Optimal. Leaf size=319 \[ -\frac{b i^2 (c+d x)^4 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 g^5 (a+b x)^4 (b c-a d)^2}-\frac{b B i^2 n (c+d x)^4 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{8 g^5 (a+b x)^4 (b c-a d)^2}+\frac{d i^2 (c+d x)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 g^5 (a+b x)^3 (b c-a d)^2}+\frac{2 B d i^2 n (c+d x)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{9 g^5 (a+b x)^3 (b c-a d)^2}-\frac{b B^2 i^2 n^2 (c+d x)^4}{32 g^5 (a+b x)^4 (b c-a d)^2}+\frac{2 B^2 d i^2 n^2 (c+d x)^3}{27 g^5 (a+b x)^3 (b c-a d)^2} \]
[Out]
________________________________________________________________________________________
Rubi [C] time = 3.78887, antiderivative size = 989, normalized size of antiderivative = 3.1, number of steps used = 98, number of rules used = 11, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.244, Rules used = {2528, 2525, 12, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ -\frac{B^2 i^2 n^2 \log ^2(a+b x) d^4}{12 b^3 (b c-a d)^2 g^5}-\frac{B^2 i^2 n^2 \log ^2(c+d x) d^4}{12 b^3 (b c-a d)^2 g^5}+\frac{7 B^2 i^2 n^2 \log (a+b x) d^4}{72 b^3 (b c-a d)^2 g^5}+\frac{B i^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) d^4}{6 b^3 (b c-a d)^2 g^5}-\frac{7 B^2 i^2 n^2 \log (c+d x) d^4}{72 b^3 (b c-a d)^2 g^5}+\frac{B^2 i^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) d^4}{6 b^3 (b c-a d)^2 g^5}-\frac{B i^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x) d^4}{6 b^3 (b c-a d)^2 g^5}+\frac{B^2 i^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right ) d^4}{6 b^3 (b c-a d)^2 g^5}+\frac{B^2 i^2 n^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right ) d^4}{6 b^3 (b c-a d)^2 g^5}+\frac{B^2 i^2 n^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right ) d^4}{6 b^3 (b c-a d)^2 g^5}+\frac{B i^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) d^3}{6 b^3 (b c-a d) g^5 (a+b x)}+\frac{7 B^2 i^2 n^2 d^3}{72 b^3 (b c-a d) g^5 (a+b x)}-\frac{i^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 d^2}{2 b^3 g^5 (a+b x)^2}-\frac{B i^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) d^2}{12 b^3 g^5 (a+b x)^2}+\frac{5 B^2 i^2 n^2 d^2}{144 b^3 g^5 (a+b x)^2}-\frac{2 (b c-a d) i^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 d}{3 b^3 g^5 (a+b x)^3}-\frac{5 B (b c-a d) i^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) d}{18 b^3 g^5 (a+b x)^3}-\frac{11 B^2 (b c-a d) i^2 n^2 d}{216 b^3 g^5 (a+b x)^3}-\frac{(b c-a d)^2 i^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4 b^3 g^5 (a+b x)^4}-\frac{B (b c-a d)^2 i^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8 b^3 g^5 (a+b x)^4}-\frac{B^2 (b c-a d)^2 i^2 n^2}{32 b^3 g^5 (a+b x)^4} \]
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{(176 c+176 d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^5} \, dx &=\int \left (\frac{30976 (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^2 g^5 (a+b x)^5}+\frac{61952 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^5 (a+b x)^4}+\frac{30976 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^5 (a+b x)^3}\right ) \, dx\\ &=\frac{\left (30976 d^2\right ) \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x)^3} \, dx}{b^2 g^5}+\frac{(61952 d (b c-a d)) \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x)^4} \, dx}{b^2 g^5}+\frac{\left (30976 (b c-a d)^2\right ) \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x)^5} \, dx}{b^2 g^5}\\ &=-\frac{7744 (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^5 (a+b x)^4}-\frac{61952 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g^5 (a+b x)^3}-\frac{15488 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^2}+\frac{\left (30976 B d^2 n\right ) \int \frac{(b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^5}+\frac{(123904 B d (b c-a d) n) \int \frac{(b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^4 (c+d x)} \, dx}{3 b^3 g^5}+\frac{\left (15488 B (b c-a d)^2 n\right ) \int \frac{(b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^5 (c+d x)} \, dx}{b^3 g^5}\\ &=-\frac{7744 (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^5 (a+b x)^4}-\frac{61952 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g^5 (a+b x)^3}-\frac{15488 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^2}+\frac{\left (30976 B d^2 (b c-a d) n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^5}+\frac{\left (123904 B d (b c-a d)^2 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^4 (c+d x)} \, dx}{3 b^3 g^5}+\frac{\left (15488 B (b c-a d)^3 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^5 (c+d x)} \, dx}{b^3 g^5}\\ &=-\frac{7744 (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^5 (a+b x)^4}-\frac{61952 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g^5 (a+b x)^3}-\frac{15488 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^2}+\frac{\left (30976 B d^2 (b c-a d) n\right ) \int \left (\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (a+b x)^3}-\frac{b d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (a+b x)^2}+\frac{b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 (a+b x)}-\frac{d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b^3 g^5}+\frac{\left (123904 B d (b c-a d)^2 n\right ) \int \left (\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (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 )}{(b c-a d)^2 (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 )}{(b c-a d)^3 (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 )}{(b c-a d)^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 )}{(b c-a d)^4 (c+d x)}\right ) \, dx}{3 b^3 g^5}+\frac{\left (15488 B (b c-a d)^3 n\right ) \int \left (\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (a+b x)^5}-\frac{b d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (a+b x)^4}+\frac{b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 (a+b x)^3}-\frac{b d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 (a+b x)^2}+\frac{b d^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^5 (a+b x)}-\frac{d^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^5 (c+d x)}\right ) \, dx}{b^3 g^5}\\ &=-\frac{7744 (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^5 (a+b x)^4}-\frac{61952 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g^5 (a+b x)^3}-\frac{15488 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^2}+\frac{\left (15488 B d^2 n\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^5}+\frac{\left (30976 B d^2 n\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^5}-\frac{\left (123904 B d^2 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{3 b^2 g^5}+\frac{\left (15488 B 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}{b^2 (b c-a d)^2 g^5}+\frac{\left (30976 B 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}{b^2 (b c-a d)^2 g^5}-\frac{\left (123904 B 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}{3 b^2 (b c-a d)^2 g^5}-\frac{\left (15488 B d^5 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}-\frac{\left (30976 B d^5 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}+\frac{\left (123904 B d^5 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{3 b^3 (b c-a d)^2 g^5}-\frac{\left (15488 B d^3 n\right ) \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 (b c-a d) g^5}-\frac{\left (30976 B d^3 n\right ) \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 (b c-a d) g^5}+\frac{\left (123904 B d^3 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{3 b^2 (b c-a d) g^5}-\frac{(15488 B d (b c-a d) n) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^4} \, dx}{b^2 g^5}+\frac{(123904 B d (b c-a d) n) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^4} \, dx}{3 b^2 g^5}+\frac{\left (15488 B (b c-a d)^2 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^5} \, dx}{b^2 g^5}\\ &=-\frac{3872 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^5 (a+b x)^4}-\frac{77440 B d (b c-a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{9 b^3 g^5 (a+b x)^3}-\frac{7744 B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 g^5 (a+b x)^2}+\frac{15488 B d^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 (b c-a d) g^5 (a+b x)}+\frac{15488 B 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 )}{3 b^3 (b c-a d)^2 g^5}-\frac{7744 (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^5 (a+b x)^4}-\frac{61952 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g^5 (a+b x)^3}-\frac{15488 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^2}-\frac{15488 B d^4 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b^3 (b c-a d)^2 g^5}+\frac{\left (7744 B^2 d^2 n^2\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^5}+\frac{\left (15488 B^2 d^2 n^2\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^5}-\frac{\left (61952 B^2 d^2 n^2\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{3 b^3 g^5}-\frac{\left (15488 B^2 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}{b^3 (b c-a d)^2 g^5}+\frac{\left (15488 B^2 