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

Optimal. Leaf size=242 \[ \frac{5 x^3}{3}-\frac{21}{512} \sqrt{22721 \sqrt{3}-34271} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{21}{512} \sqrt{22721 \sqrt{3}-34271} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{\left (835 x^2+1468\right ) x}{64 \left (x^4+2 x^2+3\right )}+\frac{25 \left (5 x^2+3\right ) x}{16 \left (x^4+2 x^2+3\right )^2}-27 x-\frac{21}{256} \sqrt{34271+22721 \sqrt{3}} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )+\frac{21}{256} \sqrt{34271+22721 \sqrt{3}} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]

[Out]

-27*x + (5*x^3)/3 + (25*x*(3 + 5*x^2))/(16*(3 + 2*x^2 + x^4)^2) - (x*(1468 + 835
*x^2))/(64*(3 + 2*x^2 + x^4)) - (21*Sqrt[34271 + 22721*Sqrt[3]]*ArcTan[(Sqrt[2*(
-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]])/256 + (21*Sqrt[34271 + 22721*Sqrt[
3]]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/256 - (21*Sqrt
[-34271 + 22721*Sqrt[3]]*Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/512 + (2
1*Sqrt[-34271 + 22721*Sqrt[3]]*Log[Sqrt[3] + Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/51
2

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Rubi [A]  time = 0.757398, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.258 \[ \frac{5 x^3}{3}-\frac{21}{512} \sqrt{22721 \sqrt{3}-34271} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{21}{512} \sqrt{22721 \sqrt{3}-34271} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{\left (835 x^2+1468\right ) x}{64 \left (x^4+2 x^2+3\right )}+\frac{25 \left (5 x^2+3\right ) x}{16 \left (x^4+2 x^2+3\right )^2}-27 x-\frac{21}{256} \sqrt{34271+22721 \sqrt{3}} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )+\frac{21}{256} \sqrt{34271+22721 \sqrt{3}} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-27*x + (5*x^3)/3 + (25*x*(3 + 5*x^2))/(16*(3 + 2*x^2 + x^4)^2) - (x*(1468 + 835
*x^2))/(64*(3 + 2*x^2 + x^4)) - (21*Sqrt[34271 + 22721*Sqrt[3]]*ArcTan[(Sqrt[2*(
-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]])/256 + (21*Sqrt[34271 + 22721*Sqrt[
3]]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/256 - (21*Sqrt
[-34271 + 22721*Sqrt[3]]*Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/512 + (2
1*Sqrt[-34271 + 22721*Sqrt[3]]*Log[Sqrt[3] + Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/51
2

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Rubi in Sympy [A]  time = 53.1182, size = 360, normalized size = 1.49 \[ \frac{5 x^{3}}{3} + \frac{x \left (96000 x^{2} + 57600\right )}{12288 \left (x^{4} + 2 x^{2} + 3\right )^{2}} - \frac{x \left (246251520 x^{2} + 432930816\right )}{18874368 \left (x^{4} + 2 x^{2} + 3\right )} - 27 x - \frac{\sqrt{6} \left (- 424230912 \sqrt{3} + 966131712\right ) \log{\left (x^{2} - \sqrt{2} x \sqrt{-1 + \sqrt{3}} + \sqrt{3} \right )}}{226492416 \sqrt{-1 + \sqrt{3}}} + \frac{\sqrt{6} \left (- 424230912 \sqrt{3} + 966131712\right ) \log{\left (x^{2} + \sqrt{2} x \sqrt{-1 + \sqrt{3}} + \sqrt{3} \right )}}{226492416 \sqrt{-1 + \sqrt{3}}} + \frac{\sqrt{3} \left (- \frac{\sqrt{2} \sqrt{-1 + \sqrt{3}} \left (- 848461824 \sqrt{3} + 1932263424\right )}{2} + 1932263424 \sqrt{2} \sqrt{-1 + \sqrt{3}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \left (x - \frac{\sqrt{-2 + 2 \sqrt{3}}}{2}\right )}{\sqrt{1 + \sqrt{3}}} \right )}}{113246208 \sqrt{-1 + \sqrt{3}} \sqrt{1 + \sqrt{3}}} + \frac{\sqrt{3} \left (- \frac{\sqrt{2} \sqrt{-1 + \sqrt{3}} \left (- 848461824 \sqrt{3} + 1932263424\right )}{2} + 1932263424 \sqrt{2} \sqrt{-1 + \sqrt{3}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \left (x + \frac{\sqrt{-2 + 2 \sqrt{3}}}{2}\right )}{\sqrt{1 + \sqrt{3}}} \right )}}{113246208 \sqrt{-1 + \sqrt{3}} \sqrt{1 + \sqrt{3}}} \]

