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

Optimal. Leaf size=198 \[ \frac{25}{7} x \left (x^4+3 x^2+4\right )^{3/2}+\frac{1}{7} x \left (38 x^2+119\right ) \sqrt{x^4+3 x^2+4}+\frac{319 x \sqrt{x^4+3 x^2+4}}{7 \left (x^2+2\right )}+\frac{81 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{\sqrt{2} \sqrt{x^4+3 x^2+4}}-\frac{319 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{7 \sqrt{x^4+3 x^2+4}} \]

[Out]

(319*x*Sqrt[4 + 3*x^2 + x^4])/(7*(2 + x^2)) + (x*(119 + 38*x^2)*Sqrt[4 + 3*x^2 +
x^4])/7 + (25*x*(4 + 3*x^2 + x^4)^(3/2))/7 - (319*Sqrt[2]*(2 + x^2)*Sqrt[(4 + 3
*x^2 + x^4)/(2 + x^2)^2]*EllipticE[2*ArcTan[x/Sqrt[2]], 1/8])/(7*Sqrt[4 + 3*x^2
+ x^4]) + (81*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticF[2*ArcTan[x
/Sqrt[2]], 1/8])/(Sqrt[2]*Sqrt[4 + 3*x^2 + x^4])

_______________________________________________________________________________________

Rubi [A]  time = 0.186034, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{25}{7} x \left (x^4+3 x^2+4\right )^{3/2}+\frac{1}{7} x \left (38 x^2+119\right ) \sqrt{x^4+3 x^2+4}+\frac{319 x \sqrt{x^4+3 x^2+4}}{7 \left (x^2+2\right )}+\frac{81 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{\sqrt{2} \sqrt{x^4+3 x^2+4}}-\frac{319 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{7 \sqrt{x^4+3 x^2+4}} \]

Antiderivative was successfully verified.

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

[Out]

(319*x*Sqrt[4 + 3*x^2 + x^4])/(7*(2 + x^2)) + (x*(119 + 38*x^2)*Sqrt[4 + 3*x^2 +
x^4])/7 + (25*x*(4 + 3*x^2 + x^4)^(3/2))/7 - (319*Sqrt[2]*(2 + x^2)*Sqrt[(4 + 3
*x^2 + x^4)/(2 + x^2)^2]*EllipticE[2*ArcTan[x/Sqrt[2]], 1/8])/(7*Sqrt[4 + 3*x^2
+ x^4]) + (81*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticF[2*ArcTan[x
/Sqrt[2]], 1/8])/(Sqrt[2]*Sqrt[4 + 3*x^2 + x^4])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 35.3093, size = 199, normalized size = 1.01 \[ \frac{x \left (\frac{570 x^{2}}{7} + 255\right ) \sqrt{x^{4} + 3 x^{2} + 4}}{15} + \frac{25 x \left (x^{4} + 3 x^{2} + 4\right )^{\frac{3}{2}}}{7} + \frac{638 x \sqrt{x^{4} + 3 x^{2} + 4}}{7 \left (2 x^{2} + 4\right )} - \frac{319 \sqrt{2} \sqrt{\frac{x^{4} + 3 x^{2} + 4}{\left (\frac{x^{2}}{2} + 1\right )^{2}}} \left (\frac{x^{2}}{2} + 1\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | \frac{1}{8}\right )}{7 \sqrt{x^{4} + 3 x^{2} + 4}} + \frac{81 \sqrt{2} \sqrt{\frac{x^{4} + 3 x^{2} + 4}{\left (\frac{x^{2}}{2} + 1\right )^{2}}} \left (\frac{x^{2}}{2} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | \frac{1}{8}\right )}{2 \sqrt{x^{4} + 3 x^{2} + 4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(570*x**2/7 + 255)*sqrt(x**4 + 3*x**2 + 4)/15 + 25*x*(x**4 + 3*x**2 + 4)**(3/2
)/7 + 638*x*sqrt(x**4 + 3*x**2 + 4)/(7*(2*x**2 + 4)) - 319*sqrt(2)*sqrt((x**4 +
3*x**2 + 4)/(x**2/2 + 1)**2)*(x**2/2 + 1)*elliptic_e(2*atan(sqrt(2)*x/2), 1/8)/(
7*sqrt(x**4 + 3*x**2 + 4)) + 81*sqrt(2)*sqrt((x**4 + 3*x**2 + 4)/(x**2/2 + 1)**2
)*(x**2/2 + 1)*elliptic_f(2*atan(sqrt(2)*x/2), 1/8)/(2*sqrt(x**4 + 3*x**2 + 4))

