Optimal. Leaf size=74 \[ -\frac{15 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 a^{7/2}}+\frac{15 x}{8 a^3}-\frac{5 x^3}{8 a^2 \left (a x^2+b\right )}-\frac{x^5}{4 a \left (a x^2+b\right )^2} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0865592, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444 \[ -\frac{15 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 a^{7/2}}+\frac{15 x}{8 a^3}-\frac{5 x^3}{8 a^2 \left (a x^2+b\right )}-\frac{x^5}{4 a \left (a x^2+b\right )^2} \]
Antiderivative was successfully verified.
[In] Int[(a + b/x^2)^(-3),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 14.5185, size = 66, normalized size = 0.89 \[ - \frac{x^{5}}{4 a \left (a x^{2} + b\right )^{2}} - \frac{5 x^{3}}{8 a^{2} \left (a x^{2} + b\right )} + \frac{15 x}{8 a^{3}} - \frac{15 \sqrt{b} \operatorname{atan}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{8 a^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/x**2)**3,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0928863, size = 66, normalized size = 0.89 \[ \frac{8 a^2 x^5+25 a b x^3+15 b^2 x}{8 a^3 \left (a x^2+b\right )^2}-\frac{15 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 a^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b/x^2)^(-3),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.008, size = 63, normalized size = 0.9 \[{\frac{x}{{a}^{3}}}+{\frac{9\,b{x}^{3}}{8\,{a}^{2} \left ( a{x}^{2}+b \right ) ^{2}}}+{\frac{7\,{b}^{2}x}{8\,{a}^{3} \left ( a{x}^{2}+b \right ) ^{2}}}-{\frac{15\,b}{8\,{a}^{3}}\arctan \left ({ax{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/x^2)^3,x)
[Out]
_______________________________________________________________________________________
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((a + b/x^2)^(-3),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.229005, size = 1, normalized size = 0.01 \[ \left [\frac{16 \, a^{2} x^{5} + 50 \, a b x^{3} + 30 \, b^{2} x + 15 \,{\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{a x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - b}{a x^{2} + b}\right )}{16 \,{\left (a^{5} x^{4} + 2 \, a^{4} b x^{2} + a^{3} b^{2}\right )}}, \frac{8 \, a^{2} x^{5} + 25 \, a b x^{3} + 15 \, b^{2} x - 15 \,{\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{x}{\sqrt{\frac{b}{a}}}\right )}{8 \,{\left (a^{5} x^{4} + 2 \, a^{4} b x^{2} + a^{3} b^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)^(-3),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 2.20351, size = 107, normalized size = 1.45 \[ \frac{15 \sqrt{- \frac{b}{a^{7}}} \log{\left (- a^{3} \sqrt{- \frac{b}{a^{7}}} + x \right )}}{16} - \frac{15 \sqrt{- \frac{b}{a^{7}}} \log{\left (a^{3} \sqrt{- \frac{b}{a^{7}}} + x \right )}}{16} + \frac{9 a b x^{3} + 7 b^{2} x}{8 a^{5} x^{4} + 16 a^{4} b x^{2} + 8 a^{3} b^{2}} + \frac{x}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b/x**2)**3,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.22085, size = 73, normalized size = 0.99 \[ -\frac{15 \, b \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{3}} + \frac{x}{a^{3}} + \frac{9 \, a b x^{3} + 7 \, b^{2} x}{8 \,{\left (a x^{2} + b\right )}^{2} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)^(-3),x, algorithm="giac")
[Out]