Optimal. Leaf size=361 \[ \frac{24 B^3 n^3 \text{PolyLog}\left (4,\frac{(c+d x) (b f-a g)}{(a+b x) (d f-c g)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{h (b f-a g)}+\frac{12 B^2 n^2 \text{PolyLog}\left (3,\frac{(c+d x) (b f-a g)}{(a+b x) (d f-c g)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{h (b f-a g)}+\frac{4 B n \text{PolyLog}\left (2,\frac{(c+d x) (b f-a g)}{(a+b x) (d f-c g)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{h (b f-a g)}+\frac{24 B^4 n^4 \text{PolyLog}\left (5,\frac{(c+d x) (b f-a g)}{(a+b x) (d f-c g)}\right )}{h (b f-a g)}-\frac{\log \left (1-\frac{(c+d x) (b f-a g)}{(a+b x) (d f-c g)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^4}{h (b f-a g)} \]
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Rubi [B] time = 1.92566, antiderivative size = 1021, normalized size of antiderivative = 2.83, number of steps used = 20, number of rules used = 9, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.209, Rules used = {6742, 36, 31, 2503, 2502, 2315, 2506, 6610, 2508} \[ \frac{\log (a+b x) A^4}{(b f-a g) h}-\frac{\log (f+g x) A^4}{(b f-a g) h}-\frac{4 B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right ) A^3}{(b f-a g) h}+\frac{4 B n \text{PolyLog}\left (2,\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}+1\right ) A^3}{(b f-a g) h}-\frac{6 B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right ) A^2}{(b f-a g) h}+\frac{12 B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{PolyLog}\left (2,\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}+1\right ) A^2}{(b f-a g) h}+\frac{12 B^2 n^2 \text{PolyLog}\left (3,\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}+1\right ) A^2}{(b f-a g) h}-\frac{4 B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right ) A}{(b f-a g) h}+\frac{12 B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \text{PolyLog}\left (2,\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}+1\right ) A}{(b f-a g) h}+\frac{24 B^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{PolyLog}\left (3,\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}+1\right ) A}{(b f-a g) h}+\frac{24 B^3 n^3 \text{PolyLog}\left (4,\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}+1\right ) A}{(b f-a g) h}-\frac{B^4 \log ^4\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac{4 B^4 n \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right ) \text{PolyLog}\left (2,\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}+1\right )}{(b f-a g) h}+\frac{12 B^4 n^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \text{PolyLog}\left (3,\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}+1\right )}{(b f-a g) h}+\frac{24 B^4 n^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{PolyLog}\left (4,\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}+1\right )}{(b f-a g) h}+\frac{24 B^4 n^4 \text{PolyLog}\left (5,\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}+1\right )}{(b f-a g) h} \]
Antiderivative was successfully verified.
