3.1005 \(\int \frac{x^2}{\sqrt{4+x^2} \sqrt{2+3 x^2}} \, dx\)

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

[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)])

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Rubi [A]  time = 0.0291869, antiderivative size = 82, normalized size of antiderivative = 1., 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 = {492, 411} \[ \frac{x \sqrt{3 x^2+2}}{3 \sqrt{x^2+4}}-\frac{\sqrt{2} \sqrt{3 x^2+2} E\left (\left .\tan ^{-1}\left (\frac{x}{2}\right )\right |-5\right )}{3 \sqrt{x^2+4} \sqrt{\frac{3 x^2+2}{x^2+4}}} \]

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 492

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(x*Sqrt[a + b*x^2])/(b*Sqr
t[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]

Rule 411

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

Mathematica [C]  time = 0.0280067, size = 38, normalized size = 0.46 \[ -\frac{1}{3} i \sqrt{2} \left (E\left (\left .i \sinh ^{-1}\left (\frac{x}{2}\right )\right |6\right )-\text{EllipticF}\left (i \sinh ^{-1}\left (\frac{x}{2}\right ),6\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.015, size = 26, normalized size = 0.3 \begin{align*}{\frac{i}{3}} \left ({\it EllipticF} \left ({\frac{i}{2}}x,\sqrt{6} \right ) -{\it EllipticE} \left ({\frac{i}{2}}x,\sqrt{6} \right ) \right ) \sqrt{2} \end{align*}

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]

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., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\sqrt{3 \, x^{2} + 2} \sqrt{x^{2} + 4}}\,{d x} \end{align*}

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]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{3 \, x^{2} + 2} \sqrt{x^{2} + 4} x^{2}}{3 \, x^{4} + 14 \, x^{2} + 8}, x\right ) \end{align*}

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]

integral(sqrt(3*x^2 + 2)*sqrt(x^2 + 4)*x^2/(3*x^4 + 14*x^2 + 8), x)

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

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., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\sqrt{3 \, x^{2} + 2} \sqrt{x^{2} + 4}}\,{d x} \end{align*}

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)