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)**3,x)

[Out]

5*x**3/3 + x*(96000*x**2 + 57600)/(12288*(x**4 + 2*x**2 + 3)**2) - x*(246251520*
x**2 + 432930816)/(18874368*(x**4 + 2*x**2 + 3)) - 27*x - sqrt(6)*(-424230912*sq
rt(3) + 966131712)*log(x**2 - sqrt(2)*x*sqrt(-1 + sqrt(3)) + sqrt(3))/(226492416
*sqrt(-1 + sqrt(3))) + sqrt(6)*(-424230912*sqrt(3) + 966131712)*log(x**2 + sqrt(
2)*x*sqrt(-1 + sqrt(3)) + sqrt(3))/(226492416*sqrt(-1 + sqrt(3))) + sqrt(3)*(-sq
rt(2)*sqrt(-1 + sqrt(3))*(-848461824*sqrt(3) + 1932263424)/2 + 1932263424*sqrt(2
)*sqrt(-1 + sqrt(3)))*atan(sqrt(2)*(x - sqrt(-2 + 2*sqrt(3))/2)/sqrt(1 + sqrt(3)
))/(113246208*sqrt(-1 + sqrt(3))*sqrt(1 + sqrt(3))) + sqrt(3)*(-sqrt(2)*sqrt(-1
+ sqrt(3))*(-848461824*sqrt(3) + 1932263424)/2 + 1932263424*sqrt(2)*sqrt(-1 + sq
rt(3)))*atan(sqrt(2)*(x + sqrt(-2 + 2*sqrt(3))/2)/sqrt(1 + sqrt(3)))/(113246208*
sqrt(-1 + sqrt(3))*sqrt(1 + sqrt(3)))

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Mathematica [C]  time = 0.391872, size = 155, normalized size = 0.64 \[ \frac{5 x^3}{3}-\frac{\left (835 x^2+1468\right ) x}{64 \left (x^4+2 x^2+3\right )}+\frac{25 \left (5 x^2+3\right ) x}{16 \left (x^4+2 x^2+3\right )^2}-27 x+\frac{21 \left (137 \sqrt{2}-175 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1-i \sqrt{2}}}\right )}{128 \sqrt{2-2 i \sqrt{2}}}+\frac{21 \left (137 \sqrt{2}+175 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt{2}}}\right )}{128 \sqrt{2+2 i \sqrt{2}}} \]

Antiderivative was successfully verified.

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

[Out]

-27*x + (5*x^3)/3 + (25*x*(3 + 5*x^2))/(16*(3 + 2*x^2 + x^4)^2) - (x*(1468 + 835
*x^2))/(64*(3 + 2*x^2 + x^4)) + (21*(-175*I + 137*Sqrt[2])*ArcTan[x/Sqrt[1 - I*S
qrt[2]]])/(128*Sqrt[2 - (2*I)*Sqrt[2]]) + (21*(175*I + 137*Sqrt[2])*ArcTan[x/Sqr
t[1 + I*Sqrt[2]]])/(128*Sqrt[2 + (2*I)*Sqrt[2]])

