Optimal. Leaf size=141 \[ \frac{g (a+b x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{2 i^3 (c+d x)^2 (b c-a d)}-\frac{B g (a+b x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 i^3 (c+d x)^2 (b c-a d)}+\frac{B^2 g (a+b x)^2}{4 i^3 (c+d x)^2 (b c-a d)} \]
[Out]
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Rubi [C] time = 1.96088, antiderivative size = 634, normalized size of antiderivative = 4.5, number of steps used = 58, number of rules used = 11, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.275, Rules used = {2528, 2525, 12, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 44} \[ \frac{b^2 B^2 g \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{d^2 i^3 (b c-a d)}+\frac{b^2 B^2 g \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{d^2 i^3 (b c-a d)}+\frac{b^2 B g \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{d^2 i^3 (b c-a d)}-\frac{b^2 B g \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{d^2 i^3 (b c-a d)}-\frac{b g \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{d^2 i^3 (c+d x)}+\frac{b B g \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{d^2 i^3 (c+d x)}+\frac{g (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{2 d^2 i^3 (c+d x)^2}-\frac{B g (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 d^2 i^3 (c+d x)^2}-\frac{b^2 B^2 g \log ^2(a+b x)}{2 d^2 i^3 (b c-a d)}-\frac{b^2 B^2 g \log ^2(c+d x)}{2 d^2 i^3 (b c-a d)}-\frac{b^2 B^2 g \log (a+b x)}{2 d^2 i^3 (b c-a d)}+\frac{b^2 B^2 g \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{d^2 i^3 (b c-a d)}+\frac{b^2 B^2 g \log (c+d x)}{2 d^2 i^3 (b c-a d)}+\frac{b^2 B^2 g \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{d^2 i^3 (b c-a d)}+\frac{B^2 g (b c-a d)}{4 d^2 i^3 (c+d x)^2}-\frac{b B^2 g}{2 d^2 i^3 (c+d x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2528
Rule 2525
Rule 12
Rule 2524
Rule 2418
Rule 2390
Rule 2301
Rule 2394
Rule 2393
Rule 2391
Rule 44
Rubi steps
\begin{align*} \int \frac{(a g+b g x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{(102 c+102 d x)^3} \, dx &=\int \left (\frac{(-b c+a d) g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{1061208 d (c+d x)^3}+\frac{b g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{1061208 d (c+d x)^2}\right ) \, dx\\ &=\frac{(b g) \int \frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{(c+d x)^2} \, dx}{1061208 d}-\frac{((b c-a d) g) \int \frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{(c+d x)^3} \, dx}{1061208 d}\\ &=\frac{(b c-a d) g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2122416 d^2 (c+d x)^2}-\frac{b g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{1061208 d^2 (c+d x)}+\frac{(b B g) \int \frac{(b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)^2} \, dx}{530604 d^2}-\frac{(B (b c-a d) g) \int \frac{(b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)^3} \, dx}{1061208 d^2}\\ &=\frac{(b c-a d) g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2122416 d^2 (c+d x)^2}-\frac{b g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{1061208 d^2 (c+d x)}+\frac{(b B (b c-a d) g) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)^2} \, dx}{530604 d^2}-\frac{\left (B (b c-a d)^2 g\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)^3} \, dx}{1061208 d^2}\\ &=\frac{(b c-a d) g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2122416 d^2 (c+d x)^2}-\frac{b g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{1061208 d^2 (c+d x)}+\frac{(b B (b c-a d) g) \int \left (\frac{b^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (a+b x)}-\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (c+d x)^2}-\frac{b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (c+d x)}\right ) \, dx}{530604 d^2}-\frac{\left (B (b c-a d)^2 g\right ) \int \left (\frac{b^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (a+b