3.742 \(\int x^{-1+n} (a+b x)^{-n} \, dx\)

Optimal. Leaf size=39 \[ \frac{x^n (a+b x)^{-n} \left (\frac{b x}{a}+1\right )^n \, _2F_1\left (n,n;n+1;-\frac{b x}{a}\right )}{n} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0313855, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{x^n (a+b x)^{-n} \left (\frac{b x}{a}+1\right )^n \, _2F_1\left (n,n;n+1;-\frac{b x}{a}\right )}{n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + n)/(a + b*x)^n,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.80863, size = 29, normalized size = 0.74 \[ \frac{x^{n} \left (1 + \frac{b x}{a}\right )^{n} \left (a + b x\right )^{- n}{{}_{2}F_{1}\left (\begin{matrix} n, n \\ n + 1 \end{matrix}\middle |{- \frac{b x}{a}} \right )}}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+n)/((b*x+a)**n),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0294176, size = 39, normalized size = 1. \[ \frac{x^n (a+b x)^{-n} \left (\frac{b x}{a}+1\right )^n \, _2F_1\left (n,n;n+1;-\frac{b x}{a}\right )}{n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 + n)/(a + b*x)^n,x]

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.073, size = 0, normalized size = 0. \[ \int{\frac{{x}^{-1+n}}{ \left ( bx+a \right ) ^{n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+n)/((b*x+a)^n),x)

[Out]

int(x^(-1+n)/((b*x+a)^n),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{-n} x^{n - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(n - 1)/(b*x + a)^n,x, algorithm="maxima")

[Out]

integrate((b*x + a)^(-n)*x^(n - 1), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{n - 1}}{{\left (b x + a\right )}^{n}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(n - 1)/(b*x + a)^n,x, algorithm="fricas")

[Out]

integral(x^(n - 1)/(b*x + a)^n, x)

_______________________________________________________________________________________

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(x**(-1+n)/((b*x+a)**n),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{n - 1}}{{\left (b x + a\right )}^{n}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(n - 1)/(b*x + a)^n,x, algorithm="giac")

[Out]

integrate(x^(n - 1)/(b*x + a)^n, x)