Optimal. Leaf size=91 \[ -\frac {e^{-\frac {A}{2 B}} (c+d x) \text {Ei}\left (\frac {A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )}{2 B}\right )}{2 B g^2 (a+b x) (b c-a d) \sqrt {\frac {e (c+d x)^2}{(a+b x)^2}}} \]
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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 = {} \[ \int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )} \, dx \]
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 (c+d x)^2}{(a+b x)^2}\right )\right )} \, dx &=\int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )} \, dx\\ \end {align*}
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Mathematica [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{A b^{2} g^{2} x^{2} + 2 \, A a b g^{2} x + A a^{2} g^{2} + {\left (B b^{2} g^{2} x^{2} + 2 \, B a b g^{2} x + B a^{2} g^{2}\right )} \log \left (\frac {d^{2} e x^{2} + 2 \, c d e x + c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )}, x\right ) \]
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 \[ \int \frac {1}{{\left (b g x + a g\right )}^{2} {\left (B \log \left (\frac {{\left (d x + c\right )}^{2} e}{{\left (b x + a\right )}^{2}}\right ) + A\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.87, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b g x +a g \right )^{2} \left (B \ln \left (\frac {\left (d x +c \right )^{2} e}{\left (b x +a \right )^{2}}\right )+A \right )}\, dx \]
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 \[ \int \frac {1}{{\left (b g x + a g\right )}^{2} {\left (B \log \left (\frac {{\left (d x + c\right )}^{2} e}{{\left (b x + a\right )}^{2}}\right ) + A\right )}}\,{d x} \]
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 \[ \int \frac {1}{{\left (a\,g+b\,g\,x\right )}^2\,\left (A+B\,\ln \left (\frac {e\,{\left (c+d\,x\right )}^2}{{\left (a+b\,x\right )}^2}\right )\right )} \,d x \]
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 \[ \frac {\int \frac {1}{A a^{2} + 2 A a b x + A b^{2} x^{2} + B a^{2} \log {\left (\frac {c^{2} e}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {2 c d e x}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {d^{2} e x^{2}}{a^{2} + 2 a b x + b^{2} x^{2}} \right )} + 2 B a b x \log {\left (\frac {c^{2} e}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {2 c d e x}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {d^{2} e x^{2}}{a^{2} + 2 a b x + b^{2} x^{2}} \right )} + B b^{2} x^{2} \log {\left (\frac {c^{2} e}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {2 c d e x}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {d^{2} e x^{2}}{a^{2} + 2 a b x + b^{2} x^{2}} \right )}}\, dx}{g^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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