\(\int (a+b x^n)^{-\frac {1-n}{n}} \, dx\) [2735]

Optimal result

Integrand size = 18, antiderivative size = 59 \[ \int \left (a+b x^n\right )^{-\frac {1-n}{n}} \, dx=x \left (a+b x^n\right )^{-\frac {1-n}{n}} \left (1+\frac {b x^n}{a}\right )^{-1+\frac {1}{n}} \operatorname {Hypergeometric2F1}\left (-1+\frac {1}{n},\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {252, 251} \[ \int \left (a+b x^n\right )^{-\frac {1-n}{n}} \, dx=x \left (a+b x^n\right )^{-\frac {1-n}{n}} \left (\frac {b x^n}{a}+1\right )^{\frac {1}{n}-1} \operatorname {Hypergeometric2F1}\left (\frac {1}{n}-1,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right ) \]

[In]

Int[(a + b*x^n)^(-((1 - n)/n)),x]

[Out]

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

Rule 251

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)],
x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p
] || GtQ[a, 0])

Rule 252

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^Fra
cPart[p]), Int[(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILt
Q[Simplify[1/n + p], 0] &&  !(IntegerQ[p] || GtQ[a, 0])

Rubi steps \begin{align*} \text {integral}& = \left (\left (a+b x^n\right )^{-\frac {1-n}{n}} \left (1+\frac {b x^n}{a}\right )^{\frac {1-n}{n}}\right ) \int \left (1+\frac {b x^n}{a}\right )^{-\frac {1-n}{n}} \, dx \\ & = x \left (a+b x^n\right )^{-\frac {1-n}{n}} \left (1+\frac {b x^n}{a}\right )^{-1+\frac {1}{n}} \, _2F_1\left (-1+\frac {1}{n},\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.90 \[ \int \left (a+b x^n\right )^{-\frac {1-n}{n}} \, dx=a x \left (a+b x^n\right )^{-1/n} \left (1+\frac {b x^n}{a}\right )^{\frac {1}{n}} \operatorname {Hypergeometric2F1}\left (-1+\frac {1}{n},\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right ) \]

[In]

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

[Out]

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

Maple [F]

\[\int \left (a +b \,x^{n}\right )^{-\frac {1-n}{n}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \left (a+b x^n\right )^{-\frac {1-n}{n}} \, dx=\int { {\left (b x^{n} + a\right )}^{\frac {n - 1}{n}} \,d x } \]

[In]

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

[Out]

integral((b*x^n + a)^((n - 1)/n), x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.22 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.81 \[ \int \left (a+b x^n\right )^{-\frac {1-n}{n}} \, dx=\frac {a^{\frac {1}{n}} a^{1 - \frac {2}{n}} x \Gamma \left (\frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{n}, -1 + \frac {1}{n} \\ 1 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (1 + \frac {1}{n}\right )} \]

[In]

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

[Out]

a**(1/n)*a**(1 - 2/n)*x*gamma(1/n)*hyper((1/n, -1 + 1/n), (1 + 1/n,), b*x**n*exp_polar(I*pi)/a)/(n*gamma(1 + 1
/n))

Maxima [F]

\[ \int \left (a+b x^n\right )^{-\frac {1-n}{n}} \, dx=\int { {\left (b x^{n} + a\right )}^{\frac {n - 1}{n}} \,d x } \]

[In]

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

[Out]

integrate((b*x^n + a)^((n - 1)/n), x)

Giac [F]

\[ \int \left (a+b x^n\right )^{-\frac {1-n}{n}} \, dx=\int { {\left (b x^{n} + a\right )}^{\frac {n - 1}{n}} \,d x } \]

[In]

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

[Out]

integrate((b*x^n + a)^((n - 1)/n), x)

Mupad [B] (verification not implemented)

Time = 6.00 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.92 \[ \int \left (a+b x^n\right )^{-\frac {1-n}{n}} \, dx=\frac {a\,x\,{\left (\frac {b\,x^n}{a}+1\right )}^{1/n}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{n}-1,\frac {1}{n};\ \frac {1}{n}+1;\ -\frac {b\,x^n}{a}\right )}{{\left (a+b\,x^n\right )}^{1/n}} \]

[In]

int((a + b*x^n)^((n - 1)/n),x)

[Out]

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