3.2681 \(\int \frac{x^{3-2 n}}{\sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=61 \[ \frac{x^{4-2 n} \sqrt{a+b x^n} \, _2F_1\left (1,\frac{1}{2} \left (\frac{8}{n}-3\right );\frac{4}{n}-1;-\frac{b x^n}{a}\right )}{2 a (2-n)} \]

[Out]

(x^(4 - 2*n)*Sqrt[a + b*x^n]*Hypergeometric2F1[1, (-3 + 8/n)/2, -1 + 4/n, -((b*x^n)/a)])/(2*a*(2 - n))

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Rubi [A]  time = 0.0263921, antiderivative size = 72, normalized size of antiderivative = 1.18, 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} \[ \frac{x^{4-2 n} \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},-2 \left (1-\frac{2}{n}\right );\frac{4}{n}-1;-\frac{b x^n}{a}\right )}{2 (2-n) \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]

Int[x^(3 - 2*n)/Sqrt[a + b*x^n],x]

[Out]

(x^(4 - 2*n)*Sqrt[1 + (b*x^n)/a]*Hypergeometric2F1[1/2, -2*(1 - 2/n), -1 + 4/n, -((b*x^n)/a)])/(2*(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^{3-2 n}}{\sqrt{a+b x^n}} \, dx &=\frac{\sqrt{1+\frac{b x^n}{a}} \int \frac{x^{3-2 n}}{\sqrt{1+\frac{b x^n}{a}}} \, dx}{\sqrt{a+b x^n}}\\ &=\frac{x^{4-2 n} \sqrt{1+\frac{b x^n}{a}} \, _2F_1\left (\frac{1}{2},-2 \left (1-\frac{2}{n}\right );-1+\frac{4}{n};-\frac{b x^n}{a}\right )}{2 (2-n) \sqrt{a+b x^n}}\\ \end{align*}

Mathematica [A]  time = 0.0229882, size = 68, normalized size = 1.11 \[ -\frac{x^{4-2 n} \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{4}{n}-2;\frac{4}{n}-1;-\frac{b x^n}{a}\right )}{2 (n-2) \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(x^(4 - 2*n)*Sqrt[1 + (b*x^n)/a]*Hypergeometric2F1[1/2, -2 + 4/n, -1 + 4/n, -((b*x^n)/a)])/(2*(-2 + n)*Sqrt[a
+ b*x^n])

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Maple [F]  time = 0.051, size = 0, normalized size = 0. \begin{align*} \int{{x}^{3-2\,n}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3-2*n)/(a+b*x^n)^(1/2),x)

[Out]

int(x^(3-2*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^{-2 \, n + 3}}{\sqrt{b x^{n} + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3-2*n)/(a+b*x^n)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^(-2*n + 3)/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^(3-2*n)/(a+b*x^n)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [C]  time = 147.809, size = 49, normalized size = 0.8 \begin{align*} \frac{x^{4} x^{- 2 n} \Gamma \left (-2 + \frac{4}{n}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, -2 + \frac{4}{n} \\ -1 + \frac{4}{n} \end{matrix}\middle |{\frac{b x^{n} e^{i \pi }}{a}} \right )}}{\sqrt{a} n \Gamma \left (-1 + \frac{4}{n}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(3-2*n)/(a+b*x**n)**(1/2),x)

[Out]

x**4*x**(-2*n)*gamma(-2 + 4/n)*hyper((1/2, -2 + 4/n), (-1 + 4/n,), b*x**n*exp_polar(I*pi)/a)/(sqrt(a)*n*gamma(
-1 + 4/n))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{-2 \, n + 3}}{\sqrt{b x^{n} + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3-2*n)/(a+b*x^n)^(1/2),x, algorithm="giac")

[Out]

integrate(x^(-2*n + 3)/sqrt(b*x^n + a), x)