Optimal. Leaf size=293 \[ -\frac {b^2 i^2 (c+d x)^5 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 g^6 (a+b x)^5 (b c-a d)^3}-\frac {d^2 i^2 (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 g^6 (a+b x)^3 (b c-a d)^3}+\frac {b d i^2 (c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^6 (a+b x)^4 (b c-a d)^3}-\frac {b^2 B i^2 n (c+d x)^5}{25 g^6 (a+b x)^5 (b c-a d)^3}-\frac {B d^2 i^2 n (c+d x)^3}{9 g^6 (a+b x)^3 (b c-a d)^3}+\frac {b B d i^2 n (c+d x)^4}{8 g^6 (a+b x)^4 (b c-a d)^3} \]
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Rubi [A] time = 0.72, antiderivative size = 375, normalized size of antiderivative = 1.28, number of steps used = 14, number of rules used = 4, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.093, Rules used = {2528, 2525, 12, 44} \[ -\frac {d^2 i^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 b^3 g^6 (a+b x)^3}-\frac {d i^2 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 b^3 g^6 (a+b x)^4}-\frac {i^2 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 b^3 g^6 (a+b x)^5}-\frac {B d^4 i^2 n}{30 b^3 g^6 (a+b x) (b c-a d)^2}+\frac {B d^3 i^2 n}{60 b^3 g^6 (a+b x)^2 (b c-a d)}-\frac {B d^5 i^2 n \log (a+b x)}{30 b^3 g^6 (b c-a d)^3}+\frac {B d^5 i^2 n \log (c+d x)}{30 b^3 g^6 (b c-a d)^3}-\frac {3 B d i^2 n (b c-a d)}{40 b^3 g^6 (a+b x)^4}-\frac {B i^2 n (b c-a d)^2}{25 b^3 g^6 (a+b x)^5}-\frac {B d^2 i^2 n}{90 b^3 g^6 (a+b x)^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 44
Rule 2525
Rule 2528
Rubi steps
\begin {align*} \int \frac {(126 c+126 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)^6} \, dx &=\int \left (\frac {15876 (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^6 (a+b x)^6}+\frac {31752 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^6 (a+b x)^5}+\frac {15876 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^6 (a+b x)^4}\right ) \, dx\\ &=\frac {\left (15876 d^2\right ) \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^6}+\frac {(31752 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)^5} \, dx}{b^2 g^6}+\frac {\left (15876 (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)^6} \, dx}{b^2 g^6}\\ &=-\frac {15876 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 g^6 (a+b x)^5}-\frac {7938 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^6 (a+b x)^4}-\frac {5292 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^6 (a+b x)^3}+\frac {\left (5292 B d^2 n\right ) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{b^3 g^6}+\frac {(7938 B d (b c-a d) n) \int \frac {b c-a d}{(a+b x)^5 (c+d x)} \, dx}{b^3 g^6}+\frac {\left (15876 B (b c-a d)^2 n\right ) \int \frac {b c-a d}{(a+b x)^6 (c+d x)} \, dx}{5 b^3 g^6}\\ &=-\frac {15876 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 g^6 (a+b x)^5}-\frac {7938 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^6 (a+b x)^4}-\frac {5292 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^6 (a+b x)^3}+\frac {\left (5292 B d^2 (b c-a d) n\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{b^3 g^6}+\frac {\left (7938 B d (b c-a