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

Optimal. Leaf size=232 \[ \frac{5 x^3}{3}-\frac{1}{32} \sqrt{\frac{1}{2} \left (26499 \sqrt{3}-14395\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{1}{32} \sqrt{\frac{1}{2} \left (26499 \sqrt{3}-14395\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{25 \left (x^2+3\right ) x}{8 \left (x^4+2 x^2+3\right )}-17 x-\frac{1}{16} \sqrt{\frac{1}{2} \left (14395+26499 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )+\frac{1}{16} \sqrt{\frac{1}{2} \left (14395+26499 \sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]

[Out]

-17*x + (5*x^3)/3 - (25*x*(3 + x^2))/(8*(3 + 2*x^2 + x^4)) - (Sqrt[(14395 + 2649
9*Sqrt[3])/2]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]])/16 +
(Sqrt[(14395 + 26499*Sqrt[3])/2]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(
1 + Sqrt[3])]])/16 - (Sqrt[(-14395 + 26499*Sqrt[3])/2]*Log[Sqrt[3] - Sqrt[2*(-1
+ Sqrt[3])]*x + x^2])/32 + (Sqrt[(-14395 + 26499*Sqrt[3])/2]*Log[Sqrt[3] + Sqrt[
2*(-1 + Sqrt[3])]*x + x^2])/32

_______________________________________________________________________________________

Rubi [A]  time = 0.701416, antiderivative size = 232, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226 \[ \frac{5 x^3}{3}-\frac{1}{32} \sqrt{\frac{1}{2} \left (26499 \sqrt{3}-14395\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{1}{32} \sqrt{\frac{1}{2} \left (26499 \sqrt{3}-14395\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{25 \left (x^2+3\right ) x}{8 \left (x^4+2 x^2+3\right )}-17 x-\frac{1}{16} \sqrt{\frac{1}{2} \left (14395+26499 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )+\frac{1}{16} \sqrt{\frac{1}{2} \left (14395+26499 \sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-17*x + (5*x^3)/3 - (25*x*(3 + x^2))/(8*(3 + 2*x^2 + x^4)) - (Sqrt[(14395 + 2649
9*Sqrt[3])/2]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]])/16 +
(Sqrt[(14395 + 26499*Sqrt[3])/2]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(
1 + Sqrt[3])]])/16 - (Sqrt[(-14395 + 26499*Sqrt[3])/2]*Log[Sqrt[3] - Sqrt[2*(-1
+ Sqrt[3])]*x + x^2])/32 + (Sqrt[(-14395 + 26499*Sqrt[3])/2]*Log[Sqrt[3] + Sqrt[
2*(-1 + Sqrt[3])]*x + x^2])/32

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 44.3172, size = 338, normalized size = 1.46 \[ \frac{5 x^{3}}{3} - \frac{x \left (4800 x^{2} + 14400\right )}{1536 \left (x^{4} + 2 x^{2} + 3\right )} - 17 x - \frac{\sqrt{6} \left (- 12192 \sqrt{3} + 46368\right ) \log{\left (x^{2} - \sqrt{2} x \sqrt{-1 + \sqrt{3}} + \sqrt{3} \right )}}{18432 \sqrt{-1 + \sqrt{3}}} + \frac{\sqrt{6} \left (- 12192 \sqrt{3} + 46368\right ) \log{\left (x^{2} + \sqrt{2} x \sqrt{-1 + \sqrt{3}} + \sqrt{3} \right )}}{18432 \sqrt{-1 + \sqrt{3}}} + \frac{\sqrt{3} \left (- \frac{\sqrt{2} \sqrt{-1 + \sqrt{3}} \left (- 24384 \sqrt{3} + 92736\right )}{2} + 92736 \sqrt{2} \sqrt{-1 + \sqrt{3}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \left (x - \frac{\sqrt{-2 + 2 \sqrt{3}}}{2}\right )}{\sqrt{1 + \sqrt{3}}} \right )}}{9216 \sqrt{-1 + \sqrt{3}} \sqrt{1 + \sqrt{3}}} + \frac{\sqrt{3} \left (- \frac{\sqrt{2} \sqrt{-1 + \sqrt{3}} \left (- 24384 \sqrt{3} + 92736\right )}{2} + 92736 \sqrt{2} \sqrt{-1 + \sqrt{3}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \left (x + \frac{\sqrt{-2 + 2 \sqrt{3}}}{2}\right )}{\sqrt{1 + \sqrt{3}}} \right )}}{9216 \sqrt{-1 + \sqrt{3}} \sqrt{1 + \sqrt{3}}} \]

