Optimal. Leaf size=35 \[ \text {Int}\left (\frac {(f+g x)^2}{B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A},x\right ) \]
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Rubi [A] time = 0.19, 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 = {} \[ \int \frac {(f+g x)^2}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {(f+g x)^2}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx &=\int \left (\frac {f^2}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}+\frac {2 f g x}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}+\frac {g^2 x^2}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}\right ) \, dx\\ &=f^2 \int \frac {1}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx+(2 f g) \int \frac {x}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx+g^2 \int \frac {x^2}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx\\ \end {align*}
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Mathematica [A] time = 0.43, size = 0, normalized size = 0.00 \[ \int \frac {(f+g x)^2}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {g^{2} x^{2} + 2 \, f g x + f^{2}}{B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A}, x\right ) \]
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 = 0.26, size = 0, normalized size = 0.00 \[ \int \frac {\left (g x +f \right )^{2}}{B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A}\, dx \]
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 \[ \int \frac {{\left (g x + f\right )}^{2}}{B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A}\,{d x} \]
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 \[ \int \frac {{\left (f+g\,x\right )}^2}{A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )} \,d x \]
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 \[ \int \frac {\left (f + g x\right )^{2}}{A + B \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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