Optimal. Leaf size=160 \[ -\frac {b g \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d^2 i^2}-\frac {A g (a+b x)}{d i^2 (c+d x)}-\frac {b B g \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^2 i^2}-\frac {B g (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{d i^2 (c+d x)}+\frac {B g (a+b x)}{d i^2 (c+d x)} \]
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Rubi [A] time = 0.40, antiderivative size = 222, normalized size of antiderivative = 1.39, number of steps used = 15, number of rules used = 11, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used = {2528, 2525, 12, 44, 2524, 2418, 2394, 2393, 2391, 2390, 2301} \[ -\frac {b B g \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{d^2 i^2}+\frac {b g \log (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d^2 i^2}+\frac {g (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d^2 i^2 (c+d x)}-\frac {B g (b c-a d)}{d^2 i^2 (c+d x)}-\frac {b B g \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{d^2 i^2}-\frac {b B g \log (a+b x)}{d^2 i^2}+\frac {b B g \log ^2(c+d x)}{2 d^2 i^2}+\frac {b B g \log (c+d x)}{d^2 i^2} \]
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 g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(41 c+41 d x)^2} \, dx &=\int \left (\frac {(-b c+a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1681 d (c+d x)^2}+\frac {b g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1681 d (c+d x)}\right ) \, dx\\ &=\frac {(b g) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{1681 d}-\frac {((b c-a d) g) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c+d x)^2} \, dx}{1681 d}\\ &=\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1681 d^2 (c+d x)}+\frac {b g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1681 d^2}-\frac {(b B g) \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}{1681 d^2}-\frac {(B (b c-a d) g) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{1681 d^2}\\ &=\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1681 d^2 (c+d x)}+\frac {b g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1681 d^2}-\frac {\left (B (b c-a d)^2 g\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{1681 d^2}-\frac {(b B g) \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}{1681 d^2 e}\\ &=\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1681 d^2 (c+d x)}+\frac {b g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1681 d^2}-\frac {\left (B (b c-a d)^2 g\right ) \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}{1681 d^2}-\frac {(b B g) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{1681 d^2 e}\\ &=-\frac {B (b c-a d) g}{1681 d^2 (c+d x)}-\frac {b B g \log (a+b x)}{1681 d^2}+\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1681 d^2 (c+d x)}+\frac {b B g \log (c+d x)}{1681 d^2}+\frac {b g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1681 d^2}-\frac {\left (b^2 B g\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{1681 d^2}+\frac {(b B g) \int \frac {\log (c+d x)}{c+d x} \, dx}{1681 d}\\ &=-\frac {B (b c-a d) g}{1681 d^2 (c+d x)}-\frac {b B g \log (a+b x)}{1681 d^2}+\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1681 d^2 (c+d x)}+\frac {b B g \log (c+d x)}{1681 d^2}-\frac {b B g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1681 d^2}+\frac {b g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1681 d^2}+\frac {(b B g) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{1681 d^2}+\frac {(b B g) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{1681 d}\\ &=-\frac {B (b c-a d) g}{1681 d^2 (c+d x)}-\frac {b B g \log (a+b x)}{1681 d^2}+\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1681 d^2 (c+d x)}+\frac {b B g \log (c+d x)}{1681 d^2}-\frac {b B g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1681 d^2}+\frac {b g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1681 d^2}+\frac {b B g \log ^2(c+d x)}{3362 d^2}+\frac {(b B g) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{1681 d^2}\\ &=-\frac {B (b c-a d) g}{1681 d^2 (c+d x)}-\frac {b B g \log (a+b x)}{1681 d^2}+\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1681 d^2 (c+d x)}+\frac {b B g \log (c+d x)}{1681 d^2}-\frac {b B