3.2685 \(\int \frac{x^{m-2 n}}{\sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=66 \[ \frac{x^{m-2 n+1} \sqrt{a+b x^n} \, _2F_1\left (1,\frac{1}{2} \left (\frac{2 (m+1)}{n}-3\right );\frac{m-n+1}{n};-\frac{b x^n}{a}\right )}{a (m-2 n+1)} \]

[Out]

(x^(1 + m - 2*n)*Sqrt[a + b*x^n]*Hypergeometric2F1[1, (-3 + (2*(1 + m))/n)/2, (1 + m - n)/n, -((b*x^n)/a)])/(a
*(1 + m - 2*n))

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Rubi [A]  time = 0.0269902, antiderivative size = 75, normalized size of antiderivative = 1.14, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {365, 364} \[ \frac{x^{m-2 n+1} \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m-2 n+1}{n};\frac{m-n+1}{n};-\frac{b x^n}{a}\right )}{(m-2 n+1) \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]

Int[x^(m - 2*n)/Sqrt[a + b*x^n],x]

[Out]

(x^(1 + m - 2*n)*Sqrt[1 + (b*x^n)/a]*Hypergeometric2F1[1/2, (1 + m - 2*n)/n, (1 + m - n)/n, -((b*x^n)/a)])/((1
+ m - 2*n)*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{x^{m-2 n}}{\sqrt{a+b x^n}} \, dx &=\frac{\sqrt{1+\frac{b x^n}{a}} \int \frac{x^{m-2 n}}{\sqrt{1+\frac{b x^n}{a}}} \, dx}{\sqrt{a+b x^n}}\\ &=\frac{x^{1+m-2 n} \sqrt{1+\frac{b x^n}{a}} \, _2F_1\left (\frac{1}{2},\frac{1+m-2 n}{n};\frac{1+m-n}{n};-\frac{b x^n}{a}\right )}{(1+m-2 n) \sqrt{a+b x^n}}\\ \end{align*}

Mathematica [A]  time = 0.0246231, size = 77, normalized size = 1.17 \[ \frac{x^{m-2 n+1} \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m-2 n+1}{n};\frac{m-2 n+1}{n}+1;-\frac{b x^n}{a}\right )}{(m-2 n+1) \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.047, size = 0, normalized size = 0. \begin{align*} \int{{x}^{m-2\,n}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [C]  time = 55.0908, size = 65, normalized size = 0.98 \begin{align*} \frac{x x^{m} x^{- 2 n} \Gamma \left (\frac{m}{n} - 2 + \frac{1}{n}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{n} - 2 + \frac{1}{n} \\ \frac{m}{n} - 1 + \frac{1}{n} \end{matrix}\middle |{\frac{b x^{n} e^{i \pi }}{a}} \right )}}{\sqrt{a} n \Gamma \left (\frac{m}{n} - 1 + \frac{1}{n}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^(m - 2*n)/sqrt(b*x^n + a), x)