Optimal. Leaf size=261 \[ -\frac {b^2 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^2 i^2 (a+b x) (b c-a d)^3}+\frac {d^2 (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^2 i^2 (c+d x) (b c-a d)^3}-\frac {2 b d \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^2 i^2 (b c-a d)^3}-\frac {b^2 B (c+d x)}{g^2 i^2 (a+b x) (b c-a d)^3}-\frac {B d^2 (a+b x)}{g^2 i^2 (c+d x) (b c-a d)^3}+\frac {b B d \log ^2\left (\frac {a+b x}{c+d x}\right )}{g^2 i^2 (b c-a d)^3} \]
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Rubi [C] time = 0.87, antiderivative size = 462, normalized size of antiderivative = 1.77, number of steps used = 28, number of rules used = 11, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.275, Rules used = {2528, 2525, 12, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ -\frac {2 b B d \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{g^2 i^2 (b c-a d)^3}-\frac {2 b B d \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{g^2 i^2 (b c-a d)^3}-\frac {2 b d \log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^2 i^2 (b c-a d)^3}-\frac {b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^2 i^2 (a+b x) (b c-a d)^2}-\frac {d \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^2 i^2 (c+d x) (b c-a d)^2}+\frac {2 b d \log (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^2 i^2 (b c-a d)^3}-\frac {b B}{g^2 i^2 (a+b x) (b c-a d)^2}+\frac {B d}{g^2 i^2 (c+d x) (b c-a d)^2}+\frac {b B d \log ^2(a+b x)}{g^2 i^2 (b c-a d)^3}+\frac {b B d \log ^2(c+d x)}{g^2 i^2 (b c-a d)^3}-\frac {2 b B d \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{g^2 i^2 (b c-a d)^3}-\frac {2 b B d \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{g^2 i^2 (b c-a d)^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 44
Rule 2301
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2418
Rule 2524
Rule 2525
Rule 2528
Rubi steps
\begin {align*} \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(44 c+44 d x)^2 (a g+b g x)^2} \, dx &=\int \left (\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1936 (b c-a d)^2 g^2 (a+b x)^2}-\frac {b^2 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{968 (b c-a d)^3 g^2 (a+b x)}+\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1936 (b c-a d)^2 g^2 (c+d x)^2}+\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{968 (b c-a d)^3 g^2 (c+d x)}\right ) \, dx\\ &=-\frac {\left (b^2 d\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{968 (b c-a d)^3 g^2}+\frac {\left (b d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{968 (b c-a d)^3 g^2}+\frac {b^2 \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{1936 (b c-a d)^2 g^2}+\frac {d^2 \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c+d x)^2} \, dx}{1936 (b c-a d)^2 g^2}\\ &=-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1936 (b c-a d)^2 g^2 (a+b x)}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1936 (b c-a d)^2 g^2 (c+d x)}-\frac {b d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{968 (b c-a d)^3 g^2}+\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{968 (b c-a d)^3 g^2}+\frac {(b B d) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{968 (b c-a d)^3 g^2}-\frac {(b B d) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{968 (b c-a d)^3 g^2}+\frac {(b B) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{1936 (b c-a d)^2 g^2}+\frac {(B d) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{1936 (b c-a d)^2 g^2}\\ &=-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1936 (b c-a d)^2 g^2 (a+b x)}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1936 (b c-a d)^2 g^2 (c+d x)}-\frac {b d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{968 (b c-a d)^3 g^2}+\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{968 (b c-a d)^3 g^2}+\frac {(b B) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{1936 (b c-a d) g^2}+\frac {(B d) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{1936 (b c-a d) g^2}+\frac {(b B d) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{968 (b c-a d)^3 e g^2}-\frac {(b B d) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{968 (b c-a d)^3 e g^2}\\ &=-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1936 (b c-a d)^2 g^2 (a+b x)}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1936 (b c-a d)^2 g^2 (c+d x)}-\frac {b d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{968 (b c-a d)^3 g^2}+\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{968 (b c-a d)^3 g^2}+\frac {(b B) \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}{1936 (b c-a d) g^2}+\frac {(B d) \int \left (\frac {b^2}{(b c-a d)^2 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^2}-\frac {b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{1936 (b c-a d) g^2}+\frac {(b B d) \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{968 (b c-a d)^3 e g^2}-\frac {(b B d) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{968 (b c-a d)^3 e g^2}\\ &=-\frac {b B}{1936 (b c-a d)^2 g^2 (a+b x)}+\frac {B d}{1936 (b c-a d)^2 g^2 (c+d x)}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1936 (b c-a d)^2 g^2 (a+b x)}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1936 (b c-a d)^2 g^2 (c+d x)}-\frac {b d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{968 (b c-a d)^3 g^2}+\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{968 (b c-a d)^3 g^2}+\frac {\left (b^2 B d\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{968 (b c-a d)^3 g^2}-\frac {\left (b^2 B d\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{968 (b c-a d)^3 g^2}-\frac {\left (b B d^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{968 (b c-a d)^3 g^2}+\frac {\left (b B d^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{968 (b c-a d)^3 g^2}\\ &=-\frac {b B}{1936 (b c-a d)^2 g^2 (a+b x)}+\frac {B d}{1936 (b c-a d)^2 g^2 (c+d x)}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1936 (b c-a d)^2 g^2 (a+b x)}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1936 (b c-a d)^2 g^2 (c+d x)}-\frac {b d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{968 (b c-a d)^3 g^2}-\frac {b B d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{968 (b c-a d)^3 g^2}+\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{968 (b c-a d)^3 g^2}-\frac {b B d \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{968 (b c-a d)^3 g^2}+\frac {(b B d) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{968 (b c-a d)^3 g^2}+\frac {(b B d) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{968 (b c-a d)^3 g^2}+\frac {\left (b^2 B d\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{968 (b c-a d)^3 g^2}+\frac {\left (b B d^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{968 (b c-a d)^3 g^2}\\ &=-\frac {b B}{1936 (b c-a d)^2 g^2 (a+b x)}+\frac {B d}{1936 (b c-a d)^2 g^2 (c+d x)}+\frac {b B d \log ^2(a+b x)}{1936 (b c-a d)^3 g^2}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1936 (b c-a d)^2 g^2 (a+b x)}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1936 (b c-a d)^2 g^2 (c+d x)}-\frac {b d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{968 (b c-a d)^3 g^2}-\frac {b B d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{968 (b c-a d)^3 g^2}+\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{968 (b c-a d)^3 g^2}+\frac {b B d \log ^2(c+d x)}{1936 (b c-a d)^3 g^2}-\frac {b B d \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{968 (b c-a d)^3 g^2}+\frac {(b B d) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{968 (b c-a d)^3 g^2}+\frac {(b B d) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{968 (b c-a d)^3 g^2}\\ &=-\frac {b B}{1936 (b c-a d)^2 g^2 (a+b x)}+\frac {B d}{1936 (b c-a d)^2 g^2 (c+d x)}+\frac {b B d \log ^2(a+b x)}{1936 (b c-a