3.2735 \(\int (a+b x^n)^{-\frac{1-n}{n}} \, dx\)

Optimal. Leaf size=59 \[ x \left (a+b x^n\right )^{-\frac{1-n}{n}} \left (\frac{b x^n}{a}+1\right )^{\frac{1}{n}-1} \, _2F_1\left (\frac{1}{n}-1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right ) \]

[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)

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Rubi [A]  time = 0.0125478, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {246, 245} \[ x \left (a+b x^n\right )^{-\frac{1-n}{n}} \left (\frac{b x^n}{a}+1\right )^{\frac{1}{n}-1} \, _2F_1\left (\frac{1}{n}-1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right ) \]

Antiderivative was successfully verified.

[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 246

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^Fr
acPart[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])

Rule 245

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])

Rubi steps

\begin{align*} \int \left (a+b x^n\right )^{-\frac{1-n}{n}} \, dx &=\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]  time = 0.0100094, size = 53, normalized size = 0.9 \[ a x \left (a+b x^n\right )^{-1/n} \left (\frac{b x^n}{a}+1\right )^{\frac{1}{n}} \, _2F_1\left (\frac{1}{n}-1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right ) \]

Antiderivative was successfully verified.

[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)

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Maple [F]  time = 0.074, size = 0, normalized size = 0. \begin{align*} \int \left ( \left ( a+b{x}^{n} \right ) ^{{\frac{1-n}{n}}} \right ) ^{-1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^n + a)^((n - 1)/n), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x^n + a)^((n - 1)/n), x)

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Sympy [C]  time = 10.7146, size = 42, normalized size = 0.71 \begin{align*} \frac{a a^{- \frac{1}{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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*a**(-1/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))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{n} + a\right )}^{\frac{n - 1}{n}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^n + a)^((n - 1)/n), x)