3.117 \(\int \frac{x^{10} \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^3} \, dx\)

Optimal. Leaf size=243 \[ x^5-9 x^3+\frac{3}{512} \sqrt{8595619+7678611 \sqrt{3}} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{3}{512} \sqrt{8595619+7678611 \sqrt{3}} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{\left (252 x^2+3305\right ) x}{64 \left (x^4+2 x^2+3\right )}-\frac{25 \left (7 x^2+15\right ) x}{16 \left (x^4+2 x^2+3\right )^2}+58 x+\frac{3}{256} \sqrt{7678611 \sqrt{3}-8595619} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )-\frac{3}{256} \sqrt{7678611 \sqrt{3}-8595619} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]

[Out]

58*x - 9*x^3 + x^5 - (25*x*(15 + 7*x^2))/(16*(3 + 2*x^2 + x^4)^2) + (x*(3305 + 2
52*x^2))/(64*(3 + 2*x^2 + x^4)) + (3*Sqrt[-8595619 + 7678611*Sqrt[3]]*ArcTan[(Sq
rt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]])/256 - (3*Sqrt[-8595619 + 767
8611*Sqrt[3]]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/256
+ (3*Sqrt[8595619 + 7678611*Sqrt[3]]*Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^
2])/512 - (3*Sqrt[8595619 + 7678611*Sqrt[3]]*Log[Sqrt[3] + Sqrt[2*(-1 + Sqrt[3])
]*x + x^2])/512

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Rubi [A]  time = 0.854242, antiderivative size = 243, 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 \[ x^5-9 x^3+\frac{3}{512} \sqrt{8595619+7678611 \sqrt{3}} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{3}{512} \sqrt{8595619+7678611 \sqrt{3}} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{\left (252 x^2+3305\right ) x}{64 \left (x^4+2 x^2+3\right )}-\frac{25 \left (7 x^2+15\right ) x}{16 \left (x^4+2 x^2+3\right )^2}+58 x+\frac{3}{256} \sqrt{7678611 \sqrt{3}-8595619} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )-\frac{3}{256} \sqrt{7678611 \sqrt{3}-8595619} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(x^10*(4 + x^2 + 3*x^4 + 5*x^6))/(3 + 2*x^2 + x^4)^3,x]

[Out]

58*x - 9*x^3 + x^5 - (25*x*(15 + 7*x^2))/(16*(3 + 2*x^2 + x^4)^2) + (x*(3305 + 2
52*x^2))/(64*(3 + 2*x^2 + x^4)) + (3*Sqrt[-8595619 + 7678611*Sqrt[3]]*ArcTan[(Sq
rt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]])/256 - (3*Sqrt[-8595619 + 767
8611*Sqrt[3]]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/256
+ (3*Sqrt[8595619 + 7678611*Sqrt[3]]*Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^
2])/512 - (3*Sqrt[8595619 + 7678611*Sqrt[3]]*Log[Sqrt[3] + Sqrt[2*(-1 + Sqrt[3])
]*x + x^2])/512

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Rubi in Sympy [A]  time = 65.0145, size = 362, normalized size = 1.49 \[ x^{5} - 9 x^{3} - \frac{x \left (268800 x^{2} + 576000\right )}{24576 \left (x^{4} + 2 x^{2} + 3\right )^{2}} + \frac{x \left (297271296 x^{2} + 3898736640\right )}{75497472 \left (x^{4} + 2 x^{2} + 3\right )} + 58 x + \frac{\sqrt{6} \left (261881856 \sqrt{3} + 8222736384\right ) \log{\left (x^{2} - \sqrt{2} x \sqrt{-1 + \sqrt{3}} + \sqrt{3} \right )}}{905969664 \sqrt{-1 + \sqrt{3}}} - \frac{\sqrt{6} \left (261881856 \sqrt{3} + 8222736384\right ) \log{\left (x^{2} + \sqrt{2} x \sqrt{-1 + \sqrt{3}} + \sqrt{3} \right )}}{905969664 \sqrt{-1 + \sqrt{3}}} - \frac{\sqrt{3} \left (- \frac{\sqrt{2} \sqrt{-1 + \sqrt{3}} \left (523763712 \sqrt{3} + 16445472768\right )}{2} + 16445472768 \sqrt{2} \sqrt{-1 + \sqrt{3}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \left (x - \frac{\sqrt{-2 + 2 \sqrt{3}}}{2}\right )}{\sqrt{1 + \sqrt{3}}} \right )}}{452984832 \sqrt{-1 + \sqrt{3}} \sqrt{1 + \sqrt{3}}} - \frac{\sqrt{3} \left (- \frac{\sqrt{2} \sqrt{-1 + \sqrt{3}} \left (523763712 \sqrt{3} + 16445472768\right )}{2} + 16445472768 \sqrt{2} \sqrt{-1 + \sqrt{3}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \left (x + \frac{\sqrt{-2 + 2 \sqrt{3}}}{2}\right )}{\sqrt{1 + \sqrt{3}}} \right )}}{452984832 \sqrt{-1 + \sqrt{3}} \sqrt{1 + \sqrt{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**10*(5*x**6+3*x**4+x**2+4)/(x**4+2*x**2+3)**3,x)