d)^3 g^2}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1936 (b c-a d)^2 g^2 (a+b x)}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1936 (b c-a d)^2 g^2 (c+d x)}-\frac {b d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{968 (b c-a d)^3 g^2}-\frac {b B d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{968 (b c-a d)^3 g^2}+\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{968 (b c-a d)^3 g^2}+\frac {b B d \log ^2(c+d x)}{1936 (b c-a d)^3 g^2}-\frac {b B d \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{968 (b c-a d)^3 g^2}-\frac {b B d \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{968 (b c-a d)^3 g^2}-\frac {b B d \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{968 (b c-a d)^3 g^2}\\ \end {align*}
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Mathematica [C] time = 0.43, size = 324, normalized size = 1.24 \[ \frac {-2 b d \log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-\frac {b (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}+2 b d \log (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+\frac {d (a d-b c) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{c+d x}-\frac {b^2 B c}{a+b x}+b B d \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )-b B d \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+\frac {a b B d}{a+b x}-\frac {a B d^2}{c+d x}+\frac {b B c d}{c+d x}}{g^2 i^2 (b c-a d)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 334, normalized size = 1.28 \[ -\frac {{\left (A + B\right )} b^{2} c^{2} - 2 \, B a b c d - {\left (A - B\right )} a^{2} d^{2} + {\left (B b^{2} d^{2} x^{2} + B a b c d + {\left (B b^{2} c d + B a b d^{2}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 2 \, {\left (A b^{2} c d - A a b d^{2}\right )} x + {\left (2 \, A b^{2} d^{2} x^{2} + B b^{2} c^{2} + 2 \, A a b c d - B a^{2} d^{2} + 2 \, {\left ({\left (A + B\right )} b^{2} c d + {\left (A - B\right )} a b d^{2}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{{\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} g^{2} i^{2} x^{2} + {\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} g^{2} i^{2} x + {\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + 3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3}\right )} g^{2} i^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [B] time = 0.05, size = 1187, normalized size = 4.55 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.41, size = 859, normalized size = 3.29 \[ -B {\left (\frac {2 \, b d x + b c + a d}{{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} g^{2} i^{2} x^{2} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} g^{2} i^{2} x + {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} g^{2} i^{2}} + \frac {2 \, b d \log \left (b x + a\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} g^{2} i^{2}} - \frac {2 \, b d \log \left (d x + c\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} g^{2} i^{2}}\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) - A {\left (\frac {2 \, b d x + b c + a d}{{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} g^{2} i^{2} x^{2} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} g^{2} i^{2} x + {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} g^{2} i^{2}} + \frac {2 \, b d \log \left (b x + a\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} g^{2} i^{2}} - \frac {2 \, b d \log \left (d x + c\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} g^{2} i^{2}}\right )} - \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2} - {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )} \log \left (b x + a\right )^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )} \log \left (b x + a\right ) \log \left (d x + c\right ) - {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )} \log \left (d x + c\right )^{2}\right )} B}{a b^{3} c^{4} g^{2} i^{2} - 