3.223 \(\int \frac{\sqrt{\frac{a}{x^4}}}{\sqrt{1+x^2}} \, dx\)

Optimal. Leaf size=21 \[ x \sqrt{x^2+1} \left (-\sqrt{\frac{a}{x^4}}\right ) \]

[Out]

-(Sqrt[a/x^4]*x*Sqrt[1 + x^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.01534, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ x \sqrt{x^2+1} \left (-\sqrt{\frac{a}{x^4}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a/x^4]/Sqrt[1 + x^2],x]

[Out]

-(Sqrt[a/x^4]*x*Sqrt[1 + x^2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 7.87076, size = 19, normalized size = 0.9 \[ - x \sqrt{\frac{a}{x^{4}}} \sqrt{x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a/x**4)**(1/2)/(x**2+1)**(1/2),x)

[Out]

-x*sqrt(a/x**4)*sqrt(x**2 + 1)

_______________________________________________________________________________________

Mathematica [A]  time = 0.00980076, size = 21, normalized size = 1. \[ x \sqrt{x^2+1} \left (-\sqrt{\frac{a}{x^4}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a/x^4]/Sqrt[1 + x^2],x]

[Out]

-(Sqrt[a/x^4]*x*Sqrt[1 + x^2])

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 18, normalized size = 0.9 \[ -x\sqrt{{\frac{a}{{x}^{4}}}}\sqrt{{x}^{2}+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a/x^4)^(1/2)/(x^2+1)^(1/2),x)

[Out]

-x*(a/x^4)^(1/2)*(x^2+1)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.775118, size = 31, normalized size = 1.48 \[ -\frac{\sqrt{a} x^{2} + \sqrt{a}}{\sqrt{x^{2} + 1} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a/x^4)/sqrt(x^2 + 1),x, algorithm="maxima")

[Out]

-(sqrt(a)*x^2 + sqrt(a))/(sqrt(x^2 + 1)*x)

_______________________________________________________________________________________

Fricas [A]  time = 0.289448, size = 66, normalized size = 3.14 \[ \frac{x^{2} \sqrt{\frac{a}{x^{4}}} - \sqrt{x^{2} + 1} x \sqrt{\frac{a}{x^{4}}}}{2 \, x^{2} - 2 \, \sqrt{x^{2} + 1} x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a/x^4)/sqrt(x^2 + 1),x, algorithm="fricas")

[Out]

(x^2*sqrt(a/x^4) - sqrt(x^2 + 1)*x*sqrt(a/x^4))/(2*x^2 - 2*sqrt(x^2 + 1)*x + 1)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\frac{a}{x^{4}}}}{\sqrt{x^{2} + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a/x**4)**(1/2)/(x**2+1)**(1/2),x)

[Out]

Integral(sqrt(a/x**4)/sqrt(x**2 + 1), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.262143, size = 30, normalized size = 1.43 \[ \frac{2 \, \sqrt{a}}{{\left (x - \sqrt{x^{2} + 1}\right )}^{2} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a/x^4)/sqrt(x^2 + 1),x, algorithm="giac")

[Out]

2*sqrt(a)/((x - sqrt(x^2 + 1))^2 - 1)