Optimal. Leaf size=118 \[ \frac{d (a+b x)^{-n} (e+f x)^n \left (-\frac{f (a+b x)}{b e-a f}\right )^n \, _2F_1\left (n,n;n+1;\frac{b (e+f x)}{b e-a f}\right )}{f^2 n}-\frac{(a+b x)^{1-n} (d e-c f) (e+f x)^{n-1}}{f (1-n) (b e-a f)} \]
[Out]
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Rubi [A] time = 0.179949, 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 \[ \frac{d (a+b x)^{-n} (e+f x)^n \left (-\frac{f (a+b x)}{b e-a f}\right )^n \, _2F_1\left (n,n;n+1;\frac{b (e+f x)}{b e-a f}\right )}{f^2 n}-\frac{(a+b x)^{1-n} (d e-c f) (e+f x)^{n-1}}{f (1-n) (b e-a f)} \]
Antiderivative was successfully verified.
[In] Int[((c + d*x)*(e + f*x)^(-2 + n))/(a + b*x)^n,x]
[Out]
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Rubi in Sympy [A] time = 22.5997, size = 88, normalized size = 0.75 \[ \frac{d \left (\frac{f \left (a + b x\right )}{a f - b e}\right )^{n} \left (a + b x\right )^{- n} \left (e + f x\right )^{n}{{}_{2}F_{1}\left (\begin{matrix} n, n \\ n + 1 \end{matrix}\middle |{\frac{b \left (- e - f x\right )}{a f - b e}} \right )}}{f^{2} n} - \frac{\left (a + b x\right )^{- n + 1} \left (e + f x\right )^{n - 1} \left (c f - d e\right )}{f \left (- n + 1\right ) \left (a f - b e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)*(f*x+e)**(-2+n)/((b*x+a)**n),x)
[Out]
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Mathematica [A] time = 0.269086, size = 125, normalized size = 1.06 \[ -\frac{(a+b x)^{-n} (e+f x)^{n-1} \left (c f^2 n (a+b x)-d (n-1) (e+f x) (b e-a f) \left (\frac{f (a+b x)}{a f-b e}\right )^n \, _2F_1\left (n,n;n+1;\frac{b (e+f x)}{b e-a f}\right )-d e f n (a+b x)\right )}{f^2 (n-1) n (b e-a f)} \]
Antiderivative was successfully verified.
[In] Integrate[((c + d*x)*(e + f*x)^(-2 + n))/(a + b*x)^n,x]
[Out]
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Maple [F] time = 0.076, size = 0, normalized size = 0. \[ \int{\frac{ \left ( dx+c \right ) \left ( fx+e \right ) ^{-2+n}}{ \left ( bx+a \right ) ^{n}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)*(f*x+e)^(-2+n)/((b*x+a)^n),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (d x + c\right )}{\left (b x + a\right )}^{-n}{\left (f x + e\right )}^{n - 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)*(f*x + e)^(n - 2)/(b*x + a)^n,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (d x + c\right )}{\left (f x + e\right )}^{n - 2}}{{\left (b x + a\right )}^{n}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)*(f*x + e)^(n - 2)/(b*x + a)^n,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x+c)*(f*x+e)**(-2+n)/((b*x+a)**n),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}{\left (f x + e\right )}^{n - 2}}{{\left (b x + a\right )}^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)*(f*x + e)^(n - 2)/(b*x + a)^n,x, algorithm="giac")
[Out]