3.2657 \(\int \frac{x^{-1-2 n}}{\sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=87 \[ -\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{4 a^{5/2} n}+\frac{3 b x^{-n} \sqrt{a+b x^n}}{4 a^2 n}-\frac{x^{-2 n} \sqrt{a+b x^n}}{2 a n} \]

[Out]

-Sqrt[a + b*x^n]/(2*a*n*x^(2*n)) + (3*b*Sqrt[a + b*x^n])/(4*a^2*n*x^n) - (3*b^2*
ArcTanh[Sqrt[a + b*x^n]/Sqrt[a]])/(4*a^(5/2)*n)

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Rubi [A]  time = 0.112258, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{4 a^{5/2} n}+\frac{3 b x^{-n} \sqrt{a+b x^n}}{4 a^2 n}-\frac{x^{-2 n} \sqrt{a+b x^n}}{2 a n} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[a + b*x^n]/(2*a*n*x^(2*n)) + (3*b*Sqrt[a + b*x^n])/(4*a^2*n*x^n) - (3*b^2*
ArcTanh[Sqrt[a + b*x^n]/Sqrt[a]])/(4*a^(5/2)*n)

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Rubi in Sympy [A]  time = 11.8329, size = 73, normalized size = 0.84 \[ - \frac{x^{- 2 n} \sqrt{a + b x^{n}}}{2 a n} + \frac{3 b x^{- n} \sqrt{a + b x^{n}}}{4 a^{2} n} - \frac{3 b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{n}}}{\sqrt{a}} \right )}}{4 a^{\frac{5}{2}} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**(-2*n)*sqrt(a + b*x**n)/(2*a*n) + 3*b*x**(-n)*sqrt(a + b*x**n)/(4*a**2*n) -
3*b**2*atanh(sqrt(a + b*x**n)/sqrt(a))/(4*a**(5/2)*n)

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Mathematica [A]  time = 0.127679, size = 85, normalized size = 0.98 \[ \frac{-3 b^2 \log \left (x^{-n} \left (2 \sqrt{a} \sqrt{a+b x^n}+2 a+b x^n\right )\right )-2 \sqrt{a} x^{-2 n} \left (2 a-3 b x^n\right ) \sqrt{a+b x^n}}{8 a^{5/2} n} \]

Antiderivative was successfully verified.

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

[Out]

((-2*Sqrt[a]*(2*a - 3*b*x^n)*Sqrt[a + b*x^n])/x^(2*n) - 3*b^2*Log[(2*a + b*x^n +
2*Sqrt[a]*Sqrt[a + b*x^n])/x^n])/(8*a^(5/2)*n)

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Maple [F]  time = 0.069, size = 0, normalized size = 0. \[ \int{{x}^{-1-2\,n}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.244004, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b^{2} x^{2 \, n} \log \left (\frac{\sqrt{a} b x^{n} - 2 \, \sqrt{b x^{n} + a} a + 2 \, a^{\frac{3}{2}}}{x^{n}}\right ) + 2 \,{\left (3 \, \sqrt{a} b x^{n} - 2 \, a^{\frac{3}{2}}\right )} \sqrt{b x^{n} + a}}{8 \, a^{\frac{5}{2}} n x^{2 \, n}}, \frac{3 \, b^{2} x^{2 \, n} \arctan \left (\frac{a}{\sqrt{b x^{n} + a} \sqrt{-a}}\right ) +{\left (3 \, \sqrt{-a} b x^{n} - 2 \, \sqrt{-a} a\right )} \sqrt{b x^{n} + a}}{4 \, \sqrt{-a} a^{2} n x^{2 \, n}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(3*b^2*x^(2*n)*log((sqrt(a)*b*x^n - 2*sqrt(b*x^n + a)*a + 2*a^(3/2))/x^n) +
2*(3*sqrt(a)*b*x^n - 2*a^(3/2))*sqrt(b*x^n + a))/(a^(5/2)*n*x^(2*n)), 1/4*(3*b^
2*x^(2*n)*arctan(a/(sqrt(b*x^n + a)*sqrt(-a))) + (3*sqrt(-a)*b*x^n - 2*sqrt(-a)*
a)*sqrt(b*x^n + a))/(sqrt(-a)*a^2*n*x^(2*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**(-1-2*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. \[ \int \frac{x^{-2 \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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