Optimal. Leaf size=156 \[ \frac {b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 B g i^2 (b c-a d)^2}-\frac {A d (a+b x)}{g i^2 (c+d x) (b c-a d)^2}-\frac {B d (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{g i^2 (c+d x) (b c-a d)^2}+\frac {B d (a+b x)}{g i^2 (c+d x) (b c-a d)^2} \]
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Rubi [C] time = 0.71, antiderivative size = 432, normalized size of antiderivative = 2.77, number of steps used = 24, number of rules used = 11, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.275, Rules used = {2528, 2524, 12, 2418, 2390, 2301, 2394, 2393, 2391, 2525, 44} \[ \frac {b B \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{g i^2 (b c-a d)^2}+\frac {b B \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{g i^2 (b c-a d)^2}+\frac {b \log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g i^2 (b c-a d)^2}+\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{g i^2 (c+d x) (b c-a d)}-\frac {b \log (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g i^2 (b c-a d)^2}-\frac {B}{g i^2 (c+d x) (b c-a d)}-\frac {b B \log ^2(a+b x)}{2 g i^2 (b c-a d)^2}-\frac {b B \log ^2(c+d x)}{2 g i^2 (b c-a d)^2}+\frac {b B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{g i^2 (b c-a d)^2}-\frac {b B \log (a+b x)}{g i^2 (b c-a d)^2}+\frac {b B \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{g i^2 (b c-a d)^2}+\frac {b B \log (c+d x)}{g i^2 (b c-a d)^2} \]
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 )}{(43 c+43 d x)^2 (a g+b g x)} \, dx &=\int \left (\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1849 (b c-a d)^2 g (a+b x)}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1849 (b c-a d) g (c+d x)^2}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1849 (b c-a d)^2 g (c+d x)}\right ) \, dx\\ &=\frac {b^2 \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{1849 (b c-a d)^2 g}-\frac {(b d) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{1849 (b c-a d)^2 g}-\frac {d \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c+d x)^2} \, dx}{1849 (b c-a d) g}\\ &=\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{1849 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1849 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1849 (b c-a d)^2 g}-\frac {(b B) \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}{1849 (b c-a d)^2 g}+\frac {(b B) \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}{1849 (b c-a d)^2 g}-\frac {B \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{1849 (b c-a d) g}\\ &=\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{1849 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1849 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1849 (b c-a d)^2 g}-\frac {B \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{1849 g}-\frac {(b B) \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}{1849 (b c-a d)^2 e g}+\frac {(b B) \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}{1849 (b c-a d)^2 e g}\\ &=\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{1849 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1849 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1849 (b c-a d)^2 g}-\frac {B \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}{1849 g}-\frac {(b B) \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{1849 (b c-a d)^2 e g}+\frac {(b B) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{1849 (b c-a d)^2 e g}\\ &=-\frac {B}{1849 (b c-a d) g (c+d x)}-\frac {b B \log (a+b x)}{1849 (b c-a d)^2 g}+\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{1849 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1849 (b c-a d)^2 g}+\frac {b B \log (c+d x)}{1849 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1849 (b c-a d)^2 g}-\frac {\left (b^2 B\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{1849 (b c-a d)^2 g}+\frac {\left (b^2 B\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{1849 (b c-a d)^2 g}+\frac {(b B d) \int \frac {\log (a+b x)}{c+d x} \, dx}{1849 (b c-a d)^2 g}-\frac {(b B d) \int \frac {\log (c+d x)}{c+d x} \, dx}{1849 (b c-a d)^2 g}\\ &=-\frac {B}{1849 (b c-a d) g (c+d x)}-\frac {b B \log (a+b x)}{1849 (b c-a d)^2 g}+\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{1849 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1849 (b c-a d)^2 g}+\frac {b B \log (c+d x)}{1849 (b c-a d)^2 