Optimal. Leaf size=103 \[ -\frac {(d+e x)^m (f+g x)^{n+1} (a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{-m} \, _2F_1\left (1,-m+n+2;n+2;\frac {c d (f+g x)}{c d f-a e g}\right )}{(n+1) (c d f-a e g)} \]
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Rubi [A] time = 0.09, antiderivative size = 107, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 3, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {891, 70, 69} \[ \frac {(d+e x)^m (f+g x)^{n+1} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{-m} \left (-\frac {g (a e+c d x)}{c d f-a e g}\right )^m \, _2F_1\left (m,n+1;n+2;\frac {c d (f+g x)}{c d f-a e g}\right )}{g (n+1)} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 891
Rubi steps
\begin {align*} \int (d+e x)^m (f+g x)^n \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx &=\left ((a e+c d x)^m (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m}\right ) \int (a e+c d x)^{-m} (f+g x)^n \, dx\\ &=\left (\left (\frac {g (a e+c d x)}{-c d f+a e g}\right )^m (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m}\right ) \int (f+g x)^n \left (-\frac {a e g}{c d f-a e g}-\frac {c d g x}{c d f-a e g}\right )^{-m} \, dx\\ &=\frac {\left (-\frac {g (a e+c d x)}{c d f-a e g}\right )^m (d+e x)^m (f+g x)^{1+n} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, _2F_1\left (m,1+n;2+n;\frac {c d (f+g x)}{c d f-a e g}\right )}{g (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 95, normalized size = 0.92 \[ \frac {(d+e x)^m (f+g x)^{n+1} ((d+e x) (a e+c d x))^{-m} \left (\frac {g (a e+c d x)}{a e g-c d f}\right )^m \, _2F_1\left (m,n+1;n+2;\frac {c d (f+g x)}{c d f-a e g}\right )}{g (n+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e x + d\right )}^{m} {\left (g x + f\right )}^{n}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{m}}, x\right ) \]
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 \[ \int \frac {{\left (e x + d\right )}^{m} {\left (g x + f\right )}^{n}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{m}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.20, size = 0, normalized size = 0.00 \[ \int \left (c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x \right )^{-m} \left (e x +d \right )^{m} \left (g x +f \right )^{n}\, dx \]
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 \[ \int \frac {{\left (e x + d\right )}^{m} {\left (g x + f\right )}^{n}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{m}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (f+g\,x\right )}^n\,{\left (d+e\,x\right )}^m}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^m} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
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