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 = 0.0089727, 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, Rules used = {195, 217, 206} \[ \frac{1}{2} x \sqrt{a+c x^2}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 \sqrt{c}} \]

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])

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \sqrt{a+c x^2} \, dx &=\frac{1}{2} x \sqrt{a+c x^2}+\frac{1}{2} a \int \frac{1}{\sqrt{a+c x^2}} \, dx\\ &=\frac{1}{2} x \sqrt{a+c x^2}+\frac{1}{2} a \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )\\ &=\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}}\\ \end{align*}

Mathematica [A]  time = 0.0205114, 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 = 0.047, size = 36, normalized size = 0.8 \begin{align*}{\frac{x}{2}\sqrt{c{x}^{2}+a}}+{\frac{a}{2}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}} \end{align*}

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.38939, size = 232, normalized size = 5.04 \begin{align*} \left [\frac{2 \, \sqrt{c x^{2} + a} c x + a \sqrt{c} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right )}{4 \, c}, \frac{\sqrt{c x^{2} + a} c x - a \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right )}{2 \, c}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.71647, size = 41, normalized size = 0.89 \begin{align*} \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}} \end{align*}

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 [A]  time = 1.24465, size = 50, normalized size = 1.09 \begin{align*} \frac{1}{2} \, \sqrt{c x^{2} + a} x - \frac{a \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{2 \, \sqrt{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(1/2),x, algorithm="giac")

[Out]

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