3.2.90 \(\int \frac {\sqrt {1+4 x^2}}{\sqrt {2-3 x^2}} \, dx\) [190]

Optimal. Leaf size=20 \[ \frac {E\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|-\frac {8}{3}\right )}{\sqrt {3}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {435} \begin {gather*} \frac {E\left (\text {ArcSin}\left (\sqrt {\frac {3}{2}} x\right )|-\frac {8}{3}\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + 4*x^2]/Sqrt[2 - 3*x^2],x]

[Out]

EllipticE[ArcSin[Sqrt[3/2]*x], -8/3]/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 {1+4 x^2}}{\sqrt {2-3 x^2}} \, dx &=\frac {E\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|-\frac {8}{3}\right )}{\sqrt {3}}\\ \end {align*}

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Mathematica [A]
time = 0.32, size = 20, normalized size = 1.00 \begin {gather*} \frac {E\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|-\frac {8}{3}\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + 4*x^2]/Sqrt[2 - 3*x^2],x]

[Out]

EllipticE[ArcSin[Sqrt[3/2]*x], -8/3]/Sqrt[3]

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Maple [A]
time = 0.09, size = 19, normalized size = 0.95

method	result	size
default	\(\frac {\EllipticE \left (\frac {x \sqrt {6}}{2}, \frac {2 i \sqrt {6}}{3}\right ) \sqrt {3}}{3}\)	\(19\)
elliptic	\(\frac {\sqrt {-\left (3 x^{2}-2\right ) \left (4 x^{2}+1\right )}\, \left (\frac {\sqrt {6}\, \sqrt {-6 x^{2}+4}\, \sqrt {4 x^{2}+1}\, \EllipticF \left (\frac {x \sqrt {6}}{2}, \frac {2 i \sqrt {6}}{3}\right )}{6 \sqrt {-12 x^{4}+5 x^{2}+2}}-\frac {\sqrt {6}\, \sqrt {-6 x^{2}+4}\, \sqrt {4 x^{2}+1}\, \left (\EllipticF \left (\frac {x \sqrt {6}}{2}, \frac {2 i \sqrt {6}}{3}\right )-\EllipticE \left (\frac {x \sqrt {6}}{2}, \frac {2 i \sqrt {6}}{3}\right )\right )}{6 \sqrt {-12 x^{4}+5 x^{2}+2}}\right )}{\sqrt {4 x^{2}+1}\, \sqrt {-3 x^{2}+2}}\)	\(155\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*EllipticE(1/2*x*6^(1/2),2/3*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((4*x^2+1)^(1/2)/(-3*x^2+2)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.22, size = 23, normalized size = 1.15 \begin {gather*} -\frac {\sqrt {4 \, x^{2} + 1} \sqrt {-3 \, x^{2} + 2}}{3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 1.60, size = 36, normalized size = 1.80 \begin {gather*} \begin {cases} \frac {\sqrt {3} E\left (\operatorname {asin}{\left (\frac {\sqrt {6} x}{2} \right )}\middle | - \frac {8}{3}\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((4*x**2+1)**(1/2)/(-3*x**2+2)**(1/2),x)

[Out]

Piecewise((sqrt(3)*elliptic_e(asin(sqrt(6)*x/2), -8/3)/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((4*x^2+1)^(1/2)/(-3*x^2+2)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(4*x^2 + 1)/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 {4\,x^2+1}}{\sqrt {2-3\,x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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