Optimal. Leaf size=82 \[ -\frac{16 b x \sqrt{a+\frac{b}{x^2}}}{3 a^4}+\frac{8 b x}{3 a^3 \sqrt{a+\frac{b}{x^2}}}+\frac{2 b x}{3 a^2 \left (a+\frac{b}{x^2}\right )^{3/2}}+\frac{x^3}{3 a \left (a+\frac{b}{x^2}\right )^{3/2}} \]
[Out]
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Rubi [A] time = 0.0787366, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{16 b x \sqrt{a+\frac{b}{x^2}}}{3 a^4}+\frac{8 b x}{3 a^3 \sqrt{a+\frac{b}{x^2}}}+\frac{2 b x}{3 a^2 \left (a+\frac{b}{x^2}\right )^{3/2}}+\frac{x^3}{3 a \left (a+\frac{b}{x^2}\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[x^2/(a + b/x^2)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 6.54573, size = 76, normalized size = 0.93 \[ \frac{x^{3}}{3 a \left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}}} + \frac{2 b x}{3 a^{2} \left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}}} + \frac{8 b x}{3 a^{3} \sqrt{a + \frac{b}{x^{2}}}} - \frac{16 b x \sqrt{a + \frac{b}{x^{2}}}}{3 a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(a+b/x**2)**(5/2),x)
[Out]
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Mathematica [A] time = 0.0377686, size = 61, normalized size = 0.74 \[ \frac{a^3 x^6-6 a^2 b x^4-24 a b^2 x^2-16 b^3}{3 a^4 x \sqrt{a+\frac{b}{x^2}} \left (a x^2+b\right )} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/(a + b/x^2)^(5/2),x]
[Out]
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Maple [A] time = 0.01, size = 60, normalized size = 0.7 \[{\frac{ \left ( a{x}^{2}+b \right ) \left ({a}^{3}{x}^{6}-6\,{a}^{2}b{x}^{4}-24\,a{b}^{2}{x}^{2}-16\,{b}^{3} \right ) }{3\,{a}^{4}{x}^{5}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(a+b/x^2)^(5/2),x)
[Out]
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Maxima [A] time = 1.43746, size = 96, normalized size = 1.17 \[ \frac{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} x^{3} - 9 \, \sqrt{a + \frac{b}{x^{2}}} b x}{3 \, a^{4}} - \frac{9 \,{\left (a + \frac{b}{x^{2}}\right )} b^{2} x^{2} - b^{3}}{3 \,{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} a^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(a + b/x^2)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.24695, size = 99, normalized size = 1.21 \[ \frac{{\left (a^{3} x^{7} - 6 \, a^{2} b x^{5} - 24 \, a b^{2} x^{3} - 16 \, b^{3} x\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{3 \,{\left (a^{6} x^{4} + 2 \, a^{5} b x^{2} + a^{4} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(a + b/x^2)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.56565, size = 337, normalized size = 4.11 \[ \frac{a^{4} b^{\frac{19}{2}} x^{8} \sqrt{\frac{a x^{2}}{b} + 1}}{3 a^{7} b^{9} x^{6} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{2} + 3 a^{4} b^{12}} - \frac{5 a^{3} b^{\frac{21}{2}} x^{6} \sqrt{\frac{a x^{2}}{b} + 1}}{3 a^{7} b^{9} x^{6} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{2} + 3 a^{4} b^{12}} - \frac{30 a^{2} b^{\frac{23}{2}} x^{4} \sqrt{\frac{a x^{2}}{b} + 1}}{3 a^{7} b^{9} x^{6} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{2} + 3 a^{4} b^{12}} - \frac{40 a b^{\frac{25}{2}} x^{2} \sqrt{\frac{a x^{2}}{b} + 1}}{3 a^{7} b^{9} x^{6} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{2} + 3 a^{4} b^{12}} - \frac{16 b^{\frac{27}{2}} \sqrt{\frac{a x^{2}}{b} + 1}}{3 a^{7} b^{9} x^{6} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{2} + 3 a^{4} b^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(a+b/x**2)**(5/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(a + b/x^2)^(5/2),x, algorithm="giac")
[Out]