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Rule 6742
Rule 36
Rule 31
Rule 2503
Rule 2502
Rule 2315
Rule 2506
Rule 6610
Rule 2508
Rubi steps
\begin{align*} \int \frac{\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^4}{(f+g x) (a h+b h x)} \, dx &=\int \left (\frac{A^4}{h (a+b x) (f+g x)}+\frac{4 A^3 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h (a+b x) (f+g x)}+\frac{6 A^2 B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{h (a+b x) (f+g x)}+\frac{4 A B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{h (a+b x) (f+g x)}+\frac{B^4 \log ^4\left (e (a+b x)^n (c+d x)^{-n}\right )}{h (a+b x) (f+g x)}\right ) \, dx\\ &=\frac{A^4 \int \frac{1}{(a+b x) (f+g x)} \, dx}{h}+\frac{\left (4 A^3 B\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (f+g x)} \, dx}{h}+\frac{\left (6 A^2 B^2\right ) \int \frac{\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (f+g x)} \, dx}{h}+\frac{\left (4 A B^3\right ) \int \frac{\log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (f+g x)} \, dx}{h}+\frac{B^4 \int \frac{\log ^4\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (f+g x)} \, dx}{h}\\ &=-\frac{4 A^3 B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac{6 A^2 B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac{4 A B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac{B^4 \log ^4\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac{\left (A^4 b\right ) \int \frac{1}{a+b x} \, dx}{(b f-a g) h}-\frac{\left (A^4 g\right ) \int \frac{1}{f+g x} \, dx}{(b f-a g) h}+\frac{\left (4 A^3 B (b c-a d) n\right ) \int \frac{\log \left (-\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{(b f-a g) h}+\frac{\left (12 A^2 B^2 (b c-a d) n\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{(b f-a g) h}+\frac{\left (12 A B^3 (b c-a d) n\right ) \int \frac{\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{(b f-a g) h}+\frac{\left (4 B^4 (b c-a d) n\right ) \int \frac{\log ^3\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{(b f-a g) h}\\ &=\frac{A^4 \log (a+b x)}{(b f-a g) h}-\frac{A^4 \log (f+g x)}{(b f-a g) h}-\frac{4 A^3 B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac{6 A^2 B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac{4 A B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac{B^4 \log ^4\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac{12 A^2 B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1+\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac{12 A B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1+\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac{4 B^4 n \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1+\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac{\left (4 A^3 B (b c-a d) n\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{(b c-a d) x}{d f-c g}\right )}{1+\frac{(b c-a d) x}{d f-c g}} \, dx,x,\frac{f+g x}{a+b x}\right )}{(b f-a g) (d f-c g) h}-\frac{\left (12 A^2 B^2 (b c-a d) n^2\right ) \int \frac{\text{Li}_2\left (1+\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{(b f-a g) h}-\frac{\left (24 A B^3 (b c-a d) n^2\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1+\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{(b f-a g) h}-\frac{\left (12 B^4 (b c-a d) n^2\right ) \int \frac{\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1+\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{(b f-a g) h}\\ &=\frac{A^4 \log (a+b x)}{(b f-a g) h}-\frac{A^4 \log (f+g x)}{(b f-a g) h}-\frac{4 A^3 B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac{6 A^2 B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac{4 A B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac{B^4 \log ^4\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac{4 A^3 B n \text{Li}_2\left (1+\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac{12 A^2 B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1+\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac{12 A B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1+\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac{4 B^4 n \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1+\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac{12 A^2 B^2 n^2 \text{Li}_3\left (1+\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac{24 A B^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_3\left (1+\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac{12 B^4 n^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_3\left (1+\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac{\left (24 A B^3 (b c-a d) n^3\right ) \int \frac{\text{Li}_3\left (1+\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{(b f-a g) h}-\frac{\left (24 B^4 (b c-a d) n^3\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_3\left (1+\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{(b f-a g) h}\\ &=\frac{A^4 \log (a+b x)}{(b f-a g) h}-\frac{A^4 \log (f+g x)}{(b f-a g) h}-\frac{4 A^3 B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac{6 A^2 B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac{4 A B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac{B^4 \log ^4\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac{4 A^3 B n \text{Li}_2\left (1+\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac{12 A^2 B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1+\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac{12 A B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1+\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac{4 