Optimal. Leaf size=36 \[ \text {Int}\left (\frac {c g+d g x}{\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2},x\right ) \]
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Rubi [A] time = 0.12, 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 = {} \[ \int \frac {c g+d g x}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {c g+d g x}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx &=\int \left (\frac {c g}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}+\frac {d g x}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}\right ) \, dx\\ &=(c g) \int \frac {1}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx+(d g) \int \frac {x}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx\\ \end {align*}
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Mathematica [A] time = 1.00, size = 0, normalized size = 0.00 \[ \int \frac {c g+d g x}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {d g x + c g}{B^{2} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \, A B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {d g x + c g}{{\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {d g x +c g}{\left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}}\, dx \]
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 \[ -\frac {b d^{2} g x^{3} + a c^{2} g + {\left (2 \, b c d g + a d^{2} g\right )} x^{2} + {\left (b c^{2} g + 2 \, a c d g\right )} x}{{\left (b c n - a d n\right )} B^{2} \log \left ({\left (b x + a\right )}^{n}\right ) - {\left (b c n - a d n\right )} B^{2} \log \left ({\left (d x + c\right )}^{n}\right ) + {\left (b c n - a d n\right )} A B + {\left (b c n \log \relax (e) - a d n \log \relax (e)\right )} B^{2}} + \int \frac {3 \, b d^{2} g x^{2} + b c^{2} g + 2 \, a c d g + 2 \, {\left (2 \, b c d g + a d^{2} g\right )} x}{{\left (b c n - a d n\right )} B^{2} \log \left ({\left (b x + a\right )}^{n}\right ) - {\left (b c n - a d n\right )} B^{2} \log \left ({\left (d x + c\right )}^{n}\right ) + {\left (b c n - a d n\right )} A B + {\left (b c n \log \relax (e) - a d n \log \relax (e)\right )} B^{2}}\,{d x} \]
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 \[ \int \frac {c\,g+d\,g\,x}{{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2} \,d x \]
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 \[ g \left (\int \frac {c}{A^{2} + 2 A B \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )} + B^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}}\, dx + \int \frac {d x}{A^{2} + 2 A B \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )} + B^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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