Optimal. Leaf size=71 \[ -\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{8 a^{3/2}}+\frac{b x^2 \sqrt{a+\frac{b}{x^2}}}{8 a}+\frac{1}{4} x^4 \sqrt{a+\frac{b}{x^2}} \]
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Rubi [A] time = 0.115383, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{8 a^{3/2}}+\frac{b x^2 \sqrt{a+\frac{b}{x^2}}}{8 a}+\frac{1}{4} x^4 \sqrt{a+\frac{b}{x^2}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b/x^2]*x^3,x]
[Out]
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Rubi in Sympy [A] time = 9.55543, size = 60, normalized size = 0.85 \[ \frac{x^{4} \sqrt{a + \frac{b}{x^{2}}}}{4} + \frac{b x^{2} \sqrt{a + \frac{b}{x^{2}}}}{8 a} - \frac{b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{2}}}}{\sqrt{a}} \right )}}{8 a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b/x**2)**(1/2)*x**3,x)
[Out]
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Mathematica [A] time = 0.0734636, size = 88, normalized size = 1.24 \[ x \sqrt{a+\frac{b}{x^2}} \left (\frac{b x}{8 a}+\frac{x^3}{4}\right )-\frac{b^2 x \sqrt{a+\frac{b}{x^2}} \log \left (\sqrt{a} \sqrt{a x^2+b}+a x\right )}{8 a^{3/2} \sqrt{a x^2+b}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b/x^2]*x^3,x]
[Out]
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Maple [A] time = 0.017, size = 82, normalized size = 1.2 \[{\frac{x}{8}\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}} \left ( 2\,x \left ( a{x}^{2}+b \right ) ^{3/2}\sqrt{a}-\sqrt{a}\sqrt{a{x}^{2}+b}xb-\ln \left ( \sqrt{a}x+\sqrt{a{x}^{2}+b} \right ){b}^{2} \right ){\frac{1}{\sqrt{a{x}^{2}+b}}}{a}^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b/x^2)^(1/2)*x^3,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(sqrt(a + b/x^2)*x^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.253864, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{a} b^{2} \log \left (2 \, a x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} -{\left (2 \, a x^{2} + b\right )} \sqrt{a}\right ) + 2 \,{\left (2 \, a^{2} x^{4} + a b x^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{16 \, a^{2}}, \frac{\sqrt{-a} b^{2} \arctan \left (\frac{\sqrt{-a}}{\sqrt{\frac{a x^{2} + b}{x^{2}}}}\right ) +{\left (2 \, a^{2} x^{4} + a b x^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{8 \, a^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a + b/x^2)*x^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 13.0924, size = 92, normalized size = 1.3 \[ \frac{a x^{5}}{4 \sqrt{b} \sqrt{\frac{a x^{2}}{b} + 1}} + \frac{3 \sqrt{b} x^{3}}{8 \sqrt{\frac{a x^{2}}{b} + 1}} + \frac{b^{\frac{3}{2}} x}{8 a \sqrt{\frac{a x^{2}}{b} + 1}} - \frac{b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{8 a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b/x**2)**(1/2)*x**3,x)
[Out]
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GIAC/XCAS [A] time = 0.231018, size = 95, normalized size = 1.34 \[ \frac{1}{8} \, \sqrt{a x^{2} + b}{\left (2 \, x^{2}{\rm sign}\left (x\right ) + \frac{b{\rm sign}\left (x\right )}{a}\right )} x - \frac{b^{2}{\rm ln}\left (\sqrt{b}\right ){\rm sign}\left (x\right )}{8 \, a^{\frac{3}{2}}} + \frac{b^{2}{\rm ln}\left ({\left | -\sqrt{a} x + \sqrt{a x^{2} + b} \right |}\right ){\rm sign}\left (x\right )}{8 \, a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a + b/x^2)*x^3,x, algorithm="giac")
[Out]