3.2505 \(\int \frac{1}{x^2 \sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=49 \[ -\frac{\sqrt{a+b x^n} \, _2F_1\left (1,\frac{1}{2}-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{a x} \]

[Out]

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

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Rubi [A]  time = 0.0185416, antiderivative size = 58, normalized size of antiderivative = 1.18, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {365, 364} \[ -\frac{\sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{x \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*Sqrt[a + b*x^n]),x]

[Out]

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

Rule 365

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])
/(1 + (b*x^n)/a)^FracPart[p], Int[(c*x)^m*(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rule 364

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

Rubi steps

\begin{align*} \int \frac{1}{x^2 \sqrt{a+b x^n}} \, dx &=\frac{\sqrt{1+\frac{b x^n}{a}} \int \frac{1}{x^2 \sqrt{1+\frac{b x^n}{a}}} \, dx}{\sqrt{a+b x^n}}\\ &=-\frac{\sqrt{1+\frac{b x^n}{a}} \, _2F_1\left (\frac{1}{2},-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{x \sqrt{a+b x^n}}\\ \end{align*}

Mathematica [A]  time = 0.0137124, size = 55, normalized size = 1.12 \[ -\frac{\sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},-\frac{1}{n};1-\frac{1}{n};-\frac{b x^n}{a}\right )}{x \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*Sqrt[a + b*x^n]),x]

[Out]

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

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Maple [F]  time = 0.038, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(a+b*x^n)^(1/2),x)

[Out]

int(1/x^2/(a+b*x^n)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x^{n} + a} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a+b*x^n)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(b*x^n + a)*x^2), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a+b*x^n)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [C]  time = 1.23866, size = 42, normalized size = 0.86 \begin{align*} \frac{\Gamma \left (- \frac{1}{n}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, - \frac{1}{n} \\ 1 - \frac{1}{n} \end{matrix}\middle |{\frac{b x^{n} e^{i \pi }}{a}} \right )}}{\sqrt{a} n x \Gamma \left (1 - \frac{1}{n}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(a+b*x**n)**(1/2),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x^{n} + a} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a+b*x^n)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(b*x^n + a)*x^2), x)