Optimal. Leaf size=152 \[ -\frac{(c+d x)^{n+1} (e+f x)^{-n} \left (\frac{d (e+f x)}{d e-c f}\right )^n (a d f-b (c f (1-n)+d e n)) \, _2F_1\left (n,n+1;n+2;-\frac{f (c+d x)}{d e-c f}\right )}{d^2 n (n+1) (d e-c f)}-\frac{(b c-a d) (c+d x)^n (e+f x)^{1-n}}{d n (d e-c f)} \]
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Rubi [A] time = 0.0740748, antiderivative size = 152, 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} \[ -\frac{(c+d x)^{n+1} (e+f x)^{-n} \left (\frac{d (e+f x)}{d e-c f}\right )^n (a d f-b c f (1-n)-b d e n) \, _2F_1\left (n,n+1;n+2;-\frac{f (c+d x)}{d e-c f}\right )}{d^2 n (n+1) (d e-c f)}-\frac{(b c-a d) (c+d x)^n (e+f x)^{1-n}}{d n (d e-c f)} \]
Antiderivative was successfully verified.
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Rule 79
Rule 70
Rule 69
Rubi steps
\begin{align*} \int (a+b x) (c+d x)^{-1+n} (e+f x)^{-n} \, dx &=-\frac{(b c-a d) (c+d x)^n (e+f x)^{1-n}}{d (d e-c f) n}-\frac{(a d f-b c f (1-n)-b d e n) \int (c+d x)^n (e+f x)^{-n} \, dx}{d (d e-c f) n}\\ &=-\frac{(b c-a d) (c+d x)^n (e+f x)^{1-n}}{d (d e-c f) n}-\frac{\left ((a d f-b c f (1-n)-b d e n) (e+f x)^{-n} \left (\frac{d (e+f x)}{d e-c f}\right )^n\right ) \int (c+d x)^n \left (\frac{d e}{d e-c f}+\frac{d f x}{d e-c f}\right )^{-n} \, dx}{d (d e-c f) n}\\ &=-\frac{(b c-a d) (c+d x)^n (e+f x)^{1-n}}{d (d e-c f) n}-\frac{(a d f-b c f (1-n)-b d e n) (c+d x)^{1+n} (e+f x)^{-n} \left (\frac{d (e+f x)}{d e-c f}\right )^n \, _2F_1\left (n,1+n;2+n;-\frac{f (c+d x)}{d e-c f}\right )}{d^2 (d e-c f) n (1+n)}\\ \end{align*}
Mathematica [A] time = 0.0950399, size = 124, normalized size = 0.82 \[ \frac{(c+d x)^n (e+f x)^{-n} \left (\frac{(c+d x) \left (\frac{d (e+f x)}{d e-c f}\right )^n (-a d f+b c (f-f n)+b d e n) \, _2F_1\left (n,n+1;n+2;\frac{f (c+d x)}{c f-d e}\right )}{n+1}+d (e+f x) (a d-b c)\right )}{d^2 n (d e-c f)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.052, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) \left ( dx+c \right ) ^{-1+n}}{ \left ( fx+e \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 (b x + a\right )}{\left (d x + c\right )}^{n - 1}}{{\left (f x + e\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 (b x + a\right )}{\left (d x + c\right )}^{n - 1}}{{\left (f x + e\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 (b x + a\right )}{\left (d x + c\right )}^{n - 1}}{{\left (f x + e\right )}^{n}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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