3.109 \(\int \frac{x^8 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^2} \, dx\)

Optimal. Leaf size=248 \[ \frac{5 x^7}{7}-\frac{17 x^5}{5}+\frac{19 x^3}{3}-\frac{1}{32} \sqrt{\frac{1}{2} \left (618291 \sqrt{3}-262771\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{1}{32} \sqrt{\frac{1}{2} \left (618291 \sqrt{3}-262771\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{25 \left (5 x^2+3\right ) x}{8 \left (x^4+2 x^2+3\right )}+38 x+\frac{1}{16} \sqrt{\frac{1}{2} \left (262771+618291 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )-\frac{1}{16} \sqrt{\frac{1}{2} \left (262771+618291 \sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]

[Out]

38*x + (19*x^3)/3 - (17*x^5)/5 + (5*x^7)/7 + (25*x*(3 + 5*x^2))/(8*(3 + 2*x^2 +
x^4)) + (Sqrt[(262771 + 618291*Sqrt[3])/2]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)
/Sqrt[2*(1 + Sqrt[3])]])/16 - (Sqrt[(262771 + 618291*Sqrt[3])/2]*ArcTan[(Sqrt[2*
(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/16 - (Sqrt[(-262771 + 618291*Sqrt
[3])/2]*Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/32 + (Sqrt[(-262771 + 618
291*Sqrt[3])/2]*Log[Sqrt[3] + Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/32

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Rubi [A]  time = 0.731425, antiderivative size = 248, 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 \[ \frac{5 x^7}{7}-\frac{17 x^5}{5}+\frac{19 x^3}{3}-\frac{1}{32} \sqrt{\frac{1}{2} \left (618291 \sqrt{3}-262771\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{1}{32} \sqrt{\frac{1}{2} \left (618291 \sqrt{3}-262771\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{25 \left (5 x^2+3\right ) x}{8 \left (x^4+2 x^2+3\right )}+38 x+\frac{1}{16} \sqrt{\frac{1}{2} \left (262771+618291 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )-\frac{1}{16} \sqrt{\frac{1}{2} \left (262771+618291 \sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(x^8*(4 + x^2 + 3*x^4 + 5*x^6))/(3 + 2*x^2 + x^4)^2,x]

[Out]

38*x + (19*x^3)/3 - (17*x^5)/5 + (5*x^7)/7 + (25*x*(3 + 5*x^2))/(8*(3 + 2*x^2 +
x^4)) + (Sqrt[(262771 + 618291*Sqrt[3])/2]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)
/Sqrt[2*(1 + Sqrt[3])]])/16 - (Sqrt[(262771 + 618291*Sqrt[3])/2]*ArcTan[(Sqrt[2*
(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/16 - (Sqrt[(-262771 + 618291*Sqrt
[3])/2]*Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/32 + (Sqrt[(-262771 + 618
291*Sqrt[3])/2]*Log[Sqrt[3] + Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/32

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Rubi in Sympy [A]  time = 62.7685, size = 352, normalized size = 1.42 \[ \frac{5 x^{7}}{7} - \frac{17 x^{5}}{5} + \frac{19 x^{3}}{3} + \frac{x \left (96000 x^{2} + 57600\right )}{6144 \left (x^{4} + 2 x^{2} + 3\right )} + 38 x + \frac{\sqrt{6} \left (- 514176 \sqrt{3} + 379008\right ) \log{\left (x^{2} - \sqrt{2} x \sqrt{-1 + \sqrt{3}} + \sqrt{3} \right )}}{73728 \sqrt{-1 + \sqrt{3}}} - \frac{\sqrt{6} \left (- 514176 \sqrt{3} + 379008\right ) \log{\left (x^{2} + \sqrt{2} x \sqrt{-1 + \sqrt{3}} + \sqrt{3} \right )}}{73728 \sqrt{-1 + \sqrt{3}}} - \frac{\sqrt{3} \left (- \frac{\sqrt{2} \sqrt{-1 + \sqrt{3}} \left (- 1028352 \sqrt{3} + 758016\right )}{2} + 758016 \sqrt{2} \sqrt{-1 + \sqrt{3}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \left (x - \frac{\sqrt{-2 + 2 \sqrt{3}}}{2}\right )}{\sqrt{1 + \sqrt{3}}} \right )}}{36864 \sqrt{-1 + \sqrt{3}} \sqrt{1 + \sqrt{3}}} - \frac{\sqrt{3} \left (- \frac{\sqrt{2} \sqrt{-1 + \sqrt{3}} \left (- 1028352 \sqrt{3} + 758016\right )}{2} + 758016 \sqrt{2} \sqrt{-1 + \sqrt{3}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \left (x + \frac{\sqrt{-2 + 2 \sqrt{3}}}{2}\right )}{\sqrt{1 + \sqrt{3}}} \right )}}{36864 \sqrt{-1 + \sqrt{3}} \sqrt{1 + \sqrt{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8*(5*x**6+3*x**4+x**2+4)/(x**4+2*x**2+3)**2,x)