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+2*x**2+3)**2,x)

[Out]

5*x**3/3 - x*(4800*x**2 + 14400)/(1536*(x**4 + 2*x**2 + 3)) - 17*x - sqrt(6)*(-1
2192*sqrt(3) + 46368)*log(x**2 - sqrt(2)*x*sqrt(-1 + sqrt(3)) + sqrt(3))/(18432*
sqrt(-1 + sqrt(3))) + sqrt(6)*(-12192*sqrt(3) + 46368)*log(x**2 + sqrt(2)*x*sqrt
(-1 + sqrt(3)) + sqrt(3))/(18432*sqrt(-1 + sqrt(3))) + sqrt(3)*(-sqrt(2)*sqrt(-1
+ sqrt(3))*(-24384*sqrt(3) + 92736)/2 + 92736*sqrt(2)*sqrt(-1 + sqrt(3)))*atan(
sqrt(2)*(x - sqrt(-2 + 2*sqrt(3))/2)/sqrt(1 + sqrt(3)))/(9216*sqrt(-1 + sqrt(3))
*sqrt(1 + sqrt(3))) + sqrt(3)*(-sqrt(2)*sqrt(-1 + sqrt(3))*(-24384*sqrt(3) + 927
36)/2 + 92736*sqrt(2)*sqrt(-1 + sqrt(3)))*atan(sqrt(2)*(x + sqrt(-2 + 2*sqrt(3))
/2)/sqrt(1 + sqrt(3)))/(9216*sqrt(-1 + sqrt(3))*sqrt(1 + sqrt(3)))

_______________________________________________________________________________________

Mathematica [C]  time = 0.318717, size = 129, normalized size = 0.56 \[ \frac{5 x^3}{3}-\frac{25 \left (x^2+3\right ) x}{8 \left (x^4+2 x^2+3\right )}-17 x+\frac{\left (127 \sqrt{2}-356 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1-i \sqrt{2}}}\right )}{16 \sqrt{2-2 i \sqrt{2}}}+\frac{\left (127 \sqrt{2}+356 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt{2}}}\right )}{16 \sqrt{2+2 i \sqrt{2}}} \]

Antiderivative was successfully verified.

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

[Out]

-17*x + (5*x^3)/3 - (25*x*(3 + x^2))/(8*(3 + 2*x^2 + x^4)) + ((-356*I + 127*Sqrt
[2])*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/(16*Sqrt[2 - (2*I)*Sqrt[2]]) + ((356*I + 127
*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/(16*Sqrt[2 + (2*I)*Sqrt[2]])

_______________________________________________________________________________________

Maple [B]  time = 0.037, size = 416, normalized size = 1.8 \[{\frac{5\,{x}^{3}}{3}}-17\,x+{\frac{1}{{x}^{4}+2\,{x}^{2}+3} \left ( -{\frac{25\,{x}^{3}}{8}}-{\frac{75\,x}{8}} \right ) }+{\frac{17\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{64}}+{\frac{89\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{32}}-{\frac{ \left ( -34+34\,\sqrt{3} \right ) \sqrt{3}}{32\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{-178+178\,\sqrt{3}}{16\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{161\,\sqrt{3}}{8\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{17\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{64}}-{\frac{89\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{32}}-{\frac{ \left ( -34+34\,\sqrt{3} \right ) \sqrt{3}}{32\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{-178+178\,\sqrt{3}}{16\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{161\,\sqrt{3}}{8\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) } \]