g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1681 d^2}+\frac {b g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1681 d^2}+\frac {b B g \log ^2(c+d x)}{3362 d^2}-\frac {b B g \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{1681 d^2}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 175, normalized size = 1.09 \[ \frac {g \left (2 b \log (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+\frac {2 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{c+d x}-b B \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-2 B \left (\frac {b c-a d}{c+d x}+b \log (a+b x)-b \log (c+d x)\right )\right )}{2 d^2 i^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {A b g x + A a g + {\left (B b g x + B a g\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{d^{2} i^{2} x^{2} + 2 \, c d i^{2} x + c^{2} i^{2}}, x\right ) \]
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 = 0.06, size = 978, normalized size = 6.11 \[ -\frac {B \,a^{2} g \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 ) \left (d x +c \right ) i^{2}}+\frac {2 B a b c g \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 ) \left (d x +c \right ) d \,i^{2}}-\frac {B a b g \ln \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right ) d}{b e}\right ) \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 ) d \,i^{2}}-\frac {B \,b^{2} c^{2} g \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 ) \left (d x +c \right ) d^{2} i^{2}}+\frac {B \,b^{2} c g \ln \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right ) d}{b e}\right ) \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 ) d^{2} i^{2}}-\frac {A \,a^{2} g}{\left (a d -b c \right ) \left (d x +c \right ) i^{2}}+\frac {2 A a b c g}{\left (a d -b c \right ) \left (d x +c \right ) d \,i^{2}}-\frac {A a b g \ln \left (-b e +\left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right ) d \right )}{\left (a d -b c \right ) d \,i^{2}}-\frac {A \,b^{2} c^{2} g}{\left (a d -b c \right ) \left (d x +c \right ) d^{2} i^{2}}+\frac {A \,b^{2} c g \ln \left (-b e +\left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right ) d \right )}{\left (a d -b c \right ) d^{2} i^{2}}+\frac {B \,a^{2} g}{\left (a d -b c \right ) \left (d x +c \right ) i^{2}}-\frac {2 B a b c g}{\left (a d -b c \right ) \left (d x +c \right ) d \,i^{2}}-\frac {B a b g \dilog \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right ) d}{b e}\right )}{\left (a d -b c \right ) d \,i^{2}}-\frac {B a b g \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 ) d \,i^{2}}+\frac {B \,b^{2} c^{2} g}{\left (a d -b c \right ) \left (d x +c \right ) d^{2} i^{2}}+\frac {B \,b^{2} c g \dilog \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right ) d}{b e}\right )}{\left (a d -b c \right ) d^{2} i^{2}}+\frac {B \,b^{2} c g \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 ) d^{2} i^{2}}-\frac {A a b g}{\left (a d -b c \right ) d \,i^{2}}+\frac {A \,b^{2} c g}{\left (a d -b c \right ) d^{2} i^{2}}+\frac {B a b g}{\left (a d -b c \right ) d \,i^{2}}-\frac {B \,b^{2} c g}{\left (a d -b c \right ) d^{2} i^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{2} \, B b g {\left (\frac {{\left (d x + c\right )} \log \left (d x + c\right )^{2} + 2 \, c \log \left (d x + c\right )}{d^{3} i^{2} x + c d^{2} i^{2}} - 2 \, \int \frac {d x \log \left (b x + a\right ) + d x \log \relax (e) + c}{d^{3} i^{2} x^{2} + 2 \, c d^{2} i^{2} x + c^{2} d i^{2}}\,{d x}\right )} + A b g {\left (\frac {c}{d^{3} i^{2} x + c d^{2} i^{2}} + \frac {\log \left (d x + c\right )}{d^{2} i^{2}}\right )} - B a g {\left (\frac {\log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{d^{2} i^{2} x + c d i^{2}} - \frac {1}{d^{2} i^{2} x + c d i^{2}} - \frac {b \log \left (b x + a\right )}{{\left (b c d - a d^{2}\right )} i^{2}} + \frac {b \log \left (d x + c\right )}{{\left (b c d - a d^{2}\right )} i^{2}}\right )} - \frac {A a g}{d^{2} i^{2} x + c d i^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (a\,g+b\,g\,x\right )\,\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}{{\left (c\,i+d\,i\,x\right )}^2} \,d x \]
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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