x)}-\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (c+d x)^3}-\frac{b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (c+d x)^2}-\frac{b^2 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (c+d x)}\right ) \, dx}{1061208 d^2}\\ &=\frac{(b c-a d) g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2122416 d^2 (c+d x)^2}-\frac{b g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{1061208 d^2 (c+d x)}+\frac{(b B g) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(c+d x)^2} \, dx}{1061208 d}-\frac{(b B g) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(c+d x)^2} \, dx}{530604 d}-\frac{\left (b^3 B g\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{1061208 d^2 (b c-a d)}+\frac{\left (b^3 B g\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{530604 d^2 (b c-a d)}+\frac{\left (b^2 B g\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{1061208 d (b c-a d)}-\frac{\left (b^2 B g\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{530604 d (b c-a d)}+\frac{(B (b c-a d) g) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(c+d x)^3} \, dx}{1061208 d}\\ &=-\frac{B (b c-a d) g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2122416 d^2 (c+d x)^2}+\frac{b B g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (c+d x)}+\frac{b^2 B g \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (b c-a d)}+\frac{(b c-a d) g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2122416 d^2 (c+d x)^2}-\frac{b g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{1061208 d^2 (c+d x)}-\frac{b^2 B g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1061208 d^2 (b c-a d)}+\frac{\left (b B^2 g\right ) \int \frac{b c-a d}{(a+b x) (c+d x)^2} \, dx}{1061208 d^2}-\frac{\left (b B^2 g\right ) \int \frac{b c-a d}{(a+b x) (c+d x)^2} \, dx}{530604 d^2}+\frac{\left (b^2 B^2 g\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{1061208 d^2 (b c-a d)}-\frac{\left (b^2 B^2 g\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{1061208 d^2 (b c-a d)}-\frac{\left (b^2 B^2 g\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{530604 d^2 (b c-a d)}+\frac{\left (b^2 B^2 g\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{530604 d^2 (b c-a d)}+\frac{\left (B^2 (b c-a d) g\right ) \int \frac{b c-a d}{(a+b x) (c+d x)^3} \, dx}{2122416 d^2}\\ &=-\frac{B (b c-a d) g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2122416 d^2 (c+d x)^2}+\frac{b B g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (c+d x)}+\frac{b^2 B g \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (b c-a d)}+\frac{(b c-a d) g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2122416 d^2 (c+d x)^2}-\frac{b g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{1061208 d^2 (c+d x)}-\frac{b^2 B g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1061208 d^2 (b c-a d)}+\frac{\left (b B^2 (b c-a d) g\right ) \int \frac{1}{(a+b x) (c+d x)^2} \, dx}{1061208 d^2}-\frac{\left (b B^2 (b c-a d) g\right ) \int \frac{1}{(a+b x) (c+d x)^2} \, dx}{530604 d^2}+\frac{\left (B^2 (b c-a d)^2 g\right ) \int \frac{1}{(a+b x) (c+d x)^3} \, dx}{2122416 d^2}+\frac{\left (b^2 B^2 g\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{1061208 d^2 (b c-a d) e}-\frac{\left (b^2 B^2 g\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{1061208 d^2 (b c-a d) e}-\frac{\left (b^2 B^2 g\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{530604 d^2 (b c-a d) e}+\frac{\left (b^2 B^2 g\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{530604 d^2 (b c-a d) e}\\ &=-\frac{B (b c-a d) g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2122416 d^2 (c+d x)^2}+\frac{b B g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (c+d x)}+\frac{b^2 B g \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (b c-a d)}+\frac{(b c-a d) g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2122416 d^2 (c+d