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+2*x^2+3)^2,x)

[Out]

5/3*x^3-17*x+(-25/8*x^3-75/8*x)/(x^4+2*x^2+3)+17/64*ln(x^2+3^(1/2)+x*(-2+2*3^(1/
2))^(1/2))*(-2+2*3^(1/2))^(1/2)*3^(1/2)+89/32*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1
/2))*(-2+2*3^(1/2))^(1/2)-17/32/(2+2*3^(1/2))^(1/2)*arctan((2*x+(-2+2*3^(1/2))^(
1/2))/(2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))*3^(1/2)-89/16/(2+2*3^(1/2))^(1/2)*arct
an((2*x+(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))+161/8/(2+2*3^(
1/2))^(1/2)*arctan((2*x+(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))*3^(1/2)-17/64
*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)*3^(1/2)-89/32*ln(x^
2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)-17/32/(2+2*3^(1/2))^(1/2)
*arctan((2*x-(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))*3^(1/2)-8
9/16/(2+2*3^(1/2))^(1/2)*arctan((2*x-(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))*
(-2+2*3^(1/2))+161/8/(2+2*3^(1/2))^(1/2)*arctan((2*x-(-2+2*3^(1/2))^(1/2))/(2+2*
3^(1/2))^(1/2))*3^(1/2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{5}{3} \, x^{3} - 17 \, x - \frac{25 \,{\left (x^{3} + 3 \, x\right )}}{8 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} + \frac{1}{8} \, \int \frac{127 \, x^{2} + 483}{x^{4} + 2 \, x^{2} + 3}\,{d x} \]

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 + 2*x^2 + 3)^2,x, algorithm="maxima")

[Out]

5/3*x^3 - 17*x - 25/8*(x^3 + 3*x)/(x^4 + 2*x^2 + 3) + 1/8*integrate((127*x^2 + 4
83)/(x^4 + 2*x^2 + 3), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.290408, size = 1013, normalized size = 4.37 \[ \text{result too large to display} \]

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 + 2*x^2 + 3)^2,x, algorithm="fricas")

[Out]

