3.84 \(\int \frac{x^4 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^2} \, dx\)

Optimal. Leaf size=56 \[ \frac{5 x^3}{3}-\frac{\left (51 x^2+50\right ) x}{2 \left (x^4+3 x^2+2\right )}-27 x+\frac{13}{2} \tan ^{-1}(x)+33 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right ) \]

[Out]

-27*x + (5*x^3)/3 - (x*(50 + 51*x^2))/(2*(2 + 3*x^2 + x^4)) + (13*ArcTan[x])/2 +
33*Sqrt[2]*ArcTan[x/Sqrt[2]]

_______________________________________________________________________________________

Rubi [A]  time = 0.121756, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129 \[ \frac{5 x^3}{3}-\frac{\left (51 x^2+50\right ) x}{2 \left (x^4+3 x^2+2\right )}-27 x+\frac{13}{2} \tan ^{-1}(x)+33 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(x^4*(4 + x^2 + 3*x^4 + 5*x^6))/(2 + 3*x^2 + x^4)^2,x]

[Out]

-27*x + (5*x^3)/3 - (x*(50 + 51*x^2))/(2*(2 + 3*x^2 + x^4)) + (13*ArcTan[x])/2 +
33*Sqrt[2]*ArcTan[x/Sqrt[2]]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 21.6324, size = 53, normalized size = 0.95 \[ \frac{5 x^{3}}{3} - \frac{x \left (24786 x^{2} + 24300\right )}{972 \left (x^{4} + 3 x^{2} + 2\right )} - 27 x + \frac{13 \operatorname{atan}{\left (x \right )}}{2} + 33 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(5*x**6+3*x**4+x**2+4)/(x**4+3*x**2+2)**2,x)

[Out]

5*x**3/3 - x*(24786*x**2 + 24300)/(972*(x**4 + 3*x**2 + 2)) - 27*x + 13*atan(x)/
2 + 33*sqrt(2)*atan(sqrt(2)*x/2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.08301, size = 57, normalized size = 1.02 \[ \frac{5 x^3}{3}+\frac{-51 x^3-50 x}{2 \left (x^4+3 x^2+2\right )}-27 x+\frac{13}{2} \tan ^{-1}(x)+33 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(x^4*(4 + x^2 + 3*x^4 + 5*x^6))/(2 + 3*x^2 + x^4)^2,x]

[Out]

-27*x + (5*x^3)/3 + (-50*x - 51*x^3)/(2*(2 + 3*x^2 + x^4)) + (13*ArcTan[x])/2 +
33*Sqrt[2]*ArcTan[x/Sqrt[2]]

_______________________________________________________________________________________

Maple [A]  time = 0.019, size = 46, normalized size = 0.8 \[{\frac{5\,{x}^{3}}{3}}-27\,x-26\,{\frac{x}{{x}^{2}+2}}+33\,\arctan \left ( 1/2\,\sqrt{2}x \right ) \sqrt{2}+{\frac{x}{2\,{x}^{2}+2}}+{\frac{13\,\arctan \left ( x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(5*x^6+3*x^4+x^2+4)/(x^4+3*x^2+2)^2,x)

[Out]

5/3*x^3-27*x-26*x/(x^2+2)+33*arctan(1/2*2^(1/2)*x)*2^(1/2)+1/2*x/(x^2+1)+13/2*ar
ctan(x)

_______________________________________________________________________________________

Maxima [A]  time = 0.797375, size = 65, normalized size = 1.16 \[ \frac{5}{3} \, x^{3} + 33 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - 27 \, x - \frac{51 \, x^{3} + 50 \, x}{2 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} + \frac{13}{2} \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^6 + 3*x^4 + x^2 + 4)*x^4/(x^4 + 3*x^2 + 2)^2,x, algorithm="maxima")

[Out]

5/3*x^3 + 33*sqrt(2)*arctan(1/2*sqrt(2)*x) - 27*x - 1/2*(51*x^3 + 50*x)/(x^4 + 3
*x^2 + 2) + 13/2*arctan(x)

_______________________________________________________________________________________

Fricas [A]  time = 0.264796, size = 93, normalized size = 1.66 \[ \frac{10 \, x^{7} - 132 \, x^{5} - 619 \, x^{3} + 198 \, \sqrt{2}{\left (x^{4} + 3 \, x^{2} + 2\right )} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + 39 \,{\left (x^{4} + 3 \, x^{2} + 2\right )} \arctan \left (x\right ) - 474 \, x}{6 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^6 + 3*x^4 + x^2 + 4)*x^4/(x^4 + 3*x^2 + 2)^2,x, algorithm="fricas")

[Out]

1/6*(10*x^7 - 132*x^5 - 619*x^3 + 198*sqrt(2)*(x^4 + 3*x^2 + 2)*arctan(1/2*sqrt(
2)*x) + 39*(x^4 + 3*x^2 + 2)*arctan(x) - 474*x)/(x^4 + 3*x^2 + 2)

_______________________________________________________________________________________

Sympy [A]  time = 0.578609, size = 53, normalized size = 0.95 \[ \frac{5 x^{3}}{3} - 27 x - \frac{51 x^{3} + 50 x}{2 x^{4} + 6 x^{2} + 4} + \frac{13 \operatorname{atan}{\left (x \right )}}{2} + 33 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(5*x**6+3*x**4+x**2+4)/(x**4+3*x**2+2)**2,x)

[Out]

5*x**3/3 - 27*x - (51*x**3 + 50*x)/(2*x**4 + 6*x**2 + 4) + 13*atan(x)/2 + 33*sqr
t(2)*atan(sqrt(2)*x/2)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.27167, size = 65, normalized size = 1.16 \[ \frac{5}{3} \, x^{3} + 33 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - 27 \, x - \frac{51 \, x^{3} + 50 \, x}{2 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} + \frac{13}{2} \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^6 + 3*x^4 + x^2 + 4)*x^4/(x^4 + 3*x^2 + 2)^2,x, algorithm="giac")

[Out]

5/3*x^3 + 33*sqrt(2)*arctan(1/2*sqrt(2)*x) - 27*x - 1/2*(51*x^3 + 50*x)/(x^4 + 3
*x^2 + 2) + 13/2*arctan(x)