\(\int \frac {x^{m-2 n}}{\sqrt {a+b x^n}} \, dx\) [2685]

Optimal result

Integrand size = 19, antiderivative size = 66 \[ \int \frac {x^{m-2 n}}{\sqrt {a+b x^n}} \, dx=\frac {x^{1+m-2 n} \sqrt {a+b x^n} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (-3+\frac {2 (1+m)}{n}\right ),\frac {1+m-n}{n},-\frac {b x^n}{a}\right )}{a (1+m-2 n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.14, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {372, 371} \[ \int \frac {x^{m-2 n}}{\sqrt {a+b x^n}} \, dx=\frac {x^{m-2 n+1} \sqrt {\frac {b x^n}{a}+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m-2 n+1}{n},\frac {m-n+1}{n},-\frac {b x^n}{a}\right )}{(m-2 n+1) \sqrt {a+b x^n}} \]

[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 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^{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] (verified)

Time = 0.07 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.17 \[ \int \frac {x^{m-2 n}}{\sqrt {a+b x^n}} \, dx=\frac {x^{1+m-2 n} \sqrt {1+\frac {b x^n}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m-2 n}{n},1+\frac {1+m-2 n}{n},-\frac {b x^n}{a}\right )}{(1+m-2 n) \sqrt {a+b x^n}} \]

[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])

Maple [F]

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

[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)

Fricas [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.89 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.21 \[ \int \frac {x^{m-2 n}}{\sqrt {a+b x^n}} \, dx=\frac {a^{- \frac {m}{n} + \frac {3}{2} - \frac {1}{n}} a^{\frac {m}{n} - 2 + \frac {1}{n}} x^{m - 2 n + 1} \Gamma \left (\frac {m}{n} - 2 + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {m}{n} - 2 + \frac {1}{n} \\ \frac {m}{n} - 1 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (\frac {m}{n} - 1 + \frac {1}{n}\right )} \]

[In]

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

[Out]

a**(-m/n + 3/2 - 1/n)*a**(m/n - 2 + 1/n)*x**(m - 2*n + 1)*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)/(n*gamma(m/n - 1 + 1/n))

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

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

[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)