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

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

[Out]

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

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Rubi [A]  time = 0.043375, 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 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x^{2-n}-b x^2}}\right )}{\sqrt{b} n} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.120053, size = 76, normalized size = 2. \[ \frac{2 x^{1-\frac{n}{2}} \sqrt{a-b x^n} \tan ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a-b x^n}}\right )}{\sqrt{b} n \sqrt{x^2 \left (a x^{-n}-b\right )}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.112, size = 0, normalized size = 0. \[ \int{\frac{1}{\sqrt{{x}^{2-n} \left ( a-b{x}^{n} \right ) }}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.251685, size = 1, normalized size = 0.03 \[ \left [-\frac{\sqrt{-b} \log \left (-\frac{2 \, \sqrt{-b} b x x^{n} - a \sqrt{-b} x + 2 \, b x^{n} \sqrt{-\frac{b x^{2} x^{n} - a x^{2}}{x^{n}}}}{x}\right )}{b n}, \frac{2 \, \arctan \left (\frac{\sqrt{b} x}{\sqrt{-\frac{b x^{2} x^{n} - a x^{2}}{x^{n}}}}\right )}{\sqrt{b} n}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-sqrt(-b)*log(-(2*sqrt(-b)*b*x*x^n - a*sqrt(-b)*x + 2*b*x^n*sqrt(-(b*x^2*x^n -
a*x^2)/x^n))/x)/(b*n), 2*arctan(sqrt(b)*x/sqrt(-(b*x^2*x^n - a*x^2)/x^n))/(sqrt(
b)*n)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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