3.58 \(\int \sqrt{a+c x^2} \, dx\)

Optimal. Leaf size=46 \[ \frac{1}{2} x \sqrt{a+c x^2}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 \sqrt{c}} \]

[Out]

(x*Sqrt[a + c*x^2])/2 + (a*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(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]

(x*Sqrt[a + c*x^2])/2 + (a*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(2*Sqrt[c])

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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]

a*atanh(sqrt(c)*x/sqrt(a + c*x**2))/(2*sqrt(c)) + x*sqrt(a + c*x**2)/2

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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]

(x*Sqrt[a + c*x^2])/2 + (a*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/(2*Sqrt[c])

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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]

1/2*x*(c*x^2+a)^(1/2)+1/2*a/c^(1/2)*ln(c^(1/2)*x+(c*x^2+a)^(1/2))

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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]

Exception raised: ValueError

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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]

[1/4*(2*sqrt(c*x^2 + a)*sqrt(c)*x + a*log(-2*sqrt(c*x^2 + a)*c*x - (2*c*x^2 + a)
*sqrt(c)))/sqrt(c), 1/2*(sqrt(c*x^2 + a)*sqrt(-c)*x + a*arctan(sqrt(-c)*x/sqrt(c
*x^2 + a)))/sqrt(-c)]

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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]

sqrt(a)*x*sqrt(1 + c*x**2/a)/2 + a*asinh(sqrt(c)*x/sqrt(a))/(2*sqrt(c))

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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]

1/2*sqrt(c*x^2 + a)*x - 1/2*a*ln(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/sqrt(c)