Optimal. Leaf size=42 \[ -\frac{2 b}{3 a^2 x^3 \left (a+\frac{b}{x^2}\right )^{3/2}}-\frac{1}{a x \left (a+\frac{b}{x^2}\right )^{3/2}} \]
[Out]
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Rubi [A] time = 0.0616614, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{2 b}{3 a^2 x^3 \left (a+\frac{b}{x^2}\right )^{3/2}}-\frac{1}{a x \left (a+\frac{b}{x^2}\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b/x^2)^(5/2)*x^2),x]
[Out]
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Rubi in Sympy [A] time = 4.58083, size = 37, normalized size = 0.88 \[ - \frac{1}{a x \left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}}} - \frac{2 b}{3 a^{2} x^{3} \left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/x**2)**(5/2)/x**2,x)
[Out]
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Mathematica [A] time = 0.041058, size = 38, normalized size = 0.9 \[ -\frac{x \sqrt{a+\frac{b}{x^2}} \left (3 a x^2+2 b\right )}{3 a^2 \left (a x^2+b\right )^2} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b/x^2)^(5/2)*x^2),x]
[Out]
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Maple [A] time = 0.01, size = 39, normalized size = 0.9 \[ -{\frac{ \left ( a{x}^{2}+b \right ) \left ( 3\,a{x}^{2}+2\,b \right ) }{3\,{x}^{5}{a}^{2}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/x^2)^(5/2)/x^2,x)
[Out]
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Maxima [A] time = 1.44415, size = 45, normalized size = 1.07 \[ -\frac{3 \,{\left (a + \frac{b}{x^{2}}\right )} x^{2} - b}{3 \,{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} a^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x^2)^(5/2)*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.254659, size = 70, normalized size = 1.67 \[ -\frac{{\left (3 \, a x^{3} + 2 \, b x\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{3 \,{\left (a^{4} x^{4} + 2 \, a^{3} b x^{2} + a^{2} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x^2)^(5/2)*x^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.98056, size = 105, normalized size = 2.5 \[ - \frac{3 a x^{2}}{3 a^{3} \sqrt{b} x^{2} \sqrt{\frac{a x^{2}}{b} + 1} + 3 a^{2} b^{\frac{3}{2}} \sqrt{\frac{a x^{2}}{b} + 1}} - \frac{2 b}{3 a^{3} \sqrt{b} x^{2} \sqrt{\frac{a x^{2}}{b} + 1} + 3 a^{2} b^{\frac{3}{2}} \sqrt{\frac{a x^{2}}{b} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b/x**2)**(5/2)/x**2,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{5}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x^2)^(5/2)*x^2),x, algorithm="giac")
[Out]