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

Optimal. Leaf size=279 \[ -\frac{23 x \left (2 x^2+\sqrt{13}+5\right )}{15 \sqrt{x^4+5 x^2+3}}+\frac{1}{15} x \left (9 x^2+25\right ) \sqrt{x^4+5 x^2+3}+\frac{\sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{\sqrt{6 \left (5+\sqrt{13}\right )} \sqrt{x^4+5 x^2+3}}+\frac{23 \sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{15 \sqrt{x^4+5 x^2+3}} \]

[Out]

(-23*x*(5 + Sqrt[13] + 2*x^2))/(15*Sqrt[3 + 5*x^2 + x^4]) + (x*(25 + 9*x^2)*Sqrt
[3 + 5*x^2 + x^4])/15 + (23*Sqrt[(5 + Sqrt[13])/6]*Sqrt[(6 + (5 - Sqrt[13])*x^2)
/(6 + (5 + Sqrt[13])*x^2)]*(6 + (5 + Sqrt[13])*x^2)*EllipticE[ArcTan[Sqrt[(5 + S
qrt[13])/6]*x], (-13 + 5*Sqrt[13])/6])/(15*Sqrt[3 + 5*x^2 + x^4]) + (Sqrt[(6 + (
5 - Sqrt[13])*x^2)/(6 + (5 + Sqrt[13])*x^2)]*(6 + (5 + Sqrt[13])*x^2)*EllipticF[
ArcTan[Sqrt[(5 + Sqrt[13])/6]*x], (-13 + 5*Sqrt[13])/6])/(Sqrt[6*(5 + Sqrt[13])]
*Sqrt[3 + 5*x^2 + x^4])

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Rubi [A]  time = 0.285305, antiderivative size = 279, 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 \[ -\frac{23 x \left (2 x^2+\sqrt{13}+5\right )}{15 \sqrt{x^4+5 x^2+3}}+\frac{1}{15} x \left (9 x^2+25\right ) \sqrt{x^4+5 x^2+3}+\frac{\sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{\sqrt{6 \left (5+\sqrt{13}\right )} \sqrt{x^4+5 x^2+3}}+\frac{23 \sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{15 \sqrt{x^4+5 x^2+3}} \]

Antiderivative was successfully verified.

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

[Out]

(-23*x*(5 + Sqrt[13] + 2*x^2))/(15*Sqrt[3 + 5*x^2 + x^4]) + (x*(25 + 9*x^2)*Sqrt
[3 + 5*x^2 + x^4])/15 + (23*Sqrt[(5 + Sqrt[13])/6]*Sqrt[(6 + (5 - Sqrt[13])*x^2)
/(6 + (5 + Sqrt[13])*x^2)]*(6 + (5 + Sqrt[13])*x^2)*EllipticE[ArcTan[Sqrt[(5 + S
qrt[13])/6]*x], (-13 + 5*Sqrt[13])/6])/(15*Sqrt[3 + 5*x^2 + x^4]) + (Sqrt[(6 + (
5 - Sqrt[13])*x^2)/(6 + (5 + Sqrt[13])*x^2)]*(6 + (5 + Sqrt[13])*x^2)*EllipticF[
ArcTan[Sqrt[(5 + Sqrt[13])/6]*x], (-13 + 5*Sqrt[13])/6])/(Sqrt[6*(5 + Sqrt[13])]
*Sqrt[3 + 5*x^2 + x^4])

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Rubi in Sympy [A]  time = 23.4639, size = 258, normalized size = 0.92 \[ \frac{x \left (9 x^{2} + 25\right ) \sqrt{x^{4} + 5 x^{2} + 3}}{15} - \frac{23 x \left (2 x^{2} + \sqrt{13} + 5\right )}{15 \sqrt{x^{4} + 5 x^{2} + 3}} + \frac{23 \sqrt{6} \sqrt{\frac{x^{2} \left (- \sqrt{13} + 5\right ) + 6}{x^{2} \left (\sqrt{13} + 5\right ) + 6}} \sqrt{\sqrt{13} + 5} \left (x^{2} \left (\sqrt{13} + 5\right ) + 6\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{6} x \sqrt{\sqrt{13} + 5}}{6} \right )}\middle | - \frac{13}{6} + \frac{5 \sqrt{13}}{6}\right )}{90 \sqrt{x^{4} + 5 x^{2} + 3}} + \frac{\sqrt{6} \sqrt{\frac{x^{2} \left (- \sqrt{13} + 5\right ) + 6}{x^{2} \left (\sqrt{13} + 5\right ) + 6}} \left (x^{2} \left (\sqrt{13} + 5\right ) + 6\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{6} x \sqrt{\sqrt{13} + 5}}{6} \right )}\middle | - \frac{13}{6} + \frac{5 \sqrt{13}}{6}\right )}{6 \sqrt{\sqrt{13} + 5} \sqrt{x^{4} + 5 x^{2} + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(9*x**2 + 25)*sqrt(x**4 + 5*x**2 + 3)/15 - 23*x*(2*x**2 + sqrt(13) + 5)/(15*sq
rt(x**4 + 5*x**2 + 3)) + 23*sqrt(6)*sqrt((x**2*(-sqrt(13) + 5) + 6)/(x**2*(sqrt(
13) + 5) + 6))*sqrt(sqrt(13) + 5)*(x**2*(sqrt(13) + 5) + 6)*elliptic_e(atan(sqrt
(6)*x*sqrt(sqrt(13) + 5)/6), -13/6 + 5*sqrt(13)/6)/(90*sqrt(x**4 + 5*x**2 + 3))
+ sqrt(6)*sqrt((x**2*(-sqrt(13) + 5) + 6)/(x**2*(sqrt(13) + 5) + 6))*(x**2*(sqrt
(13) + 5) + 6)*elliptic_f(atan(sqrt(6)*x*sqrt(sqrt(13) + 5)/6), -13/6 + 5*sqrt(1
3)/6)/(6*sqrt(sqrt(13) + 5)*sqrt(x**4 + 5*x**2 + 3))

