3.1006 \(\int \frac{x^2}{\sqrt{2+3 x^2} \sqrt{1+4 x^2}} \, dx\)

Optimal. Leaf size=88 \[ \frac{x \sqrt{3 x^2+2}}{3 \sqrt{4 x^2+1}}-\frac{\sqrt{3 x^2+2} E\left (\tan ^{-1}(2 x)|\frac{5}{8}\right )}{3 \sqrt{2} \sqrt{\frac{3 x^2+2}{4 x^2+1}} \sqrt{4 x^2+1}} \]

[Out]

(x*Sqrt[2 + 3*x^2])/(3*Sqrt[1 + 4*x^2]) - (Sqrt[2 + 3*x^2]*EllipticE[ArcTan[2*x], 5/8])/(3*Sqrt[2]*Sqrt[(2 + 3
*x^2)/(1 + 4*x^2)]*Sqrt[1 + 4*x^2])

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Rubi [A]  time = 0.0307442, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {492, 411} \[ \frac{x \sqrt{3 x^2+2}}{3 \sqrt{4 x^2+1}}-\frac{\sqrt{3 x^2+2} E\left (\tan ^{-1}(2 x)|\frac{5}{8}\right )}{3 \sqrt{2} \sqrt{\frac{3 x^2+2}{4 x^2+1}} \sqrt{4 x^2+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[2 + 3*x^2])/(3*Sqrt[1 + 4*x^2]) - (Sqrt[2 + 3*x^2]*EllipticE[ArcTan[2*x], 5/8])/(3*Sqrt[2]*Sqrt[(2 + 3
*x^2)/(1 + 4*x^2)]*Sqrt[1 + 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{2+3 x^2} \sqrt{1+4 x^2}} \, dx &=\frac{x \sqrt{2+3 x^2}}{3 \sqrt{1+4 x^2}}-\frac{1}{3} \int \frac{\sqrt{2+3 x^2}}{\left (1+4 x^2\right )^{3/2}} \, dx\\ &=\frac{x \sqrt{2+3 x^2}}{3 \sqrt{1+4 x^2}}-\frac{\sqrt{2+3 x^2} E\left (\tan ^{-1}(2 x)|\frac{5}{8}\right )}{3 \sqrt{2} \sqrt{\frac{2+3 x^2}{1+4 x^2}} \sqrt{1+4 x^2}}\\ \end{align*}

Mathematica [C]  time = 0.0325587, size = 50, normalized size = 0.57 \[ -\frac{i \left (E\left (i \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|\frac{8}{3}\right )-\text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ),\frac{8}{3}\right )\right )}{4 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-I/4)*(EllipticE[I*ArcSinh[Sqrt[3/2]*x], 8/3] - EllipticF[I*ArcSinh[Sqrt[3/2]*x], 8/3]))/Sqrt[3]

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Maple [C]  time = 0.017, size = 36, normalized size = 0.4 \begin{align*}{\frac{i}{12}} \left ({\it EllipticF} \left ({\frac{i}{2}}x\sqrt{6},{\frac{2\,\sqrt{6}}{3}} \right ) -{\it EllipticE} \left ({\frac{i}{2}}x\sqrt{6},{\frac{2\,\sqrt{6}}{3}} \right ) \right ) \sqrt{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*I*(EllipticF(1/2*I*x*6^(1/2),2/3*6^(1/2))-EllipticE(1/2*I*x*6^(1/2),2/3*6^(1/2)))*3^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\sqrt{4 \, x^{2} + 1} \sqrt{3 \, x^{2} + 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{4 \, x^{2} + 1} \sqrt{3 \, x^{2} + 2} x^{2}}{12 \, x^{4} + 11 \, x^{2} + 2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(4*x^2 + 1)*sqrt(3*x^2 + 2)*x^2/(12*x^4 + 11*x^2 + 2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\sqrt{3 x^{2} + 2} \sqrt{4 x^{2} + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2/(sqrt(3*x**2 + 2)*sqrt(4*x**2 + 1)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\sqrt{4 \, x^{2} + 1} \sqrt{3 \, x^{2} + 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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