Optimal. Leaf size=89 \[ -\frac {i (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^3 (a+b x)^2 (b c-a d)}-\frac {B i n (c+d x)^2}{4 g^3 (a+b x)^2 (b c-a d)} \]
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Rubi [B] time = 0.29, antiderivative size = 201, normalized size of antiderivative = 2.26, number of steps used = 10, number of rules used = 4, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {2528, 2525, 12, 44} \[ -\frac {d i \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 g^3 (a+b x)}-\frac {i (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 b^2 g^3 (a+b x)^2}-\frac {B d^2 i n \log (a+b x)}{2 b^2 g^3 (b c-a d)}+\frac {B d^2 i n \log (c+d x)}{2 b^2 g^3 (b c-a d)}-\frac {B i n (b c-a d)}{4 b^2 g^3 (a+b x)^2}-\frac {B d i n}{2 b^2 g^3 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 44
Rule 2525
Rule 2528
Rubi steps
\begin {align*} \int \frac {(114 c+114 d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^3} \, dx &=\int \left (\frac {114 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b g^3 (a+b x)^3}+\frac {114 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b g^3 (a+b x)^2}\right ) \, dx\\ &=\frac {(114 d) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{b g^3}+\frac {(114 (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)^3} \, dx}{b g^3}\\ &=-\frac {57 (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^3 (a+b x)^2}-\frac {114 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^3 (a+b x)}+\frac {(114 B d n) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^3}+\frac {(57 B (b c-a d) n) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^2 g^3}\\ &=-\frac {57 (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^3 (a+b x)^2}-\frac {114 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^3 (a+b x)}+\frac {(114 B d (b c-a d) n) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^3}+\frac {\left (57 B (b c-a d)^2 n\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{b^2 g^3}\\ &=-\frac {57 (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^3 (a+b x)^2}-\frac {114 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^3 (a+b x)}+\frac {(114 B d (b c-a d) n) \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^2 g^3}+\frac {\left (57 B (b c-a d)^2 n\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^2 g^3}\\ &=-\frac {57 B (b c-a d) n}{2 b^2 g^3 (a+b x)^2}-\frac {57 B d n}{b^2 g^3 (a+b x)}-\frac {57 B d^2 n \log (a+b x)}{b^2 (b c-a d) g^3}-\frac {57 (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^3 (a+b x)^2}-\frac {114 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^3 (a+b x)}+\frac {57 B d^2 n \log (c+d x)}{b^2 (b c-a d) g^3}\\ \end {align*}
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Mathematica [B] time = 0.17, size = 216, normalized size = 2.43 \[ \frac {i \left (-\frac {d \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 (a+b x)}-\frac {(b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 b^2 (a+b x)^2}-\frac {B n \left (-\frac {2 d^2 \log (a+b x)}{b c-a d}+\frac {2 d^2 \log (c+d x)}{b c-a d}+\frac {b c-a d}{(a+b x)^2}-\frac {2 d}{a+b x}\right )}{4 b^2}-\frac {B d n \left (\frac {d \log (a+b x)}{b c-a d}-\frac {d \log (c+d x)}{b c-a d}+\frac {1}{a+b x}\right )}{b^2}\right )}{g^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.86, size = 250, normalized size = 2.81 \[ -\frac {{\left (B b^{2} c^{2} - B a^{2} d^{2}\right )} i n + 2 \, {\left (A b^{2} c^{2} - A a^{2} d^{2}\right )} i + 2 \, {\left ({\left (B b^{2} c d - B a b d^{2}\right )} i n + 2 \, {\left (A b^{2} c d - A a b d^{2}\right )} i\right )} x + 2 \, {\left (2 \, {\left (B b^{2} c d - B a b d^{2}\right )} i x + {\left (B b^{2} c^{2} - B a^{2} d^{2}\right )} i\right )} \log \relax (e) + 2 \, {\left (B b^{2} d^{2} i n x^{2} + 2 \, B b^{2} c d i n x + B b^{2} c^{2} i n\right )} \log \left (\frac {b x + a}{d x + c}\right )}{4 \, {\left ({\left (b^{5} c - a b^{4} d\right )} g^{3} x^{2} + 2 \, {\left (a b^{4} c - a^{2} b^{3} d\right )} g^{3} x + {\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} g^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 7.18, size = 98, normalized size = 1.10 \[ -\frac {1}{4} \, {\left (\frac {2 \, {\left (d x + c\right )}^{2} B i n \log \left (\frac {b x + a}{d x + c}\right )}{{\left (b x + a\right )}^{2} g^{3}} + \frac {{\left (B i n + 2 \, A i + 2 \, B i\right )} {\left (d x + c\right )}^{2}}{{\left (b x + a\right )}^{2} g^{3}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.31, size = 0, normalized size = 0.00 \[ \int \frac {\left (d i x +c i \right ) \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )}{\left (b g x +a g \right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.25, size = 582, normalized size = 6.54 \[ -\frac {1}{4} \, B d i n {\left (\frac {3 \, a b c - a^{2} d + 2 \, {\left (2 \, b^{2} c - a b d\right )} x}{{\left (b^{5} c - a b^{4} d\right )} g^{3} x^{2} + 2 \, {\left (a b^{4} c - a^{2} b^{3} d\right )} g^{3} x + {\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} g^{3}} + \frac {2 \, {\left (2 \, b c d - a d^{2}\right )} \log \left (b x + a\right )}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} g^{3}} - \frac {2 \, {\left (2 \, b c d - a d^{2}\right )} \log \left (d x + c\right )}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} g^{3}}\right )} + \frac {1}{4} \, B c i n {\left (\frac {2 \, b d x - b c + 3 \, a d}{{\left (b^{4} c - a b^{3} d\right )} g^{3} x^{2} + 2 \, {\left (a b^{3} c - a^{2} b^{2} d\right )} g^{3} x + {\left (a^{2} b^{2} c - a^{3} b d\right )} g^{3}} + \frac {2 \, d^{2} \log \left (b x + a\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}} - \frac {2 \, d^{2} \log \left (d x + c\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}}\right )} - \frac {{\left (2 \, b x + a\right )} B d i \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right )}{2 \, {\left (b^{4} g^{3} x^{2} + 2 \, a b^{3} g^{3} x + a^{2} b^{2} g^{3}\right )}} - \frac {{\left (2 \, b x + a\right )} A d i}{2 \, {\left (b^{4} g^{3} x^{2} + 2 \, a b^{3} g^{3} x + a^{2} b^{2} g^{3}\right )}} - \frac {B c i \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right )}{2 \, {\left (b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}\right )}} - \frac {A c i}{2 \, {\left (b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.25, size = 204, normalized size = 2.29 \[ -\frac {x\,\left (2\,A\,b\,d\,i+B\,b\,d\,i\,n\right )+A\,a\,d\,i+A\,b\,c\,i+\frac {B\,a\,d\,i\,n}{2}+\frac {B\,b\,c\,i\,n}{2}}{2\,a^2\,b^2\,g^3+4\,a\,b^3\,g^3\,x+2\,b^4\,g^3\,x^2}-\frac {\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (\frac {B\,c\,i}{2\,b}+\frac {B\,a\,d\,i}{2\,b^2}+\frac {B\,d\,i\,x}{b}\right )}{a^2\,g^3+2\,a\,b\,g^3\,x+b^2\,g^3\,x^2}-\frac {B\,d^2\,i\,n\,\mathrm {atan}\left (\frac {b\,c\,2{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{b^2\,g^3\,\left (a\,d-b\,c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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