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

Optimal. Leaf size=237 \[ x^5-\frac{17 x^3}{3}+\frac{3}{32} \sqrt{\frac{3}{2} \left (8669+5011 \sqrt{3}\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{3}{32} \sqrt{\frac{3}{2} \left (8669+5011 \sqrt{3}\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{25 \left (3-x^2\right ) x}{8 \left (x^4+2 x^2+3\right )}+19 x+\frac{3}{16} \sqrt{\frac{3}{2} \left (5011 \sqrt{3}-8669\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )-\frac{3}{16} \sqrt{\frac{3}{2} \left (5011 \sqrt{3}-8669\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]

[Out]

19*x - (17*x^3)/3 + x^5 + (25*x*(3 - x^2))/(8*(3 + 2*x^2 + x^4)) + (3*Sqrt[(3*(-
8669 + 5011*Sqrt[3]))/2]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[
3])]])/16 - (3*Sqrt[(3*(-8669 + 5011*Sqrt[3]))/2]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])]
+ 2*x)/Sqrt[2*(1 + Sqrt[3])]])/16 + (3*Sqrt[(3*(8669 + 5011*Sqrt[3]))/2]*Log[Sq
rt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/32 - (3*Sqrt[(3*(8669 + 5011*Sqrt[3]))/
2]*Log[Sqrt[3] + Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/32

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Rubi [A]  time = 0.725377, antiderivative size = 237, 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 \[ x^5-\frac{17 x^3}{3}+\frac{3}{32} \sqrt{\frac{3}{2} \left (8669+5011 \sqrt{3}\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{3}{32} \sqrt{\frac{3}{2} \left (8669+5011 \sqrt{3}\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{25 \left (3-x^2\right ) x}{8 \left (x^4+2 x^2+3\right )}+19 x+\frac{3}{16} \sqrt{\frac{3}{2} \left (5011 \sqrt{3}-8669\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )-\frac{3}{16} \sqrt{\frac{3}{2} \left (5011 \sqrt{3}-8669\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

19*x - (17*x^3)/3 + x^5 + (25*x*(3 - x^2))/(8*(3 + 2*x^2 + x^4)) + (3*Sqrt[(3*(-
8669 + 5011*Sqrt[3]))/2]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[
3])]])/16 - (3*Sqrt[(3*(-8669 + 5011*Sqrt[3]))/2]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])]
+ 2*x)/Sqrt[2*(1 + Sqrt[3])]])/16 + (3*Sqrt[(3*(8669 + 5011*Sqrt[3]))/2]*Log[Sq
rt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/32 - (3*Sqrt[(3*(8669 + 5011*Sqrt[3]))/
2]*Log[Sqrt[3] + Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/32

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Rubi in Sympy [A]  time = 50.0979, size = 342, normalized size = 1.44 \[ x^{5} - \frac{17 x^{3}}{3} + \frac{x \left (- 9600 x^{2} + 28800\right )}{3072 \left (x^{4} + 2 x^{2} + 3\right )} + 19 x + \frac{\sqrt{6} \left (53568 \sqrt{3} + 101952\right ) \log{\left (x^{2} - \sqrt{2} x \sqrt{-1 + \sqrt{3}} + \sqrt{3} \right )}}{36864 \sqrt{-1 + \sqrt{3}}} - \frac{\sqrt{6} \left (53568 \sqrt{3} + 101952\right ) \log{\left (x^{2} + \sqrt{2} x \sqrt{-1 + \sqrt{3}} + \sqrt{3} \right )}}{36864 \sqrt{-1 + \sqrt{3}}} - \frac{\sqrt{3} \left (- \frac{\sqrt{2} \sqrt{-1 + \sqrt{3}} \left (107136 \sqrt{3} + 203904\right )}{2} + 203904 \sqrt{2} \sqrt{-1 + \sqrt{3}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \left (x - \frac{\sqrt{-2 + 2 \sqrt{3}}}{2}\right )}{\sqrt{1 + \sqrt{3}}} \right )}}{18432 \sqrt{-1 + \sqrt{3}} \sqrt{1 + \sqrt{3}}} - \frac{\sqrt{3} \left (- \frac{\sqrt{2} \sqrt{-1 + \sqrt{3}} \left (107136 \sqrt{3} + 203904\right )}{2} + 203904 \sqrt{2} \sqrt{-1 + \sqrt{3}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \left (x + \frac{\sqrt{-2 + 2 \sqrt{3}}}{2}\right )}{\sqrt{1 + \sqrt{3}}} \right )}}{18432 \sqrt{-1 + \sqrt{3}} \sqrt{1 + \sqrt{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**5 - 17*x**3/3 + x*(-9600*x**2 + 28800)/(3072*(x**4 + 2*x**2 + 3)) + 19*x + sq
rt(6)*(53568*sqrt(3) + 101952)*log(x**2 - sqrt(2)*x*sqrt(-1 + sqrt(3)) + sqrt(3)
)/(36864*sqrt(-1 + sqrt(3))) - sqrt(6)*(53568*sqrt(3) + 101952)*log(x**2 + sqrt(
2)*x*sqrt(-1 + sqrt(3)) + sqrt(3))/(36864*sqrt(-1 + sqrt(3))) - sqrt(3)*(-sqrt(2
)*sqrt(-1 + sqrt(3))*(107136*sqrt(3) + 203904)/2 + 203904*sqrt(2)*sqrt(-1 + sqrt
(3)))*atan(sqrt(2)*(x - sqrt(-2 + 2*sqrt(3))/2)/sqrt(1 + sqrt(3)))/(18432*sqrt(-
1 + sqrt(3))*sqrt(1 + sqrt(3))) - sqrt(3)*(-sqrt(2)*sqrt(-1 + sqrt(3))*(107136*s
qrt(3) + 203904)/2 + 203904*sqrt(2)*sqrt(-1 + sqrt(3)))*atan(sqrt(2)*(x + sqrt(-
2 + 2*sqrt(3))/2)/sqrt(1 + sqrt(3)))/(18432*sqrt(-1 + sqrt(3))*sqrt(1 + sqrt(3))
)