[Out]

x**5 - 9*x**3 - x*(268800*x**2 + 576000)/(24576*(x**4 + 2*x**2 + 3)**2) + x*(297
271296*x**2 + 3898736640)/(75497472*(x**4 + 2*x**2 + 3)) + 58*x + sqrt(6)*(26188
1856*sqrt(3) + 8222736384)*log(x**2 - sqrt(2)*x*sqrt(-1 + sqrt(3)) + sqrt(3))/(9
05969664*sqrt(-1 + sqrt(3))) - sqrt(6)*(261881856*sqrt(3) + 8222736384)*log(x**2
+ sqrt(2)*x*sqrt(-1 + sqrt(3)) + sqrt(3))/(905969664*sqrt(-1 + sqrt(3))) - sqrt
(3)*(-sqrt(2)*sqrt(-1 + sqrt(3))*(523763712*sqrt(3) + 16445472768)/2 + 164454727
68*sqrt(2)*sqrt(-1 + sqrt(3)))*atan(sqrt(2)*(x - sqrt(-2 + 2*sqrt(3))/2)/sqrt(1
+ sqrt(3)))/(452984832*sqrt(-1 + sqrt(3))*sqrt(1 + sqrt(3))) - sqrt(3)*(-sqrt(2)
*sqrt(-1 + sqrt(3))*(523763712*sqrt(3) + 16445472768)/2 + 16445472768*sqrt(2)*sq
rt(-1 + sqrt(3)))*atan(sqrt(2)*(x + sqrt(-2 + 2*sqrt(3))/2)/sqrt(1 + sqrt(3)))/(
452984832*sqrt(-1 + sqrt(3))*sqrt(1 + sqrt(3)))

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Mathematica [C]  time = 0.422754, size = 156, normalized size = 0.64 \[ x^5-9 x^3+\frac{\left (252 x^2+3305\right ) x}{64 \left (x^4+2 x^2+3\right )}-\frac{25 \left (7 x^2+15\right ) x}{16 \left (x^4+2 x^2+3\right )^2}+58 x+\frac{3 \left (148 \sqrt{2}+4795 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1-i \sqrt{2}}}\right )}{128 \sqrt{2-2 i \sqrt{2}}}+\frac{3 \left (148 \sqrt{2}-4795 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt{2}}}\right )}{128 \sqrt{2+2 i \sqrt{2}}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^10*(4 + x^2 + 3*x^4 + 5*x^6))/(3 + 2*x^2 + x^4)^3,x]

[Out]

58*x - 9*x^3 + x^5 - (25*x*(15 + 7*x^2))/(16*(3 + 2*x^2 + x^4)^2) + (x*(3305 + 2
52*x^2))/(64*(3 + 2*x^2 + x^4)) + (3*(4795*I + 148*Sqrt[2])*ArcTan[x/Sqrt[1 - I*
Sqrt[2]]])/(128*Sqrt[2 - (2*I)*Sqrt[2]]) + (3*(-4795*I + 148*Sqrt[2])*ArcTan[x/S
qrt[1 + I*Sqrt[2]]])/(128*Sqrt[2 + (2*I)*Sqrt[2]])

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Maple [B]  time = 0.033, size = 429, normalized size = 1.8 \[{x}^{5}-9\,{x}^{3}+58\,x+{\frac{1}{ \left ({x}^{4}+2\,{x}^{2}+3 \right ) ^{2}} \left ({\frac{63\,{x}^{7}}{16}}+{\frac{3809\,{x}^{5}}{64}}+{\frac{3333\,{x}^{3}}{32}}+{\frac{8415\,x}{64}} \right ) }-{\frac{5091\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{1024}}-{\frac{14385\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{1024}}+{\frac{ \left ( -10182+10182\,\sqrt{3} \right ) \sqrt{3}}{512\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{-28770+28770\,\sqrt{3}}{512\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{4647\,\sqrt{3}}{64\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{5091\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{1024}}+{\frac{14385\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{1024}}+{\frac{ \left ( -10182+10182\,\sqrt{3} \right ) \sqrt{3}}{512\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{-28770+28770\,\sqrt{3}}{512\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{4647\,\sqrt{3}}{64\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^10*(5*x^6+3*x^4+x^2+4)/(x^4+2*x^2+3)^3,x)

