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

Optimal. Leaf size=37 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x^{2-n}+b x^2}}\right )}{\sqrt{b} n} \]

[Out]

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

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Rubi [A]  time = 0.0477664, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x^{2-n}+b x^2}}\right )}{\sqrt{b} n} \]

Antiderivative was successfully verified.

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

[Out]

(2*ArcTanh[(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 [B]  time = 0.101867, size = 76, normalized size = 2.05 \[ \frac{2 x^{\frac{2-n}{2}} \sqrt{a+b x^n} \tanh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a+b x^n}}\right )}{\sqrt{b} n \sqrt{x^{2-n} \left (a+b x^n\right )}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x^((2 - n)/2)*Sqrt[a + b*x^n]*ArcTanh[(Sqrt[b]*x^(n/2))/Sqrt[a + b*x^n]])/(Sq
rt[b]*n*Sqrt[x^(2 - n)*(a + b*x^n)])

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Maple [F]  time = 0.077, 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 = 0.239073, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (\frac{2 \, b^{\frac{3}{2}} x x^{n} + a \sqrt{b} x + 2 \, b x^{n} \sqrt{\frac{b x^{2} x^{n} + a x^{2}}{x^{n}}}}{x}\right )}{\sqrt{b} n}, \frac{2 \, \sqrt{-b} \arctan \left (\frac{b x}{\sqrt{-b} \sqrt{\frac{b x^{2} x^{n} + a x^{2}}{x^{n}}}}\right )}{b n}\right ] \]

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]

[log((2*b^(3/2)*x*x^n + a*sqrt(b)*x + 2*b*x^n*sqrt((b*x^2*x^n + a*x^2)/x^n))/x)/
(sqrt(b)*n), 2*sqrt(-b)*arctan(b*x/(sqrt(-b)*sqrt((b*x^2*x^n + a*x^2)/x^n)))/(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)