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Maple [B]  time = 0.033, size = 426, normalized size = 1.8 \[{\frac{5\,{x}^{3}}{3}}-27\,x+{\frac{1}{ \left ({x}^{4}+2\,{x}^{2}+3 \right ) ^{2}} \left ( -{\frac{835\,{x}^{7}}{64}}-{\frac{1569\,{x}^{5}}{32}}-{\frac{4941\,{x}^{3}}{64}}-{\frac{513\,x}{8}} \right ) }-{\frac{693\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{1024}}+{\frac{3675\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{1024}}+{\frac{ \left ( -1386+1386\,\sqrt{3} \right ) \sqrt{3}}{512\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{-7350+7350\,\sqrt{3}}{512\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{273\,\sqrt{3}}{8\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{693\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{1024}}-{\frac{3675\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{1024}}+{\frac{ \left ( -1386+1386\,\sqrt{3} \right ) \sqrt{3}}{512\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{-7350+7350\,\sqrt{3}}{512\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{273\,\sqrt{3}}{8\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) } \]

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)^3,x)

[Out]

5/3*x^3-27*x+(-835/64*x^7-1569/32*x^5-4941/64*x^3-513/8*x)/(x^4+2*x^2+3)^2-693/1
024*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)*3^(1/2)+3675/102
4*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)+693/512/(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)-3675/512/(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))+273/8/(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)+693/1024*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/
2))*(-2+2*3^(1/2))^(1/2)*3^(1/2)-3675/1024*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2)
)*(-2+2*3^(1/2))^(1/2)+693/512/(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)-3675/512/(2+2*3^(1/2))^(1/2)*ar
ctan((2*x-(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))+273/8/(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. \[ \frac{5}{3} \, x^{3} - 27 \, x - \frac{835 \, x^{7} + 3138 \, x^{5} + 4941 \, x^{3} + 4104 \, x}{64 \,{\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )}} + \frac{21}{64} \, \int \frac{137 \, x^{2} + 312}{x^{4} + 2 \, x^{2} + 3}\,{d x} \]

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)^3,x, algorithm="maxima")

[Out]

5/3*x^3 - 27*x - 1/64*(835*x^7 + 3138*x^5 + 4941*x^3 + 4104*x)/(x^8 + 4*x^6 + 10
*x^4 + 12*x^2 + 9) + 21/64*integrate((137*x^2 + 312)/(x^4 + 2*x^2 + 3), x)

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Fricas [A]  time = 0.316244, size = 1188, normalized size = 4.91 \[ \text{result too large to display} \]

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)^3,x, algorithm="fricas")

[Out]

