Optimal. Leaf size=35 \[ \text {Int}\left (\frac {(f+g x)^2}{\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2},x\right ) \]
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Rubi [A] time = 0.21, 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 = {} \[ \int \frac {(f+g x)^2}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {(f+g x)^2}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx &=\int \left (\frac {f^2}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}+\frac {2 f g x}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}+\frac {g^2 x^2}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}\right ) \, dx\\ &=f^2 \int \frac {1}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx+(2 f g) \int \frac {x}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx+g^2 \int \frac {x^2}{\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 = 0.94, size = 0, normalized size = 0.00 \[ \int \frac {(f+g x)^2}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {g^{2} x^{2} + 2 \, f g x + f^{2}}{B^{2} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \, A B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A^{2}}, x\right ) \]
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 \[ \int \frac {{\left (g x + f\right )}^{2}}{{\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.27, size = 0, normalized size = 0.00 \[ \int \frac {\left (g x +f \right )^{2}}{\left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}}\, dx \]
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 \[ -\frac {b d g^{2} x^{4} + a c f^{2} + {\left (a d g^{2} + {\left (2 \, d f g + c g^{2}\right )} b\right )} x^{3} + {\left ({\left (2 \, d f g + c g^{2}\right )} a + {\left (d f^{2} + 2 \, c f g\right )} b\right )} x^{2} + {\left (b c f^{2} + {\left (d f^{2} + 2 \, c f g\right )} a\right )} x}{{\left (b c n - a d n\right )} B^{2} \log \left ({\left (b x + a\right )}^{n}\right ) - {\left (b c n - a d n\right )} B^{2} \log \left ({\left (d x + c\right )}^{n}\right ) + {\left (b c n - a d n\right )} A B + {\left (b c n \log \relax (e) - a d n \log \relax (e)\right )} B^{2}} + \int \frac {4 \, b d g^{2} x^{3} + b c f^{2} + 3 \, {\left (a d g^{2} + {\left (2 \, d f g + c g^{2}\right )} b\right )} x^{2} + {\left (d f^{2} + 2 \, c f g\right )} a + 2 \, {\left ({\left (2 \, d f g + c g^{2}\right )} a + {\left (d f^{2} + 2 \, c f g\right )} b\right )} x}{{\left (b c n - a d n\right )} B^{2} \log \left ({\left (b x + a\right )}^{n}\right ) - {\left (b c n - a d n\right )} B^{2} \log \left ({\left (d x + c\right )}^{n}\right ) + {\left (b c n - a d n\right )} A B + {\left (b c n \log \relax (e) - a d n \log \relax (e)\right )} B^{2}}\,{d x} \]
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 \[ \int \frac {{\left (f+g\,x\right )}^2}{{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2} \,d x \]
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 \[ \int \frac {\left (f + g x\right )^{2}}{\left (A + B \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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