[Out]

x^5-9*x^3+58*x+(63/16*x^7+3809/64*x^5+3333/32*x^3+8415/64*x)/(x^4+2*x^2+3)^2-509
1/1024*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)*3^(1/2)-14385
/1024*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)+5091/512/(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)+14385/512/(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))-4647/64/(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)+5091/1024*ln(x^2+3^(1/2)-x*(-2+2*3^(
1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)*3^(1/2)+14385/1024*ln(x^2+3^(1/2)-x*(-2+2*3^(1
/2))^(1/2))*(-2+2*3^(1/2))^(1/2)+5091/512/(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)+14385/512/(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))-
4647/64/(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. \[ x^{5} - 9 \, x^{3} + 58 \, x + \frac{252 \, x^{7} + 3809 \, x^{5} + 6666 \, x^{3} + 8415 \, x}{64 \,{\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )}} + \frac{3}{64} \, \int \frac{148 \, x^{2} - 4647}{x^{4} + 2 \, x^{2} + 3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^6 + 3*x^4 + x^2 + 4)*x^10/(x^4 + 2*x^2 + 3)^3,x, algorithm="maxima")

[Out]

x^5 - 9*x^3 + 58*x + 1/64*(252*x^7 + 3809*x^5 + 6666*x^3 + 8415*x)/(x^8 + 4*x^6
+ 10*x^4 + 12*x^2 + 9) + 3/64*integrate((148*x^2 - 4647)/(x^4 + 2*x^2 + 3), x)

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Fricas [A]  time = 0.297943, size = 1202, normalized size = 4.95 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^6 + 3*x^4 + x^2 + 4)*x^10/(x^4 + 2*x^2 + 3)^3,x, algorithm="fricas")

[Out]

