}-\frac{45387 B d (b c-a d) n}{2 b^3 g^3 (a+b x)}-\frac{45387 B d^2 n \log (a+b x)}{2 b^3 g^3}-\frac{15129 B d^2 n \log ^2(a+b x)}{2 b^3 g^3}-\frac{15129 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b^3 g^3 (a+b x)^2}-\frac{30258 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^3 (a+b x)}+\frac{15129 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^3}+\frac{45387 B d^2 n \log (c+d x)}{2 b^3 g^3}+\frac{15129 B d^2 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g^3}+\frac{15129 B d^2 n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g^3}\\ \end{align*}
Mathematica [A] time = 0.354789, size = 258, normalized size = 1.07 \[ \frac{i^2 \left (-2 B d^2 n \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+4 d^2 \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+\frac{8 d (a d-b c) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{a+b x}-\frac{2 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{(a+b x)^2}+\frac{6 B d n (a d-b c)}{a+b x}-\frac{B n (b c-a d)^2}{(a+b x)^2}-6 B d^2 n \log (a+b x)+6 B d^2 n \log (c+d x)\right )}{4 b^3 g^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.671, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dix+ci \right ) ^{2}}{ \left ( bgx+ag \right ) ^{3}} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \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 (\frac{A d^{2} i^{2} x^{2} + 2 \, A c d i^{2} x + A c^{2} i^{2} +{\left (B d^{2} i^{2} x^{2} + 2 \, B c d i^{2} x + B c^{2} i^{2}\right )} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )}{b^{3} g^{3} x^{3} + 3 \, a b^{2} g^{3} x^{2} + 3 \, a^{2} b g^{3} x + a^{3} g^{3}}, 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 \frac{{\left (d i x + c i\right )}^{2}{\left (B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}}{{\left (b g x + a g\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]