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Mathematica [C]  time = 0.299269, size = 132, normalized size = 0.56 \[ x^5-\frac{17 x^3}{3}-\frac{25 \left (x^2-3\right ) x}{8 \left (x^4+2 x^2+3\right )}+19 x+\frac{9 \left (31 \sqrt{2}+90 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1-i \sqrt{2}}}\right )}{16 \sqrt{2-2 i \sqrt{2}}}+\frac{9 \left (31 \sqrt{2}-90 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt{2}}}\right )}{16 \sqrt{2+2 i \sqrt{2}}} \]

Antiderivative was successfully verified.

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

[Out]

19*x - (17*x^3)/3 + x^5 - (25*x*(-3 + x^2))/(8*(3 + 2*x^2 + x^4)) + (9*(90*I + 3
1*Sqrt[2])*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/(16*Sqrt[2 - (2*I)*Sqrt[2]]) + (9*(-90
*I + 31*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/(16*Sqrt[2 + (2*I)*Sqrt[2]])

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Maple [B]  time = 0.043, size = 419, normalized size = 1.8 \[{x}^{5}-{\frac{17\,{x}^{3}}{3}}+19\,x+{\frac{1}{{x}^{4}+2\,{x}^{2}+3} \left ( -{\frac{25\,{x}^{3}}{8}}+{\frac{75\,x}{8}} \right ) }-{\frac{57\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{16}}-{\frac{405\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{64}}+{\frac{ \left ( -114+114\,\sqrt{3} \right ) \sqrt{3}}{8\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{-810+810\,\sqrt{3}}{32\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{177\,\sqrt{3}}{8\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{57\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{16}}+{\frac{405\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{64}}+{\frac{ \left ( -114+114\,\sqrt{3} \right ) \sqrt{3}}{8\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{-810+810\,\sqrt{3}}{32\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{177\,\sqrt{3}}{8\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x^5-17/3*x^3+19*x+(-25/8*x^3+75/8*x)/(x^4+2*x^2+3)-57/16*ln(x^2+3^(1/2)+x*(-2+2*
3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)*3^(1/2)-405/64*ln(x^2+3^(1/2)+x*(-2+2*3^(1/
2))^(1/2))*(-2+2*3^(1/2))^(1/2)+57/8/(2+2*3^(1/2))^(1/2)*arctan((2*x+(-2+2*3^(1/
2))^(1/2))/(2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))*3^(1/2)+405/32/(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))-177/8/(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)
+57/16*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)*3^(1/2)+405/6
4*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)+57/8/(2+2*3^(1/2))
^(1/2)*arctan((2*x-(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))*3^(
1/2)+405/32/(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))-177/8/(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. \[ x^{5} - \frac{17}{3} \, x^{3} + 19 \, x - \frac{25 \,{\left (x^{3} - 3 \, x\right )}}{8 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} + \frac{9}{8} \, \int \frac{31 \, x^{2} - 59}{x^{4} + 2 \, x^{2} + 3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x^5 - 17/3*x^3 + 19*x - 25/8*(x^3 - 3*x)/(x^4 + 2*x^2 + 3) + 9/8*integrate((31*x
^2 - 59)/(x^4 + 2*x^2 + 3), x)