1/873655296*sqrt(853179)*4^(3/4)*(4*sqrt(853179)*4^(1/4)*(7678611*sqrt(3)*sqrt(2
)*(64*x^13 - 320*x^11 + 2048*x^9 + 10108*x^7 + 34593*x^5 + 46026*x^3 + 41823*x)
+ 8595619*sqrt(2)*(64*x^13 - 320*x^11 + 2048*x^9 + 10108*x^7 + 34593*x^5 + 46026
*x^3 + 41823*x))*sqrt((8595619*sqrt(3) + 23035833)/(66002414605209*sqrt(3) + 125
383933330562)) + 172231176*2183743218123^(1/4)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 +
9)*arctan(2*2183743218123^(1/4)*(1697*sqrt(3) + 4795)/(sqrt(853179)*4^(1/4)*sqrt
(1/853179)*(7678611*sqrt(3)*sqrt(2) + 8595619*sqrt(2))*sqrt((2216706744410990259
940368525*sqrt(3)*x^2 + 2183743218123^(1/4)*sqrt(853179)*4^(1/4)*(42510258596843
07632934719*sqrt(3)*x + 7494932476650732136107491*x)*sqrt((8595619*sqrt(3) + 230
35833)/(66002414605209*sqrt(3) + 125383933330562)) + 391636307122244572929688736
1*x^2 + 853179*sqrt(3)*(2598173120073267461975*sqrt(3) + 4590318176165195966259)
)/(2598173120073267461975*sqrt(3) + 4590318176165195966259))*sqrt((8595619*sqrt(
3) + 23035833)/(66002414605209*sqrt(3) + 125383933330562)) + sqrt(853179)*4^(1/4
)*(7678611*sqrt(3)*sqrt(2)*x + 8595619*sqrt(2)*x)*sqrt((8595619*sqrt(3) + 230358
33)/(66002414605209*sqrt(3) + 125383933330562)) + 2*2183743218123^(1/4)*(1549*sq
rt(3)*sqrt(2) + 148*sqrt(2)))) + 172231176*2183743218123^(1/4)*(x^8 + 4*x^6 + 10
*x^4 + 12*x^2 + 9)*arctan(2*2183743218123^(1/4)*(1697*sqrt(3) + 4795)/(sqrt(8531
79)*4^(1/4)*sqrt(1/853179)*(7678611*sqrt(3)*sqrt(2) + 8595619*sqrt(2))*sqrt((221
6706744410990259940368525*sqrt(3)*x^2 - 2183743218123^(1/4)*sqrt(853179)*4^(1/4)
*(4251025859684307632934719*sqrt(3)*x + 7494932476650732136107491*x)*sqrt((85956
19*sqrt(3) + 23035833)/(66002414605209*sqrt(3) + 125383933330562)) + 39163630712
22445729296887361*x^2 + 853179*sqrt(3)*(2598173120073267461975*sqrt(3) + 4590318
176165195966259))/(2598173120073267461975*sqrt(3) + 4590318176165195966259))*sqr
t((8595619*sqrt(3) + 23035833)/(66002414605209*sqrt(3) + 125383933330562)) + sqr
t(853179)*4^(1/4)*(7678611*sqrt(3)*sqrt(2)*x + 8595619*sqrt(2)*x)*sqrt((8595619*
sqrt(3) + 23035833)/(66002414605209*sqrt(3) + 125383933330562)) - 2*218374321812
3^(1/4)*(1549*sqrt(3)*sqrt(2) + 148*sqrt(2)))) - 3*2183743218123^(1/4)*(7678611*
sqrt(3)*sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9) + 8595619*sqrt(2)*(x^8 + 4*x
^6 + 10*x^4 + 12*x^2 + 9))*log(13300240466465941559642211150*sqrt(3)*x^2 + 6*218
3743218123^(1/4)*sqrt(853179)*4^(1/4)*(4251025859684307632934719*sqrt(3)*x + 749
4932476650732136107491*x)*sqrt((8595619*sqrt(3) + 23035833)/(66002414605209*sqrt
(3) + 125383933330562)) + 23498178427334674375781324166*x^2 + 5119074*sqrt(3)*(2
598173120073267461975*sqrt(3) + 4590318176165195966259)) + 3*2183743218123^(1/4)
*(7678611*sqrt(3)*sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9) + 8595619*sqrt(2)*
(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9))*log(13300240466465941559642211150*sqrt(3)*x
^2 - 6*2183743218123^(1/4)*sqrt(853179)*4^(1/4)*(4251025859684307632934719*sqrt(
3)*x + 7494932476650732136107491*x)*sqrt((8595619*sqrt(3) + 23035833)/(660024146
05209*sqrt(3) + 125383933330562)) + 23498178427334674375781324166*x^2 + 5119074*
sqrt(3)*(2598173120073267461975*sqrt(3) + 4590318176165195966259)))/((7678611*sq
rt(3)*sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9) + 8595619*sqrt(2)*(x^8 + 4*x^6
+ 10*x^4 + 12*x^2 + 9))*sqrt((8595619*sqrt(3) + 23035833)/(66002414605209*sqrt(
3) + 125383933330562)))

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Sympy [A]  time = 2.15214, size = 82, normalized size = 0.34 \[ x^{5} - 9 x^{3} + 58 x + \frac{252 x^{7} + 3809 x^{5} + 6666 x^{3} + 8415 x}{64 x^{8} + 256 x^{6} + 640 x^{4} + 768 x^{2} + 576} + 3 \operatorname{RootSum}{\left (17179869184 t^{4} - 2253289947136 t^{2} + 176883200667963, \left ( t \mapsto t \log{\left (- \frac{56941871104 t^{3}}{55104008440689} - \frac{1957224667904 t}{55104008440689} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**10*(5*x**6+3*x**4+x**2+4)/(x**4+2*x**2+3)**3,x)

[Out]

x**5 - 9*x**3 + 58*x + (252*x**7 + 3809*x**5 + 6666*x**3 + 8415*x)/(64*x**8 + 25
6*x**6 + 640*x**4 + 768*x**2 + 576) + 3*RootSum(17179869184*_t**4 - 225328994713
6*_t**2 + 176883200667963, Lambda(_t, _t*log(-56941871104*_t**3/55104008440689 -
1957224667904*_t/55104008440689 + x)))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x^{6} + 3 \, x^{4} + x^{2} + 4\right )} x^{10}}{{\left (x^{4} + 2 \, x^{2} + 3\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^6 + 3*x^4 + x^2 + 4)*x^10/(x^4 + 2*x^2 + 3)^3,x, algorithm="giac")

[Out]

integrate((5*x^6 + 3*x^4 + x^2 + 4)*x^10/(x^4 + 2*x^2 + 3)^3, x)