Optimal. Leaf size=71 \[ -\frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{8 \sqrt{b}}-\frac{3 a \sqrt{a+\frac{b}{x^2}}}{8 x}-\frac{\left (a+\frac{b}{x^2}\right )^{3/2}}{4 x} \]
[Out]
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Rubi [A] time = 0.0829662, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{8 \sqrt{b}}-\frac{3 a \sqrt{a+\frac{b}{x^2}}}{8 x}-\frac{\left (a+\frac{b}{x^2}\right )^{3/2}}{4 x} \]
Antiderivative was successfully verified.
[In] Int[(a + b/x^2)^(3/2)/x^2,x]
[Out]
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Rubi in Sympy [A] time = 6.22149, size = 61, normalized size = 0.86 \[ - \frac{3 a^{2} \operatorname{atanh}{\left (\frac{\sqrt{b}}{x \sqrt{a + \frac{b}{x^{2}}}} \right )}}{8 \sqrt{b}} - \frac{3 a \sqrt{a + \frac{b}{x^{2}}}}{8 x} - \frac{\left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}}}{4 x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b/x**2)**(3/2)/x**2,x)
[Out]
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Mathematica [A] time = 0.0971036, size = 101, normalized size = 1.42 \[ \frac{\sqrt{a+\frac{b}{x^2}} \left (-3 a^2 x^4 \log \left (\sqrt{b} \sqrt{a x^2+b}+b\right )+3 a^2 x^4 \log (x)-\sqrt{b} \sqrt{a x^2+b} \left (5 a x^2+2 b\right )\right )}{8 \sqrt{b} x^3 \sqrt{a x^2+b}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b/x^2)^(3/2)/x^2,x]
[Out]
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Maple [B] time = 0.012, size = 125, normalized size = 1.8 \[ -{\frac{1}{8\,{b}^{2}x} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{{\frac{3}{2}}} \left ( 3\,{b}^{3/2}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{2}+b}+b}{x}} \right ){x}^{4}{a}^{2}- \left ( a{x}^{2}+b \right ) ^{{\frac{3}{2}}}{x}^{4}{a}^{2}+ \left ( a{x}^{2}+b \right ) ^{{\frac{5}{2}}}{x}^{2}a-3\,\sqrt{a{x}^{2}+b}{x}^{4}{a}^{2}b+2\, \left ( a{x}^{2}+b \right ) ^{5/2}b \right ) \left ( a{x}^{2}+b \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b/x^2)^(3/2)/x^2,x)
[Out]
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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)^(3/2)/x^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.257626, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a^{2} \sqrt{b} x^{3} \log \left (\frac{2 \, b x \sqrt{\frac{a x^{2} + b}{x^{2}}} -{\left (a x^{2} + 2 \, b\right )} \sqrt{b}}{x^{2}}\right ) - 2 \,{\left (5 \, a b x^{2} + 2 \, b^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{16 \, b x^{3}}, \frac{3 \, a^{2} \sqrt{-b} x^{3} \arctan \left (\frac{\sqrt{-b}}{x \sqrt{\frac{a x^{2} + b}{x^{2}}}}\right ) -{\left (5 \, a b x^{2} + 2 \, b^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{8 \, b x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)^(3/2)/x^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 11.4192, size = 71, normalized size = 1. \[ - \frac{5 a^{\frac{3}{2}} \sqrt{1 + \frac{b}{a x^{2}}}}{8 x} - \frac{\sqrt{a} b \sqrt{1 + \frac{b}{a x^{2}}}}{4 x^{3}} - \frac{3 a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x} \right )}}{8 \sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b/x**2)**(3/2)/x**2,x)
[Out]
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GIAC/XCAS [A] time = 0.249742, size = 85, normalized size = 1.2 \[ \frac{1}{8} \, a^{2}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{a x^{2} + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \frac{5 \,{\left (a x^{2} + b\right )}^{\frac{3}{2}} - 3 \, \sqrt{a x^{2} + b} b}{a^{2} x^{4}}\right )}{\rm sign}\left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)^(3/2)/x^2,x, algorithm="giac")
[Out]