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]

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

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Rubi [A]  time = 0.01, antiderivative size = 46, normalized size of antiderivative = 1.00, 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 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])

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]

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*}

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Mathematica [A]  time = 0.02, 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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fricas [A]  time = 0.93, size = 94, normalized size = 2.04 \[ \left [\frac {2 \, \sqrt {c x^{2} + a} c x + a \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right )}{4 \, c}, \frac {\sqrt {c x^{2} + a} c x - a \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right )}{2 \, c}\right ] \]

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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giac [A]  time = 0.40, size = 37, normalized size = 0.80 \[ \frac {1}{2} \, \sqrt {c x^{2} + a} x - \frac {a \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{2 \, \sqrt {c}} \]

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)

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maple [A]  time = 0.04, size = 36, normalized size = 0.78 \[ \frac {a \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}+\frac {\sqrt {c \,x^{2}+a}\, x}{2} \]

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 [A]  time = 1.36, size = 28, normalized size = 0.61 \[ \frac {1}{2} \, \sqrt {c x^{2} + a} x + \frac {a \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.13, size = 35, normalized size = 0.76 \[ \frac {x\,\sqrt {c\,x^2+a}}{2}+\frac {a\,\ln \left (\sqrt {c}\,x+\sqrt {c\,x^2+a}\right )}{2\,\sqrt {c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^(1/2),x)

[Out]

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

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sympy [A]  time = 1.91, 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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