Optimal. Leaf size=103 \[ -\frac {e e^{A/B} \text {Ei}\left (-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{B}\right )}{B^2 g^2 (b c-a d)}-\frac {c+d x}{B g^2 (a+b x) (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )} \]
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Rubi [F] time = 0.09, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2} \, dx &=\int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2} \, dx\\ \end {align*}
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Mathematica [A] time = 0.18, size = 87, normalized size = 0.84 \[ \frac {e e^{A/B} \text {Ei}\left (-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{B}\right )+\frac {B (c+d x)}{(a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}}{B^2 g^2 (a d-b c)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 199, normalized size = 1.93 \[ -\frac {B d x + B c + {\left ({\left (B b e x + B a e\right )} e^{\frac {A}{B}} \log \left (\frac {b e x + a e}{d x + c}\right ) + {\left (A b e x + A a e\right )} e^{\frac {A}{B}}\right )} \operatorname {log\_integral}\left (\frac {{\left (d x + c\right )} e^{\left (-\frac {A}{B}\right )}}{b e x + a e}\right )}{{\left (A B^{2} b^{2} c - A B^{2} a b d\right )} g^{2} x + {\left (A B^{2} a b c - A B^{2} a^{2} d\right )} g^{2} + {\left ({\left (B^{3} b^{2} c - B^{3} a b d\right )} g^{2} x + {\left (B^{3} a b c - B^{3} a^{2} d\right )} g^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b g x + a g\right )}^{2} {\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.34, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b g x +a g \right )^{2} \left (B \ln \left (\frac {\left (b x +a \right ) e}{d x +c}\right )+A \right )^{2}}\, dx \]
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 \[ -\frac {d x + c}{{\left (a b c g^{2} - a^{2} d g^{2}\right )} A B + {\left (a b c g^{2} \log \relax (e) - a^{2} d g^{2} \log \relax (e)\right )} B^{2} + {\left ({\left (b^{2} c g^{2} - a b d g^{2}\right )} A B + {\left (b^{2} c g^{2} \log \relax (e) - a b d g^{2} \log \relax (e)\right )} B^{2}\right )} x + {\left ({\left (b^{2} c g^{2} - a b d g^{2}\right )} B^{2} x + {\left (a b c g^{2} - a^{2} d g^{2}\right )} B^{2}\right )} \log \left (b x + a\right ) - {\left ({\left (b^{2} c g^{2} - a b d g^{2}\right )} B^{2} x + {\left (a b c g^{2} - a^{2} d g^{2}\right )} B^{2}\right )} \log \left (d x + c\right )} + \int -\frac {1}{B^{2} a^{2} g^{2} \log \relax (e) + A B a^{2} g^{2} + {\left (B^{2} b^{2} g^{2} \log \relax (e) + A B b^{2} g^{2}\right )} x^{2} + 2 \, {\left (B^{2} a b g^{2} \log \relax (e) + A B a b g^{2}\right )} x + {\left (B^{2} b^{2} g^{2} x^{2} + 2 \, B^{2} a b g^{2} x + B^{2} a^{2} g^{2}\right )} \log \left (b x + a\right ) - {\left (B^{2} b^{2} g^{2} x^{2} + 2 \, B^{2} a b g^{2} x + B^{2} a^{2} g^{2}\right )} \log \left (d x + c\right )}\,{d x} \]
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 \[ \int \frac {1}{{\left (a\,g+b\,g\,x\right )}^2\,{\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {c + d x}{A B a^{2} d g^{2} - A B a b c g^{2} + A B a b d g^{2} x - A B b^{2} c g^{2} x + \left (B^{2} a^{2} d g^{2} - B^{2} a b c g^{2} + B^{2} a b d g^{2} x - B^{2} b^{2} c g^{2} x\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}} - \frac {\int \frac {1}{A a^{2} + 2 A a b x + A b^{2} x^{2} + B a^{2} \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )} + 2 B a b x \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )} + B b^{2} x^{2} \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}\, dx}{B g^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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