Optimal. Leaf size=314 \[ -\frac {2 b e^{\frac {2 A}{B n}} (c+d x)^2 \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{2/n} \text {Ei}\left (-\frac {2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{B n}\right )}{B^2 g^3 n^2 (a+b x)^2 (b c-a d)^2}+\frac {d e^{\frac {A}{B n}} (c+d x) \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{\frac {1}{n}} \text {Ei}\left (-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{B n}\right )}{B^2 g^3 n^2 (a+b x) (b c-a d)^2}-\frac {b (c+d x)^2}{B g^3 n (a+b x)^2 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}+\frac {d (c+d x)}{B g^3 n (a+b x) (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )} \]
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Rubi [F] time = 0.09, 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 = {} \[ \int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{(a g+b g x)^3 \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)^3 \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 = 0.62, size = 254, normalized size = 0.81 \[ \frac {(c+d x) \left (-2 b e^{\frac {2 A}{B n}} (c+d x) \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{2/n} \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right ) \text {Ei}\left (-\frac {2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{B n}\right )+d (a+b x) e^{\frac {A}{B n}} \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{\frac {1}{n}} \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right ) \text {Ei}\left (-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{B n}\right )+B n (a d-b c)\right )}{B^2 g^3 n^2 (a+b x)^2 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.93, size = 755, normalized size = 2.40 \[ -\frac {{\left (B b c d - B a d^{2}\right )} n x - {\left (A b^{2} d x^{2} + 2 \, A a b d x + A a^{2} d + {\left (B b^{2} d x^{2} + 2 \, B a b d x + B a^{2} d\right )} \log \relax (e) + {\left (B b^{2} d n x^{2} + 2 \, B a b d n x + B a^{2} d 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 ) + 2 \, {\left (A b^{3} x^{2} + 2 \, A a b^{2} x + A a^{2} b + {\left (B b^{3} x^{2} + 2 \, B a b^{2} x + B a^{2} b\right )} \log \relax (e) + {\left (B b^{3} n x^{2} + 2 \, B a b^{2} n x + B a^{2} b n\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} e^{\left (\frac {2 \, {\left (B \log \relax (e) + A\right )}}{B n}\right )} \operatorname {log\_integral}\left (\frac {{\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} e^{\left (-\frac {2 \, {\left (B \log \relax (e) + A\right )}}{B n}\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) + {\left (B b c^{2} - B a c d\right )} n}{{\left (A B^{2} b^{4} c^{2} - 2 \, A B^{2} a b^{3} c d + A B^{2} a^{2} b^{2} d^{2}\right )} g^{3} n^{2} x^{2} + 2 \, {\left (A B^{2} a b^{3} c^{2} - 2 \, A B^{2} a^{2} b^{2} c d + A B^{2} a^{3} b d^{2}\right )} g^{3} n^{2} x + {\left (A B^{2} a^{2} b^{2} c^{2} - 2 \, A B^{2} a^{3} b c d + A B^{2} a^{4} d^{2}\right )} g^{3} n^{2} + {\left ({\left (B^{3} b^{4} c^{2} - 2 \, B^{3} a b^{3} c d + B^{3} a^{2} b^{2} d^{2}\right )} g^{3} n^{2} x^{2} + 2 \, {\left (B^{3} a b^{3} c^{2} - 2 \, B^{3} a^{2} b^{2} c d + B^{3} a^{3} b d^{2}\right )} g^{3} n^{2} x + {\left (B^{3} a^{2} b^{2} c^{2} - 2 \, B^{3} a^{3} b c d + B^{3} a^{4} d^{2}\right )} g^{3} n^{2}\right )} \log \relax (e) + {\left ({\left (B^{3} b^{4} c^{2} - 2 \, B^{3} a b^{3} c d + B^{3} a^{2} b^{2} d^{2}\right )} g^{3} n^{3} x^{2} + 2 \, {\left (B^{3} a b^{3} c^{2} - 2 \, B^{3} a^{2} b^{2} c d + B^{3} a^{3} b d^{2}\right )} g^{3} n^{3} x + {\left (B^{3} a^{2} b^{2} c^{2} - 2 \, B^{3} a^{3} b c d + B^{3} a^{4} d^{2}\right )} g^{3} 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 \[ \int \frac {1}{{\left (b g x + a g\right )}^{3} {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.29, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b g x +a g \right )^{3} \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}}\, dx \]
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^{2} b c g^{3} n - a^{3} d g^{3} n\right )} A B + {\left (a^{2} b c g^{3} n \log \relax (e) - a^{3} d g^{3} n \log \relax (e)\right )} B^{2} + {\left ({\left (b^{3} c g^{3} n - a b^{2} d g^{3} n\right )} A B + {\left (b^{3} c g^{3} n \log \relax (e) - a b^{2} d g^{3} n \log \relax (e)\right )} B^{2}\right )} x^{2} + 2 \, {\left ({\left (a b^{2} c g^{3} n - a^{2} b d g^{3} n\right )} A B + {\left (a b^{2} c g^{3} n \log \relax (e) - a^{2} b d g^{3} n \log \relax (e)\right )} B^{2}\right )} x + {\left ({\left (b^{3} c g^{3} n - a b^{2} d g^{3} n\right )} B^{2} x^{2} + 2 \, {\left (a b^{2} c g^{3} n - a^{2} b d g^{3} n\right )} B^{2} x + {\left (a^{2} b c g^{3} n - a^{3} d g^{3} n\right )} B^{2}\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - {\left ({\left (b^{3} c g^{3} n - a b^{2} d g^{3} n\right )} B^{2} x^{2} + 2 \, {\left (a b^{2} c g^{3} n - a^{2} b d g^{3} n\right )} B^{2} x + {\left (a^{2} b c g^{3} n - a^{3} d g^{3} n\right )} B^{2}\right )} \log \left ({\left (d x + c\right )}^{n}\right )} - \int \frac {b d x + 2 \, b c - a d}{{\left ({\left (b^{4} c g^{3} n - a b^{3} d g^{3} n\right )} A B + {\left (b^{4} c g^{3} n \log \relax (e) - a b^{3} d g^{3} n \log \relax (e)\right )} B^{2}\right )} x^{3} + {\left (a^{3} b c g^{3} n - a^{4} d g^{3} n\right )} A B + {\left (a^{3} b c g^{3} n \log \relax (e) - a^{4} d g^{3} n \log \relax (e)\right )} B^{2} + 3 \, {\left ({\left (a b^{3} c g^{3} n - a^{2} b^{2} d g^{3} n\right )} A B + {\left (a b^{3} c g^{3} n \log \relax (e) - a^{2} b^{2} d g^{3} n \log \relax (e)\right )} B^{2}\right )} x^{2} + 3 \, {\left ({\left (a^{2} b^{2} c g^{3} n - a^{3} b d g^{3} n\right )} A B + {\left (a^{2} b^{2} c g^{3} n \log \relax (e) - a^{3} b d g^{3} n \log \relax (e)\right )} B^{2}\right )} x + {\left ({\left (b^{4} c g^{3} n - a b^{3} d g^{3} n\right )} B^{2} x^{3} + 3 \, {\left (a b^{3} c g^{3} n - a^{2} b^{2} d g^{3} n\right )} B^{2} x^{2} + 3 \, {\left (a^{2} b^{2} c g^{3} n - a^{3} b d g^{3} n\right )} B^{2} x + {\left (a^{3} b c g^{3} n - a^{4} d g^{3} n\right )} B^{2}\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - {\left ({\left (b^{4} c g^{3} n - a b^{3} d g^{3} n\right )} B^{2} x^{3} + 3 \, {\left (a b^{3} c g^{3} n - a^{2} b^{2} d g^{3} n\right )} B^{2} x^{2} + 3 \, {\left (a^{2} b^{2} c g^{3} n - a^{3} b d g^{3} n\right )} B^{2} x + {\left (a^{3} b c g^{3} n - a^{4} d g^{3} n\right )} B^{2}\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 = -0.00 \[ \int \frac {1}{{\left (a\,g+b\,g\,x\right )}^3\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2} \,d x \]
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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