Optimal. Leaf size=58 \[ \frac{8 x \sqrt{a+\frac{b}{x^2}}}{3 a^3}-\frac{4 x}{3 a^2 \sqrt{a+\frac{b}{x^2}}}-\frac{x}{3 a \left (a+\frac{b}{x^2}\right )^{3/2}} \]
[Out]
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Rubi [A] time = 0.0386351, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{8 x \sqrt{a+\frac{b}{x^2}}}{3 a^3}-\frac{4 x}{3 a^2 \sqrt{a+\frac{b}{x^2}}}-\frac{x}{3 a \left (a+\frac{b}{x^2}\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(a + b/x^2)^(-5/2),x]
[Out]
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Rubi in Sympy [A] time = 3.21847, size = 51, normalized size = 0.88 \[ - \frac{x}{3 a \left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}}} - \frac{4 x}{3 a^{2} \sqrt{a + \frac{b}{x^{2}}}} + \frac{8 x \sqrt{a + \frac{b}{x^{2}}}}{3 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/x**2)**(5/2),x)
[Out]
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Mathematica [A] time = 0.0341844, size = 51, normalized size = 0.88 \[ \frac{3 a^2 x^4+12 a b x^2+8 b^2}{3 a^3 x \sqrt{a+\frac{b}{x^2}} \left (a x^2+b\right )} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b/x^2)^(-5/2),x]
[Out]
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Maple [A] time = 0.007, size = 50, normalized size = 0.9 \[{\frac{ \left ( a{x}^{2}+b \right ) \left ( 3\,{x}^{4}{a}^{2}+12\,ab{x}^{2}+8\,{b}^{2} \right ) }{3\,{a}^{3}{x}^{5}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/x^2)^(5/2),x)
[Out]
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Maxima [A] time = 1.44322, size = 69, normalized size = 1.19 \[ \frac{\sqrt{a + \frac{b}{x^{2}}} x}{a^{3}} + \frac{6 \,{\left (a + \frac{b}{x^{2}}\right )} b x^{2} - b^{2}}{3 \,{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} a^{3} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)^(-5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.233599, size = 85, normalized size = 1.47 \[ \frac{{\left (3 \, a^{2} x^{5} + 12 \, a b x^{3} + 8 \, b^{2} x\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{3 \,{\left (a^{5} x^{4} + 2 \, a^{4} b x^{2} + a^{3} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)^(-5/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.47467, size = 163, normalized size = 2.81 \[ \frac{3 a^{2} b^{\frac{9}{2}} x^{4} \sqrt{\frac{a x^{2}}{b} + 1}}{3 a^{5} b^{4} x^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6}} + \frac{12 a b^{\frac{11}{2}} x^{2} \sqrt{\frac{a x^{2}}{b} + 1}}{3 a^{5} b^{4} x^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6}} + \frac{8 b^{\frac{13}{2}} \sqrt{\frac{a x^{2}}{b} + 1}}{3 a^{5} b^{4} x^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b/x**2)**(5/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)^(-5/2),x, algorithm="giac")
[Out]