Optimal. Leaf size=46 \[ \frac{1}{2} x \sqrt{a+c x^2}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 \sqrt{c}} \]
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Rubi [A] time = 0.0242806, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{1}{2} x \sqrt{a+c x^2}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 \sqrt{c}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + c*x^2],x]
[Out]
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Rubi in Sympy [A] time = 3.17443, size = 39, normalized size = 0.85 \[ \frac{a \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{2 \sqrt{c}} + \frac{x \sqrt{a + c x^{2}}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.023936, size = 49, normalized size = 1.07 \[ \frac{1}{2} x \sqrt{a+c x^2}+\frac{a \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{2 \sqrt{c}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + c*x^2],x]
[Out]
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Maple [A] time = 0.003, size = 36, normalized size = 0.8 \[{\frac{x}{2}\sqrt{c{x}^{2}+a}}+{\frac{a}{2}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+a)^(1/2),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(sqrt(c*x^2 + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.236027, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, \sqrt{c x^{2} + a} \sqrt{c} x + a \log \left (-2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right )}{4 \, \sqrt{c}}, \frac{\sqrt{c x^{2} + a} \sqrt{-c} x + a \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right )}{2 \, \sqrt{-c}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.22564, size = 41, normalized size = 0.89 \[ \frac{\sqrt{a} x \sqrt{1 + \frac{c x^{2}}{a}}}{2} + \frac{a \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2 \sqrt{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+a)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.210636, size = 50, normalized size = 1.09 \[ \frac{1}{2} \, \sqrt{c x^{2} + a} x - \frac{a{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{2 \, \sqrt{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + a),x, algorithm="giac")
[Out]