[Out]

5*x**7/7 - 17*x**5/5 + 19*x**3/3 + x*(96000*x**2 + 57600)/(6144*(x**4 + 2*x**2 +
3)) + 38*x + sqrt(6)*(-514176*sqrt(3) + 379008)*log(x**2 - sqrt(2)*x*sqrt(-1 +
sqrt(3)) + sqrt(3))/(73728*sqrt(-1 + sqrt(3))) - sqrt(6)*(-514176*sqrt(3) + 3790
08)*log(x**2 + sqrt(2)*x*sqrt(-1 + sqrt(3)) + sqrt(3))/(73728*sqrt(-1 + sqrt(3))
) - sqrt(3)*(-sqrt(2)*sqrt(-1 + sqrt(3))*(-1028352*sqrt(3) + 758016)/2 + 758016*
sqrt(2)*sqrt(-1 + sqrt(3)))*atan(sqrt(2)*(x - sqrt(-2 + 2*sqrt(3))/2)/sqrt(1 + s
qrt(3)))/(36864*sqrt(-1 + sqrt(3))*sqrt(1 + sqrt(3))) - sqrt(3)*(-sqrt(2)*sqrt(-
1 + sqrt(3))*(-1028352*sqrt(3) + 758016)/2 + 758016*sqrt(2)*sqrt(-1 + sqrt(3)))*
atan(sqrt(2)*(x + sqrt(-2 + 2*sqrt(3))/2)/sqrt(1 + sqrt(3)))/(36864*sqrt(-1 + sq
rt(3))*sqrt(1 + sqrt(3)))

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Mathematica [C]  time = 0.337006, size = 145, normalized size = 0.58 \[ \frac{5 x^7}{7}-\frac{17 x^5}{5}+\frac{19 x^3}{3}+\frac{25 \left (5 x^2+3\right ) x}{8 \left (x^4+2 x^2+3\right )}+38 x-\frac{\left (1339 \sqrt{2}+352 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1-i \sqrt{2}}}\right )}{16 \sqrt{2-2 i \sqrt{2}}}-\frac{\left (1339 \sqrt{2}-352 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt{2}}}\right )}{16 \sqrt{2+2 i \sqrt{2}}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^8*(4 + x^2 + 3*x^4 + 5*x^6))/(3 + 2*x^2 + x^4)^2,x]

[Out]

38*x + (19*x^3)/3 - (17*x^5)/5 + (5*x^7)/7 + (25*x*(3 + 5*x^2))/(8*(3 + 2*x^2 +
x^4)) - ((352*I + 1339*Sqrt[2])*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/(16*Sqrt[2 - (2*I
)*Sqrt[2]]) - ((-352*I + 1339*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/(16*Sqrt[2
+ (2*I)*Sqrt[2]])

