Optimal. Leaf size=47 \[ -\frac{\sqrt{a+b x^2}}{2 x^2}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 \sqrt{a}} \]
[Out]
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Rubi [A] time = 0.078623, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{\sqrt{a+b x^2}}{2 x^2}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 \sqrt{a}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b*x^2]/x^3,x]
[Out]
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Rubi in Sympy [A] time = 7.78796, size = 41, normalized size = 0.87 \[ - \frac{\sqrt{a + b x^{2}}}{2 x^{2}} - \frac{b \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{2 \sqrt{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**(1/2)/x**3,x)
[Out]
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Mathematica [A] time = 0.0459745, size = 58, normalized size = 1.23 \[ \frac{1}{2} \left (-\frac{\sqrt{a+b x^2}}{x^2}-\frac{b \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )}{\sqrt{a}}+\frac{b \log (x)}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b*x^2]/x^3,x]
[Out]
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Maple [A] time = 0.006, size = 63, normalized size = 1.3 \[ -{\frac{1}{2\,a{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{b}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{b}{2\,a}\sqrt{b{x}^{2}+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^(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(b*x^2 + a)/x^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.236683, size = 1, normalized size = 0.02 \[ \left [\frac{b x^{2} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right ) - 2 \, \sqrt{b x^{2} + a} \sqrt{a}}{4 \, \sqrt{a} x^{2}}, -\frac{b x^{2} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) + \sqrt{b x^{2} + a} \sqrt{-a}}{2 \, \sqrt{-a} x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^2 + a)/x^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.45371, size = 42, normalized size = 0.89 \[ - \frac{\sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{2 x} - \frac{b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2 \sqrt{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**(1/2)/x**3,x)
[Out]
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GIAC/XCAS [A] time = 0.208675, size = 58, normalized size = 1.23 \[ \frac{1}{2} \, b{\left (\frac{\arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{\sqrt{b x^{2} + a}}{b x^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^2 + a)/x^3,x, algorithm="giac")
[Out]