x)^2}-\frac{b g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{1061208 d^2 (c+d x)}-\frac{b^2 B g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1061208 d^2 (b c-a d)}+\frac{\left (b B^2 (b c-a d) g\right ) \int \left (\frac{b^2}{(b c-a d)^2 (a+b x)}-\frac{d}{(b c-a d) (c+d x)^2}-\frac{b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{1061208 d^2}-\frac{\left (b B^2 (b c-a d) g\right ) \int \left (\frac{b^2}{(b c-a d)^2 (a+b x)}-\frac{d}{(b c-a d) (c+d x)^2}-\frac{b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{530604 d^2}+\frac{\left (B^2 (b c-a d)^2 g\right ) \int \left (\frac{b^3}{(b c-a d)^3 (a+b x)}-\frac{d}{(b c-a d) (c+d x)^3}-\frac{b d}{(b c-a d)^2 (c+d x)^2}-\frac{b^2 d}{(b c-a d)^3 (c+d x)}\right ) \, dx}{2122416 d^2}+\frac{\left (b^2 B^2 g\right ) \int \left (\frac{b e \log (a+b x)}{a+b x}-\frac{d e \log (a+b x)}{c+d x}\right ) \, dx}{1061208 d^2 (b c-a d) e}-\frac{\left (b^2 B^2 g\right ) \int \left (\frac{b e \log (c+d x)}{a+b x}-\frac{d e \log (c+d x)}{c+d x}\right ) \, dx}{1061208 d^2 (b c-a d) e}-\frac{\left (b^2 B^2 g\right ) \int \left (\frac{b e \log (a+b x)}{a+b x}-\frac{d e \log (a+b x)}{c+d x}\right ) \, dx}{530604 d^2 (b c-a d) e}+\frac{\left (b^2 B^2 g\right ) \int \left (\frac{b e \log (c+d x)}{a+b x}-\frac{d e \log (c+d x)}{c+d x}\right ) \, dx}{530604 d^2 (b c-a d) e}\\ &=\frac{B^2 (b c-a d) g}{4244832 d^2 (c+d x)^2}-\frac{b B^2 g}{2122416 d^2 (c+d x)}-\frac{b^2 B^2 g \log (a+b x)}{2122416 d^2 (b c-a d)}-\frac{B (b c-a d) g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2122416 d^2 (c+d x)^2}+\frac{b B g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (c+d x)}+\frac{b^2 B g \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (b c-a d)}+\frac{(b c-a d) g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2122416 d^2 (c+d x)^2}-\frac{b g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{1061208 d^2 (c+d x)}+\frac{b^2 B^2 g \log (c+d x)}{2122416 d^2 (b c-a d)}-\frac{b^2 B g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1061208 d^2 (b c-a d)}+\frac{\left (b^3 B^2 g\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{1061208 d^2 (b c-a d)}-\frac{\left (b^3 B^2 g\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{1061208 d^2 (b c-a d)}-\frac{\left (b^3 B^2 g\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{530604 d^2 (b c-a d)}+\frac{\left (b^3 B^2 g\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{530604 d^2 (b c-a d)}-\frac{\left (b^2 B^2 g\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{1061208 d (b c-a d)}+\frac{\left (b^2 B^2 g\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{1061208 d (b c-a d)}+\frac{\left (b^2 B^2 g\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{530604 d (b c-a d)}-\frac{\left (b^2 B^2 g\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{530604 d (b c-a d)}\\ &=\frac{B^2 (b c-a d) g}{4244832 d^2 (c+d x)^2}-\frac{b B^2 g}{2122416 d^2 (c+d x)}-\frac{b^2 B^2 g \log (a+b x)}{2122416 d^2 (b c-a d)}-\frac{B (b c-a d) g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2122416 d^2 (c+d x)^2}+\frac{b B g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (c+d x)}+\frac{b^2 B g \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (b c-a d)}+\frac{(b c-a d) g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2122416 d^2 (c+d x)^2}-\frac{b g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{1061208 d^2 (c+d x)}+\frac{b^2 B^2 g \log (c+d x)}{2122416 d^2 (b c-a d)}+\frac{b^2 B^2 g \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1061208 d^2 (b c-a d)}-\frac{b^2 B g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1061208 d^2 (b c-a d)}+\frac{b^2 B^2 g \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{1061208 d^2 (b c-a d)}+\frac{\left (b^2 B^2 g\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{1061208 d^2 (b c-a d)}+\frac{\left (b^2 B^2 g\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{1061208 d^2 (b c-a d)}-\frac{\left (b^2 B^2 g\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{530604 d^2 (b c-a d)}-\frac{\left (b^2 B^2 g\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{530604 d^2 (b c-a d)}+\frac{\left (b^3 B^2 g\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{1061208 d^2 (b c-a d)}-\frac{\left (b^3 B^2 g\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{530604 d^2 (b c-a d)}+\frac{\left (b^2 B^2 g\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{1061208 d (b c-a d)}-\frac{\left (b^2 B^2 g\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{530604 d (b c-a d)}\\ &=\frac{B^2 (b c-a d) g}{4244832 d^2 (c+d x)^2}-\frac{b B^2 g}{2122416 d^2 (c+d x)}-\frac{b^2 B^2 g \log (a+b x)}{2122416 d^2 (b c-a d)}-\frac{b^2 B^2 g \log ^2(a+b x)}{2122416 d^2 (b c-a d)}-\frac{B (b c-a d) g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2122416 d^2 (c+d x)^2}+\frac{b B g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (c+d x)}+\frac{b^2 B g \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (b c-a d)}+\frac{(b c-a d) g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2122416 d^2 (c+d x)^2}-\frac{b g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{1061208 d^2 (c+d x)}+\frac{b^2 B^2 g \log (c+d x)}{2122416 d^2 (b c-a d)}+\frac{b^2 B^2 g \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1061208 d^2 (b c-a d)}-\frac{b^2 B g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1061208 d^2 (b c-a d)}-\frac{b^2 B^2 g \log ^2(c+d x)}{2122416 d^2 (b c-a d)}+\frac{b^2 B^2 g \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{1061208 d^2 (b c-a d)}+\frac{\left (b^2 B^2 g\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{1061208 d^2 (b c-a d)}+\frac{\left (b^2 B^2 g\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{1061208 d^2 (b c-a d)}-\frac{\left (b^2 B^2 g\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{530604 d^2 (b c-a d)}-\frac{\left (b^2 B^2 g\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{530604 d^2 (b c-a d)}\\ &=\frac{B^2 (b c-a d) g}{4244832 d^2 (c+d x)^2}-\frac{b B^2 g}{2122416 d^2 (c+d x)}-\frac{b^2 B^2 g \log (a+b x)}{2122416 d^2 (b c-a d)}-\frac{b^2 B^2 g \log ^2(a+b x)}{2122416 d^2 (b c-a d)}-\frac{B (b c-a d) g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2122416 d^2 (c+d x)^2}+\frac{b B g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (c+d x)}+\frac{b^2 B g \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (b c-a d)}+\frac{(b c-a d) g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2122416 d^2 (c+d x)^2}-\frac{b g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{1061208 d^2 (c+d x)}+\frac{b^2 B^2 g \log (c+d x)}{2122416 d^2 (b c-a d)}+\frac{b^2 B^2 g \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1061208 d^2 (b c-a d)}-\frac{b^2 B g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1061208 d^2 (b c-a d)}-\frac{b^2 B^2 g \log ^2(c+d x)}{2122416 d^2 (b c-a d)}+\frac{b^2 B^2 g \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{1061208 d^2 (b c-a d)}+\frac{b^2 B^2 g \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{1061208 d^2 (b c-a d)}+\frac{b^2 B^2 g \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{1061208 d^2 (b c-a d)}\\ \end{align*}
