3.2466 \(\int \frac{1}{a+b x^n} \, dx\)

Optimal. Leaf size=24 \[ \frac{x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0195887, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^n)^(-1),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 1.77875, size = 17, normalized size = 0.71 \[ \frac{x{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b*x**n),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.00778167, size = 24, normalized size = 1. \[ \frac{x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^n)^(-1),x]

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.046, size = 0, normalized size = 0. \[ \int \left ( a+b{x}^{n} \right ) ^{-1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*x^n),x)

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{b x^{n} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(b*x^n + a), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/(b*x^n + a), x)

_______________________________________________________________________________________

Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b*x**n),x)

[Out]

Exception raised: TypeError

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(b*x^n + a), x)