3 \, a^{2} b^{2} c^{3} d g^{2} i^{2} + 3 \, a^{3} b c^{2} d^{2} g^{2} i^{2} - a^{4} c d^{3} g^{2} i^{2} + {\left (b^{4} c^{3} d g^{2} i^{2} - 3 \, a b^{3} c^{2} d^{2} g^{2} i^{2} + 3 \, a^{2} b^{2} c d^{3} g^{2} i^{2} - a^{3} b d^{4} g^{2} i^{2}\right )} x^{2} + {\left (b^{4} c^{4} g^{2} i^{2} - 2 \, a b^{3} c^{3} d g^{2} i^{2} + 2 \, a^{3} b c d^{3} g^{2} i^{2} - a^{4} d^{4} g^{2} i^{2}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.18, size = 415, normalized size = 1.59 \[ \frac {B\,b\,d\,{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2}{g^2\,i^2\,{\left (a\,d-b\,c\right )}^3}-\frac {A\,a\,d}{g^2\,i^2\,{\left (a\,d-b\,c\right )}^2\,\left (a+b\,x\right )\,\left (c+d\,x\right )}-\frac {A\,b\,c}{g^2\,i^2\,{\left (a\,d-b\,c\right )}^2\,\left (a+b\,x\right )\,\left (c+d\,x\right )}+\frac {B\,a\,d}{g^2\,i^2\,{\left (a\,d-b\,c\right )}^2\,\left (a+b\,x\right )\,\left (c+d\,x\right )}-\frac {B\,b\,c}{g^2\,i^2\,{\left (a\,d-b\,c\right )}^2\,\left (a+b\,x\right )\,\left (c+d\,x\right )}-\frac {2\,A\,b\,d\,x}{g^2\,i^2\,{\left (a\,d-b\,c\right )}^2\,\left (a+b\,x\right )\,\left (c+d\,x\right )}-\frac {B\,a\,d\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^2\,i^2\,{\left (a\,d-b\,c\right )}^2\,\left (a+b\,x\right )\,\left (c+d\,x\right )}-\frac {B\,b\,c\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^2\,i^2\,{\left (a\,d-b\,c\right )}^2\,\left (a+b\,x\right )\,\left (c+d\,x\right )}-\frac {2\,B\,b\,d\,x\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^2\,i^2\,{\left (a\,d-b\,c\right )}^2\,\left (a+b\,x\right )\,\left (c+d\,x\right )}-\frac {A\,b\,d\,\mathrm {atan}\left (\frac {a\,d\,1{}\mathrm {i}+b\,c\,1{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}\right )\,4{}\mathrm {i}}{g^2\,i^2\,{\left (a\,d-b\,c\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 5.84, size = 828, normalized size = 3.17 \[ - \frac {2 A b d \log {\left (x + \frac {- \frac {2 A a^{4} b d^{5}}{\left (a d - b c\right )^{3}} + \frac {8 A a^{3} b^{2} c d^{4}}{\left (a d - b c\right )^{3}} - \frac {12 A a^{2} b^{3} c^{2} d^{3}}{\left (a d - b c\right )^{3}} + \frac {8 A a b^{4} c^{3} d^{2}}{\left (a d - b c\right )^{3}} + 2 A a b d^{2} - \frac {2 A b^{5} c^{4} d}{\left (a d - b c\right )^{3}} + 2 A b^{2} c d}{4 A b^{2} d^{2}} \right )}}{g^{2} i^{2} \left (a d - b c\right )^{3}} + \frac {2 A b d \log {\left (x + \frac {\frac {2 A a^{4} b d^{5}}{\left (a d - b c\right )^{3}} - \frac {8 A a^{3} b^{2} c d^{4}}{\left (a d - b c\right )^{3}} + \frac {12 A a^{2} b^{3} c^{2} d^{3}}{\left (a d - b c\right )^{3}} - \frac {8 A a b^{4} c^{3} d^{2}}{\left (a d - b c\right )^{3}} + 2 A a b d^{2} + \frac {2 A b^{5} c^{4} d}{\left (a d - b c\right )^{3}} + 2 A b^{2} c d}{4 A b^{2} d^{2}} \right )}}{g^{2} i^{2} \left (a d - b c\right )^{3}} + \frac {B b d \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{a^{3} d^{3} g^{2} i^{2} - 3 a^{2} b c d^{2} g^{2} i^{2} + 3 a b^{2} c^{2} d g^{2} i^{2} - b^{3} c^{3} g^{2} i^{2}} + \frac {\left (- B a d - B b c - 2 B b d x\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{a^{3} c d^{2} g^{2} i^{2} + a^{3} d^{3} g^{2} i^{2} x - 2 a^{2} b c^{2} d g^{2} i^{2} - a^{2} b c d^{2} g^{2} i^{2} x + a^{2} b d^{3} g^{2} i^{2} x^{2} + a b^{2} c^{3} g^{2} i^{2} - a b^{2} c^{2} d g^{2} i^{2} x - 2 a b^{2} c d^{2} g^{2} i^{2} x^{2} + b^{3} c^{3} g^{2} i^{2} x + b^{3} c^{2} d g^{2} i^{2} x^{2}} - \frac {A a d + A b c + 2 A b d x - B a d + B b c}{a^{3} c d^{2} g^{2} i^{2} - 2 a^{2} b c^{2} d g^{2} i^{2} + a b^{2} c^{3} g^{2} i^{2} + x^{2} \left (a^{2} b d^{3} g^{2} i^{2} - 2 a b^{2} c d^{2} g^{2} i^{2} + b^{3} c^{2} d g^{2} i^{2}\right ) + x \left (a^{3} d^{3} g^{2} i^{2} - a^{2} b c d^{2} g^{2} i^{2} - a b^{2} c^{2} d g^{2} i^{2} + b^{3} c^{3} g^{2} i^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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