3.2.96 \(\int \frac {\sqrt {4+x^2}}{\sqrt {2+3 x^2}} \, dx\) [196]

Optimal. Leaf size=136 \[ \frac {x \sqrt {2+3 x^2}}{3 \sqrt {4+x^2}}-\frac {\sqrt {2} \sqrt {2+3 x^2} E\left (\left .\tan ^{-1}\left (\frac {x}{2}\right )\right |-5\right )}{3 \sqrt {4+x^2} \sqrt {\frac {2+3 x^2}{4+x^2}}}+\frac {2 \sqrt {2} \sqrt {2+3 x^2} F\left (\left .\tan ^{-1}\left (\frac {x}{2}\right )\right |-5\right )}{\sqrt {4+x^2} \sqrt {\frac {2+3 x^2}{4+x^2}}} \]

[Out]

1/3*x*(3*x^2+2)^(1/2)/(x^2+4)^(1/2)-1/3*(1/(x^2+4))^(1/2)*EllipticE(x/(x^2+4)^(1/2),I*5^(1/2))*2^(1/2)*(3*x^2+
2)^(1/2)/((3*x^2+2)/(x^2+4))^(1/2)+2*(1/(x^2+4))^(1/2)*EllipticF(x/(x^2+4)^(1/2),I*5^(1/2))*2^(1/2)*(3*x^2+2)^
(1/2)/((3*x^2+2)/(x^2+4))^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {433, 429, 506, 422} \begin {gather*} \frac {2 \sqrt {2} \sqrt {3 x^2+2} F\left (\left .\text {ArcTan}\left (\frac {x}{2}\right )\right |-5\right )}{\sqrt {x^2+4} \sqrt {\frac {3 x^2+2}{x^2+4}}}-\frac {\sqrt {2} \sqrt {3 x^2+2} E\left (\left .\text {ArcTan}\left (\frac {x}{2}\right )\right |-5\right )}{3 \sqrt {x^2+4} \sqrt {\frac {3 x^2+2}{x^2+4}}}+\frac {\sqrt {3 x^2+2} x}{3 \sqrt {x^2+4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[4 + x^2]/Sqrt[2 + 3*x^2],x]

[Out]

(x*Sqrt[2 + 3*x^2])/(3*Sqrt[4 + x^2]) - (Sqrt[2]*Sqrt[2 + 3*x^2]*EllipticE[ArcTan[x/2], -5])/(3*Sqrt[4 + x^2]*
Sqrt[(2 + 3*x^2)/(4 + x^2)]) + (2*Sqrt[2]*Sqrt[2 + 3*x^2]*EllipticF[ArcTan[x/2], -5])/(Sqrt[4 + x^2]*Sqrt[(2 +
3*x^2)/(4 + x^2)])

Rule 422

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sq
rt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /;
FreeQ[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]

Rule 429

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*
Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /
; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]

Rule 433

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[a, Int[1/(Sqrt[a + b*x^2]*Sqrt[c +
d*x^2]), x], x] + Dist[b, Int[x^2/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && PosQ[
d/c] && PosQ[b/a]

Rule 506

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt
[c + d*x^2])), x] - Dist[c/b, Int[Sqrt[a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b
*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]

Rubi steps

\begin {align*} \int \frac {\sqrt {4+x^2}}{\sqrt {2+3 x^2}} \, dx &=4 \int \frac {1}{\sqrt {4+x^2} \sqrt {2+3 x^2}} \, dx+\int \frac {x^2}{\sqrt {4+x^2} \sqrt {2+3 x^2}} \, dx\\ &=\frac {x \sqrt {2+3 x^2}}{3 \sqrt {4+x^2}}+\frac {2 \sqrt {2} \sqrt {2+3 x^2} F\left (\left .\tan ^{-1}\left (\frac {x}{2}\right )\right |-5\right )}{\sqrt {4+x^2} \sqrt {\frac {2+3 x^2}{4+x^2}}}-\frac {4}{3} \int \frac {\sqrt {2+3 x^2}}{\left (4+x^2\right )^{3/2}} \, dx\\ &=\frac {x \sqrt {2+3 x^2}}{3 \sqrt {4+x^2}}-\frac {\sqrt {2} \sqrt {2+3 x^2} E\left (\left .\tan ^{-1}\left (\frac {x}{2}\right )\right |-5\right )}{3 \sqrt {4+x^2} \sqrt {\frac {2+3 x^2}{4+x^2}}}+\frac {2 \sqrt {2} \sqrt {2+3 x^2} F\left (\left .\tan ^{-1}\left (\frac {x}{2}\right )\right |-5\right )}{\sqrt {4+x^2} \sqrt {\frac {2+3 x^2}{4+x^2}}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.28, size = 27, normalized size = 0.20 \begin {gather*} -\frac {2 i E\left (i \sinh ^{-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*I)*EllipticE[I*ArcSinh[Sqrt[3/2]*x], 1/6])/Sqrt[3]

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Maple [A]
time = 0.08, size = 26, normalized size = 0.19

method	result	size
default	\(-\frac {i \left (5 \EllipticF \left (\frac {i x}{2}, \sqrt {6}\right )+\EllipticE \left (\frac {i x}{2}, \sqrt {6}\right )\right ) \sqrt {2}}{3}\)	\(26\)
elliptic	\(\frac {\sqrt {\left (3 x^{2}+2\right ) \left (x^{2}+4\right )}\, \left (-\frac {2 i \sqrt {x^{2}+4}\, \sqrt {6 x^{2}+4}\, \EllipticF \left (\frac {i x}{2}, \sqrt {6}\right )}{\sqrt {3 x^{4}+14 x^{2}+8}}+\frac {i \sqrt {x^{2}+4}\, \sqrt {6 x^{2}+4}\, \left (\EllipticF \left (\frac {i x}{2}, \sqrt {6}\right )-\EllipticE \left (\frac {i x}{2}, \sqrt {6}\right )\right )}{3 \sqrt {3 x^{4}+14 x^{2}+8}}\right )}{\sqrt {3 x^{2}+2}\, \sqrt {x^{2}+4}}\)	\(127\)

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]

-1/3*I*(5*EllipticF(1/2*I*x,6^(1/2))+EllipticE(1/2*I*x,6^(1/2)))*2^(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 [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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]

Exception raised: TypeError >> Symbolic function elliptic_ec takes exactly 1 arguments (2 given)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{2} + 4}}{\sqrt {3 x^{2} + 2}}\, dx \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]

Integral(sqrt(x**2 + 4)/sqrt(3*x**2 + 2), x)

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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 = -0.01 \begin {gather*} \int \frac {\sqrt {x^2+4}}{\sqrt {3\,x^2+2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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