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 (c+d x)}{a+b x} \, dx}{b^3 (b c-a d)^2 g^5}-\frac{\left (30976 B^2 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}{b^3 (b c-a d)^2 g^5}+\frac{\left (30976 B^2 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 (c+d x)}{a+b x} \, dx}{b^3 (b c-a d)^2 g^5}+\frac{\left (123904 B^2 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}{3 b^3 (b c-a d)^2 g^5}-\frac{\left (123904 B^2 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 (c+d x)}{a+b x} \, dx}{3 b^3 (b c-a d)^2 g^5}-\frac{\left (15488 B^2 d^3 n^2\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^3 (b c-a d) g^5}-\frac{\left (30976 B^2 d^3 n^2\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^3 (b c-a d) g^5}+\frac{\left (123904 B^2 d^3 n^2\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{3 b^3 (b c-a d) g^5}-\frac{\left (15488 B^2 d (b c-a d) n^2\right ) \int \frac{b c-a d}{(a+b x)^4 (c+d x)} \, dx}{3 b^3 g^5}+\frac{\left (123904 B^2 d (b c-a d) n^2\right ) \int \frac{b c-a d}{(a+b x)^4 (c+d x)} \, dx}{9 b^3 g^5}+\frac{\left (3872 B^2 (b c-a d)^2 n^2\right ) \int \frac{b c-a d}{(a+b x)^5 (c+d x)} \, dx}{b^3 g^5}\\ &=-\frac{3872 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^5 (a+b x)^4}-\frac{77440 B d (b c-a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{9 b^3 g^5 (a+b x)^3}-\frac{7744 B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 g^5 (a+b x)^2}+\frac{15488 B d^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 (b c-a d) g^5 (a+b x)}+\frac{15488 B 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 )}{3 b^3 (b c-a d)^2 g^5}-\frac{7744 (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^5 (a+b x)^4}-\frac{61952 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g^5 (a+b x)^3}-\frac{15488 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^2}-\frac{15488 B d^4 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b^3 (b c-a d)^2 g^5}-\frac{\left (15488 B^2 d^3 n^2\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^5}-\frac{\left (30976 B^2 d^3 n^2\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^5}+\frac{\left (123904 B^2 d^3 n^2\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{3 b^3 g^5}-\frac{\left (15488 B^2 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}{b^3 (b c-a d)^2 g^5}+\frac{\left (15488 B^2 d^4 n^2\right ) \int \left (\frac{b \log (c+d x)}{a+b x}-\frac{d \log (c+d x)}{c+d x}\right ) \, dx}{b^3 (b c-a d)^2 g^5}-\frac{\left (30976 B^2 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}{b^3 (b c-a d)^2 g^5}+\frac{\left (30976 B^2 d^4 n^2\right ) \int \left (\frac{b \log (c+d x)}{a+b x}-\frac{d \log (c+d x)}{c+d x}\right ) \, dx}{b^3 (b c-a d)^2 g^5}+\frac{\left (123904 B^2 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}{3 b^3 (b c-a d)^2 g^5}-\frac{\left (123904 B^2 d^4 n^2\right ) \int \left (\frac{b \log (c+d x)}{a+b x}-\frac{d \log (c+d x)}{c+d x}\right ) \, dx}{3 b^3 (b c-a d)^2 g^5}+\frac{\left (7744 B^2 d^2 (b c-a d) n^2\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^5}+\frac{\left (15488 B^2 d^2 (b c-a d) n^2\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^5}-\frac{\left (61952 B^2 d^2 (b c-a d) n^2\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{3 b^3 g^5}-\frac{\left (15488 B^2 d (b c-a d)^2 n^2\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{3 b^3 g^5}+\frac{\left (123904 B^2 d (b c-a d)^2 n^2\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{9 b^3 g^5}+\frac{\left (3872 B^2 (b c-a d)^3 n^2\right ) \int \frac{1}{(a+b x)^5 (c+d x)} \, dx}{b^3 g^5}\\ &=-\frac{3872 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^5 (a+b x)^4}-\frac{77440 B d (b c-a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{9 b^3 g^5 (a+b x)^3}-\frac{7744 B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 g^5 (a+b x)^2}+\frac{15488 B d^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 (b c-a d) g^5 (a+b x)}+\frac{15488 B 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 )}{3 b^3 (b c-a d)^2 g^5}-\frac{7744 (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^5 (a+b x)^4}-\frac{61952 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g^5 (a+b x)^3}-\frac{15488 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^2}-\frac{15488 B d^4 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b^3 (b c-a d)^2 g^5}-\frac{\left (15488 B^2 d^3 n^2\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^5}-\frac{\left (30976 B^2 d^3 n^2\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^5}+\frac{\left (123904 B^2 d^3 n^2\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}{3 b^3 g^5}-\frac{\left (15488 B^2 d^4 n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}+\frac{\left (15488 B^2 d^4 n^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}-\frac{\left (30976 B^2 d^4 n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}+\frac{\left (30976 B^2 d^4 n^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}+\frac{\left (123904 B^2 d^4 n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{3 b^2 (b c-a d)^2 g^5}-\frac{\left (123904 B^2 d^4 n^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{3 b^2 (b c-a d)^2 g^5}+\frac{\left (15488 B^2 d^5 n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}-\frac{\left (15488 B^2 d^5 n^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}+\frac{\left (30976 B^2 d^5 n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}-\frac{\left (30976 B^2 d^5 n^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}-\frac{\left (123904 B^2 d^5 n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{3 b^3 (b c-a d)^2 g^5}+\frac{\left (123904 B^2 d^5 n^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{3 b^3 (b c-a d)^2 g^5}+\frac{\left (7744 B^2 d^2 (b c-a d) n^2\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}{b^3 g^5}+\frac{\left (15488 B^2 d^2 (b c-a d) n^2\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}{b^3 g^5}-\frac{\left (61952 B^2 d^2 (b c-a d) n^2\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}{3 b^3 g^5}-\frac{\left (15488 B^2 d (b c-a d)^2 n^2\right ) \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}{3 b^3 g^5}+\frac{\left (123904 B^2 d (b c-a d)^2 n^2\right ) \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}{9 b^3 g^5}+\frac{\left (3872 B^2 (b c-a d)^3 n^2\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^5}-\frac{b d}{(b c-a d)^2 (a+b x)^4}+\frac{b d^2}{(b c-a d)^3 (a+b x)^3}-\frac{b d^3}{(b c-a d)^4 (a+b x)^2}+\frac{b d^4}{(b c-a d)^5 (a+b x)}-\frac{d^5}{(b c-a d)^5 (c+d x)}\right ) \, dx}{b^3 g^5}\\ &=-\frac{968 B^2 (b c-a d)^2 n^2}{b^3 g^5 (a+b x)^4}-\frac{42592 B^2 d (b c-a d) n^2}{27 b^3 g^5 (a+b x)^3}+\frac{9680 B^2 d^2 n^2}{9 b^3 g^5 (a+b x)^2}+\frac{27104 B^2 d^3 n^2}{9 b^3 (b c-a d) g^5 (a+b x)}+\frac{27104 B^2 d^4 n^2 \log (a+b x)}{9 b^3 (b c-a d)^2 g^5}-\frac{3872 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^5 (a+b x)^4}-\frac{77440 B d (b c-a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{9 b^3 g^5 (a+b x)^3}-\frac{7744 B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 g^5 (a+b x)^2}+\frac{15488 B d^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 (b c-a d) g^5 (a+b x)}+\frac{15488 B 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 )}{3 b^3 (b c-a d)^2 g^5}-\frac{7744 (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^5 (a+b x)^4}-\frac{61952 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g^5 (a+b x)^3}-\frac{15488 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^2}-\frac{27104 B^2 d^4 n^2 \log (c+d x)}{9 b^3 (b c-a d)^2 g^5}+\frac{15488 B^2 d^4 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^3 (b c-a d)^2 g^5}-\frac{15488 B d^4 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b^3 (b c-a d)^2 g^5}+\frac{15488 B^2 d^4 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{3 b^3 (b c-a d)^2 g^5}-\frac{\left (15488 B^2 d^4 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^3 (b c-a d)^2 g^5}-\frac{\left (15488 B^2 d^4 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{b^3 (b c-a d)^2 g^5}-\frac{\left (30976 B^2 d^4 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^3 (b c-a d)^2 g^5}-\frac{\left (30976 B^2 d^4 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{b^3 (b c-a d)^2 g^5}+\frac{\left (123904 B^2 d^4 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{3 b^3 (b c-a d)^2 g^5}+\frac{\left (123904 B^2 d^4 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{3 b^3 (b c-a d)^2 g^5}-\frac{\left (15488 B^2 d^4 n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}-\frac{\left (30976 B^2 d^4 n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}+\frac{\left (123904 B^2 d^4 n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{3 b^2 (b c-a d)^2 g^5}-\frac{\left (15488 B^2 d^5 n^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}-\frac{\left (30976 B^2 d^5 n^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}+\frac{\left (123904 B^2 d^5 n^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{3 