Mathematica [C] time = 0.897672, size = 767, normalized size = 5.44 \[ \frac{g \left (-B \left (-2 b^2 B (c+d x)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+2 b^2 B (c+d x)^2 \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+4 b^2 (c+d x)^2 \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-4 b^2 (c+d x)^2 \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+2 (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+4 b (c+d x) (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-B \left (2 b^2 (c+d x)^2 \log (a+b x)+2 b (c+d x) (b c-a d)+(b c-a d)^2-2 b^2 (c+d x)^2 \log (c+d x)\right )-4 b B (c+d x) (b (c+d x) \log (a+b x)-a d-b (c+d x) \log (c+d x)+b c)\right )+4 b B (c+d x) \left (-b B (c+d x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+b B (c+d x) \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+2 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+2 b (c+d x) \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-2 b (c+d x) \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-2 B (b (c+d x) \log (a+b x)-a d-b (c+d x) \log (c+d x)+b c)\right )+2 (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2-4 b (c+d x) (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2\right )}{4 d^2 i^3 (c+d x)^2 (b c-a d)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.059, size = 2449, normalized size = 17.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.9975, size = 2654, normalized size = 18.82 \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 = 0.495467, size = 602, normalized size = 4.27 \begin{align*} -\frac{2 \,{\left ({\left (2 \, A^{2} - 2 \, A B + B^{2}\right )} b^{2} c d -{\left (2 \, A^{2} - 2 \, A B + B^{2}\right )} a b d^{2}\right )} g x - 2 \,{\left (B^{2} b^{2} d^{2} g x^{2} + 2 \, B^{2} a b d^{2} g x + B^{2} a^{2} d^{2} g\right )} \log \left (\frac{b e x + a e}{d x + c}\right )^{2} +{\left ({\left (2 \, A^{2} - 2 \, A B + B^{2}\right )} b^{2} c^{2} -{\left (2 \, A^{2} - 2 \, A B + B^{2}\right )} a^{2} d^{2}\right )} g - 2 \,{\left ({\left (2 \, A B - B^{2}\right )} b^{2} d^{2} g x^{2} + 2 \,{\left (2 \, A B - B^{2}\right )} a b d^{2} g x +{\left (2 \, A B - B^{2}\right )} a^{2} d^{2} g\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{4 \,{\left ({\left (b c d^{4} - a d^{5}\right )} i^{3} x^{2} + 2 \,{\left (b c^{2} d^{3} - a c d^{4}\right )} i^{3} x +{\left (b c^{3} d^{2} - a c^{2} d^{3}\right )} i^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 14.2331, size = 712, normalized size = 5.05 \begin{align*} \frac{B b^{2} g \left (2 A - B\right ) \log{\left (x + \frac{2 A B a b^{2} d g + 2 A B b^{3} c g - B^{2} a b^{2} d g - B^{2} b^{3} c g - \frac{B a^{2} b^{2} d^{2} g \left (2 A - B\right )}{a d - b c} + \frac{2 B a b^{3} c d g \left (2 A - B\right )}{a d - b c} - \frac{B b^{4} c^{2} g \left (2 A - B\right )}{a d - b c}}{4 A B b^{3} d g - 2 B^{2} b^{3} d g} \right )}}{2 d^{2} i^{3} \left (a d - b c\right )} - \frac{B b^{2} g \left (2 A - B\right ) \log{\left (x + \frac{2 A B a b^{2} d g + 2 A B b^{3} c g - B^{2} a b^{2} d g - B^{2} b^{3} c g + \frac{B a^{2} b^{2} d^{2} g \left (2 A - B\right )}{a d - b c} - \frac{2 B a b^{3} c d g \left (2 A - B\right )}{a d - b c} + \frac{B b^{4} c^{2} g \left (2 A - B\right )}{a d - b c}}{4 A B b^{3} d g - 2 B^{2} b^{3} d g} \right )}}{2 d^{2} i^{3} \left (a d - b c\right )} + \frac{\left (- B^{2} a^{2} g - 2 B^{2} a b g x - B^{2} b^{2} g x^{2}\right ) \log{\left (\frac{e \left (a + b x\right )}{c + d x} \right )}^{2}}{2 a c^{2} d i^{3} + 4 a c d^{2} i^{3} x + 2 a d^{3} i^{3} x^{2} - 2 b c^{3} i^{3} - 4 b c^{2} d i^{3} x - 2 b c d^{2} i^{3} x^{2}} - \frac{2 A^{2} a d g + 2 A^{2} b c g - 2 A B a d g - 2 A B b c g + B^{2} a d g + B^{2} b c g + x \left (4 A^{2} b d g - 4 A B b d g + 2 B^{2} b d g\right )}{4 c^{2} d^{2} i^{3} + 8 c d^{3} i^{3} x + 4 d^{4} i^{3} x^{2}} + \frac{\left (- 2 A B a d g - 2 A B b c g - 4 A B b d g x + B^{2} a d g + B^{2} b c g + 2 B^{2} b d g x\right ) \log{\left (\frac{e \left (a + b x\right )}{c + d x} \right )}}{2 c^{2} d^{2} i^{3} + 4 c d^{3} i^{3} x + 2 d^{4} i^{3} x^{2}} \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{{\left (b g x + a g\right )}{\left (B \log \left (\frac{{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}}{{\left (d i x + c i\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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