Optimal. Leaf size=154 \[ \frac {e^{-\frac {A}{B n}} (a+b x) \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-1/n} \text {Ei}\left (\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{B n}\right )}{B^2 (b c-a d) g^2 n^2 (c+d x)}-\frac {a+b x}{B (b c-a d) g^2 n (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )} \]
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Rubi [A]
time = 0.08, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {2551, 2334,
2337, 2209} \begin {gather*} \frac {(a+b x) e^{-\frac {A}{B n}} \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 g^2 n^2 (c+d x) (b c-a d)}-\frac {a+b x}{B g^2 n (c+d x) (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 2209
Rule 2334
Rule 2337
Rule 2551
Rubi steps
\begin {align*} \int \frac {1}{(c g+d 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}{(c g+d 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 = 0.12, size = 180, normalized size = 1.17 \begin {gather*} -\frac {e^{-\frac {A}{B n}} (a+b x) \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-1/n} \left (B e^{\frac {A}{B n}} n \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 ) \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 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (d g x +c g \right )^{2} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )^{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 254, normalized size = 1.65 \begin {gather*} -\frac {{\left (B b n x + B a n\right )} e^{\left (\frac {A + B}{B n}\right )} - {\left ({\left (A + B\right )} d x + {\left (A + B\right )} c + {\left (B d n x + B c n\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} \operatorname {log\_integral}\left (\frac {{\left (b x + a\right )} e^{\left (\frac {A + B}{B n}\right )}}{d x + c}\right )}{{\left ({\left (B^{3} b c d - B^{3} a d^{2}\right )} g^{2} n^{3} x + {\left (B^{3} b c^{2} - B^{3} a c d\right )} g^{2} n^{3}\right )} e^{\left (\frac {A + B}{B n}\right )} \log \left (\frac {b x + a}{d x + c}\right ) + {\left ({\left ({\left (A B^{2} + B^{3}\right )} b c d - {\left (A B^{2} + B^{3}\right )} a d^{2}\right )} g^{2} n^{2} x + {\left ({\left (A B^{2} + B^{3}\right )} b c^{2} - {\left (A B^{2} + B^{3}\right )} a c d\right )} g^{2} n^{2}\right )} e^{\left (\frac {A + B}{B n}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.88, size = 140, normalized size = 0.91 \begin {gather*} -{\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} {\left (\frac {b x + a}{{\left (B^{2} g^{2} n^{2} \log \left (\frac {b x + a}{d x + c}\right ) + A B g^{2} n + B^{2} g^{2} n\right )} {\left (d x + c\right )}} - \frac {{\rm Ei}\left (\frac {A}{B n} + \frac {1}{n} + \log \left (\frac {b x + a}{d x + c}\right )\right ) e^{\left (-\frac {A}{B n} - \frac {1}{n}\right )}}{B^{2} g^{2} n^{2}}\right )} \end {gather*}
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 \begin {gather*} \int \frac {1}{{\left (c\,g+d\,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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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