\(\int \frac {\sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx\) [296]

Optimal result

Integrand size = 23, antiderivative size = 19 \[ \int \frac {\sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=-\frac {E\left (\left .\arccos \left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{\sqrt {3}} \]

[Out]

-1/3*(x^2)^(1/2)/x*EllipticE(1/2*(-6*x^2+4)^(1/2),2^(1/2))*3^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {436} \[ \int \frac {\sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=-\frac {E\left (\left .\arccos \left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{\sqrt {3}} \]

[In]

Int[Sqrt[-1 + 3*x^2]/Sqrt[2 - 3*x^2],x]

[Out]

-(EllipticE[ArcCos[Sqrt[3/2]*x], 2]/Sqrt[3])

Rule 436

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(-Sqrt[a - b*(c/d)]/(Sqrt[c]*Rt[-d/
c, 2]))*EllipticE[ArcCos[Rt[-d/c, 2]*x], b*(c/(b*c - a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[
c, 0] && GtQ[a - b*(c/d), 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {E\left (\left .\cos ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{\sqrt {3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.84 \[ \int \frac {\sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=\frac {\sqrt {-1+3 x^2} E\left (\left .\arcsin \left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{\sqrt {3-9 x^2}} \]

[In]

Integrate[Sqrt[-1 + 3*x^2]/Sqrt[2 - 3*x^2],x]

[Out]

(Sqrt[-1 + 3*x^2]*EllipticE[ArcSin[Sqrt[3/2]*x], 2])/Sqrt[3 - 9*x^2]

Maple [A] (verified)

Time = 2.50 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.95

method	result	size
default	\(-\frac {E\left (\frac {x \sqrt {2}\, \sqrt {3}}{2}, \sqrt {2}\right ) \sqrt {-3 x^{2}+1}\, \sqrt {3}}{3 \sqrt {3 x^{2}-1}}\)	\(37\)
elliptic	\(\frac {\sqrt {-\left (3 x^{2}-2\right ) \left (3 x^{2}-1\right )}\, \left (-\frac {\sqrt {6}\, \sqrt {-6 x^{2}+4}\, \sqrt {-3 x^{2}+1}\, F\left (\frac {x \sqrt {6}}{2}, \sqrt {2}\right )}{6 \sqrt {-9 x^{4}+9 x^{2}-2}}+\frac {\sqrt {6}\, \sqrt {-6 x^{2}+4}\, \sqrt {-3 x^{2}+1}\, \left (F\left (\frac {x \sqrt {6}}{2}, \sqrt {2}\right )-E\left (\frac {x \sqrt {6}}{2}, \sqrt {2}\right )\right )}{6 \sqrt {-9 x^{4}+9 x^{2}-2}}\right )}{\sqrt {3 x^{2}-1}\, \sqrt {-3 x^{2}+2}}\)	\(146\)

[In]

int((3*x^2-1)^(1/2)/(-3*x^2+2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/3*EllipticE(1/2*x*2^(1/2)*3^(1/2),2^(1/2))*(-3*x^2+1)^(1/2)*3^(1/2)/(3*x^2-1)^(1/2)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 72, normalized size of antiderivative = 3.79 \[ \int \frac {\sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=\frac {-4 i \, \sqrt {3} \sqrt {2} x E(\arcsin \left (\frac {\sqrt {3} \sqrt {2}}{3 \, x}\right )\,|\,\frac {1}{2}) + i \, \sqrt {3} \sqrt {2} x F(\arcsin \left (\frac {\sqrt {3} \sqrt {2}}{3 \, x}\right )\,|\,\frac {1}{2}) - 6 \, \sqrt {3 \, x^{2} - 1} \sqrt {-3 \, x^{2} + 2}}{18 \, x} \]

[In]

integrate((3*x^2-1)^(1/2)/(-3*x^2+2)^(1/2),x, algorithm="fricas")

[Out]

1/18*(-4*I*sqrt(3)*sqrt(2)*x*elliptic_e(arcsin(1/3*sqrt(3)*sqrt(2)/x), 1/2) + I*sqrt(3)*sqrt(2)*x*elliptic_f(a
rcsin(1/3*sqrt(3)*sqrt(2)/x), 1/2) - 6*sqrt(3*x^2 - 1)*sqrt(-3*x^2 + 2))/x

Sympy [F]

\[ \int \frac {\sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=\int \frac {\sqrt {3 x^{2} - 1}}{\sqrt {2 - 3 x^{2}}}\, dx \]

[In]

integrate((3*x**2-1)**(1/2)/(-3*x**2+2)**(1/2),x)

[Out]

Integral(sqrt(3*x**2 - 1)/sqrt(2 - 3*x**2), x)

Maxima [F]

\[ \int \frac {\sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=\int { \frac {\sqrt {3 \, x^{2} - 1}}{\sqrt {-3 \, x^{2} + 2}} \,d x } \]

[In]

integrate((3*x^2-1)^(1/2)/(-3*x^2+2)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(3*x^2 - 1)/sqrt(-3*x^2 + 2), x)

Giac [F]

\[ \int \frac {\sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=\int { \frac {\sqrt {3 \, x^{2} - 1}}{\sqrt {-3 \, x^{2} + 2}} \,d x } \]

[In]

integrate((3*x^2-1)^(1/2)/(-3*x^2+2)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(3*x^2 - 1)/sqrt(-3*x^2 + 2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=\int \frac {\sqrt {3\,x^2-1}}{\sqrt {2-3\,x^2}} \,d x \]

[In]

int((3*x^2 - 1)^(1/2)/(2 - 3*x^2)^(1/2),x)

[Out]

int((3*x^2 - 1)^(1/2)/(2 - 3*x^2)^(1/2), x)