Optimal. Leaf size=153 \[ -\frac {e^{\frac {A}{B n}} (c+d x) \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{\frac {1}{n}} \text {Ei}\left (-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{B n}\right )}{B^2 g^2 n^2 (a+b x) (b c-a d)}-\frac {c+d x}{B g^2 n (a+b x) (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )} \]
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Rubi [F] time = 0.10, 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 = {} \[ \int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx &=\int \frac {1}{(a g+b g 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 = 0.19, size = 146, normalized size = 0.95 \[ -\frac {(c+d x) \left (e^{\frac {A}{B n}} \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{\frac {1}{n}} \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right ) \text {Ei}\left (-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{B n}\right )+B n\right )}{B^2 g^2 n^2 (a+b x) (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 274, normalized size = 1.79 \[ -\frac {B d n x + B c n + {\left (A b x + A a + {\left (B b x + B a\right )} \log \relax (e) + {\left (B b n x + B a n\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} e^{\left (\frac {B \log \relax (e) + A}{B n}\right )} \operatorname {log\_integral}\left (\frac {{\left (d x + c\right )} e^{\left (-\frac {B \log \relax (e) + A}{B n}\right )}}{b x + a}\right )}{{\left (A B^{2} b^{2} c - A B^{2} a b d\right )} g^{2} n^{2} x + {\left (A B^{2} a b c - A B^{2} a^{2} d\right )} g^{2} n^{2} + {\left ({\left (B^{3} b^{2} c - B^{3} a b d\right )} g^{2} n^{2} x + {\left (B^{3} a b c - B^{3} a^{2} d\right )} g^{2} n^{2}\right )} \log \relax (e) + {\left ({\left (B^{3} b^{2} c - B^{3} a b d\right )} g^{2} n^{3} x + {\left (B^{3} a b c - B^{3} a^{2} d\right )} g^{2} n^{3}\right )} \log \left (\frac {b x + a}{d x + c}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b g x + a g\right )}^{2} {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.28, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b g x +a g \right )^{2} \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {d x + c}{{\left (a b c g^{2} n - a^{2} d g^{2} n\right )} A B + {\left (a b c g^{2} n \log \relax (e) - a^{2} d g^{2} n \log \relax (e)\right )} B^{2} + {\left ({\left (b^{2} c g^{2} n - a b d g^{2} n\right )} A B + {\left (b^{2} c g^{2} n \log \relax (e) - a b d g^{2} n \log \relax (e)\right )} B^{2}\right )} x + {\left ({\left (b^{2} c g^{2} n - a b d g^{2} n\right )} B^{2} x + {\left (a b c g^{2} n - a^{2} d g^{2} n\right )} B^{2}\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - {\left ({\left (b^{2} c g^{2} n - a b d g^{2} n\right )} B^{2} x + {\left (a b c g^{2} n - a^{2} d g^{2} n\right )} B^{2}\right )} \log \left ({\left (d x + c\right )}^{n}\right )} + \int -\frac {1}{B^{2} a^{2} g^{2} n \log \relax (e) + A B a^{2} g^{2} n + {\left (B^{2} b^{2} g^{2} n \log \relax (e) + A B b^{2} g^{2} n\right )} x^{2} + 2 \, {\left (B^{2} a b g^{2} n \log \relax (e) + A B a b g^{2} n\right )} x + {\left (B^{2} b^{2} g^{2} n x^{2} + 2 \, B^{2} a b g^{2} n x + B^{2} a^{2} g^{2} n\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - {\left (B^{2} b^{2} g^{2} n x^{2} + 2 \, B^{2} a b g^{2} n x + B^{2} a^{2} g^{2} n\right )} \log \left ({\left (d x + c\right )}^{n}\right )}\,{d x} \]
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 \[ \int \frac {1}{{\left (a\,g+b\,g\,x\right )}^2\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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