3.2.89 \(\int \frac {\sqrt {4+x^2}}{\sqrt {2-3 x^2}} \, dx\) [189]

Optimal. Leaf size=21 \[ \frac {2 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|-\frac {1}{6}\right )}{\sqrt {3}} \]

[Out]

2/3*EllipticE(1/2*x*6^(1/2),1/6*I*6^(1/2))*3^(1/2)

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Rubi [A]
time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {435} \begin {gather*} \frac {2 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{2}} x\right )|-\frac {1}{6}\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[4 + x^2]/Sqrt[2 - 3*x^2],x]

[Out]

(2*EllipticE[ArcSin[Sqrt[3/2]*x], -1/6])/Sqrt[3]

Rule 435

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

Rubi steps

\begin {align*} \int \frac {\sqrt {4+x^2}}{\sqrt {2-3 x^2}} \, dx &=\frac {2 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|-\frac {1}{6}\right )}{\sqrt {3}}\\ \end {align*}

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Mathematica [A]
time = 0.30, size = 21, normalized size = 1.00 \begin {gather*} \frac {2 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|-\frac {1}{6}\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[4 + x^2]/Sqrt[2 - 3*x^2],x]

[Out]

(2*EllipticE[ArcSin[Sqrt[3/2]*x], -1/6])/Sqrt[3]

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Maple [A]
time = 0.09, size = 19, normalized size = 0.90

method	result	size
default	\(\frac {2 \EllipticE \left (\frac {x \sqrt {6}}{2}, \frac {i \sqrt {6}}{6}\right ) \sqrt {3}}{3}\)	\(19\)
elliptic	\(\frac {\sqrt {-\left (3 x^{2}-2\right ) \left (x^{2}+4\right )}\, \left (\frac {\sqrt {6}\, \sqrt {-6 x^{2}+4}\, \sqrt {x^{2}+4}\, \EllipticF \left (\frac {x \sqrt {6}}{2}, \frac {i \sqrt {6}}{6}\right )}{3 \sqrt {-3 x^{4}-10 x^{2}+8}}-\frac {\sqrt {6}\, \sqrt {-6 x^{2}+4}\, \sqrt {x^{2}+4}\, \left (\EllipticF \left (\frac {x \sqrt {6}}{2}, \frac {i \sqrt {6}}{6}\right )-\EllipticE \left (\frac {x \sqrt {6}}{2}, \frac {i \sqrt {6}}{6}\right )\right )}{3 \sqrt {-3 x^{4}-10 x^{2}+8}}\right )}{\sqrt {x^{2}+4}\, \sqrt {-3 x^{2}+2}}\)	\(147\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+4)^(1/2)/(-3*x^2+2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/3*EllipticE(1/2*x*6^(1/2),1/6*I*6^(1/2))*3^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+4)^(1/2)/(-3*x^2+2)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(x^2 + 4)/sqrt(-3*x^2 + 2), x)

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Fricas [A]
time = 0.27, size = 21, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {x^{2} + 4} \sqrt {-3 \, x^{2} + 2}}{3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+4)^(1/2)/(-3*x^2+2)^(1/2),x, algorithm="fricas")

[Out]

-1/3*sqrt(x^2 + 4)*sqrt(-3*x^2 + 2)/x

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Sympy [A]
time = 1.62, size = 37, normalized size = 1.76 \begin {gather*} \begin {cases} \frac {2 \sqrt {3} E\left (\operatorname {asin}{\left (\frac {\sqrt {6} x}{2} \right )}\middle | - \frac {1}{6}\right )}{3} & \text {for}\: x > - \frac {\sqrt {6}}{3} \wedge x < \frac {\sqrt {6}}{3} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2+4)**(1/2)/(-3*x**2+2)**(1/2),x)

[Out]

Piecewise((2*sqrt(3)*elliptic_e(asin(sqrt(6)*x/2), -1/6)/3, (x > -sqrt(6)/3) & (x < sqrt(6)/3)))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+4)^(1/2)/(-3*x^2+2)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(x^2 + 4)/sqrt(-3*x^2 + 2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {\sqrt {x^2+4}}{\sqrt {2-3\,x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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