Optimal. Leaf size=88 \[ -\frac{5}{2} a b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )-\frac{5 b^2 \sqrt{a+\frac{b}{x^2}}}{2 x}+\frac{5}{3} b x \left (a+\frac{b}{x^2}\right )^{3/2}+\frac{1}{3} x^3 \left (a+\frac{b}{x^2}\right )^{5/2} \]
[Out]
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Rubi [A] time = 0.124649, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{5}{2} a b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )-\frac{5 b^2 \sqrt{a+\frac{b}{x^2}}}{2 x}+\frac{5}{3} b x \left (a+\frac{b}{x^2}\right )^{3/2}+\frac{1}{3} x^3 \left (a+\frac{b}{x^2}\right )^{5/2} \]
Antiderivative was successfully verified.
[In] Int[(a + b/x^2)^(5/2)*x^2,x]
[Out]
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Rubi in Sympy [A] time = 10.6446, size = 78, normalized size = 0.89 \[ - \frac{5 a b^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b}}{x \sqrt{a + \frac{b}{x^{2}}}} \right )}}{2} - \frac{5 b^{2} \sqrt{a + \frac{b}{x^{2}}}}{2 x} + \frac{5 b x \left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}}}{3} + \frac{x^{3} \left (a + \frac{b}{x^{2}}\right )^{\frac{5}{2}}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b/x**2)**(5/2)*x**2,x)
[Out]
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Mathematica [A] time = 0.1091, size = 107, normalized size = 1.22 \[ \frac{\sqrt{a+\frac{b}{x^2}} \left (\sqrt{a x^2+b} \left (2 a^2 x^4+14 a b x^2-3 b^2\right )+15 a b^{3/2} x^2 \log (x)-15 a b^{3/2} x^2 \log \left (\sqrt{b} \sqrt{a x^2+b}+b\right )\right )}{6 x \sqrt{a x^2+b}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b/x^2)^(5/2)*x^2,x]
[Out]
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Maple [A] time = 0.012, size = 122, normalized size = 1.4 \[ -{\frac{{x}^{3}}{6\,b} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{{\frac{5}{2}}} \left ( -3\, \left ( a{x}^{2}+b \right ) ^{5/2}{x}^{2}a+15\,{b}^{5/2}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{2}+b}+b}{x}} \right ){x}^{2}a+3\, \left ( a{x}^{2}+b \right ) ^{7/2}-5\, \left ( a{x}^{2}+b \right ) ^{3/2}{x}^{2}ab-15\,\sqrt{a{x}^{2}+b}{x}^{2}a{b}^{2} \right ) \left ( a{x}^{2}+b \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b/x^2)^(5/2)*x^2,x)
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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)^(5/2)*x^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.259164, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a b^{\frac{3}{2}} x \log \left (-\frac{a x^{2} - 2 \, \sqrt{b} x \sqrt{\frac{a x^{2} + b}{x^{2}}} + 2 \, b}{x^{2}}\right ) + 2 \,{\left (2 \, a^{2} x^{4} + 14 \, a b x^{2} - 3 \, b^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{12 \, x}, -\frac{15 \, a \sqrt{-b} b x \arctan \left (\frac{b}{\sqrt{-b} x \sqrt{\frac{a x^{2} + b}{x^{2}}}}\right ) -{\left (2 \, a^{2} x^{4} + 14 \, a b x^{2} - 3 \, b^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{6 \, x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)^(5/2)*x^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 17.9147, size = 112, normalized size = 1.27 \[ \frac{a^{2} \sqrt{b} x^{2} \sqrt{\frac{a x^{2}}{b} + 1}}{3} + \frac{7 a b^{\frac{3}{2}} \sqrt{\frac{a x^{2}}{b} + 1}}{3} + \frac{5 a b^{\frac{3}{2}} \log{\left (\frac{a x^{2}}{b} \right )}}{4} - \frac{5 a b^{\frac{3}{2}} \log{\left (\sqrt{\frac{a x^{2}}{b} + 1} + 1 \right )}}{2} - \frac{b^{\frac{5}{2}} \sqrt{\frac{a x^{2}}{b} + 1}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b/x**2)**(5/2)*x**2,x)
[Out]
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GIAC/XCAS [A] time = 0.262812, size = 101, normalized size = 1.15 \[ \frac{1}{6} \,{\left (\frac{15 \, b^{2} \arctan \left (\frac{\sqrt{a x^{2} + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} + 2 \,{\left (a x^{2} + b\right )}^{\frac{3}{2}} + 12 \, \sqrt{a x^{2} + b} b - \frac{3 \, \sqrt{a x^{2} + b} b^{2}}{a x^{2}}\right )} a{\rm sign}\left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)^(5/2)*x^2,x, algorithm="giac")
[Out]