3.2674 \(\int \frac{x^{3-2 n}}{\sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=61 \[ \frac{x^{4-2 n} \sqrt{a+b x^n} \, _2F_1\left (1,\frac{1}{2} \left (\frac{8}{n}-3\right );\frac{4}{n}-1;-\frac{b x^n}{a}\right )}{2 a (2-n)} \]

[Out]

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

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Rubi [A]  time = 0.0795379, antiderivative size = 72, normalized size of antiderivative = 1.18, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{x^{4-2 n} \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},-2 \left (1-\frac{2}{n}\right );\frac{4}{n}-1;-\frac{b x^n}{a}\right )}{2 (2-n) \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[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])

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Rubi in Sympy [A]  time = 8.14306, size = 54, normalized size = 0.89 \[ \frac{x^{- 2 n + 4} \sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, -2 + \frac{4}{n} \\ - \frac{n - 4}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{2 a \sqrt{1 + \frac{b x^{n}}{a}} \left (- n + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(3-2*n)/(a+b*x**n)**(1/2),x)

[Out]

x**(-2*n + 4)*sqrt(a + b*x**n)*hyper((1/2, -2 + 4/n), (-(n - 4)/n,), -b*x**n/a)/
(2*a*sqrt(1 + b*x**n/a)*(-n + 2))

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Mathematica [A]  time = 0.418328, size = 116, normalized size = 1.9 \[ \frac{x^4 \left (b^2 \left (3 n^2-32 n+64\right ) \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{4}{n};\frac{n+4}{n};-\frac{b x^n}{a}\right )+8 x^{-2 n} \left (a+b x^n\right ) \left (b (3 n-8) x^n-2 a (n-4)\right )\right )}{32 a^2 (n-4) (n-2) \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

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

[Out]

(x^4*((8*(a + b*x^n)*(-2*a*(-4 + n) + b*(-8 + 3*n)*x^n))/x^(2*n) + b^2*(64 - 32*
n + 3*n^2)*Sqrt[1 + (b*x^n)/a]*Hypergeometric2F1[1/2, 4/n, (4 + n)/n, -((b*x^n)/
a)]))/(32*a^2*(-4 + n)*(-2 + n)*Sqrt[a + b*x^n])

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Maple [F]  time = 0.061, size = 0, normalized size = 0. \[ \int{{x}^{3-2\,n}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(-2*n + 3)/sqrt(b*x^n + a),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. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(-2*n + 3)/sqrt(b*x^n + a),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{-2 \, n + 3}}{\sqrt{b x^{n} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(-2*n + 3)/sqrt(b*x^n + a),x, algorithm="giac")

[Out]

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