Optimal. Leaf size=135 \[ \frac{(a+b x)^{-n} (e+f x)^{n+1} \left (-\frac{f (a+b x)}{b e-a f}\right )^n (b (2 c f-d e (1-n))-a d f (n+1)) \, _2F_1\left (n,n+1;n+2;\frac{b (e+f x)}{b e-a f}\right )}{2 b f^2 (n+1)}+\frac{d (a+b x)^{1-n} (e+f x)^{n+1}}{2 b f} \]
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Rubi [A] time = 0.215309, antiderivative size = 134, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{(a+b x)^{-n} (e+f x)^{n+1} \left (-\frac{f (a+b x)}{b e-a f}\right )^n (-a d f (n+1)+2 b c f-b d e (1-n)) \, _2F_1\left (n,n+1;n+2;\frac{b (e+f x)}{b e-a f}\right )}{2 b f^2 (n+1)}+\frac{d (a+b x)^{1-n} (e+f x)^{n+1}}{2 b f} \]
Antiderivative was successfully verified.
[In] Int[((c + d*x)*(e + f*x)^n)/(a + b*x)^n,x]
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Rubi in Sympy [A] time = 21.8258, size = 105, normalized size = 0.78 \[ \frac{d \left (a + b x\right )^{- n + 1} \left (e + f x\right )^{n + 1}}{2 b f} - \frac{\left (\frac{f \left (a + b x\right )}{a f - b e}\right )^{n} \left (a + b x\right )^{- n} \left (e + f x\right )^{n + 1} \left (- 2 b c f + d \left (a f \left (n + 1\right ) + b e \left (- n + 1\right )\right )\right ){{}_{2}F_{1}\left (\begin{matrix} n, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{b \left (- e - f x\right )}{a f - b e}} \right )}}{2 b f^{2} \left (n + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)*(f*x+e)**n/((b*x+a)**n),x)
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Mathematica [C] time = 0.607406, size = 192, normalized size = 1.42 \[ (a+b x)^{-n} (e+f x)^n \left (\frac{3 a d e x^2 F_1\left (2;n,-n;3;-\frac{b x}{a},-\frac{f x}{e}\right )}{6 a e F_1\left (2;n,-n;3;-\frac{b x}{a},-\frac{f x}{e}\right )+2 n x \left (a f F_1\left (3;n,1-n;4;-\frac{b x}{a},-\frac{f x}{e}\right )-b e F_1\left (3;n+1,-n;4;-\frac{b x}{a},-\frac{f x}{e}\right )\right )}+\frac{c (e+f x) \left (\frac{f (a+b x)}{a f-b e}\right )^n \, _2F_1\left (n,n+1;n+2;\frac{b (e+f x)}{b e-a f}\right )}{f (n+1)}\right ) \]
Warning: Unable to verify antiderivative.
[In] Integrate[((c + d*x)*(e + f*x)^n)/(a + b*x)^n,x]
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Maple [F] time = 0.086, size = 0, normalized size = 0. \[ \int{\frac{ \left ( dx+c \right ) \left ( fx+e \right ) ^{n}}{ \left ( bx+a \right ) ^{n}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)*(f*x+e)^n/((b*x+a)^n),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (d x + c\right )}{\left (b x + a\right )}^{-n}{\left (f x + e\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)*(f*x + e)^n/(b*x + a)^n,x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (d x + c\right )}{\left (f x + e\right )}^{n}}{{\left (b x + a\right )}^{n}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)*(f*x + e)^n/(b*x + a)^n,x, algorithm="fricas")
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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)**n/((b*x+a)**n),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}{\left (f x + e\right )}^{n}}{{\left (b x + a\right )}^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)*(f*x + e)^n/(b*x + a)^n,x, algorithm="giac")
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