3.292 \(\int \left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4} \, dx\)

Optimal. Leaf size=149 \[ \frac{5 x \left (x^2+2\right )}{\sqrt{x^4+3 x^2+2}}+\frac{1}{3} x \left (3 x^2+10\right ) \sqrt{x^4+3 x^2+2}+\frac{11 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{3 \sqrt{x^4+3 x^2+2}}-\frac{5 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{x^4+3 x^2+2}} \]

[Out]

(5*x*(2 + x^2))/Sqrt[2 + 3*x^2 + x^4] + (x*(10 + 3*x^2)*Sqrt[2 + 3*x^2 + x^4])/3
- (5*Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticE[ArcTan[x], 1/2])/Sqr
t[2 + 3*x^2 + x^4] + (11*Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticF[A
rcTan[x], 1/2])/(3*Sqrt[2 + 3*x^2 + x^4])

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Rubi [A]  time = 0.114905, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{5 x \left (x^2+2\right )}{\sqrt{x^4+3 x^2+2}}+\frac{1}{3} x \left (3 x^2+10\right ) \sqrt{x^4+3 x^2+2}+\frac{11 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{3 \sqrt{x^4+3 x^2+2}}-\frac{5 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{x^4+3 x^2+2}} \]

Antiderivative was successfully verified.

[In]  Int[(7 + 5*x^2)*Sqrt[2 + 3*x^2 + x^4],x]

[Out]

(5*x*(2 + x^2))/Sqrt[2 + 3*x^2 + x^4] + (x*(10 + 3*x^2)*Sqrt[2 + 3*x^2 + x^4])/3
- (5*Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticE[ArcTan[x], 1/2])/Sqr
t[2 + 3*x^2 + x^4] + (11*Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticF[A
rcTan[x], 1/2])/(3*Sqrt[2 + 3*x^2 + x^4])

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Rubi in Sympy [A]  time = 17.9323, size = 139, normalized size = 0.93 \[ \frac{5 x \left (2 x^{2} + 4\right )}{2 \sqrt{x^{4} + 3 x^{2} + 2}} + \frac{x \left (15 x^{2} + 50\right ) \sqrt{x^{4} + 3 x^{2} + 2}}{15} - \frac{5 \sqrt{\frac{2 x^{2} + 4}{x^{2} + 1}} \left (4 x^{2} + 4\right ) E\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{4 \sqrt{x^{4} + 3 x^{2} + 2}} + \frac{11 \sqrt{\frac{2 x^{2} + 4}{x^{2} + 1}} \left (4 x^{2} + 4\right ) F\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{12 \sqrt{x^{4} + 3 x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5*x**2+7)*(x**4+3*x**2+2)**(1/2),x)

[Out]

5*x*(2*x**2 + 4)/(2*sqrt(x**4 + 3*x**2 + 2)) + x*(15*x**2 + 50)*sqrt(x**4 + 3*x*
*2 + 2)/15 - 5*sqrt((2*x**2 + 4)/(x**2 + 1))*(4*x**2 + 4)*elliptic_e(atan(x), 1/
2)/(4*sqrt(x**4 + 3*x**2 + 2)) + 11*sqrt((2*x**2 + 4)/(x**2 + 1))*(4*x**2 + 4)*e
lliptic_f(atan(x), 1/2)/(12*sqrt(x**4 + 3*x**2 + 2))

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Mathematica [C]  time = 0.0671692, size = 109, normalized size = 0.73 \[ \frac{3 x^7+19 x^5+36 x^3-7 i \sqrt{x^2+1} \sqrt{x^2+2} F\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )-15 i \sqrt{x^2+1} \sqrt{x^2+2} E\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )+20 x}{3 \sqrt{x^4+3 x^2+2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(7 + 5*x^2)*Sqrt[2 + 3*x^2 + x^4],x]

[Out]

(20*x + 36*x^3 + 19*x^5 + 3*x^7 - (15*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticE[I
*ArcSinh[x/Sqrt[2]], 2] - (7*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticF[I*ArcSinh[
x/Sqrt[2]], 2])/(3*Sqrt[2 + 3*x^2 + x^4])

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Maple [C]  time = 0.009, size = 137, normalized size = 0.9 \[{\frac{10\,x}{3}\sqrt{{x}^{4}+3\,{x}^{2}+2}}-{{\frac{11\,i}{3}}\sqrt{2}{\it EllipticF} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}+{{\frac{5\,i}{2}}\sqrt{2} \left ({\it EllipticF} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}+{x}^{3}\sqrt{{x}^{4}+3\,{x}^{2}+2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5*x^2+7)*(x^4+3*x^2+2)^(1/2),x)

[Out]

10/3*x*(x^4+3*x^2+2)^(1/2)-11/3*I*2^(1/2)*(2*x^2+4)^(1/2)*(x^2+1)^(1/2)/(x^4+3*x
^2+2)^(1/2)*EllipticF(1/2*I*2^(1/2)*x,2^(1/2))+5/2*I*2^(1/2)*(2*x^2+4)^(1/2)*(x^
2+1)^(1/2)/(x^4+3*x^2+2)^(1/2)*(EllipticF(1/2*I*2^(1/2)*x,2^(1/2))-EllipticE(1/2
*I*2^(1/2)*x,2^(1/2)))+x^3*(x^4+3*x^2+2)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x^{4} + 3 \, x^{2} + 2}{\left (5 \, x^{2} + 7\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^4 + 3*x^2 + 2)*(5*x^2 + 7),x, algorithm="maxima")

[Out]

integrate(sqrt(x^4 + 3*x^2 + 2)*(5*x^2 + 7), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{x^{4} + 3 \, x^{2} + 2}{\left (5 \, x^{2} + 7\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^4 + 3*x^2 + 2)*(5*x^2 + 7),x, algorithm="fricas")

[Out]

integral(sqrt(x^4 + 3*x^2 + 2)*(5*x^2 + 7), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\left (x^{2} + 1\right ) \left (x^{2} + 2\right )} \left (5 x^{2} + 7\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x**2+7)*(x**4+3*x**2+2)**(1/2),x)

[Out]

Integral(sqrt((x**2 + 1)*(x**2 + 2))*(5*x**2 + 7), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x^{4} + 3 \, x^{2} + 2}{\left (5 \, x^{2} + 7\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^4 + 3*x^2 + 2)*(5*x^2 + 7),x, algorithm="giac")

[Out]

integrate(sqrt(x^4 + 3*x^2 + 2)*(5*x^2 + 7), x)