3.418 \(\int \frac{1}{\sqrt{x^n \left (a-b x^{2-n}\right )}} \, dx\)

Optimal. Leaf size=38 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x^n-b x^2}}\right )}{\sqrt{b} (2-n)} \]

[Out]

(2*ArcTan[(Sqrt[b]*x)/Sqrt[-(b*x^2) + a*x^n]])/(Sqrt[b]*(2 - n))

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Rubi [A]  time = 0.0410929, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x^n-b x^2}}\right )}{\sqrt{b} (2-n)} \]

Antiderivative was successfully verified.

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

[Out]

(2*ArcTan[(Sqrt[b]*x)/Sqrt[-(b*x^2) + a*x^n]])/(Sqrt[b]*(2 - n))

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Rubi in Sympy [A]  time = 3.25418, size = 31, normalized size = 0.82 \[ \frac{2 \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a x^{n} - b x^{2}}} \right )}}{\sqrt{b} \left (- n + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*atan(sqrt(b)*x/sqrt(a*x**n - b*x**2))/(sqrt(b)*(-n + 2))

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Mathematica [B]  time = 0.128499, size = 80, normalized size = 2.11 \[ -\frac{2 \sqrt{a} x^{n/2} \sqrt{1-\frac{b x^{2-n}}{a}} \sin ^{-1}\left (\frac{\sqrt{b} x^{1-\frac{n}{2}}}{\sqrt{a}}\right )}{\sqrt{b} (n-2) \sqrt{a x^n-b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.078, size = 0, normalized size = 0. \[ \int{\frac{1}{\sqrt{{x}^{n} \left ( a-b{x}^{2-n} \right ) }}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-{\left (b x^{-n + 2} - a\right )} x^{n}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-{\left (b x^{-n + 2} - a\right )} x^{n}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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