Optimal. Leaf size=33 \[ \text {Int}\left (\frac {a g+b g x}{\left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )^2},x\right ) \]
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Rubi [A] time = 0.11, 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 = {} \[ \int \frac {a g+b g x}{\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {a g+b g x}{\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2} \, dx &=\int \left (\frac {a g}{\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}+\frac {b g x}{\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}\right ) \, dx\\ &=(a g) \int \frac {1}{\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2} \, dx+(b g) \int \frac {x}{\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 = 0.97, size = 0, normalized size = 0.00 \[ \int \frac {a g+b g x}{\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b g x + a g}{B^{2} \log \left (\frac {d e x + c e}{b x + a}\right )^{2} + 2 \, A B \log \left (\frac {d e x + c e}{b x + a}\right ) + A^{2}}, x\right ) \]
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 [A] time = 1.12, size = 0, normalized size = 0.00 \[ \int \frac {b g x +a g}{\left (B \ln \left (\frac {\left (d x +c \right ) e}{b x +a}\right )+A \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {b^{2} d g x^{3} + a^{2} c g + {\left (b^{2} c g + 2 \, a b d g\right )} x^{2} + {\left (2 \, a b c g + a^{2} d g\right )} x}{{\left (b c - a d\right )} B^{2} \log \left (b x + a\right ) - {\left (b c - a d\right )} B^{2} \log \left (d x + c\right ) - {\left (b c - a d\right )} A B - {\left (b c \log \relax (e) - a d \log \relax (e)\right )} B^{2}} + \int \frac {3 \, b^{2} d g x^{2} + 2 \, a b c g + a^{2} d g + 2 \, {\left (b^{2} c g + 2 \, a b d g\right )} x}{{\left (b c - a d\right )} B^{2} \log \left (b x + a\right ) - {\left (b c - a d\right )} B^{2} \log \left (d x + c\right ) - {\left (b c - a d\right )} A B - {\left (b c \log \relax (e) - a d \log \relax (e)\right )} B^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {a\,g+b\,g\,x}{{\left (A+B\,\ln \left (\frac {e\,\left (c+d\,x\right )}{a+b\,x}\right )\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {- a^{2} c g - a^{2} d g x - 2 a b c g x - 2 a b d g x^{2} - b^{2} c g x^{2} - b^{2} d g x^{3}}{A B a d - A B b c + \left (B^{2} a d - B^{2} b c\right ) \log {\left (\frac {e \left (c + d x\right )}{a + b x} \right )}} + \frac {g \left (\int \frac {a^{2} d}{A + B \log {\left (\frac {c e}{a + b x} + \frac {d e x}{a + b x} \right )}}\, dx + \int \frac {2 a b c}{A + B \log {\left (\frac {c e}{a + b x} + \frac {d e x}{a + b x} \right )}}\, dx + \int \frac {2 b^{2} c x}{A + B \log {\left (\frac {c e}{a + b x} + \frac {d e x}{a + b x} \right )}}\, dx + \int \frac {3 b^{2} d x^{2}}{A + B \log {\left (\frac {c e}{a + b x} + \frac {d e x}{a + b x} \right )}}\, dx + \int \frac {4 a b d x}{A + B \log {\left (\frac {c e}{a + b x} + \frac {d e x}{a + b x} \right )}}\, dx\right )}{B \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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