Optimal. Leaf size=183 \[ -\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 b g^4 (a+b x)^3}-\frac {B d^3 n \log (a+b x)}{3 b g^4 (b c-a d)^3}+\frac {B d^3 n \log (c+d x)}{3 b g^4 (b c-a d)^3}-\frac {B d^2 n}{3 b g^4 (a+b x) (b c-a d)^2}+\frac {B d n}{6 b g^4 (a+b x)^2 (b c-a d)}-\frac {B n}{9 b g^4 (a+b x)^3} \]
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Rubi [A] time = 0.15, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2525, 12, 44} \[ -\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 b g^4 (a+b x)^3}-\frac {B d^2 n}{3 b g^4 (a+b x) (b c-a d)^2}-\frac {B d^3 n \log (a+b x)}{3 b g^4 (b c-a d)^3}+\frac {B d^3 n \log (c+d x)}{3 b g^4 (b c-a d)^3}+\frac {B d n}{6 b g^4 (a+b x)^2 (b c-a d)}-\frac {B n}{9 b g^4 (a+b x)^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 44
Rule 2525
Rubi steps
\begin {align*} \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^4} \, dx &=-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 b g^4 (a+b x)^3}+\frac {(B n) \int \frac {b c-a d}{g^3 (a+b x)^4 (c+d x)} \, dx}{3 b g}\\ &=-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 b g^4 (a+b x)^3}+\frac {(B (b c-a d) n) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{3 b g^4}\\ &=-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 b g^4 (a+b x)^3}+\frac {(B (b c-a d) 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}{3 b g^4}\\ &=-\frac {B n}{9 b g^4 (a+b x)^3}+\frac {B d n}{6 b (b c-a d) g^4 (a+b x)^2}-\frac {B d^2 n}{3 b (b c-a d)^2 g^4 (a+b x)}-\frac {B d^3 n \log (a+b x)}{3 b (b c-a d)^3 g^4}-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 b g^4 (a+b x)^3}+\frac {B d^3 n \log (c+d x)}{3 b (b c-a d)^3 g^4}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 145, normalized size = 0.79 \[ -\frac {\frac {B n \left ((b c-a d) \left (11 a^2 d^2+a b d (15 d x-7 c)+b^2 \left (2 c^2-3 c d x+6 d^2 x^2\right )\right )-6 d^3 (a+b x)^3 \log (c+d x)+6 d^3 (a+b x)^3 \log (a+b x)\right )}{(b c-a d)^3}+6 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{18 b g^4 (a+b x)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.94, size = 482, normalized size = 2.63 \[ -\frac {6 \, A b^{3} c^{3} - 18 \, A a b^{2} c^{2} d + 18 \, A a^{2} b c d^{2} - 6 \, A a^{3} d^{3} + 6 \, {\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} n x^{2} - 3 \, {\left (B b^{3} c^{2} d - 6 \, B a b^{2} c d^{2} + 5 \, B a^{2} b d^{3}\right )} n x + {\left (2 \, B b^{3} c^{3} - 9 \, B a b^{2} c^{2} d + 18 \, B a^{2} b c d^{2} - 11 \, B a^{3} d^{3}\right )} n + 6 \, {\left (B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d + 3 \, B a^{2} b c d^{2} - B a^{3} d^{3}\right )} \log \relax (e) + 6 \, {\left (B b^{3} d^{3} n x^{3} + 3 \, B a b^{2} d^{3} n x^{2} + 3 \, B a^{2} b d^{3} n x + {\left (B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d + 3 \, B a^{2} b c d^{2}\right )} n\right )} \log \left (\frac {b x + a}{d x + c}\right )}{18 \, {\left ({\left (b^{7} c^{3} - 3 \, a b^{6} c^{2} d + 3 \, a^{2} b^{5} c d^{2} - a^{3} b^{4} d^{3}\right )} g^{4} x^{3} + 3 \, {\left (a b^{6} c^{3} - 3 \, a^{2} b^{5} c^{2} d + 3 \, a^{3} b^{4} c d^{2} - a^{4} b^{3} d^{3}\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{5} c^{3} - 3 \, a^{3} b^{4} c^{2} d + 3 \, a^{4} b^{3} c d^{2} - a^{5} b^{2} d^{3}\right )} g^{4} x + {\left (a^{3} b^{4} c^{3} - 3 \, a^{4} b^{3} c^{2} d + 3 \, a^{5} b^{2} c d^{2} - a^{6} b d^{3}\right )} g^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 5.22, size = 375, normalized size = 2.05 \[ -\frac {1}{18} \, {\left (\frac {6 \, {\left (B