b^3 (b c-a d)^2 g^5}\\ &=-\frac{968 B^2 (b c-a d)^2 n^2}{b^3 g^5 (a+b x)^4}-\frac{42592 B^2 d (b c-a d) n^2}{27 b^3 g^5 (a+b x)^3}+\frac{9680 B^2 d^2 n^2}{9 b^3 g^5 (a+b x)^2}+\frac{27104 B^2 d^3 n^2}{9 b^3 (b c-a d) g^5 (a+b x)}+\frac{27104 B^2 d^4 n^2 \log (a+b x)}{9 b^3 (b c-a d)^2 g^5}-\frac{7744 B^2 d^4 n^2 \log ^2(a+b x)}{3 b^3 (b c-a d)^2 g^5}-\frac{3872 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^5 (a+b x)^4}-\frac{77440 B d (b c-a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{9 b^3 g^5 (a+b x)^3}-\frac{7744 B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 g^5 (a+b x)^2}+\frac{15488 B d^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 (b c-a d) g^5 (a+b x)}+\frac{15488 B 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 )}{3 b^3 (b c-a d)^2 g^5}-\frac{7744 (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^5 (a+b x)^4}-\frac{61952 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g^5 (a+b x)^3}-\frac{15488 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^2}-\frac{27104 B^2 d^4 n^2 \log (c+d x)}{9 b^3 (b c-a d)^2 g^5}+\frac{15488 B^2 d^4 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^3 (b c-a d)^2 g^5}-\frac{15488 B d^4 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b^3 (b c-a d)^2 g^5}-\frac{7744 B^2 d^4 n^2 \log ^2(c+d x)}{3 b^3 (b c-a d)^2 g^5}+\frac{15488 B^2 d^4 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{3 b^3 (b c-a d)^2 g^5}-\frac{\left (15488 B^2 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 )}{b^3 (b c-a d)^2 g^5}-\frac{\left (15488 B^2 d^4 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b^3 (b c-a d)^2 g^5}-\frac{\left (30976 B^2 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 )}{b^3 (b c-a d)^2 g^5}-\frac{\left (30976 B^2 d^4 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b^3 (b c-a d)^2 g^5}+\frac{\left (123904 B^2 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 )}{3 b^3 (b c-a d)^2 g^5}+\frac{\left (123904 B^2 d^4 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{3 b^3 (b c-a d)^2 g^5}\\ &=-\frac{968 B^2 (b c-a d)^2 n^2}{b^3 g^5 (a+b x)^4}-\frac{42592 B^2 d (b c-a d) n^2}{27 b^3 g^5 (a+b x)^3}+\frac{9680 B^2 d^2 n^2}{9 b^3 g^5 (a+b x)^2}+\frac{27104 B^2 d^3 n^2}{9 b^3 (b c-a d) g^5 (a+b x)}+\frac{27104 B^2 d^4 n^2 \log (a+b x)}{9 b^3 (b c-a d)^2 g^5}-\frac{7744 B^2 d^4 n^2 \log ^2(a+b x)}{3 b^3 (b c-a d)^2 g^5}-\frac{3872 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^5 (a+b x)^4}-\frac{77440 B d (b c-a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{9 b^3 g^5 (a+b x)^3}-\frac{7744 B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 g^5 (a+b x)^2}+\frac{15488 B d^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 (b c-a d) g^5 (a+b x)}+\frac{15488 B 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 )}{3 b^3 (b c-a d)^2 g^5}-\frac{7744 (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^5 (a+b x)^4}-\frac{61952 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g^5 (a+b x)^3}-\frac{15488 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^2}-\frac{27104 B^2 d^4 n^2 \log (c+d x)}{9 b^3 (b c-a d)^2 g^5}+\frac{15488 B^2 d^4 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^3 (b c-a d)^2 g^5}-\frac{15488 B d^4 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b^3 (b c-a d)^2 g^5}-\frac{7744 B^2 d^4 n^2 \log ^2(c+d x)}{3 b^3 (b c-a d)^2 g^5}+\frac{15488 B^2 d^4 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{3 b^3 (b c-a d)^2 g^5}+\frac{15488 B^2 d^4 n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{3 b^3 (b c-a d)^2 g^5}+\frac{15488 B^2 d^4 n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{3 b^3 (b c-a d)^2 g^5}\\ \end{align*}
Mathematica [C] time = 3.14642, size = 1860, normalized size = 5.83 \[ \text{result too large to display} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.703, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dix+ci \right ) ^{2}}{ \left ( bgx+ag \right ) ^{5}} \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.
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Maxima [B] time = 4.47387, size = 10917, normalized size = 34.22 \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 = 0.715539, size = 3484, normalized size = 10.92 \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{{\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 )}^{2}}{{\left (b g x + a g\right )}^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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