Optimal. Leaf size=139 \[ \frac{(b e-a f) (a+b x)^{m+1} (e+f x)^n (c+d x)^{-m-n} \left (\frac{b (e+f x)}{b e-a f}\right )^{-n} \left (\frac{b (c+d x)}{b c-a d}\right )^{m+n} F_1\left (m+1;m+n,-n-1;m+2;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b^2 (m+1)} \]
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Rubi [A] time = 0.093506, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {140, 139, 138} \[ \frac{(b e-a f) (a+b x)^{m+1} (e+f x)^n (c+d x)^{-m-n} \left (\frac{b (e+f x)}{b e-a f}\right )^{-n} \left (\frac{b (c+d x)}{b c-a d}\right )^{m+n} F_1\left (m+1;m+n,-n-1;m+2;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b^2 (m+1)} \]
Antiderivative was successfully verified.
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Rule 140
Rule 139
Rule 138
Rubi steps
\begin{align*} \int (a+b x)^m (c+d x)^{-m-n} (e+f x)^{1+n} \, dx &=\left ((c+d x)^{-m-n} \left (\frac{b (c+d x)}{b c-a d}\right )^{m+n}\right ) \int (a+b x)^m \left (\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}\right )^{-m-n} (e+f x)^{1+n} \, dx\\ &=\frac{\left ((b e-a f) (c+d x)^{-m-n} \left (\frac{b (c+d x)}{b c-a d}\right )^{m+n} (e+f x)^n \left (\frac{b (e+f x)}{b e-a f}\right )^{-n}\right ) \int (a+b x)^m \left (\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}\right )^{-m-n} \left (\frac{b e}{b e-a f}+\frac{b f x}{b e-a f}\right )^{1+n} \, dx}{b}\\ &=\frac{(b e-a f) (a+b x)^{1+m} (c+d x)^{-m-n} \left (\frac{b (c+d x)}{b c-a d}\right )^{m+n} (e+f x)^n \left (\frac{b (e+f x)}{b e-a f}\right )^{-n} F_1\left (1+m;m+n,-1-n;2+m;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b^2 (1+m)}\\ \end{align*}
Mathematica [A] time = 0.228209, size = 133, normalized size = 0.96 \[ \frac{(a+b x)^{m+1} (e+f x)^{n+1} (c+d x)^{-m-n} \left (\frac{b (e+f x)}{b e-a f}\right )^{-n-1} \left (\frac{b (c+d x)}{b c-a d}\right )^{m+n} F_1\left (m+1;m+n,-n-1;m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )}{b (m+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.147, size = 0, normalized size = 0. \begin{align*} \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{-n-m} \left ( fx+e \right ) ^{1+n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - n}{\left (f x + e\right )}^{n + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - n}{\left (f x + e\right )}^{n + 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - n}{\left (f x + e\right )}^{n + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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