Optimal. Leaf size=389 \[ \frac{3 b^2 d (c+d x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^4 i (a+b x)^2 (b c-a d)^4}-\frac{b^3 (c+d x)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{3 g^4 i (a+b x)^3 (b c-a d)^4}-\frac{d^3 \log \left (\frac{a+b x}{c+d x}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^4 i (b c-a d)^4}-\frac{3 b d^2 (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^4 i (a+b x) (b c-a d)^4}+\frac{3 b^2 B d n (c+d x)^2}{4 g^4 i (a+b x)^2 (b c-a d)^4}-\frac{b^3 B n (c+d x)^3}{9 g^4 i (a+b x)^3 (b c-a d)^4}-\frac{3 b B d^2 n (c+d x)}{g^4 i (a+b x) (b c-a d)^4}+\frac{B d^3 n \log ^2\left (\frac{a+b x}{c+d x}\right )}{2 g^4 i (b c-a d)^4} \]
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Rubi [C] time = 1.07713, antiderivative size = 646, normalized size of antiderivative = 1.66, number of steps used = 30, 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} \[ -\frac{B d^3 n \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{g^4 i (b c-a d)^4}-\frac{B d^3 n \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{g^4 i (b c-a d)^4}-\frac{d^3 \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^4 i (b c-a d)^4}+\frac{d^3 \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^4 i (b c-a d)^4}-\frac{d^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^4 i (a+b x) (b c-a d)^3}+\frac{d \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^4 i (a+b x)^2 (b c-a d)^2}-\frac{B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A}{3 g^4 i (a+b x)^3 (b c-a d)}-\frac{11 B d^2 n}{6 g^4 i (a+b x) (b c-a d)^3}+\frac{B d^3 n \log ^2(a+b x)}{2 g^4 i (b c-a d)^4}+\frac{B d^3 n \log ^2(c+d x)}{2 g^4 i (b c-a d)^4}-\frac{11 B d^3 n \log (a+b x)}{6 g^4 i (b c-a d)^4}+\frac{11 B d^3 n \log (c+d x)}{6 g^4 i (b c-a d)^4}-\frac{B d^3 n \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{g^4 i (b c-a d)^4}-\frac{B d^3 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{g^4 i (b c-a d)^4}+\frac{5 B d n}{12 g^4 i (a+b x)^2 (b c-a d)^2}-\frac{B n}{9 g^4 i (a+b x)^3 (b c-a d)} \]
Antiderivative was successfully verified.
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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{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(142 c+142 d x) (a g+b g x)^4} \, dx &=\int \left (\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d) g^4 (a+b x)^4}-\frac{b d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^2 g^4 (a+b x)^3}+\frac{b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^3 g^4 (a+b x)^2}-\frac{b d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^4 g^4 (a+b x)}+\frac{d^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^4 g^4 (c+d x)}\right ) \, dx\\ &=-\frac{\left (b d^3\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{142 (b c-a d)^4 g^4}+\frac{d^4 \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{142 (b c-a d)^4 g^4}+\frac{\left (b d^2\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{142 (b c-a d)^3 g^4}-\frac{(b d) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{142 (b c-a d)^2 g^4}+\frac{b \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^4} \, dx}{142 (b c-a d) g^4}\\ &=-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{426 (b c-a d) g^4 (a+b x)^3}+\frac{d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{284 (b c-a d)^2 g^4 (a+b x)^2}-\frac{d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^3 g^4 (a+b x)}-\frac{d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^4 g^4}+\frac{d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{142 (b c-a d)^4 g^4}+\frac{\left (B d^3 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}{142 (b c-a d)^4 g^4}-\frac{\left (B d^3 n\right ) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{142 (b c-a d)^4 g^4}+\frac{\left (B d^2 n\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{142 (b c-a d)^3 g^4}-\frac{(B d n) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{284 (b c-a d)^2 g^4}+\frac{(B n) \int \frac{b c-a d}{(a+b x)^4 (c+d x)} \, dx}{426 (b c-a d) g^4}\\ &=-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{426 (b c-a d) g^4 (a+b x)^3}+\frac{d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{284 (b c-a d)^2 g^4 (a+b x)^2}-\frac{d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^3 g^4 (a+b x)}-\frac{d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^4 g^4}+\frac{d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{142 (b c-a d)^4 g^4}+\frac{(B n) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{426 g^4}+\frac{\left (B d^3 n\right ) \int \left (\frac{b \log (a+b x)}{a+b x}-\frac{d \log (a+b x)}{c+d x}\right ) \, dx}{142 (b c-a d)^4 g^4}-\frac{\left (B d^3 n\right ) \int \left (\frac{b \log (c+d x)}{a+b x}-\frac{d \log (c+d x)}{c+d x}\right ) \, dx}{142 (b c-a d)^4 g^4}+\frac{\left (B d^2 n\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{142 (b c-a d)^2 g^4}-\frac{(B d n) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{284 (b c-a d) g^4}\\ &=-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{426 (b c-a d) g^4 (a+b x)^3}+\frac{d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{284 (b c-a d)^2 g^4 (a+b x)^2}-\frac{d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^3 g^4 (a+b x)}-\frac{d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^4 g^4}+\frac{d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{142 (b c-a d)^4 g^4}+\frac{(B n) \int \left (\frac{b}{(b c-a d) (a+b x)^4}-\frac{b d}{(b c-a d)^2 (a+b x)^3}+\frac{b d^2}{(b c-a d)^3 (a+b x)^2}-\frac{b d^3}{(b c-a d)^4 (a+b x)}+\frac{d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{426 g^4}+\frac{\left (b B d^3 n\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{142 (b