3.2678 \(\int \frac{x^{3+2 n}}{\sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=59 \[ \frac{x^{2 (n+2)} \sqrt{a+b x^n} \, _2F_1\left (1,\frac{1}{2} \left (5+\frac{8}{n}\right );3+\frac{4}{n};-\frac{b x^n}{a}\right )}{2 a (n+2)} \]

[Out]

(x^(2*(2 + n))*Sqrt[a + b*x^n]*Hypergeometric2F1[1, (5 + 8/n)/2, 3 + 4/n, -((b*x^n)/a)])/(2*a*(2 + n))

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Rubi [A]  time = 0.0241425, antiderivative size = 70, normalized size of antiderivative = 1.19, 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} \[ \frac{x^{2 (n+2)} \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},2 \left (1+\frac{2}{n}\right );3+\frac{4}{n};-\frac{b x^n}{a}\right )}{2 (n+2) \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]

Int[x^(3 + 2*n)/Sqrt[a + b*x^n],x]

[Out]

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

Mathematica [A]  time = 0.0231649, size = 68, normalized size = 1.15 \[ \frac{x^{2 n+4} \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},2+\frac{4}{n};3+\frac{4}{n};-\frac{b x^n}{a}\right )}{2 (n+2) \sqrt{a+b x^n}} \]

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

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Maple [F]  time = 0.062, 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 = 126.673, size = 49, normalized size = 0.83 \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} \\ 3 + \frac{4}{n} \end{matrix}\middle |{\frac{b x^{n} e^{i \pi }}{a}} \right )}}{\sqrt{a} n \Gamma \left (3 + \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), (3 + 4/n,), b*x**n*exp_polar(I*pi)/a)/(sqrt(a)*n*gamma(3 +
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)