B^4 n \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1+\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac{12 A^2 B^2 n^2 \text{Li}_3\left (1+\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac{24 A B^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_3\left (1+\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac{12 B^4 n^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_3\left (1+\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac{24 A B^3 n^3 \text{Li}_4\left (1+\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac{24 B^4 n^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_4\left (1+\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac{\left (24 B^4 (b c-a d) n^4\right ) \int \frac{\text{Li}_4\left (1+\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{(b f-a g) h}\\ &=\frac{A^4 \log (a+b x)}{(b f-a g) h}-\frac{A^4 \log (f+g x)}{(b f-a g) h}-\frac{4 A^3 B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac{6 A^2 B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac{4 A B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac{B^4 \log ^4\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac{4 A^3 B n \text{Li}_2\left (1+\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac{12 A^2 B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1+\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac{12 A B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1+\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac{4 B^4 n \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1+\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac{12 A^2 B^2 n^2 \text{Li}_3\left (1+\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac{24 A B^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_3\left (1+\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac{12 B^4 n^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_3\left (1+\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac{24 A B^3 n^3 \text{Li}_4\left (1+\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac{24 B^4 n^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_4\left (1+\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac{24 B^4 n^4 \text{Li}_5\left (1+\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}\\ \end{align*}
Mathematica [F] time = 4.20214, size = 0, normalized size = 0. \[ \int \frac{\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^4}{(f+g x) (a h+b h x)} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 5.059, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( gx+f \right ) \left ( bhx+ah \right ) } \left ( A+B\ln \left ({\frac{e \left ( bx+a \right ) ^{n}}{ \left ( dx+c \right ) ^{n}}} \right ) \right ) ^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} A^{4}{\left (\frac{\log \left (b x + a\right )}{{\left (b f - a g\right )} h} - \frac{\log \left (g x + f\right )}{{\left (b f - a g\right )} h}\right )} + \int \frac{B^{4} \log \left ({\left (b x + a\right )}^{n}\right )^{4} + B^{4} \log \left ({\left (d x + c\right )}^{n}\right )^{4} + B^{4} \log \left (e\right )^{4} + 4 \, A B^{3} \log \left (e\right )^{3} + 6 \, A^{2} B^{2} \log \left (e\right )^{2} + 4 \, A^{3} B \log \left (e\right ) + 4 \,{\left (B^{4} \log \left (e\right ) + A B^{3}\right )} \log \left ({\left (b x + a\right )}^{n}\right )^{3} - 4 \,{\left (B^{4} \log \left ({\left (b x + a\right )}^{n}\right ) + B^{4} \log \left (e\right ) + A B^{3}\right )} \log \left ({\left (d x + c\right )}^{n}\right )^{3} + 6 \,{\left (B^{4} \log \left (e\right )^{2} + 2 \, A B^{3} \log \left (e\right ) + A^{2} B^{2}\right )} \log \left ({\left (b x + a\right )}^{n}\right )^{2} + 6 \,{\left (B^{4} \log \left ({\left (b x + a\right )}^{n}\right )^{2} + B^{4} \log \left (e\right )^{2} + 2 \, A B^{3} \log \left (e\right ) + A^{2} B^{2} + 2 \,{\left (B^{4} \log \left (e\right ) + A B^{3}\right )} \log \left ({\left (b x + a\right )}^{n}\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )^{2} + 4 \,{\left (B^{4} \log \left (e\right )^{3} + 3 \, A B^{3} \log \left (e\right )^{2} + 3 \, A^{2} B^{2} \log \left (e\right ) + A^{3} B\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - 4 \,{\left (B^{4} \log \left ({\left (b x + a\right )}^{n}\right )^{3} + B^{4} \log \left (e\right )^{3} + 3 \, A B^{3} \log \left (e\right )^{2} + 3 \, A^{2} B^{2} \log \left (e\right ) + A^{3} B + 3 \,{\left (B^{4} \log \left (e\right ) + A B^{3}\right )} \log \left ({\left (b x + a\right )}^{n}\right )^{2} + 3 \,{\left (B^{4} \log \left (e\right )^{2} + 2 \, A B^{3} \log \left (e\right ) + A^{2} B^{2}\right )} \log \left ({\left (b x + a\right )}^{n}\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{b g h x^{2} + a f h +{\left (b f h + a g h\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{B^{4} \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{4} + 4 \, A B^{3} \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{3} + 6 \, A^{2} B^{2} \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{2} + 4 \, A^{3} B \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A^{4}}{b g h x^{2} + a f h +{\left (b f + a g\right )} h x}, x\right ) \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 (B \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{4}}{{\left (b h x + a h\right )}{\left (g x + f\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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