c-a d)^4 g^4}-\frac{\left (b B d^3 n\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{142 (b c-a d)^4 g^4}-\frac{\left (B d^4 n\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{142 (b c-a d)^4 g^4}+\frac{\left (B d^4 n\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{142 (b c-a d)^4 g^4}+\frac{\left (B 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}{142 (b c-a d)^2 g^4}-\frac{(B d n) \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}{284 (b c-a d) g^4}\\ &=-\frac{B n}{1278 (b c-a d) g^4 (a+b x)^3}+\frac{5 B d n}{1704 (b c-a d)^2 g^4 (a+b x)^2}-\frac{11 B d^2 n}{852 (b c-a d)^3 g^4 (a+b x)}-\frac{11 B d^3 n \log (a+b x)}{852 (b c-a d)^4 g^4}-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{426 (b c-a d) g^4 (a+b x)^3}+\frac{d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{284 (b c-a d)^2 g^4 (a+b x)^2}-\frac{d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^3 g^4 (a+b x)}-\frac{d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^4 g^4}+\frac{11 B d^3 n \log (c+d x)}{852 (b c-a d)^4 g^4}-\frac{B d^3 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{142 (b c-a d)^4 g^4}+\frac{d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{142 (b c-a d)^4 g^4}-\frac{B d^3 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{142 (b c-a d)^4 g^4}+\frac{\left (B d^3 n\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{142 (b c-a d)^4 g^4}+\frac{\left (B d^3 n\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{142 (b c-a d)^4 g^4}+\frac{\left (b B d^3 n\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{142 (b c-a d)^4 g^4}+\frac{\left (B d^4 n\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{142 (b c-a d)^4 g^4}\\ &=-\frac{B n}{1278 (b c-a d) g^4 (a+b x)^3}+\frac{5 B d n}{1704 (b c-a d)^2 g^4 (a+b x)^2}-\frac{11 B d^2 n}{852 (b c-a d)^3 g^4 (a+b x)}-\frac{11 B d^3 n \log (a+b x)}{852 (b c-a d)^4 g^4}+\frac{B d^3 n \log ^2(a+b x)}{284 (b c-a d)^4 g^4}-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{426 (b c-a d) g^4 (a+b x)^3}+\frac{d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{284 (b c-a d)^2 g^4 (a+b x)^2}-\frac{d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^3 g^4 (a+b x)}-\frac{d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^4 g^4}+\frac{11 B d^3 n \log (c+d x)}{852 (b c-a d)^4 g^4}-\frac{B d^3 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{142 (b c-a d)^4 g^4}+\frac{d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{142 (b c-a d)^4 g^4}+\frac{B d^3 n \log ^2(c+d x)}{284 (b c-a d)^4 g^4}-\frac{B d^3 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{142 (b c-a d)^4 g^4}+\frac{\left (B d^3 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 )}{142 (b c-a d)^4 g^4}+\frac{\left (B d^3 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{142 (b c-a d)^4 g^4}\\ &=-\frac{B n}{1278 (b c-a d) g^4 (a+b x)^3}+\frac{5 B d n}{1704 (b c-a d)^2 g^4 (a+b x)^2}-\frac{11 B d^2 n}{852 (b c-a d)^3 g^4 (a+b x)}-\frac{11 B d^3 n \log (a+b x)}{852 (b c-a d)^4 g^4}+\frac{B d^3 n \log ^2(a+b x)}{284 (b c-a d)^4 g^4}-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{426 (b c-a d) g^4 (a+b x)^3}+\frac{d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{284 (b c-a d)^2 g^4 (a+b x)^2}-\frac{d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^3 g^4 (a+b x)}-\frac{d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^4 g^4}+\frac{11 B d^3 n \log (c+d x)}{852 (b c-a d)^4 g^4}-\frac{B d^3 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{142 (b c-a d)^4 g^4}+\frac{d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{142 (b c-a d)^4 g^4}+\frac{B d^3 n \log ^2(c+d x)}{284 (b c-a d)^4 g^4}-\frac{B d^3 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{142 (b c-a d)^4 g^4}-\frac{B d^3 n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{142 (b c-a d)^4 g^4}-\frac{B d^3 n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{142 (b c-a d)^4 g^4}\\ \end{align*}
Mathematica [C] time = 0.754456, size = 518, normalized size = 1.33 \[ \frac{-36 B d^3 n \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )-36 B d^3 n \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\frac{36 A d^2 (a d-b c)}{a+b x}+\frac{18 A d (b c-a d)^2}{(a+b x)^2}-\frac{12 A (b c-a d)^3}{(a+b x)^3}-36 A d^3 \log (a+b x)-36 B d^3 \log (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+36 B d^3 \log (c+d x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\frac{36 B d^2 (a d-b c) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x}+\frac{66 B d^2 n (a d-b c)}{a+b x}-36 B d^3 n \log (c+d x) \log \left (\frac{d (a+b x)}{a d-b c}\right )-36 B d^3 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )+\frac{18 B d (b c-a d)^2 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^2}-\frac{12 B (b c-a d)^3 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^3}+\frac{15 B d n (b c-a d)^2}{(a+b x)^2}-\frac{4 B n (b c-a d)^3}{(a+b x)^3}+18 B d^3 n \log ^2(a+b x)-66 B d^3 n \log (a+b x)+36 A d^3 \log (c+d x)+18 B d^3 n \log ^2(c+d x)+66 B d^3 n \log (c+d x)}{36 g^4 i (b c-a d)^4} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.773, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( bgx+ag \right ) ^{4} \left ( dix+ci \right ) } \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.
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Maxima [B] time = 1.79299, size = 1987, normalized size = 5.11 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.599218, size = 1809, normalized size = 4.65 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A}{{\left (b g x + a g\right )}^{4}{\left (d i x + c i\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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