Optimal. Leaf size=118 \[ \frac{d (a+b x)^{-n} (e+f x)^n \left (-\frac{f (a+b x)}{b e-a f}\right )^n \, _2F_1\left (n,n;n+1;\frac{b (e+f x)}{b e-a f}\right )}{f^2 n}-\frac{(a+b x)^{1-n} (d e-c f) (e+f x)^{n-1}}{f (1-n) (b e-a f)} \]
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Rubi [A] time = 0.0539389, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {79, 70, 69} \[ \frac{d (a+b x)^{-n} (e+f x)^n \left (-\frac{f (a+b x)}{b e-a f}\right )^n \, _2F_1\left (n,n;n+1;\frac{b (e+f x)}{b e-a f}\right )}{f^2 n}-\frac{(a+b x)^{1-n} (d e-c f) (e+f x)^{n-1}}{f (1-n) (b e-a f)} \]
Antiderivative was successfully verified.
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Rule 79
Rule 70
Rule 69
Rubi steps
\begin{align*} \int (a+b x)^{-n} (c+d x) (e+f x)^{-2+n} \, dx &=-\frac{(d e-c f) (a+b x)^{1-n} (e+f x)^{-1+n}}{f (b e-a f) (1-n)}+\frac{d \int (a+b x)^{-n} (e+f x)^{-1+n} \, dx}{f}\\ &=-\frac{(d e-c f) (a+b x)^{1-n} (e+f x)^{-1+n}}{f (b e-a f) (1-n)}+\frac{\left (d (a+b x)^{-n} \left (\frac{f (a+b x)}{-b e+a f}\right )^n\right ) \int (e+f x)^{-1+n} \left (-\frac{a f}{b e-a f}-\frac{b f x}{b e-a f}\right )^{-n} \, dx}{f}\\ &=-\frac{(d e-c f) (a+b x)^{1-n} (e+f x)^{-1+n}}{f (b e-a f) (1-n)}+\frac{d (a+b x)^{-n} \left (-\frac{f (a+b x)}{b e-a f}\right )^n (e+f x)^n \, _2F_1\left (n,n;1+n;\frac{b (e+f x)}{b e-a f}\right )}{f^2 n}\\ \end{align*}
Mathematica [A] time = 0.157032, size = 105, normalized size = 0.89 \[ \frac{(a+b x)^{-n} (e+f x)^n \left (\frac{f (a+b x) (d e-c f)}{(n-1) (e+f x) (b e-a f)}+\frac{d \left (\frac{f (a+b x)}{a f-b e}\right )^n \, _2F_1\left (n,n;n+1;\frac{b (e+f x)}{b e-a f}\right )}{n}\right )}{f^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.051, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx+c \right ) \left ( fx+e \right ) ^{-2+n}}{ \left ( bx+a \right ) ^{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 \frac{{\left (d x + c\right )}{\left (f x + e\right )}^{n - 2}}{{\left (b x + a\right )}^{n}}\,{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 (\frac{{\left (d x + c\right )}{\left (f x + e\right )}^{n - 2}}{{\left (b x + a\right )}^{n}}, 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 \frac{{\left (d x + c\right )}{\left (f x + e\right )}^{n - 2}}{{\left (b x + a\right )}^{n}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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