g}+\frac {b B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1849 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1849 (b c-a d)^2 g}+\frac {b B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{1849 (b c-a d)^2 g}-\frac {(b B) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{1849 (b c-a d)^2 g}-\frac {(b B) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{1849 (b c-a d)^2 g}-\frac {\left (b^2 B\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{1849 (b c-a d)^2 g}-\frac {(b B d) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{1849 (b c-a d)^2 g}\\ &=-\frac {B}{1849 (b c-a d) g (c+d x)}-\frac {b B \log (a+b x)}{1849 (b c-a d)^2 g}-\frac {b B \log ^2(a+b x)}{3698 (b c-a d)^2 g}+\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{1849 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1849 (b c-a d)^2 g}+\frac {b B \log (c+d x)}{1849 (b c-a d)^2 g}+\frac {b B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1849 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1849 (b c-a d)^2 g}-\frac {b B \log ^2(c+d x)}{3698 (b c-a d)^2 g}+\frac {b B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{1849 (b c-a d)^2 g}-\frac {(b B) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{1849 (b c-a d)^2 g}-\frac {(b B) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{1849 (b c-a d)^2 g}\\ &=-\frac {B}{1849 (b c-a d) g (c+d x)}-\frac {b B \log (a+b x)}{1849 (b c-a d)^2 g}-\frac {b B \log ^2(a+b x)}{3698 (b c-a d)^2 g}+\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{1849 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1849 (b c-a d)^2 g}+\frac {b B \log (c+d x)}{1849 (b c-a d)^2 g}+\frac {b B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1849 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1849 (b c-a d)^2 g}-\frac {b B \log ^2(c+d x)}{3698 (b c-a d)^2 g}+\frac {b B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{1849 (b c-a d)^2 g}+\frac {b B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{1849 (b c-a d)^2 g}+\frac {b B \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{1849 (b c-a d)^2 g}\\ \end {align*}
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Mathematica [C] time = 0.28, size = 292, normalized size = 1.87 \[ \frac {2 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+2 b (c+d x) \log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-2 b (c+d x) \log (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-b B (c+d x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )+b B (c+d x) \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-2 B (b (c+d x) \log (a+b x)-a d-b (c+d x) \log (c+d x)+b c)}{2 g i^2 (c+d x) (b c-a d)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 151, normalized size = 0.97 \[ \frac {2 \, {\left (A - B\right )} b c - 2 \, {\left (A - B\right )} a d + {\left (B b d x + B b c\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 2 \, {\left ({\left (A - B\right )} b d x + A b c - B a d\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{2 \, {\left ({\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} g i^{2} x + {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} g i^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.00, size = 204, normalized size = 1.31 \[ -\frac {{\left (B b e \log \left (\frac {b x e + a e}{d x + c}\right )^{2} + 2 \, A b e \log \left (\frac {b x e + a e}{d x + c}\right ) - \frac {2 \, {\left (b x e + a e\right )} B d \log \left (\frac {b x e + a e}{d x + c}\right )}{d x + c} - \frac {2 \, {\left (b x e + a e\right )} A d}{d x + c} + \frac {2 \, {\left (b x e + a e\right )} B d}{d x + c}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}}{2 \, {\left (b c g - a d g\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 759, normalized size = 4.87 \[ -\frac {B \,a^{2} d^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right )^{3} \left (d x +c \right ) g \,i^{2}}+\frac {2 B a b c d \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right )^{3} \left (d x +c \right ) g \,i^{2}}+\frac {B a b d \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )^{2}}{2 \left (a d -b c \right )^{3} g \,i^{2}}-\frac {B \,b^{2} c^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right )^{3} \left (d x +c \right ) g \,i^{2}}-\frac {B \,b^{2} c \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )^{2}}{2 \left (a d -b c \right )^{3} g \,i^{2}}-\frac {A \,a^{2} d^{2}}{\left (a d -b c \right )^{3} \left (d x +c \right ) g \,i^{2}}+\frac {2 A a b c d}{\left (a d -b c \right )^{3} \left (d x +c \right ) g \,i^{2}}+\frac {A a b d \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right )^{3} g \,i^{2}}-\frac {A \,b^{2} c^{2}}{\left (a d -b c \right )^{3} \left (d x +c \right ) g \,i^{2}}-\frac {A \,b^{2} c \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right )^{3} g \,i^{2}}+\frac {B \,a^{2} d^{2}}{\left (a d -b c \right )^{3} \left (d x +c \right ) g \,i^{2}}-\frac {2 B a b c d}{\left (a d -b c \right )^{3} \left (d x +c \right ) g \,i^{2}}-\frac {B a b d \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right )^{3} g \,i^{2}}+\frac {B \,b^{2} c^{2}}{\left (a d -b c \right )^{3} \left (d x +c \right ) g \,i^{2}}+\frac {B \,b^{2} c \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right )^{3} g \,i^{2}}-\frac {A a b d}{\left (a d -b c \right )^{3} g \,i^{2}}+\frac {A \,b^{2} c}{\left (a d -b c \right )^{3} g \,i^{2}}+\frac {B a b d}{\left (a d -b c \right )^{3} g \,i^{2}}-\frac {B \,b^{2} c}{\left (a d -b c \right )^{3} g \,i^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.32, size = 421, normalized size = 2.70 \[ B {\left (\frac {1}{{\left (b c d - a d^{2}\right )} g i^{2} x + {\left (b c^{2} - a c d\right )} g i^{2}} + \frac {b \log \left (b x + a\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g i^{2}} - \frac {b \log \left (d x + c\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g i^{2}}\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) + A {\left (\frac {1}{{\left (b c d - a d^{2}\right )} g i^{2} x + {\left (b c^{2} - a c d\right )} g i^{2}} + \frac {b \log \left (b x + a\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g i^{2}} - \frac {b \log \left (d x + c\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g i^{2}}\right )} - \frac {{\left ({\left (b d x + b c\right )} \log \left (b x + a\right )^{2} + {\left (b d x + b c\right )} \log \left (d x + c\right )^{2} + 2 \, b c - 2 \, a d + 2 \, {\left (b d x + b c\right )} \log \left (b x + a\right ) - 2 \, {\left (b d x + b c + {\left (b d x + b c\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} B}{2 \, {\left (b^{2} c^{3} g i^{2} - 2 \, a b c^{2} d g i^{2} + a^{2} c d^{2} g i^{2} + {\left (b^{2} c^{2} d g i^{2} - 2 \, a b c d^{2} g i^{2} + a^{2} d^{3} g i^{2}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.81, size = 247, normalized size = 1.58 \[ \frac {B\,b\,{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2}{2\,g\,i^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {A-B}{\left (a\,d-b\,c\right )\,\left (c\,g\,i^2+d\,g\,i^2\,x\right )}-\frac {B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (a\,d-b\,c\right )}{b\,d\,g\,i^2\,\left (\frac {x}{b}+\frac {c}{b\,d}\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {b\,\mathrm {atan}\left (\frac {\left (2\,b\,d\,x+\frac {a^2\,d^2\,g\,i^2-b^2\,c^2\,g\,i^2}{g\,i^2\,\left (a\,d-b\,c\right )}\right )\,1{}\mathrm {i}}{a\,d-b\,c}\right )\,\left (A-B\right )\,2{}\mathrm {i}}{g\,i^2\,{\left (a\,d-b\,c\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.50, size = 386, normalized size = 2.47 \[ \frac {B b \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{2 a^{2} d^{2} g i^{2} - 4 a b c d g i^{2} + 2 b^{2} c^{2} g i^{2}} - \frac {B \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{a c d g i^{2} + a d^{2} g i^{2} x - b c^{2} g i^{2} - b c d g i^{2} x} + \left (A - B\right ) \left (- \frac {b \log {\left (x + \frac {- \frac {a^{3} b d^{3}}{\left (a d - b c\right )^{2}} + \frac {3 a^{2} b^{2} c d^{2}}{\left (a d - b c\right )^{2}} - \frac {3 a b^{3} c^{2} d}{\left (a d - b c\right )^{2}} + a b d + \frac {b^{4} c^{3}}{\left (a d - b c\right )^{2}} + b^{2} c}{2 b^{2} d} \right )}}{g i^{2} \left (a d - b c\right )^{2}} + \frac {b \log {\left (x + \frac {\frac {a^{3} b d^{3}}{\left (a d - b c\right )^{2}} - \frac {3 a^{2} b^{2} c d^{2}}{\left (a d - b c\right )^{2}} + \frac {3 a b^{3} c^{2} d}{\left (a d - b c\right )^{2}} + a b d - \frac {b^{4} c^{3}}{\left (a d - b c\right )^{2}} + b^{2} c}{2 b^{2} d} \right )}}{g i^{2} \left (a d - b c\right )^{2}} - \frac {1}{a c d g i^{2} - b c^{2} g i^{2} + x \left (a d^{2} g i^{2} - b c d g i^{2}\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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