Optimal. Leaf size=35 \[ \frac{2 x \sqrt{a+\frac{b}{x^2}}}{a^2}-\frac{x}{a \sqrt{a+\frac{b}{x^2}}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0241494, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{2 x \sqrt{a+\frac{b}{x^2}}}{a^2}-\frac{x}{a \sqrt{a+\frac{b}{x^2}}} \]
Antiderivative was successfully verified.
[In] Int[(a + b/x^2)^(-3/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 1.96324, size = 29, normalized size = 0.83 \[ - \frac{x}{a \sqrt{a + \frac{b}{x^{2}}}} + \frac{2 x \sqrt{a + \frac{b}{x^{2}}}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/x**2)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0218164, size = 27, normalized size = 0.77 \[ \frac{a x^2+2 b}{a^2 x \sqrt{a+\frac{b}{x^2}}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b/x^2)^(-3/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.006, size = 37, normalized size = 1.1 \[{\frac{ \left ( a{x}^{2}+b \right ) \left ( a{x}^{2}+2\,b \right ) }{{x}^{3}{a}^{2}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/x^2)^(3/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.43021, size = 43, normalized size = 1.23 \[ \frac{\sqrt{a + \frac{b}{x^{2}}} x}{a^{2}} + \frac{b}{\sqrt{a + \frac{b}{x^{2}}} a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)^(-3/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.23137, size = 53, normalized size = 1.51 \[ \frac{{\left (a x^{3} + 2 \, b x\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{a^{3} x^{2} + a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)^(-3/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 2.94914, size = 42, normalized size = 1.2 \[ \frac{x^{2}}{a \sqrt{b} \sqrt{\frac{a x^{2}}{b} + 1}} + \frac{2 \sqrt{b}}{a^{2} \sqrt{\frac{a x^{2}}{b} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b/x**2)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)^(-3/2),x, algorithm="giac")
[Out]