d)^2 n\right ) \int \frac {1}{(a+b x)^5 (c+d x)} \, dx}{b^3 g^6}+\frac {\left (15876 B (b c-a d)^3 n\right ) \int \frac {1}{(a+b x)^6 (c+d x)} \, dx}{5 b^3 g^6}\\ &=-\frac {15876 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 g^6 (a+b x)^5}-\frac {7938 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^6 (a+b x)^4}-\frac {5292 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^6 (a+b x)^3}+\frac {\left (5292 B d^2 (b c-a d) n\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}{b^3 g^6}+\frac {\left (7938 B d (b c-a d)^2 n\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^6}+\frac {\left (15876 B (b c-a d)^3 n\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^6}-\frac {b d}{(b c-a d)^2 (a+b x)^5}+\frac {b d^2}{(b c-a d)^3 (a+b x)^4}-\frac {b d^3}{(b c-a d)^4 (a+b x)^3}+\frac {b d^4}{(b c-a d)^5 (a+b x)^2}-\frac {b d^5}{(b c-a d)^6 (a+b x)}+\frac {d^6}{(b c-a d)^6 (c+d x)}\right ) \, dx}{5 b^3 g^6}\\ &=-\frac {15876 B (b c-a d)^2 n}{25 b^3 g^6 (a+b x)^5}-\frac {11907 B d (b c-a d) n}{10 b^3 g^6 (a+b x)^4}-\frac {882 B d^2 n}{5 b^3 g^6 (a+b x)^3}+\frac {1323 B d^3 n}{5 b^3 (b c-a d) g^6 (a+b x)^2}-\frac {2646 B d^4 n}{5 b^3 (b c-a d)^2 g^6 (a+b x)}-\frac {2646 B d^5 n \log (a+b x)}{5 b^3 (b c-a d)^3 g^6}-\frac {15876 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 g^6 (a+b x)^5}-\frac {7938 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^6 (a+b x)^4}-\frac {5292 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^6 (a+b x)^3}+\frac {2646 B d^5 n \log (c+d x)}{5 b^3 (b c-a d)^3 g^6}\\ \end {align*}
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Mathematica [A] time = 1.05, size = 357, normalized size = 1.22 \[ \frac {i^2 \left (-\frac {360 a^2 A d^2}{(a+b x)^5}-\frac {60 B \left (a^2 d^2+a b d (3 c+5 d x)+b^2 \left (6 c^2+15 c d x+10 d^2 x^2\right )\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^5}-\frac {72 a^2 B d^2 n}{(a+b x)^5}-\frac {360 A b^2 c^2}{(a+b x)^5}-\frac {900 A b c d}{(a+b x)^4}+\frac {720 a A b c d}{(a+b x)^5}-\frac {600 A d^2}{(a+b x)^3}+\frac {900 a A d^2}{(a+b x)^4}-\frac {72 b^2 B c^2 n}{(a+b x)^5}-\frac {60 B d^5 n \log (a+b x)}{(b c-a d)^3}+\frac {60 B d^5 n \log (c+d x)}{(b c-a d)^3}-\frac {60 B d^4 n}{(a+b x) (b c-a d)^2}+\frac {30 B d^3 n}{(a+b x)^2 (b c-a d)}-\frac {135 b B c d n}{(a+b x)^4}+\frac {144 a b B c d n}{(a+b x)^5}-\frac {20 B d^2 n}{(a+b x)^3}+\frac {135 a B d^2 n}{(a+b x)^4}\right )}{1800 b^3 g^6} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.00, size = 1087, normalized size = 3.71 \[ -\frac {60 \, {\left (B b^{5} c d^{4} - B a b^{4} d^{5}\right )} i^{2} n x^{4} - 30 \, {\left (B b^{5} c^{2} d^{3} - 10 \, B a b^{4} c d^{4} + 9 \, B a^{2} b^{3} d^{5}\right )} i^{2} n x^{3} + {\left (72 \, B b^{5} c^{5} - 225 \, B a b^{4} c^{4} d + 200 \, B a^{2} b^{3} c^{3} d^{2} - 47 \, B a^{5} d^{5}\right )} i^{2} n + 60 \, {\left (6 \, A b^{5} c^{5} - 15 \, A a b^{4} c^{4} d + 10 \, A a^{2} b^{3} c^{3} d^{2} - A a^{5} d^{5}\right )} i^{2} + 10 \, {\left ({\left (2 \, B b^{5} c^{3} d^{2} - 15 \, B a b^{4} c^{2} d^{3} + 60 \, B a^{2} b^{3} c d^{4} - 47 \, B a^{3} b^{2} d^{5}\right )} i^{2} n + 60 \, {\left (A b^{5} c^{3} d^{2} - 3 \, A a b^{4} c^{2} d^{3} + 3 \, A a^{2} b^{3} c d^{4} - A a^{3} b^{2} d^{5}\right )} i^{2}\right )} x^{2} + 5 \, {\left ({\left (27 \, B b^{5} c^{4} d - 100 \, B a b^{4} c^{3} d^{2} + 120 \, B a^{2} b^{3} c^{2} d^{3} - 47 \, B a^{4} b d^{5}\right )} i^{2} n + 60 \, {\left (3 \, A b^{5} c^{4} d - 8 \, A a b^{4} c^{3} d^{2} + 6 \, A a^{2} b^{3} c^{2} d^{3} - A a^{4} b d^{5}\right )} i^{2}\right )} x + 60 \, {\left (10 \, {\left (B b^{5} c^{3} d^{2} - 3 \, B a b^{4} c^{2} d^{3} + 3 \, B a^{2} b^{3} c d^{4} - B a^{3} b^{2} d^{5}\right )} i^{2} x^{2} + 5 \, {\left (3 \, B b^{5} c^{4} d - 8 \, B a b^{4} c^{3} d^{2} + 6 \, B a^{2} b^{3} c^{2} d^{3} - B a^{4} b d^{5}\right )} i^{2} x + {\left (6 \, B b^{5} c^{5} - 15 \, B a b^{4} c^{4} d + 10 \, B a^{2} b^{3} c^{3} d^{2} - B a^{5} d^{5}\right )} i^{2}\right )} \log \relax (e) + 60 \, {\left (B b^{5} d^{5} i^{2} n x^{5} + 5 \, B a b^{4} d^{5} i^{2} n x^{4} + 10 \, B a^{2} b^{3} d^{5} i^{2} n x^{3} + 10 \, {\left (B b^{5} c^{3} d^{2} - 3 \, B a b^{4} c^{2} d^{3} + 3 \, B a^{2} b^{3} c d^{4}\right )} i^{2} n x^{2} + 5 \, {\left (3 \, B b^{5} c^{4} d - 8 \, B a b^{4} c^{3} d^{2} + 6 \, B a^{2} b^{3} c^{2} d^{3}\right )} i^{2} n x + {\left (6 \, B b^{5} c^{5} - 15 \, B a b^{4} c^{4} d + 10 \, B a^{2} b^{3} c^{3} d^{2}\right )} i^{2} n\right )} \log \left (\frac {b x + a}{d x + c}\right )}{1800 \, {\left ({\left (b^{11} c^{3} - 3 \, a b^{10} c^{2} d + 3 \, a^{2} b^{9} c d^{2} - a^{3} b^{8} d^{3}\right )} g^{6} x^{5} + 5 \, {\left (a b^{10} c^{3} - 3 \, a^{2} b^{9} c^{2} d + 3 \, a^{3} b^{8} c d^{2} - a^{4} b^{7} d^{3}\right )} g^{6} x^{4} + 10 \, {\left (a^{2} b^{9} c^{3} - 3 \, a^{3} b^{8} c^{2} d + 3 \, a^{4} b^{7} c d^{2} - a^{5} b^{6} d^{3}\right )} g^{6} x^{3} + 10 \, {\left (a^{3} b^{8} c^{3} - 3 \, a^{4} b^{7} c^{2} d + 3 \, a^{5} b^{6} c d^{2} - a^{6} b^{5} d^{3}\right )} g^{6} x^{2} + 5 \, {\left (a^{4} b^{7} c^{3} - 3 \, a^{5} b^{6} c^{2} d + 3 \, a^{6} b^{5} c d^{2} - a^{7} b^{4} d^{3}\right )} g^{6} x + {\left (a^{5} b^{6} c^{3} - 3 \, a^{6} b^{5} c^{2} d + 3 \, a^{7} b^{4} c d^{2} - a^{8} b^{3} d^{3}\right )} g^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 110.70, size = 376, normalized size = 1.28 \[ \frac {1}{1800} \, {\left (\frac {60 \, {\left (6 \, B b^{2} n - \frac {15 \, {\left (b x + a\right )} B b d n}{d x + c} + \frac {10 \, {\left (b x + a\right )}^{2} B d^{2} n}{{\left (d x + c\right )}^{2}}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{\frac {{\left (b x + a\right )}^{5} b^{2} c^{2} g^{6}}{{\left (d x + c\right )}^{5}} - \frac {2 \, {\left (b x + a\right )}^{5} a b c d g^{6}}{{\left (d x + c\right )}^{5}} + \frac {{\left (b x + a\right )}^{5} a^{2} d^{2} g^{6}}{{\left (d x + c\right )}^{5}}} + \frac {72 \, B b^{2} n - \frac {225 \, {\left (b x + a\right )} B b d n}{d x + c} + \frac {200 \, {\left (b x + a\right )}^{2} B d^{2} n}{{\left (d x + c\right )}^{2}} + 360 \, A b^{2} + 360 \, B b^{2} - \frac {900 \, {\left (b x + a\right )} A b d}{d x + c} - \frac {900 \, {\left (b x + a\right )} B b d}{d x + c} + \frac {600 \, {\left (b x + a\right )}^{2} A d^{2}}{{\left (d x + c\right )}^{2}} + \frac {600 \, {\left (b x + a\right )}^{2} B d^{2}}{{\left (d x + c\right )}^{2}}}{\frac {{\left (b x + a\right )}^{5} b^{2} c^{2} g^{6}}{{\left (d x + c\right )}^{5}} - \frac {2 \, {\left (b x + a\right )}^{5} a b c d g^{6}}{{\left (d x + c\right )}^{5}} + \frac {{\left (b x + a\right )}^{5} a^{2} d^{2} g^{6}}{{\left (d x + c\right )}^{5}}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.45, size = 0, normalized size = 0.00 \[ \int \frac {\left (d i x +c i \right )^{2} \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )}{\left (b g x +a g \right )^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 3.18, size = 3058, normalized size = 10.44 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.71, size = 954, normalized size = 3.26 \[ \frac {B\,d^5\,i^2\,n\,\mathrm {atanh}\left (\frac {30\,a^3\,b^3\,d^3\,g^6-30\,a^2\,b^4\,c\,d^2\,g^6-30\,a\,b^5\,c^2\,d\,g^6+30\,b^6\,c^3\,g^6}{30\,b^3\,g^6\,{\left (a\,d-b\,c\right )}^3}+\frac {2\,b\,d\,x\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^3}\right )}{15\,b^3\,g^6\,{\left (a\,d-b\,c\right )}^3}-\frac {\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (a\,\left (\frac {B\,a\,d^2\,i^2}{30\,b^3}+\frac {B\,c\,d\,i^2}{10\,b^2}\right )+x\,\left (b\,\left (\frac {B\,a\,d^2\,i^2}{30\,b^3}+\frac {B\,c\,d\,i^2}{10\,b^2}\right )+\frac {2\,B\,a\,d^2\,i^2}{15\,b^2}+\frac {2\,B\,c\,d\,i^2}{5\,b}\right )+\frac {B\,c^2\,i^2}{5\,b}+\frac {B\,d^2\,i^2\,x^2}{3\,b}\right )}{a^5\,g^6+5\,a^4\,b\,g^6\,x+10\,a^3\,b^2\,g^6\,x^2+10\,a^2\,b^3\,g^6\,x^3+5\,a\,b^4\,g^6\,x^4+b^5\,g^6\,x^5}-\frac {\frac {60\,A\,a^4\,d^4\,i^2+360\,A\,b^4\,c^4\,i^2+47\,B\,a^4\,d^4\,i^2\,n+72\,B\,b^4\,c^4\,i^2\,n+60\,A\,a^2\,b^2\,c^2\,d^2\,i^2-540\,A\,a\,b^3\,c^3\,d\,i^2+60\,A\,a^3\,b\,c\,d^3\,i^2-153\,B\,a\,b^3\,c^3\,d\,i^2\,n+47\,B\,a^3\,b\,c\,d^3\,i^2\,n+47\,B\,a^2\,b^2\,c^2\,d^2\,i^2\,n}{60\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {x^2\,\left (60\,A\,a^2\,b^2\,d^4\,i^2+60\,A\,b^4\,c^2\,d^2\,i^2+47\,B\,a^2\,b^2\,d^4\,i^2\,n+2\,B\,b^4\,c^2\,d^2\,i^2\,n-120\,A\,a\,b^3\,c\,d^3\,i^2-13\,B\,a\,b^3\,c\,d^3\,i^2\,n\right )}{6\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {x\,\left (60\,A\,a^3\,b\,d^4\,i^2+180\,A\,b^4\,c^3\,d\,i^2-300\,A\,a\,b^3\,c^2\,d^2\,i^2+60\,A\,a^2\,b^2\,c\,d^3\,i^2+47\,B\,a^3\,b\,d^4\,i^2\,n+27\,B\,b^4\,c^3\,d\,i^2\,n-73\,B\,a\,b^3\,c^2\,d^2\,i^2\,n+47\,B\,a^2\,b^2\,c\,d^3\,i^2\,n\right )}{12\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {d\,x^3\,\left (9\,B\,a\,b^3\,d^3\,i^2\,n-B\,b^4\,c\,d^2\,i^2\,n\right )}{2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {B\,b^4\,d^4\,i^2\,n\,x^4}{a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}}{30\,a^5\,b^3\,g^6+150\,a^4\,b^4\,g^6\,x+300\,a^3\,b^5\,g^6\,x^2+300\,a^2\,b^6\,g^6\,x^3+150\,a\,b^7\,g^6\,x^4+30\,b^8\,g^6\,x^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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