\(\int \frac {x^{3-2 n}}{\sqrt {a+b x^n}} \, dx\) [2681]

Optimal result

Integrand size = 19, antiderivative size = 61 \[ \int \frac {x^{3-2 n}}{\sqrt {a+b x^n}} \, dx=\frac {x^{4-2 n} \sqrt {a+b x^n} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (-3+\frac {8}{n}\right ),-1+\frac {4}{n},-\frac {b x^n}{a}\right )}{2 a (2-n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.18, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {372, 371} \[ \int \frac {x^{3-2 n}}{\sqrt {a+b x^n}} \, dx=\frac {x^{4-2 n} \sqrt {\frac {b x^n}{a}+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-2 \left (1-\frac {2}{n}\right ),\frac {4}{n}-1,-\frac {b x^n}{a}\right )}{2 (2-n) \sqrt {a+b x^n}} \]

[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 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 372

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])

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.06 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.11 \[ \int \frac {x^{3-2 n}}{\sqrt {a+b x^n}} \, dx=-\frac {x^{4-2 n} \sqrt {1+\frac {b x^n}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-2+\frac {4}{n},-1+\frac {4}{n},-\frac {b x^n}{a}\right )}{2 (-2+n) \sqrt {a+b x^n}} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {x^{3-2 n}}{\sqrt {a +b \,x^{n}}}d x\]

[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)

Fricas [F(-2)]

Exception generated. \[ \int \frac {x^{3-2 n}}{\sqrt {a+b x^n}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.55 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.16 \[ \int \frac {x^{3-2 n}}{\sqrt {a+b x^n}} \, dx=\frac {a^{-2 + \frac {4}{n}} a^{\frac {3}{2} - \frac {4}{n}} b^{-2 + \frac {4}{n}} b^{2 - \frac {4}{n}} x^{4 - 2 n} \Gamma \left (-2 + \frac {4}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, -2 + \frac {4}{n} \\ -1 + \frac {4}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (-1 + \frac {4}{n}\right )} \]

[In]

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

[Out]

a**(-2 + 4/n)*a**(3/2 - 4/n)*b**(-2 + 4/n)*b**(2 - 4/n)*x**(4 - 2*n)*gamma(-2 + 4/n)*hyper((1/2, -2 + 4/n), (-
1 + 4/n,), b*x**n*exp_polar(I*pi)/a)/(n*gamma(-1 + 4/n))

Maxima [F]

\[ \int \frac {x^{3-2 n}}{\sqrt {a+b x^n}} \, dx=\int { \frac {x^{-2 \, n + 3}}{\sqrt {b x^{n} + a}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {x^{3-2 n}}{\sqrt {a+b x^n}} \, dx=\int { \frac {x^{-2 \, n + 3}}{\sqrt {b x^{n} + a}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{3-2 n}}{\sqrt {a+b x^n}} \, dx=\int \frac {x^{3-2\,n}}{\sqrt {a+b\,x^n}} \,d x \]

[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)