1/69798912*sqrt(22721)*4^(3/4)*(4*sqrt(22721)*4^(1/4)*(22721*sqrt(3)*sqrt(2)*(32
0*x^11 - 3904*x^9 - 20041*x^7 - 57414*x^5 - 74151*x^3 - 58968*x) - 34271*sqrt(2)
*(320*x^11 - 3904*x^9 - 20041*x^7 - 57414*x^5 - 74151*x^3 - 58968*x))*sqrt((3427
1*sqrt(3) - 68163)/(778671391*sqrt(3) - 1361616482)) + 6894216*1548731523^(1/4)*
(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9)*arctan(2*1548731523^(1/4)*(33*sqrt(3) - 175)
/(sqrt(22721)*4^(1/4)*sqrt(1/22721)*(22721*sqrt(3)*sqrt(2) - 34271*sqrt(2))*sqrt
((2266204729347424955*sqrt(3)*x^2 + 1548731523^(1/4)*sqrt(22721)*4^(1/4)*(182673
85333855091*sqrt(3)*x - 31642871720158331*x)*sqrt((34271*sqrt(3) - 68163)/(77867
1391*sqrt(3) - 1361616482)) - 3927765773395386729*x^2 + 22721*sqrt(3)*(997405364
79355*sqrt(3) - 172869405985449))/(99740536479355*sqrt(3) - 172869405985449))*sq
rt((34271*sqrt(3) - 68163)/(778671391*sqrt(3) - 1361616482)) + sqrt(22721)*4^(1/
4)*(22721*sqrt(3)*sqrt(2)*x - 34271*sqrt(2)*x)*sqrt((34271*sqrt(3) - 68163)/(778
671391*sqrt(3) - 1361616482)) + 2*1548731523^(1/4)*(104*sqrt(3)*sqrt(2) - 137*sq
rt(2)))) + 6894216*1548731523^(1/4)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9)*arctan(2
*1548731523^(1/4)*(33*sqrt(3) - 175)/(sqrt(22721)*4^(1/4)*sqrt(1/22721)*(22721*s
qrt(3)*sqrt(2) - 34271*sqrt(2))*sqrt((2266204729347424955*sqrt(3)*x^2 - 15487315
23^(1/4)*sqrt(22721)*4^(1/4)*(18267385333855091*sqrt(3)*x - 31642871720158331*x)
*sqrt((34271*sqrt(3) - 68163)/(778671391*sqrt(3) - 1361616482)) - 39277657733953
86729*x^2 + 22721*sqrt(3)*(99740536479355*sqrt(3) - 172869405985449))/(997405364
79355*sqrt(3) - 172869405985449))*sqrt((34271*sqrt(3) - 68163)/(778671391*sqrt(3
) - 1361616482)) + sqrt(22721)*4^(1/4)*(22721*sqrt(3)*sqrt(2)*x - 34271*sqrt(2)*
x)*sqrt((34271*sqrt(3) - 68163)/(778671391*sqrt(3) - 1361616482)) - 2*1548731523
^(1/4)*(104*sqrt(3)*sqrt(2) - 137*sqrt(2)))) + 63*1548731523^(1/4)*(22721*sqrt(3
)*sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9) - 34271*sqrt(2)*(x^8 + 4*x^6 + 10*
x^4 + 12*x^2 + 9))*log(4532409458694849910*sqrt(3)*x^2 + 2*1548731523^(1/4)*sqrt
(22721)*4^(1/4)*(18267385333855091*sqrt(3)*x - 31642871720158331*x)*sqrt((34271*
sqrt(3) - 68163)/(778671391*sqrt(3) - 1361616482)) - 7855531546790773458*x^2 + 4
5442*sqrt(3)*(99740536479355*sqrt(3) - 172869405985449)) - 63*1548731523^(1/4)*(
22721*sqrt(3)*sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9) - 34271*sqrt(2)*(x^8 +
4*x^6 + 10*x^4 + 12*x^2 + 9))*log(4532409458694849910*sqrt(3)*x^2 - 2*154873152
3^(1/4)*sqrt(22721)*4^(1/4)*(18267385333855091*sqrt(3)*x - 31642871720158331*x)*
sqrt((34271*sqrt(3) - 68163)/(778671391*sqrt(3) - 1361616482)) - 785553154679077
3458*x^2 + 45442*sqrt(3)*(99740536479355*sqrt(3) - 172869405985449)))/((22721*sq
rt(3)*sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9) - 34271*sqrt(2)*(x^8 + 4*x^6 +
10*x^4 + 12*x^2 + 9))*sqrt((34271*sqrt(3) - 68163)/(778671391*sqrt(3) - 1361616
482)))

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Sympy [A]  time = 2.19193, size = 80, normalized size = 0.33 \[ \frac{5 x^{3}}{3} - 27 x - \frac{835 x^{7} + 3138 x^{5} + 4941 x^{3} + 4104 x}{64 x^{8} + 256 x^{6} + 640 x^{4} + 768 x^{2} + 576} + 21 \operatorname{RootSum}{\left (17179869184 t^{4} + 8983937024 t^{2} + 1548731523, \left ( t \mapsto t \log{\left (- \frac{1107296256 t^{3}}{310800559} + \frac{438857984 t}{310800559} + x \right )} \right )\right )} \]

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)**3,x)

[Out]

5*x**3/3 - 27*x - (835*x**7 + 3138*x**5 + 4941*x**3 + 4104*x)/(64*x**8 + 256*x**
6 + 640*x**4 + 768*x**2 + 576) + 21*RootSum(17179869184*_t**4 + 8983937024*_t**2
+ 1548731523, Lambda(_t, _t*log(-1107296256*_t**3/310800559 + 438857984*_t/3108
00559 + x)))

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

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)^3,x, algorithm="giac")

[Out]

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