Integrand size = 35, antiderivative size = 256 \[ \int \frac {1}{(c g+d g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx=\frac {b e^{-\frac {A}{B n}} (a+b x) \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{B n}\right )}{B^2 (b c-a d)^2 g^3 n^2 (c+d x)}-\frac {2 d e^{-\frac {2 A}{B n}} (a+b x)^2 \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{B n}\right )}{B^2 (b c-a d)^2 g^3 n^2 (c+d x)^2}-\frac {a+b x}{B (b c-a d) g^3 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )} \]
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Time = 0.22 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {2551, 2357, 2367, 2337, 2209, 2347} \[ \int \frac {1}{(c g+d g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx=-\frac {2 d (a+b x)^2 e^{-\frac {2 A}{B n}} \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{B n}\right )}{B^2 g^3 n^2 (c+d x)^2 (b c-a d)^2}+\frac {b (a+b x) e^{-\frac {A}{B n}} \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{B n}\right )}{B^2 g^3 n^2 (c+d x) (b c-a d)^2}-\frac {a+b x}{B g^3 n (c+d x)^2 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )} \]
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Rule 2209
Rule 2337
Rule 2347
Rule 2357
Rule 2367
Rule 2551
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {b-d x}{\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)^2 g^3} \\ & = -\frac {a+b x}{B (b c-a d) g^3 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}+\frac {2 \text {Subst}\left (\int \frac {b-d x}{A+B \log \left (e x^n\right )} \, dx,x,\frac {a+b x}{c+d x}\right )}{B (b c-a d)^2 g^3 n}-\frac {b \text {Subst}\left (\int \frac {1}{A+B \log \left (e x^n\right )} \, dx,x,\frac {a+b x}{c+d x}\right )}{B (b c-a d)^2 g^3 n} \\ & = -\frac {a+b x}{B (b c-a d) g^3 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}+\frac {2 \text {Subst}\left (\int \left (\frac {b}{A+B \log \left (e x^n\right )}-\frac {d x}{A+B \log \left (e x^n\right )}\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{B (b c-a d)^2 g^3 n}-\frac {\left (b (a+b x) \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-1/n}\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)^2 g^3 n^2 (c+d x)} \\ & = -\frac {b e^{-\frac {A}{B n}} (a+b x) \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-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 (b c-a d)^2 g^3 n^2 (c+d x)}-\frac {a+b x}{B (b c-a d) g^3 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}+\frac {(2 b) \text {Subst}\left (\int \frac {1}{A+B \log \left (e x^n\right )} \, dx,x,\frac {a+b x}{c+d x}\right )}{B (b c-a d)^2 g^3 n}-\frac {(2 d) \text {Subst}\left (\int \frac {x}{A+B \log \left (e x^n\right )} \, dx,x,\frac {a+b x}{c+d x}\right )}{B (b c-a d)^2 g^3 n} \\ & = -\frac {b e^{-\frac {A}{B n}} (a+b x) \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-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 (b c-a d)^2 g^3 n^2 (c+d x)}-\frac {a+b x}{B (b c-a d) g^3 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}-\frac {\left (2 d (a+b x)^2 \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-2/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {2 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)^2 g^3 n^2 (c+d x)^2}+\frac {\left (2 b (a+b x) \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-1/n}\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)^2 g^3 n^2 (c+d x)} \\ & = \frac {b e^{-\frac {A}{B n}} (a+b x) \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-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 (b c-a d)^2 g^3 n^2 (c+d x)}-\frac {2 d e^{-\frac {2 A}{B n}} (a+b 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 (b c-a d)^2 g^3 n^2 (c+d x)^2}-\frac {a+b x}{B (b c-a d) g^3 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )} \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.12 \[ \int \frac {1}{(c g+d g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx=\frac {e^{-\frac {2 A}{B n}} (a+b x) \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-2/n} \left (-B (b c-a d) e^{\frac {2 A}{B n}} n \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{2/n}+b e^{\frac {A}{B n}} \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{\frac {1}{n}} (c+d x) \operatorname {ExpIntegralEi}\left (\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{B n}\right ) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-2 d (a+b x) \operatorname {ExpIntegralEi}\left (\frac {2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{B n}\right ) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )\right )}{B^2 (b c-a d)^2 g^3 n^2 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )} \]
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\[\int \frac {1}{\left (d g x +c g \right )^{3} {\left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}^{2}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 770 vs. \(2 (256) = 512\).
