Optimal. Leaf size=306 \[ \frac {(m+1)^2 (a+b x) e^{\frac {A (m+1)}{B n}} (g (a+b x))^{-m-2} (i (c+d x))^{m+2} \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{\frac {m+1}{n}} \text {Ei}\left (-\frac {(m+1) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{B n}\right )}{2 B^3 i^2 n^3 (c+d x) (b c-a d)}+\frac {(m+1) (a+b x) (g (a+b x))^{-m-2} (i (c+d x))^{m+2}}{2 B^2 i^2 n^2 (c+d x) (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}-\frac {(a+b x) (g (a+b x))^{-m-2} (i (c+d x))^{m+2}}{2 B i^2 n (c+d x) (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2} \]
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Rubi [F] time = 0.76, 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 = {} \[ \int \frac {(a g+b g x)^{-2-m} (c i+d i x)^m}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {(223 c+223 d x)^m (a g+b g x)^{-2-m}}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3} \, dx &=\int \frac {(223 c+223 d x)^m (a g+b g x)^{-2-m}}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3} \, dx\\ \end {align*}
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Mathematica [F] time = 0.33, size = 0, normalized size = 0.00 \[ \int \frac {(a g+b g x)^{-2-m} (c i+d i x)^m}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3} \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 0.93, size = 815, normalized size = 2.66 \[ -\frac {{\left (B^{2} a c g^{2} n^{2} + {\left (B^{2} b d g^{2} n^{2} - {\left (A B b d g^{2} m + A B b d g^{2}\right )} n\right )} x^{2} - {\left (A B a c g^{2} m + A B a c g^{2}\right )} n + {\left ({\left (B^{2} b c + B^{2} a d\right )} g^{2} n^{2} - {\left ({\left (A B b c + A B a d\right )} g^{2} m + {\left (A B b c + A B a d\right )} g^{2}\right )} n\right )} x - {\left ({\left (B^{2} b d g^{2} m + B^{2} b d g^{2}\right )} n x^{2} + {\left ({\left (B^{2} b c + B^{2} a d\right )} g^{2} m + {\left (B^{2} b c + B^{2} a d\right )} g^{2}\right )} n x + {\left (B^{2} a c g^{2} m + B^{2} a c g^{2}\right )} n\right )} \log \relax (e) - {\left ({\left (B^{2} b d g^{2} m + B^{2} b d g^{2}\right )} n^{2} x^{2} + {\left ({\left (B^{2} b c + B^{2} a d\right )} g^{2} m + {\left (B^{2} b c + B^{2} a d\right )} g^{2}\right )} n^{2} x + {\left (B^{2} a c g^{2} m + B^{2} a c g^{2}\right )} n^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} {\left (b g x + a g\right )}^{-m - 2} e^{\left (m \log \left (b g x + a g\right ) - m \log \left (\frac {b x + a}{d x + c}\right ) + m \log \left (\frac {i}{g}\right )\right )} - {\left ({\left (B^{2} m^{2} + 2 \, B^{2} m + B^{2}\right )} n^{2} \log \left (\frac {b x + a}{d x + c}\right )^{2} + A^{2} m^{2} + 2 \, A^{2} m + {\left (B^{2} m^{2} + 2 \, B^{2} m + B^{2}\right )} \log \relax (e)^{2} + 2 \, {\left (A B m^{2} + 2 \, A B m + A B\right )} n \log \left (\frac {b x + a}{d x + c}\right ) + A^{2} + 2 \, {\left (A B m^{2} + 2 \, A B m + {\left (B^{2} m^{2} + 2 \, B^{2} m + B^{2}\right )} n \log \left (\frac {b x + a}{d x + c}\right ) + A B\right )} \log \relax (e)\right )} {\rm Ei}\left (-\frac {{\left (B m + B\right )} n \log \left (\frac {b x + a}{d x + c}\right ) + A m + {\left (B m + B\right )} \log \relax (e) + A}{B n}\right ) e^{\left (\frac {B m n \log \left (\frac {i}{g}\right ) + A m + {\left (B m + B\right )} \log \relax (e) + A}{B n}\right )}}{2 \, {\left ({\left (B^{5} b c - B^{5} a d\right )} g^{2} n^{5} \log \left (\frac {b x + a}{d x + c}\right )^{2} + {\left (B^{5} b c - B^{5} a d\right )} g^{2} n^{3} \log \relax (e)^{2} + 2 \, {\left (A B^{4} b c - A B^{4} a d\right )} g^{2} n^{4} \log \left (\frac {b x + a}{d x + c}\right ) + {\left (A^{2} B^{3} b c - A^{2} B^{3} a d\right )} g^{2} n^{3} + 2 \, {\left ({\left (B^{5} b c - B^{5} a d\right )} g^{2} n^{4} \log \left (\frac {b x + a}{d x + c}\right ) + {\left (A B^{4} b c - A B^{4} a d\right )} g^{2} n^{3}\right )} \log \relax (e)\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 \[ \int \frac {{\left (b g x + a g\right )}^{-m - 2} {\left (d i x + c i\right )}^{m}}{{\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-1)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (b g x +a g \right )^{-m -2} \left (d i x +c i \right )^{m}}{\left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{3}}\, dx \]
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 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (c\,i+d\,i\,x\right )}^m}{{\left (a\,g+b\,g\,x\right )}^{m+2}\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^3} \,d x \]
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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