Integrand size = 35, antiderivative size = 153 \[ \int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx=-\frac {e^{\frac {A}{B n}} \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{\frac {1}{n}} (c+d x) \operatorname {ExpIntegralEi}\left (-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{B n}\right )}{B^2 (b c-a d) g^2 n^2 (a+b x)}-\frac {c+d x}{B (b c-a d) g^2 n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )} \]
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Time = 0.10 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {2549, 2343, 2347, 2209} \[ \int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx=-\frac {e^{\frac {A}{B n}} (c+d x) \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{B n}\right )}{B^2 g^2 n^2 (a+b x) (b c-a d)}-\frac {c+d x}{B g^2 n (a+b x) (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )} \]
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Rule 2209
Rule 2343
Rule 2347
Rule 2549
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{x^2 \left (A+B \log \left (e x^n\right )\right )^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d) g^2} \\ & = -\frac {c+d x}{B (b c-a d) g^2 n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}-\frac {\text {Subst}\left (\int \frac {1}{x^2 \left (A+B \log \left (e x^n\right )\right )} \, dx,x,\frac {a+b x}{c+d x}\right )}{B (b c-a d) g^2 n} \\ & = -\frac {c+d x}{B (b c-a d) g^2 n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}-\frac {\left (\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{\frac {1}{n}} (c+d x)\right ) \text {Subst}\left (\int \frac {e^{-\frac {x}{n}}}{A+B x} \, dx,x,\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{B (b c-a d) g^2 n^2 (a+b x)} \\ & = -\frac {e^{\frac {A}{B n}} \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{\frac {1}{n}} (c+d x) \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 (b c-a d) g^2 n^2 (a+b x)}-\frac {c+d x}{B (b c-a d) g^2 n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx=-\frac {(c+d x) \left (B n+e^{\frac {A}{B n}} \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{B n}\right ) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )\right )}{B^2 (b c-a d) g^2 n^2 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )} \]
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\[\int \frac {1}{\left (b g x +a g \right )^{2} {\left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}^{2}}d x\]
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none
Time = 0.27 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.79 \[ \int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx=-\frac {B d n x + B c n + {\left (A b x + A a + {\left (B b x + B a\right )} \log \left (e\right ) + {\left (B b n x + B a n\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} e^{\left (\frac {B \log \left (e\right ) + A}{B n}\right )} \operatorname {log\_integral}\left (\frac {{\left (d x + c\right )} e^{\left (-\frac {B \log \left (e\right ) + A}{B n}\right )}}{b x + a}\right )}{{\left (A B^{2} b^{2} c - A B^{2} a b d\right )} g^{2} n^{2} x + {\left (A B^{2} a b c - A B^{2} a^{2} d\right )} g^{2} n^{2} + {\left ({\left (B^{3} b^{2} c - B^{3} a b d\right )} g^{2} n^{2} x + {\left (B^{3} a b c - B^{3} a^{2} d\right )} g^{2} n^{2}\right )} \log \left (e\right ) + {\left ({\left (B^{3} b^{2} c - B^{3} a b d\right )} g^{2} n^{3} x + {\left (B^{3} a b c - B^{3} a^{2} d\right )} g^{2} n^{3}\right )} \log \left (\frac {b x + a}{d x + c}\right )} \]
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Timed out. \[ \int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx=\int { \frac {1}{{\left (b g x + a g\right )}^{2} {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}} \,d x } \]
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\[ \int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx=\int { \frac {1}{{\left (b g x + a g\right )}^{2} {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx=\int \frac {1}{{\left (a\,g+b\,g\,x\right )}^2\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2} \,d x \]
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