Optimal. Leaf size=104 \[ \frac {c+d x}{B g^2 (a+b x) (b c-a d) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}-\frac {e^{-\frac {A}{B}} \text {Ei}\left (\frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{B}\right )}{B^2 e g^2 (b c-a d)} \]
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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 = {} \[ \int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2} \, dx \]
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 (c+d x)}{a+b x}\right )\right )^2} \, dx &=\int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2} \, dx\\ \end {align*}
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Mathematica [A] time = 0.12, size = 88, normalized size = 0.85 \[ \frac {\frac {e^{-\frac {A}{B}} \text {Ei}\left (\frac {A}{B}+\log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{e}-\frac {B (c+d x)}{(a+b x) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}}{B^2 g^2 (a d-b c)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.57, size = 208, normalized size = 2.00 \[ \frac {{\left (B d e x + B c e\right )} e^{\frac {A}{B}} - {\left (A b x + A a + {\left (B b x + B a\right )} \log \left (\frac {d e x + c e}{b x + a}\right )\right )} \operatorname {log\_integral}\left (\frac {{\left (d e x + c e\right )} e^{\frac {A}{B}}}{b x + a}\right )}{{\left ({\left (B^{3} b^{2} c - B^{3} a b d\right )} e g^{2} x + {\left (B^{3} a b c - B^{3} a^{2} d\right )} e g^{2}\right )} e^{\frac {A}{B}} \log \left (\frac {d e x + c e}{b x + a}\right ) + {\left ({\left (A B^{2} b^{2} c - A B^{2} a b d\right )} e g^{2} x + {\left (A B^{2} a b c - A B^{2} a^{2} d\right )} e g^{2}\right )} e^{\frac {A}{B}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.34, size = 152, normalized size = 1.46 \[ {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} {\left (\frac {d x e + c e}{{\left (B^{2} g^{2} \log \left (\frac {d x e + c e}{b x + a}\right ) + A B g^{2}\right )} {\left (b x + a\right )}} - \frac {{\rm Ei}\left (\frac {A}{B} + \log \left (\frac {d x e + c e}{b x + a}\right )\right ) e^{\left (-\frac {A}{B}\right )}}{B^{2} g^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.45, size = 258, normalized size = 2.48 \[ \frac {a d}{\left (a d -b c \right ) \left (\ln \left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right )+\frac {A}{B}\right ) \left (b x +a \right ) B^{2} b \,g^{2}}-\frac {c}{\left (a d -b c \right ) \left (\ln \left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right )+\frac {A}{B}\right ) \left (b x +a \right ) B^{2} g^{2}}-\frac {\Ei \left (1, -\ln \left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right )-\frac {A}{B}\right ) {\mathrm e}^{-\frac {A}{B}}}{\left (a d -b c \right ) B^{2} e \,g^{2}}-\frac {d}{\left (a d -b c \right ) \left (\ln \left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right )+\frac {A}{B}\right ) B^{2} b \,g^{2}} \]
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 \[ \frac {d x + c}{{\left (a b c g^{2} - a^{2} d g^{2}\right )} A B + {\left (a b c g^{2} \log \relax (e) - a^{2} d g^{2} \log \relax (e)\right )} B^{2} + {\left ({\left (b^{2} c g^{2} - a b d g^{2}\right )} A B + {\left (b^{2} c g^{2} \log \relax (e) - a b d g^{2} \log \relax (e)\right )} B^{2}\right )} x - {\left ({\left (b^{2} c g^{2} - a b d g^{2}\right )} B^{2} x + {\left (a b c g^{2} - a^{2} d g^{2}\right )} B^{2}\right )} \log \left (b x + a\right ) + {\left ({\left (b^{2} c g^{2} - a b d g^{2}\right )} B^{2} x + {\left (a b c g^{2} - a^{2} d g^{2}\right )} B^{2}\right )} \log \left (d x + c\right )} + \int \frac {1}{B^{2} a^{2} g^{2} \log \relax (e) + A B a^{2} g^{2} + {\left (B^{2} b^{2} g^{2} \log \relax (e) + A B b^{2} g^{2}\right )} x^{2} + 2 \, {\left (B^{2} a b g^{2} \log \relax (e) + A B a b g^{2}\right )} x - {\left (B^{2} b^{2} g^{2} x^{2} + 2 \, B^{2} a b g^{2} x + B^{2} a^{2} g^{2}\right )} \log \left (b x + a\right ) + {\left (B^{2} b^{2} g^{2} x^{2} + 2 \, B^{2} a b g^{2} x + B^{2} a^{2} g^{2}\right )} \log \left (d x + c\right )}\,{d x} \]
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 \[ \int \frac {1}{{\left (a\,g+b\,g\,x\right )}^2\,{\left (A+B\,\ln \left (\frac {e\,\left (c+d\,x\right )}{a+b\,x}\right )\right )}^2} \,d x \]
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 \[ \frac {- c - d x}{A B a^{2} d g^{2} - A B a b c g^{2} + A B a b d g^{2} x - A B b^{2} c g^{2} x + \left (B^{2} a^{2} d g^{2} - B^{2} a b c g^{2} + B^{2} a b d g^{2} x - B^{2} b^{2} c g^{2} x\right ) \log {\left (\frac {e \left (c + d x\right )}{a + b x} \right )}} + \frac {\int \frac {1}{A a^{2} + 2 A a b x + A b^{2} x^{2} + B a^{2} \log {\left (\frac {c e}{a + b x} + \frac {d e x}{a + b x} \right )} + 2 B a b x \log {\left (\frac {c e}{a + b x} + \frac {d e x}{a + b x} \right )} + B b^{2} x^{2} \log {\left (\frac {c e}{a + b x} + \frac {d e x}{a + b x} \right )}}\, dx}{B g^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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