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

Optimal. Leaf size=148 \[ \frac{4 \sqrt{3 x^2+2} x}{3 \sqrt{4 x^2+1}}+\frac{\sqrt{3 x^2+2} F\left (\tan ^{-1}(2 x)|\frac{5}{8}\right )}{2 \sqrt{2} \sqrt{\frac{3 x^2+2}{4 x^2+1}} \sqrt{4 x^2+1}}-\frac{2 \sqrt{2} \sqrt{3 x^2+2} E\left (\tan ^{-1}(2 x)|\frac{5}{8}\right )}{3 \sqrt{\frac{3 x^2+2}{4 x^2+1}} \sqrt{4 x^2+1}} \]

[Out]

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

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Rubi [A]  time = 0.16047, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ \frac{4 \sqrt{3 x^2+2} x}{3 \sqrt{4 x^2+1}}+\frac{\sqrt{3 x^2+2} F\left (\tan ^{-1}(2 x)|\frac{5}{8}\right )}{2 \sqrt{2} \sqrt{\frac{3 x^2+2}{4 x^2+1}} \sqrt{4 x^2+1}}-\frac{2 \sqrt{2} \sqrt{3 x^2+2} E\left (\tan ^{-1}(2 x)|\frac{5}{8}\right )}{3 \sqrt{\frac{3 x^2+2}{4 x^2+1}} \sqrt{4 x^2+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 20.2569, size = 121, normalized size = 0.82 \[ \frac{4 x \sqrt{3 x^{2} + 2}}{3 \sqrt{4 x^{2} + 1}} - \frac{2 \sqrt{3 x^{2} + 2} E\left (\operatorname{atan}{\left (2 x \right )}\middle | \frac{5}{8}\right )}{3 \sqrt{\frac{3 x^{2} + 2}{8 x^{2} + 2}} \sqrt{4 x^{2} + 1}} + \frac{\sqrt{3 x^{2} + 2} F\left (\operatorname{atan}{\left (2 x \right )}\middle | \frac{5}{8}\right )}{4 \sqrt{\frac{3 x^{2} + 2}{8 x^{2} + 2}} \sqrt{4 x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((4*x**2+1)**(1/2)/(3*x**2+2)**(1/2),x)

[Out]

4*x*sqrt(3*x**2 + 2)/(3*sqrt(4*x**2 + 1)) - 2*sqrt(3*x**2 + 2)*elliptic_e(atan(2
*x), 5/8)/(3*sqrt((3*x**2 + 2)/(8*x**2 + 2))*sqrt(4*x**2 + 1)) + sqrt(3*x**2 + 2
)*elliptic_f(atan(2*x), 5/8)/(4*sqrt((3*x**2 + 2)/(8*x**2 + 2))*sqrt(4*x**2 + 1)
)

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Mathematica [C]  time = 0.0252582, size = 27, normalized size = 0.18 \[ -\frac{i E\left (i \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|\frac{8}{3}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.113, size = 26, normalized size = 0.2 \[ -{\frac{i}{3}}{\it EllipticE} \left ({\frac{i}{2}}\sqrt{3}\sqrt{2}x,{\frac{2\,\sqrt{3}\sqrt{2}}{3}} \right ) \sqrt{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{4 \, x^{2} + 1}}{\sqrt{3 \, x^{2} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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