Optimal. Leaf size=52 \[ -\frac{(a c-b c x)^{n+1} \, _2F_1\left (2,n+1;n+2;\frac{a-b x}{2 a}\right )}{4 a^2 b c (n+1)} \]
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Rubi [A] time = 0.0389448, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ -\frac{(a c-b c x)^{n+1} \, _2F_1\left (2,n+1;n+2;\frac{a-b x}{2 a}\right )}{4 a^2 b c (n+1)} \]
Antiderivative was successfully verified.
[In] Int[(a*c - b*c*x)^n/(a + b*x)^2,x]
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Rubi in Sympy [A] time = 8.17525, size = 39, normalized size = 0.75 \[ - \frac{\left (a c - b c x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 2, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\frac{a}{2} - \frac{b x}{2}}{a}} \right )}}{4 a^{2} b c \left (n + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-b*c*x+a*c)**n/(b*x+a)**2,x)
[Out]
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Mathematica [A] time = 0.0358064, size = 52, normalized size = 1. \[ -\frac{(a-b x) (c (a-b x))^n \, _2F_1\left (2,n+1;n+2;\frac{a-b x}{2 a}\right )}{4 a^2 b (n+1)} \]
Antiderivative was successfully verified.
[In] Integrate[(a*c - b*c*x)^n/(a + b*x)^2,x]
[Out]
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Maple [F] time = 0.069, size = 0, normalized size = 0. \[ \int{\frac{ \left ( -bcx+ac \right ) ^{n}}{ \left ( bx+a \right ) ^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-b*c*x+a*c)^n/(b*x+a)^2,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-b c x + a c\right )}^{n}}{{\left (b x + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-b*c*x + a*c)^n/(b*x + a)^2,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (-b c x + a c\right )}^{n}}{b^{2} x^{2} + 2 \, a b x + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-b*c*x + a*c)^n/(b*x + a)^2,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (- c \left (- a + b x\right )\right )^{n}}{\left (a + b x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-b*c*x+a*c)**n/(b*x+a)**2,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-b c x + a c\right )}^{n}}{{\left (b x + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-b*c*x + a*c)^n/(b*x + a)^2,x, algorithm="giac")
[Out]