Time = 0.28 (sec) , antiderivative size = 770, normalized size of antiderivative = 3.01 \[ \int \frac {1}{(c g+d g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx=\frac {{\left ({\left (A b d^{2} x^{2} + 2 \, A b c d x + A b c^{2} + {\left (B b d^{2} x^{2} + 2 \, B b c d x + B b c^{2}\right )} \log \left (e\right ) + {\left (B b d^{2} n x^{2} + 2 \, B b c d n x + B b c^{2} 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 (b x + a\right )} e^{\left (\frac {B \log \left (e\right ) + A}{B n}\right )}}{d x + c}\right ) - {\left ({\left (B b^{2} c - B a b d\right )} n x + {\left (B a b c - B a^{2} d\right )} n\right )} e^{\left (\frac {2 \, {\left (B \log \left (e\right ) + A\right )}}{B n}\right )} - 2 \, {\left (A d^{3} x^{2} + 2 \, A c d^{2} x + A c^{2} d + {\left (B d^{3} x^{2} + 2 \, B c d^{2} x + B c^{2} d\right )} \log \left (e\right ) + {\left (B d^{3} n x^{2} + 2 \, B c d^{2} n x + B c^{2} d n\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} \operatorname {log\_integral}\left (\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} e^{\left (\frac {2 \, {\left (B \log \left (e\right ) + A\right )}}{B n}\right )}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )\right )} e^{\left (-\frac {2 \, {\left (B \log \left (e\right ) + A\right )}}{B n}\right )}}{{\left (A B^{2} b^{2} c^{2} d^{2} - 2 \, A B^{2} a b c d^{3} + A B^{2} a^{2} d^{4}\right )} g^{3} n^{2} x^{2} + 2 \, {\left (A B^{2} b^{2} c^{3} d - 2 \, A B^{2} a b c^{2} d^{2} + A B^{2} a^{2} c d^{3}\right )} g^{3} n^{2} x + {\left (A B^{2} b^{2} c^{4} - 2 \, A B^{2} a b c^{3} d + A B^{2} a^{2} c^{2} d^{2}\right )} g^{3} n^{2} + {\left ({\left (B^{3} b^{2} c^{2} d^{2} - 2 \, B^{3} a b c d^{3} + B^{3} a^{2} d^{4}\right )} g^{3} n^{2} x^{2} + 2 \, {\left (B^{3} b^{2} c^{3} d - 2 \, B^{3} a b c^{2} d^{2} + B^{3} a^{2} c d^{3}\right )} g^{3} n^{2} x + {\left (B^{3} b^{2} c^{4} - 2 \, B^{3} a b c^{3} d + B^{3} a^{2} c^{2} d^{2}\right )} g^{3} n^{2}\right )} \log \left (e\right ) + {\left ({\left (B^{3} b^{2} c^{2} d^{2} - 2 \, B^{3} a b c d^{3} + B^{3} a^{2} d^{4}\right )} g^{3} n^{3} x^{2} + 2 \, {\left (B^{3} b^{2} c^{3} d - 2 \, B^{3} a b c^{2} d^{2} + B^{3} a^{2} c d^{3}\right )} g^{3} n^{3} x + {\left (B^{3} b^{2} c^{4} - 2 \, B^{3} a b c^{3} d + B^{3} a^{2} c^{2} d^{2}\right )} g^{3} n^{3}\right )} \log \left (\frac {b x + a}{d x + c}\right )} \]
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Timed out. \[ \int \frac {1}{(c g+d g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(c g+d g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx=\int { \frac {1}{{\left (d g x + c g\right )}^{3} {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}} \,d x } \]
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Time = 0.42 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.27 \[ \int \frac {1}{(c g+d g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx={\left (\frac {b {\rm Ei}\left (\frac {\log \left (e\right )}{n} + \frac {A}{B n} + \log \left (\frac {b x + a}{d x + c}\right )\right ) e^{\left (-\frac {A}{B n}\right )}}{{\left (B^{2} b c g^{3} n^{2} - B^{2} a d g^{3} n^{2}\right )} e^{\left (\frac {1}{n}\right )}} - \frac {2 \, d {\rm Ei}\left (\frac {2 \, \log \left (e\right )}{n} + \frac {2 \, A}{B n} + 2 \, \log \left (\frac {b x + a}{d x + c}\right )\right ) e^{\left (-\frac {2 \, A}{B n}\right )}}{{\left (B^{2} b c g^{3} n^{2} - B^{2} a d g^{3} n^{2}\right )} e^{\frac {2}{n}}} - \frac {\frac {{\left (b x + a\right )} b}{d x + c} - \frac {{\left (b x + a\right )}^{2} d}{{\left (d x + c\right )}^{2}}}{B^{2} b c g^{3} n^{2} \log \left (\frac {b x + a}{d x + c}\right ) - B^{2} a d g^{3} n^{2} \log \left (\frac {b x + a}{d x + c}\right ) + B^{2} b c g^{3} n \log \left (e\right ) - B^{2} a d g^{3} n \log \left (e\right ) + A B b c g^{3} n - A B a d g^{3} n}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \]
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Timed out. \[ \int \frac {1}{(c g+d g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx=\int \frac {1}{{\left (c\,g+d\,g\,x\right )}^3\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2} \,d x \]
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