3.2684 \(\int \frac{x^{-1-\frac{n}{2}}}{\sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=26 \[ -\frac{2 x^{-n/2} \sqrt{a+b x^n}}{a n} \]

[Out]

(-2*Sqrt[a + b*x^n])/(a*n*x^(n/2))

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Rubi [A]  time = 0.0270315, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ -\frac{2 x^{-n/2} \sqrt{a+b x^n}}{a n} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[a + b*x^n])/(a*n*x^(n/2))

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Rubi in Sympy [A]  time = 3.09779, size = 20, normalized size = 0.77 \[ - \frac{2 x^{- \frac{n}{2}} \sqrt{a + b x^{n}}}{a n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*x**(-n/2)*sqrt(a + b*x**n)/(a*n)

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Mathematica [A]  time = 0.0348794, size = 26, normalized size = 1. \[ -\frac{2 x^{-n/2} \sqrt{a+b x^n}}{a n} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[a + b*x^n])/(a*n*x^(n/2))

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Maple [F]  time = 0.083, size = 0, normalized size = 0. \[ \int{1{x}^{-1-{\frac{n}{2}}}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.53584, size = 30, normalized size = 1.15 \[ -\frac{2 \, \sqrt{b x^{n} + a} x^{-\frac{1}{2} \, n}}{a n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(-1/2*n - 1)/sqrt(b*x^n + a),x, algorithm="maxima")

[Out]

-2*sqrt(b*x^n + a)*x^(-1/2*n)/(a*n)

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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^(-1/2*n - 1)/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**(-1-1/2*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. \[ \int \frac{x^{-\frac{1}{2} \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(-1/2*n - 1)/sqrt(b*x^n + a),x, algorithm="giac")

[Out]

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