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]

1/3*x*(3*x^2+2)^(1/2)/(4*x^2+1)^(1/2)-1/6*(1/(4*x^2+1))^(1/2)*EllipticE(2*x/(4*x^2+1)^(1/2),1/4*10^(1/2))*2^(1
/2)*(3*x^2+2)^(1/2)/((3*x^2+2)/(4*x^2+1))^(1/2)

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Rubi [A]  time = 0.03, antiderivative size = 88, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [C]  time = 0.04, size = 50, normalized size = 0.57 \[ -\frac {i \left (E\left (i \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|\frac {8}{3}\right )-F\left (i \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|\frac {8}{3}\right )\right )}{4 \sqrt {3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-1/4*I)*(EllipticE[I*ArcSinh[Sqrt[3/2]*x], 8/3] - EllipticF[I*ArcSinh[Sqrt[3/2]*x], 8/3]))/Sqrt[3]

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fricas [F]  time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {4 \, x^{2} + 1} \sqrt {3 \, x^{2} + 2} x^{2}}{12 \, x^{4} + 11 \, x^{2} + 2}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt {4 \, x^{2} + 1} \sqrt {3 \, x^{2} + 2}}\,{d x} \]

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)

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maple [C]  time = 0.04, size = 36, normalized size = 0.41 \[ \frac {i \left (-\EllipticE \left (\frac {i \sqrt {6}\, x}{2}, \frac {2 \sqrt {6}}{3}\right )+\EllipticF \left (\frac {i \sqrt {6}\, x}{2}, \frac {2 \sqrt {6}}{3}\right )\right ) \sqrt {3}}{12} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt {4 \, x^{2} + 1} \sqrt {3 \, x^{2} + 2}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2}{\sqrt {3\,x^2+2}\,\sqrt {4\,x^2+1}} \,d x \]

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]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt {3 x^{2} + 2} \sqrt {4 x^{2} + 1}}\, dx \]

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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