3.2684 \(\int \frac {x^{m-n}}{\sqrt {a+b x^n}} \, dx\)

Optimal. Leaf size=63 \[ \frac {x^{m-n+1} \sqrt {a+b x^n} \, _2F_1\left (1,\frac {1}{2} \left (\frac {2 (m+1)}{n}-1\right );\frac {m+1}{n};-\frac {b x^n}{a}\right )}{a (m-n+1)} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 72, 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 = {365, 364} \[ \frac {x^{m-n+1} \sqrt {\frac {b x^n}{a}+1} \, _2F_1\left (\frac {1}{2},\frac {m-n+1}{n};\frac {m+1}{n};-\frac {b x^n}{a}\right )}{(m-n+1) \sqrt {a+b x^n}} \]

Antiderivative was successfully verified.

[In]

Int[x^(m - n)/Sqrt[a + b*x^n],x]

[Out]

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

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])

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])

Rubi steps

\begin {align*} \int \frac {x^{m-n}}{\sqrt {a+b x^n}} \, dx &=\frac {\sqrt {1+\frac {b x^n}{a}} \int \frac {x^{m-n}}{\sqrt {1+\frac {b x^n}{a}}} \, dx}{\sqrt {a+b x^n}}\\ &=\frac {x^{1+m-n} \sqrt {1+\frac {b x^n}{a}} \, _2F_1\left (\frac {1}{2},\frac {1+m-n}{n};\frac {1+m}{n};-\frac {b x^n}{a}\right )}{(1+m-n) \sqrt {a+b x^n}}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 72, normalized size = 1.14 \[ \frac {x^{m-n+1} \sqrt {\frac {b x^n}{a}+1} \, _2F_1\left (\frac {1}{2},\frac {m-n+1}{n};\frac {m+1}{n};-\frac {b x^n}{a}\right )}{(m-n+1) \sqrt {a+b x^n}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(m-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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m - n}}{\sqrt {b x^{n} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.20, size = 0, normalized size = 0.00 \[ \int \frac {x^{m -n}}{\sqrt {b \,x^{n}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m - n}}{\sqrt {b x^{n} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^{m-n}}{\sqrt {a+b\,x^n}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(m - n)/(a + b*x^n)^(1/2),x)

[Out]

int(x^(m - n)/(a + b*x^n)^(1/2), x)

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sympy [C]  time = 9.56, size = 60, normalized size = 0.95 \[ \frac {x x^{m} x^{- n} \Gamma \left (\frac {m}{n} - 1 + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {m}{n} - 1 + \frac {1}{n} \\ \frac {m}{n} + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{\sqrt {a} n \Gamma \left (\frac {m}{n} + \frac {1}{n}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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