b^{2} n - \frac {3 \, {\left (b x + a\right )} B b d n}{d x + c} + \frac {3 \, {\left (b x + a\right )}^{2} B d^{2} n}{{\left (d x + c\right )}^{2}}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{\frac {{\left (b x + a\right )}^{3} b^{2} c^{2} g^{4}}{{\left (d x + c\right )}^{3}} - \frac {2 \, {\left (b x + a\right )}^{3} a b c d g^{4}}{{\left (d x + c\right )}^{3}} + \frac {{\left (b x + a\right )}^{3} a^{2} d^{2} g^{4}}{{\left (d x + c\right )}^{3}}} + \frac {2 \, B b^{2} n - \frac {9 \, {\left (b x + a\right )} B b d n}{d x + c} + \frac {18 \, {\left (b x + a\right )}^{2} B d^{2} n}{{\left (d x + c\right )}^{2}} + 6 \, A b^{2} + 6 \, B b^{2} - \frac {18 \, {\left (b x + a\right )} A b d}{d x + c} - \frac {18 \, {\left (b x + a\right )} B b d}{d x + c} + \frac {18 \, {\left (b x + a\right )}^{2} A d^{2}}{{\left (d x + c\right )}^{2}} + \frac {18 \, {\left (b x + a\right )}^{2} B d^{2}}{{\left (d x + c\right )}^{2}}}{\frac {{\left (b x + a\right )}^{3} b^{2} c^{2} g^{4}}{{\left (d x + c\right )}^{3}} - \frac {2 \, {\left (b x + a\right )}^{3} a b c d g^{4}}{{\left (d x + c\right )}^{3}} + \frac {{\left (b x + a\right )}^{3} a^{2} d^{2} g^{4}}{{\left (d x + c\right )}^{3}}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.29, size = 0, normalized size = 0.00 \[ \int \frac {B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A}{\left (b g x +a g \right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.35, size = 432, normalized size = 2.36 \[ -\frac {1}{18} \, B n {\left (\frac {6 \, b^{2} d^{2} x^{2} + 2 \, b^{2} c^{2} - 7 \, a b c d + 11 \, a^{2} d^{2} - 3 \, {\left (b^{2} c d - 5 \, a b d^{2}\right )} x}{{\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} g^{4} x^{3} + 3 \, {\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2}\right )} g^{4} x + {\left (a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2}\right )} g^{4}} + \frac {6 \, d^{3} \log \left (b x + a\right )}{{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} g^{4}} - \frac {6 \, d^{3} \log \left (d x + c\right )}{{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} g^{4}}\right )} - \frac {B \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right )}{3 \, {\left (b^{4} g^{4} x^{3} + 3 \, a b^{3} g^{4} x^{2} + 3 \, a^{2} b^{2} g^{4} x + a^{3} b g^{4}\right )}} - \frac {A}{3 \, {\left (b^{4} g^{4} x^{3} + 3 \, a b^{3} g^{4} x^{2} + 3 \, a^{2} b^{2} g^{4} x + a^{3} b g^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.85, size = 349, normalized size = 1.91 \[ \frac {2\,A\,a\,c\,d}{3\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {A\,b\,c^2}{3\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {A\,a^2\,d^2}{3\,b\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {B\,b\,c^2\,n}{9\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{3\,b\,g^4\,{\left (a+b\,x\right )}^3}-\frac {B\,b\,d^2\,n\,x^2}{3\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}+\frac {7\,B\,a\,c\,d\,n}{18\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {11\,B\,a^2\,d^2\,n}{18\,b\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {5\,B\,a\,d^2\,n\,x}{6\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}+\frac {B\,b\,c\,d\,n\,x}{6\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {B\,d^3\,n\,\mathrm {atan}\left (\frac {a\,d\,1{}\mathrm {i}+b\,c\,1{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}\right )\,2{}\mathrm {i}}{3\,b\,g^4\,{\left (a\,d-b\,c\right )}^3} \]
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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