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Maple [B]  time = 0.106, size = 427, normalized size = 1.7 \[{\frac{5\,{x}^{7}}{7}}-{\frac{17\,{x}^{5}}{5}}+{\frac{19\,{x}^{3}}{3}}+38\,x-{\frac{1}{{x}^{4}+2\,{x}^{2}+3} \left ( -{\frac{125\,{x}^{3}}{8}}-{\frac{75\,x}{8}} \right ) }+{\frac{505\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{64}}+{\frac{11\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{4}}-{\frac{ \left ( -1010+1010\,\sqrt{3} \right ) \sqrt{3}}{32\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{-22+22\,\sqrt{3}}{2\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{329\,\sqrt{3}}{8\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{505\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{64}}-{\frac{11\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{4}}-{\frac{ \left ( -1010+1010\,\sqrt{3} \right ) \sqrt{3}}{32\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{-22+22\,\sqrt{3}}{2\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{329\,\sqrt{3}}{8\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8*(5*x^6+3*x^4+x^2+4)/(x^4+2*x^2+3)^2,x)

[Out]

5/7*x^7-17/5*x^5+19/3*x^3+38*x-(-125/8*x^3-75/8*x)/(x^4+2*x^2+3)+505/64*ln(x^2+3
^(1/2)+x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)*3^(1/2)+11/4*ln(x^2+3^(1/2)+
x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)-505/32/(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)-11/2/(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))-329/8/(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)-505/64*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)
*3^(1/2)-11/4*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)-505/32
/(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)-11/2/(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))-329/8/(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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{5}{7} \, x^{7} - \frac{17}{5} \, x^{5} + \frac{19}{3} \, x^{3} + 38 \, x + \frac{25 \,{\left (5 \, x^{3} + 3 \, x\right )}}{8 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} - \frac{1}{8} \, \int \frac{1339 \, x^{2} + 987}{x^{4} + 2 \, x^{2} + 3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^6 + 3*x^4 + x^2 + 4)*x^8/(x^4 + 2*x^2 + 3)^2,x, algorithm="maxima")

[Out]

5/7*x^7 - 17/5*x^5 + 19/3*x^3 + 38*x + 25/8*(5*x^3 + 3*x)/(x^4 + 2*x^2 + 3) - 1/
8*integrate((1339*x^2 + 987)/(x^4 + 2*x^2 + 3), x)

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Fricas [A]  time = 0.312552, size = 1040, normalized size = 4.19 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^6 + 3*x^4 + x^2 + 4)*x^8/(x^4 + 2*x^2 + 3)^2,x, algorithm="fricas")

[Out]