-1/21024*sqrt(219)*(739608*143883^(1/4)*(x^4 + 2*x^2 + 3)*arctan(2*143883^(1/4)*
(17*sqrt(3) + 178)/(sqrt(219)*sqrt(1/219)*(26499*sqrt(3)*sqrt(2) - 14395*sqrt(2)
)*sqrt((10288182782315085*sqrt(3)*x^2 + 2*143883^(1/4)*sqrt(219)*(60771148202248
31*sqrt(3)*x - 11712797811682234*x)*sqrt((14395*sqrt(3) - 79497)/(381453105*sqrt
(3) - 1156903514)) - 23749107007222377*x^2 + 219*sqrt(3)*(46978003572215*sqrt(3)
- 108443410991883))/(46978003572215*sqrt(3) - 108443410991883))*sqrt((14395*sqr
t(3) - 79497)/(381453105*sqrt(3) - 1156903514)) + sqrt(219)*(26499*sqrt(3)*sqrt(
2)*x - 14395*sqrt(2)*x)*sqrt((14395*sqrt(3) - 79497)/(381453105*sqrt(3) - 115690
3514)) + 143883^(1/4)*(161*sqrt(3)*sqrt(2) - 127*sqrt(2)))) + 739608*143883^(1/4
)*(x^4 + 2*x^2 + 3)*arctan(2*143883^(1/4)*(17*sqrt(3) + 178)/(sqrt(219)*sqrt(1/2
19)*(26499*sqrt(3)*sqrt(2) - 14395*sqrt(2))*sqrt((10288182782315085*sqrt(3)*x^2
- 2*143883^(1/4)*sqrt(219)*(6077114820224831*sqrt(3)*x - 11712797811682234*x)*sq
rt((14395*sqrt(3) - 79497)/(381453105*sqrt(3) - 1156903514)) - 23749107007222377
*x^2 + 219*sqrt(3)*(46978003572215*sqrt(3) - 108443410991883))/(46978003572215*s
qrt(3) - 108443410991883))*sqrt((14395*sqrt(3) - 79497)/(381453105*sqrt(3) - 115
6903514)) + sqrt(219)*(26499*sqrt(3)*sqrt(2)*x - 14395*sqrt(2)*x)*sqrt((14395*sq
rt(3) - 79497)/(381453105*sqrt(3) - 1156903514)) - 143883^(1/4)*(161*sqrt(3)*sqr
t(2) - 127*sqrt(2)))) - 3*143883^(1/4)*(26499*sqrt(3)*sqrt(2)*(x^4 + 2*x^2 + 3)
- 14395*sqrt(2)*(x^4 + 2*x^2 + 3))*log(113170010605465935*sqrt(3)*x^2 + 22*14388
3^(1/4)*sqrt(219)*(6077114820224831*sqrt(3)*x - 11712797811682234*x)*sqrt((14395
*sqrt(3) - 79497)/(381453105*sqrt(3) - 1156903514)) - 261240177079446147*x^2 + 2
409*sqrt(3)*(46978003572215*sqrt(3) - 108443410991883)) + 3*143883^(1/4)*(26499*
sqrt(3)*sqrt(2)*(x^4 + 2*x^2 + 3) - 14395*sqrt(2)*(x^4 + 2*x^2 + 3))*log(1131700
10605465935*sqrt(3)*x^2 - 22*143883^(1/4)*sqrt(219)*(6077114820224831*sqrt(3)*x
- 11712797811682234*x)*sqrt((14395*sqrt(3) - 79497)/(381453105*sqrt(3) - 1156903
514)) - 261240177079446147*x^2 + 2409*sqrt(3)*(46978003572215*sqrt(3) - 10844341
0991883)) - 4*sqrt(219)*(26499*sqrt(3)*sqrt(2)*(40*x^7 - 328*x^5 - 771*x^3 - 144
9*x) - 14395*sqrt(2)*(40*x^7 - 328*x^5 - 771*x^3 - 1449*x))*sqrt((14395*sqrt(3)
- 79497)/(381453105*sqrt(3) - 1156903514)))/((26499*sqrt(3)*sqrt(2)*(x^4 + 2*x^2
+ 3) - 14395*sqrt(2)*(x^4 + 2*x^2 + 3))*sqrt((14395*sqrt(3) - 79497)/(381453105
*sqrt(3) - 1156903514)))

_______________________________________________________________________________________

Sympy [A]  time = 1.97303, size = 58, normalized size = 0.25 \[ \frac{5 x^{3}}{3} - 17 x - \frac{25 x^{3} + 75 x}{8 x^{4} + 16 x^{2} + 24} + \operatorname{RootSum}{\left (1048576 t^{4} + 29480960 t^{2} + 2106591003, \left ( t \mapsto t \log{\left (\frac{557056 t^{3}}{816619683} + \frac{166600064 t}{816619683} + x \right )} \right )\right )} \]

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+2*x**2+3)**2,x)

[Out]

5*x**3/3 - 17*x - (25*x**3 + 75*x)/(8*x**4 + 16*x**2 + 24) + RootSum(1048576*_t*
*4 + 29480960*_t**2 + 2106591003, Lambda(_t, _t*log(557056*_t**3/816619683 + 166
600064*_t/816619683 + x)))

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x^{6} + 3 \, x^{4} + x^{2} + 4\right )} x^{4}}{{\left (x^{4} + 2 \, x^{2} + 3\right )}^{2}}\,{d x} \]

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 + 2*x^2 + 3)^2,x, algorithm="giac")

[Out]

integrate((5*x^6 + 3*x^4 + x^2 + 4)*x^4/(x^4 + 2*x^2 + 3)^2, x)