3.2636 \(\int \frac{x^{-1-\frac{n}{2}}}{a+b x^n} \, dx\)

Optimal. Leaf size=50 \[ \frac{2 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} x^{-n/2}}{\sqrt{b}}\right )}{a^{3/2} n}-\frac{2 x^{-n/2}}{a n} \]

[Out]

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

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Rubi [A]  time = 0.0663002, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{2 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} x^{-n/2}}{\sqrt{b}}\right )}{a^{3/2} n}-\frac{2 x^{-n/2}}{a n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - n/2)/(a + b*x^n),x]

[Out]

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

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Rubi in Sympy [A]  time = 11.6626, size = 39, normalized size = 0.78 \[ - \frac{2 x^{- \frac{n}{2}}}{a n} + \frac{2 \sqrt{b} \operatorname{atan}{\left (\frac{\sqrt{a} x^{- \frac{n}{2}}}{\sqrt{b}} \right )}}{a^{\frac{3}{2}} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*x**(-n/2)/(a*n) + 2*sqrt(b)*atan(sqrt(a)*x**(-n/2)/sqrt(b))/(a**(3/2)*n)

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Mathematica [A]  time = 0.0140553, size = 50, normalized size = 1. \[ \frac{2 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} x^{-n/2}}{\sqrt{b}}\right )}{a^{3/2} n}-\frac{2 x^{-n/2}}{a n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0., size = 79, normalized size = 1.6 \[ -2\,{\frac{1}{an{x}^{n/2}}}+{\frac{1}{{a}^{2}n}\sqrt{-ab}\ln \left ({x}^{{\frac{n}{2}}}-{\frac{1}{b}\sqrt{-ab}} \right ) }-{\frac{1}{{a}^{2}n}\sqrt{-ab}\ln \left ({x}^{{\frac{n}{2}}}+{\frac{1}{b}\sqrt{-ab}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-1/2*n)/(a+b*x^n),x)

[Out]

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

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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(x^(-1/2*n - 1)/(b*x^n + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240631, size = 1, normalized size = 0.02 \[ \left [-\frac{2 \, x x^{-\frac{1}{2} \, n - 1} - \sqrt{-\frac{b}{a}} \log \left (\frac{a x^{2} x^{-n - 2} + 2 \, a x x^{-\frac{1}{2} \, n - 1} \sqrt{-\frac{b}{a}} - b}{a x^{2} x^{-n - 2} + b}\right )}{a n}, -\frac{2 \,{\left (x x^{-\frac{1}{2} \, n - 1} + \sqrt{\frac{b}{a}} \arctan \left (\frac{\sqrt{\frac{b}{a}}}{x x^{-\frac{1}{2} \, n - 1}}\right )\right )}}{a n}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-(2*x*x^(-1/2*n - 1) - sqrt(-b/a)*log((a*x^2*x^(-n - 2) + 2*a*x*x^(-1/2*n - 1)*
sqrt(-b/a) - b)/(a*x^2*x^(-n - 2) + b)))/(a*n), -2*(x*x^(-1/2*n - 1) + sqrt(b/a)
*arctan(sqrt(b/a)/(x*x^(-1/2*n - 1))))/(a*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(x**(-1-1/2*n)/(a+b*x**n),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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