1/230828640*sqrt(68699)*(616643160*14158657803^(1/4)*(x^4 + 2*x^2 + 3)*arctan(2*
14158657803^(1/4)*(505*sqrt(3) + 176)/(sqrt(68699)*sqrt(1/68699)*(618291*sqrt(3)
*sqrt(2) - 262771*sqrt(2))*sqrt((31677855794226273142765*sqrt(3)*x^2 + 2*1415865
7803^(1/4)*sqrt(68699)*(356093456721312589541*sqrt(3)*x - 515283996859113983188*
x)*sqrt((262771*sqrt(3) - 1854873)/(162468944361*sqrt(3) - 607949940242)) - 8626
8515781092897641041*x^2 + 68699*sqrt(3)*(461110871981051735*sqrt(3) - 1255746310
442552259))/(461110871981051735*sqrt(3) - 1255746310442552259))*sqrt((262771*sqr
t(3) - 1854873)/(162468944361*sqrt(3) - 607949940242)) + sqrt(68699)*(618291*sqr
t(3)*sqrt(2)*x - 262771*sqrt(2)*x)*sqrt((262771*sqrt(3) - 1854873)/(162468944361
*sqrt(3) - 607949940242)) + 14158657803^(1/4)*(329*sqrt(3)*sqrt(2) - 1339*sqrt(2
)))) + 616643160*14158657803^(1/4)*(x^4 + 2*x^2 + 3)*arctan(2*14158657803^(1/4)*
(505*sqrt(3) + 176)/(sqrt(68699)*sqrt(1/68699)*(618291*sqrt(3)*sqrt(2) - 262771*
sqrt(2))*sqrt((31677855794226273142765*sqrt(3)*x^2 - 2*14158657803^(1/4)*sqrt(68
699)*(356093456721312589541*sqrt(3)*x - 515283996859113983188*x)*sqrt((262771*sq
rt(3) - 1854873)/(162468944361*sqrt(3) - 607949940242)) - 8626851578109289764104
1*x^2 + 68699*sqrt(3)*(461110871981051735*sqrt(3) - 1255746310442552259))/(46111
0871981051735*sqrt(3) - 1255746310442552259))*sqrt((262771*sqrt(3) - 1854873)/(1
62468944361*sqrt(3) - 607949940242)) + sqrt(68699)*(618291*sqrt(3)*sqrt(2)*x - 2
62771*sqrt(2)*x)*sqrt((262771*sqrt(3) - 1854873)/(162468944361*sqrt(3) - 6079499
40242)) - 14158657803^(1/4)*(329*sqrt(3)*sqrt(2) - 1339*sqrt(2)))) - 105*1415865
7803^(1/4)*(618291*sqrt(3)*sqrt(2)*(x^4 + 2*x^2 + 3) - 262771*sqrt(2)*(x^4 + 2*x
^2 + 3))*log(95033567382678819428295*sqrt(3)*x^2 + 6*14158657803^(1/4)*sqrt(6869
9)*(356093456721312589541*sqrt(3)*x - 515283996859113983188*x)*sqrt((262771*sqrt
(3) - 1854873)/(162468944361*sqrt(3) - 607949940242)) - 258805547343278692923123
*x^2 + 206097*sqrt(3)*(461110871981051735*sqrt(3) - 1255746310442552259)) + 105*
14158657803^(1/4)*(618291*sqrt(3)*sqrt(2)*(x^4 + 2*x^2 + 3) - 262771*sqrt(2)*(x^
4 + 2*x^2 + 3))*log(95033567382678819428295*sqrt(3)*x^2 - 6*14158657803^(1/4)*sq
rt(68699)*(356093456721312589541*sqrt(3)*x - 515283996859113983188*x)*sqrt((2627
71*sqrt(3) - 1854873)/(162468944361*sqrt(3) - 607949940242)) - 25880554734327869
2923123*x^2 + 206097*sqrt(3)*(461110871981051735*sqrt(3) - 1255746310442552259))
+ 4*sqrt(68699)*(618291*sqrt(3)*sqrt(2)*(600*x^11 - 1656*x^9 + 1408*x^7 + 33992
*x^5 + 92925*x^3 + 103635*x) - 262771*sqrt(2)*(600*x^11 - 1656*x^9 + 1408*x^7 +
33992*x^5 + 92925*x^3 + 103635*x))*sqrt((262771*sqrt(3) - 1854873)/(162468944361
*sqrt(3) - 607949940242)))/((618291*sqrt(3)*sqrt(2)*(x^4 + 2*x^2 + 3) - 262771*s
qrt(2)*(x^4 + 2*x^2 + 3))*sqrt((262771*sqrt(3) - 1854873)/(162468944361*sqrt(3)
- 607949940242)))

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Sympy [A]  time = 1.95431, size = 71, normalized size = 0.29 \[ \frac{5 x^{7}}{7} - \frac{17 x^{5}}{5} + \frac{19 x^{3}}{3} + 38 x + \frac{125 x^{3} + 75 x}{8 x^{4} + 16 x^{2} + 24} + \operatorname{RootSum}{\left (1048576 t^{4} + 538155008 t^{2} + 1146851282043, \left ( t \mapsto t \log{\left (- \frac{16547840 t^{3}}{453886804809} - \frac{11974973632 t}{453886804809} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8*(5*x**6+3*x**4+x**2+4)/(x**4+2*x**2+3)**2,x)

[Out]

5*x**7/7 - 17*x**5/5 + 19*x**3/3 + 38*x + (125*x**3 + 75*x)/(8*x**4 + 16*x**2 +
24) + RootSum(1048576*_t**4 + 538155008*_t**2 + 1146851282043, Lambda(_t, _t*log
(-16547840*_t**3/453886804809 - 11974973632*_t/453886804809 + x)))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x^{6} + 3 \, x^{4} + x^{2} + 4\right )} x^{8}}{{\left (x^{4} + 2 \, x^{2} + 3\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^6 + 3*x^4 + x^2 + 4)*x^8/(x^4 + 2*x^2 + 3)^2,x, algorithm="giac")

[Out]

integrate((5*x^6 + 3*x^4 + x^2 + 4)*x^8/(x^4 + 2*x^2 + 3)^2, x)