3.1232 \(\int \frac{(a c-b c x)^n}{(a+b x)^2} \, dx\)

Optimal. Leaf size=52 \[ -\frac{(a c-b c x)^{n+1} \, _2F_1\left (2,n+1;n+2;\frac{a-b x}{2 a}\right )}{4 a^2 b c (n+1)} \]

[Out]

-((a*c - b*c*x)^(1 + n)*Hypergeometric2F1[2, 1 + n, 2 + n, (a - b*x)/(2*a)])/(4*
a^2*b*c*(1 + n))

_______________________________________________________________________________________

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]

[Out]

-((a*c - b*c*x)^(1 + n)*Hypergeometric2F1[2, 1 + n, 2 + n, (a - b*x)/(2*a)])/(4*
a^2*b*c*(1 + n))

_______________________________________________________________________________________

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]

-(a*c - b*c*x)**(n + 1)*hyper((2, n + 1), (n + 2,), (a/2 - b*x/2)/a)/(4*a**2*b*c
*(n + 1))

_______________________________________________________________________________________

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]

-((a - b*x)*(c*(a - b*x))^n*Hypergeometric2F1[2, 1 + n, 2 + n, (a - b*x)/(2*a)])
/(4*a^2*b*(1 + n))

_______________________________________________________________________________________

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]

int((-b*c*x+a*c)^n/(b*x+a)^2,x)

_______________________________________________________________________________________

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]

integrate((-b*c*x + a*c)^n/(b*x + a)^2, x)

_______________________________________________________________________________________

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]

integral((-b*c*x + a*c)^n/(b^2*x^2 + 2*a*b*x + a^2), x)

_______________________________________________________________________________________

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]

Integral((-c*(-a + b*x))**n/(a + b*x)**2, x)

_______________________________________________________________________________________

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]

integrate((-b*c*x + a*c)^n/(b*x + a)^2, x)