_______________________________________________________________________________________

Mathematica [C]  time = 1.037, size = 343, normalized size = 1.73 \[ \frac{\sqrt{2} \left (319 \sqrt{7}-35 i\right ) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} F\left (i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )-319 \sqrt{2} \left (\sqrt{7}+3 i\right ) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} E\left (i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )+4 \sqrt{-\frac{i}{\sqrt{7}-3 i}} x \left (25 x^8+188 x^6+658 x^4+1109 x^2+876\right )}{28 \sqrt{-\frac{i}{\sqrt{7}-3 i}} \sqrt{x^4+3 x^2+4}} \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[(-I)/(-3*I + Sqrt[7])]*x*(876 + 1109*x^2 + 658*x^4 + 188*x^6 + 25*x^8) -
319*Sqrt[2]*(3*I + Sqrt[7])*Sqrt[(-3*I + Sqrt[7] - (2*I)*x^2)/(-3*I + Sqrt[7])]
*Sqrt[(3*I + Sqrt[7] + (2*I)*x^2)/(3*I + Sqrt[7])]*EllipticE[I*ArcSinh[Sqrt[(-2*
I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] + Sqrt[2]*(-35*I + 319
*Sqrt[7])*Sqrt[(-3*I + Sqrt[7] - (2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[(3*I + Sqrt[7
] + (2*I)*x^2)/(3*I + Sqrt[7])]*EllipticF[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])
]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])])/(28*Sqrt[(-I)/(-3*I + Sqrt[7])]*Sqrt[4 +
3*x^2 + x^4])

_______________________________________________________________________________________

Maple [C]  time = 0.011, size = 258, normalized size = 1.3 \[{\frac{219\,x}{7}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{1984}{7\,\sqrt{-6+2\,i\sqrt{7}}}\sqrt{1- \left ( -{\frac{3}{8}}+{\frac{i}{8}}\sqrt{7} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{3}{8}}-{\frac{i}{8}}\sqrt{7} \right ){x}^{2}}{\it EllipticF} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}}-{\frac{10208}{7\,\sqrt{-6+2\,i\sqrt{7}} \left ( i\sqrt{7}+3 \right ) }\sqrt{1- \left ( -{\frac{3}{8}}+{\frac{i}{8}}\sqrt{7} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{3}{8}}-{\frac{i}{8}}\sqrt{7} \right ){x}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ) \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}}+{\frac{113\,{x}^{3}}{7}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{25\,{x}^{5}}{7}\sqrt{{x}^{4}+3\,{x}^{2}+4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

219/7*x*(x^4+3*x^2+4)^(1/2)+1984/7/(-6+2*I*7^(1/2))^(1/2)*(1-(-3/8+1/8*I*7^(1/2)
)*x^2)^(1/2)*(1-(-3/8-1/8*I*7^(1/2))*x^2)^(1/2)/(x^4+3*x^2+4)^(1/2)*EllipticF(1/
4*x*(-6+2*I*7^(1/2))^(1/2),1/4*(2+6*I*7^(1/2))^(1/2))-10208/7/(-6+2*I*7^(1/2))^(
1/2)*(1-(-3/8+1/8*I*7^(1/2))*x^2)^(1/2)*(1-(-3/8-1/8*I*7^(1/2))*x^2)^(1/2)/(x^4+
3*x^2+4)^(1/2)/(I*7^(1/2)+3)*(EllipticF(1/4*x*(-6+2*I*7^(1/2))^(1/2),1/4*(2+6*I*
7^(1/2))^(1/2))-EllipticE(1/4*x*(-6+2*I*7^(1/2))^(1/2),1/4*(2+6*I*7^(1/2))^(1/2)
))+113/7*x^3*(x^4+3*x^2+4)^(1/2)+25/7*x^5*(x^4+3*x^2+4)^(1/2)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (25 \, x^{4} + 70 \, x^{2} + 49\right )} \sqrt{x^{4} + 3 \, x^{2} + 4}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((25*x^4 + 70*x^2 + 49)*sqrt(x^4 + 3*x^2 + 4), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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