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Fricas [A]  time = 0.303413, size = 1031, normalized size = 4.35 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/481056*sqrt(5011)*(21528*677973267^(1/4)*(x^4 + 2*x^2 + 3)*arctan(2*677973267^
(1/4)*(76*sqrt(3) + 135)/(3*sqrt(5011)*sqrt(1/15033)*(5011*sqrt(3)*sqrt(2) + 866
9*sqrt(2))*sqrt((19622617239317025*sqrt(3)*x^2 + 2*677973267^(1/4)*sqrt(5011)*(7
3549620457552*sqrt(3)*x + 127391679512403*x)*sqrt((8669*sqrt(3) + 15033)/(434403
59*sqrt(3) + 75240962)) + 33987370050227547*x^2 + 15033*sqrt(3)*(1305302816425*s
qrt(3) + 2260850798259))/(1305302816425*sqrt(3) + 2260850798259))*sqrt((8669*sqr
t(3) + 15033)/(43440359*sqrt(3) + 75240962)) + 3*sqrt(5011)*(5011*sqrt(3)*sqrt(2
)*x + 8669*sqrt(2)*x)*sqrt((8669*sqrt(3) + 15033)/(43440359*sqrt(3) + 75240962))
+ 677973267^(1/4)*(59*sqrt(3)*sqrt(2) + 93*sqrt(2)))) + 21528*677973267^(1/4)*(
x^4 + 2*x^2 + 3)*arctan(2*677973267^(1/4)*(76*sqrt(3) + 135)/(3*sqrt(5011)*sqrt(
1/15033)*(5011*sqrt(3)*sqrt(2) + 8669*sqrt(2))*sqrt((19622617239317025*sqrt(3)*x
^2 - 2*677973267^(1/4)*sqrt(5011)*(73549620457552*sqrt(3)*x + 127391679512403*x)
*sqrt((8669*sqrt(3) + 15033)/(43440359*sqrt(3) + 75240962)) + 33987370050227547*
x^2 + 15033*sqrt(3)*(1305302816425*sqrt(3) + 2260850798259))/(1305302816425*sqrt
(3) + 2260850798259))*sqrt((8669*sqrt(3) + 15033)/(43440359*sqrt(3) + 75240962))
+ 3*sqrt(5011)*(5011*sqrt(3)*sqrt(2)*x + 8669*sqrt(2)*x)*sqrt((8669*sqrt(3) + 1
5033)/(43440359*sqrt(3) + 75240962)) - 677973267^(1/4)*(59*sqrt(3)*sqrt(2) + 93*
sqrt(2)))) - 9*677973267^(1/4)*(5011*sqrt(3)*sqrt(2)*(x^4 + 2*x^2 + 3) + 8669*sq
rt(2)*(x^4 + 2*x^2 + 3))*log(19622617239317025*sqrt(3)*x^2 + 2*677973267^(1/4)*s
qrt(5011)*(73549620457552*sqrt(3)*x + 127391679512403*x)*sqrt((8669*sqrt(3) + 15
033)/(43440359*sqrt(3) + 75240962)) + 33987370050227547*x^2 + 15033*sqrt(3)*(130
5302816425*sqrt(3) + 2260850798259)) + 9*677973267^(1/4)*(5011*sqrt(3)*sqrt(2)*(
x^4 + 2*x^2 + 3) + 8669*sqrt(2)*(x^4 + 2*x^2 + 3))*log(19622617239317025*sqrt(3)
*x^2 - 2*677973267^(1/4)*sqrt(5011)*(73549620457552*sqrt(3)*x + 127391679512403*
x)*sqrt((8669*sqrt(3) + 15033)/(43440359*sqrt(3) + 75240962)) + 3398737005022754
7*x^2 + 15033*sqrt(3)*(1305302816425*sqrt(3) + 2260850798259)) + 4*sqrt(5011)*(5
011*sqrt(3)*sqrt(2)*(24*x^9 - 88*x^7 + 256*x^5 + 429*x^3 + 1593*x) + 8669*sqrt(2
)*(24*x^9 - 88*x^7 + 256*x^5 + 429*x^3 + 1593*x))*sqrt((8669*sqrt(3) + 15033)/(4
3440359*sqrt(3) + 75240962)))/((5011*sqrt(3)*sqrt(2)*(x^4 + 2*x^2 + 3) + 8669*sq
rt(2)*(x^4 + 2*x^2 + 3))*sqrt((8669*sqrt(3) + 15033)/(43440359*sqrt(3) + 7524096
2)))

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Sympy [A]  time = 1.96872, size = 63, normalized size = 0.27 \[ x^{5} - \frac{17 x^{3}}{3} + 19 x - \frac{25 x^{3} - 75 x}{8 x^{4} + 16 x^{2} + 24} + 3 \operatorname{RootSum}{\left (1048576 t^{4} - 53262336 t^{2} + 677973267, \left ( t \mapsto t \log{\left (- \frac{2490368 t^{3}}{13484601} + \frac{20518496 t}{4494867} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**5 - 17*x**3/3 + 19*x - (25*x**3 - 75*x)/(8*x**4 + 16*x**2 + 24) + 3*RootSum(1
048576*_t**4 - 53262336*_t**2 + 677973267, Lambda(_t, _t*log(-2490368*_t**3/1348
4601 + 20518496*_t/4494867 + x)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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