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

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

[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)

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Rubi [A]
time = 0.02, 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 = {372, 371} \begin {gather*} \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}} \end {gather*}

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 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*} \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*}

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Mathematica [A]
time = 0.04, size = 77, normalized size = 1.17 \begin {gather*} \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};1+\frac {1+m-2 n}{n};-\frac {b x^n}{a}\right )}{(1+m-2 n) \sqrt {a+b x^n}} \end {gather*}

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 = 0.04, size = 0, normalized size = 0.00 \[\int \frac {x^{m -2 n}}{\sqrt {a +b \,x^{n}}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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

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Sympy [C] Result contains complex when optimal does not.
time = 7.65, size = 65, normalized size = 0.98 \begin {gather*} \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 {gather*}

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*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
*x**(2*n)*gamma(m/n - 1 + 1/n))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^{m-2\,n}}{\sqrt {a+b\,x^n}} \,d x \end {gather*}

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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