3.11.5 \(\int \frac {x^2}{\sqrt {4+x^2} \sqrt {2+3 x^2}} \, dx\) [1005]

Optimal. Leaf size=82 \[ \frac {x \sqrt {2+3 x^2}}{3 \sqrt {4+x^2}}-\frac {\sqrt {2} \sqrt {2+3 x^2} E\left (\left .\tan ^{-1}\left (\frac {x}{2}\right )\right |-5\right )}{3 \sqrt {4+x^2} \sqrt {\frac {2+3 x^2}{4+x^2}}} \]

[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)

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Rubi [A]
time = 0.02, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {506, 422} \begin {gather*} \frac {x \sqrt {3 x^2+2}}{3 \sqrt {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}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/(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)])

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 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 {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 {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}}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.27, size = 38, normalized size = 0.46 \begin {gather*} -\frac {1}{3} i \sqrt {2} \left (E\left (\left .i \sinh ^{-1}\left (\frac {x}{2}\right )\right |6\right )-F\left (\left .i \sinh ^{-1}\left (\frac {x}{2}\right )\right |6\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1/3*I)*Sqrt[2]*(EllipticE[I*ArcSinh[x/2], 6] - EllipticF[I*ArcSinh[x/2], 6])

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Maple [A]
time = 0.11, size = 26, normalized size = 0.32

method	result	size
default	\(\frac {i \left (\EllipticF \left (\frac {i x}{2}, \sqrt {6}\right )-\EllipticE \left (\frac {i x}{2}, \sqrt {6}\right )\right ) \sqrt {2}}{3}\)	\(26\)
elliptic	\(\frac {i \sqrt {\left (3 x^{2}+2\right ) \left (x^{2}+4\right )}\, \sqrt {6 x^{2}+4}\, \left (\EllipticF \left (\frac {i x}{2}, \sqrt {6}\right )-\EllipticE \left (\frac {i x}{2}, \sqrt {6}\right )\right )}{3 \sqrt {3 x^{2}+2}\, \sqrt {3 x^{4}+14 x^{2}+8}}\)	\(70\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2}{\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/((x^2 + 4)^(1/2)*(3*x^2 + 2)^(1/2)),x)

[Out]

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

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