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Mathematica [C]  time = 0.516216, size = 229, normalized size = 0.82 \[ \frac{i \sqrt{2} \left (23 \sqrt{13}-130\right ) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} F\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{13}}} x\right )|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right )-23 i \sqrt{2} \left (\sqrt{13}-5\right ) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} E\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{13}}} x\right )|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right )+2 x \left (9 x^6+70 x^4+152 x^2+75\right )}{30 \sqrt{x^4+5 x^2+3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*x*(75 + 152*x^2 + 70*x^4 + 9*x^6) - (23*I)*Sqrt[2]*(-5 + Sqrt[13])*Sqrt[(-5 +
Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticE[I*ArcSi
nh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6] + I*Sqrt[2]*(-130 + 23*Sqrt
[13])*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*E
llipticF[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6])/(30*Sqrt[3
+ 5*x^2 + x^4])

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Maple [A]  time = 0.016, size = 226, normalized size = 0.8 \[{\frac{5\,x}{3}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+6\,{\frac{\sqrt{1- \left ( -5/6+1/6\,\sqrt{13} \right ){x}^{2}}\sqrt{1- \left ( -5/6-1/6\,\sqrt{13} \right ){x}^{2}}{\it EllipticF} \left ( 1/6\,x\sqrt{-30+6\,\sqrt{13}},5/6\,\sqrt{3}+1/6\,\sqrt{39} \right ) }{\sqrt{-30+6\,\sqrt{13}}\sqrt{{x}^{4}+5\,{x}^{2}+3}}}+{\frac{552}{5\,\sqrt{-30+6\,\sqrt{13}} \left ( 5+\sqrt{13} \right ) }\sqrt{1- \left ( -{\frac{5}{6}}+{\frac{\sqrt{13}}{6}} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{5}{6}}-{\frac{\sqrt{13}}{6}} \right ){x}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{-30+6\,\sqrt{13}}}{6}},{\frac{5\,\sqrt{3}}{6}}+{\frac{\sqrt{39}}{6}} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{-30+6\,\sqrt{13}}}{6}},{\frac{5\,\sqrt{3}}{6}}+{\frac{\sqrt{39}}{6}} \right ) \right ){\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}}}+{\frac{3\,{x}^{3}}{5}\sqrt{{x}^{4}+5\,{x}^{2}+3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/3*x*(x^4+5*x^2+3)^(1/2)+6/(-30+6*13^(1/2))^(1/2)*(1-(-5/6+1/6*13^(1/2))*x^2)^(
1/2)*(1-(-5/6-1/6*13^(1/2))*x^2)^(1/2)/(x^4+5*x^2+3)^(1/2)*EllipticF(1/6*x*(-30+
6*13^(1/2))^(1/2),5/6*3^(1/2)+1/6*39^(1/2))+552/5/(-30+6*13^(1/2))^(1/2)*(1-(-5/
6+1/6*13^(1/2))*x^2)^(1/2)*(1-(-5/6-1/6*13^(1/2))*x^2)^(1/2)/(x^4+5*x^2+3)^(1/2)
/(5+13^(1/2))*(EllipticF(1/6*x*(-30+6*13^(1/2))^(1/2),5/6*3^(1/2)+1/6*39^(1/2))-
EllipticE(1/6*x*(-30+6*13^(1/2))^(1/2),5/6*3^(1/2)+1/6*39^(1/2)))+3/5*x^3*(x^4+5
*x^2+3)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((3*x**2 + 2)*sqrt(x**4 + 5*x**2 + 3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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