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

Optimal. Leaf size=80 \[ -\frac{1}{9 x^4}+\frac{13}{54 x^2}+\frac{125 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{216 \sqrt{2}}+\frac{25 \left (5 x^2+7\right )}{216 \left (x^4+2 x^2+3\right )}-\frac{13}{108} \log \left (x^4+2 x^2+3\right )+\frac{13 \log (x)}{27} \]

[Out]

-1/(9*x^4) + 13/(54*x^2) + (25*(7 + 5*x^2))/(216*(3 + 2*x^2 + x^4)) + (125*ArcTa
n[(1 + x^2)/Sqrt[2]])/(216*Sqrt[2]) + (13*Log[x])/27 - (13*Log[3 + 2*x^2 + x^4])
/108

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Rubi [A]  time = 0.230945, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226 \[ -\frac{1}{9 x^4}+\frac{13}{54 x^2}+\frac{125 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{216 \sqrt{2}}+\frac{25 \left (5 x^2+7\right )}{216 \left (x^4+2 x^2+3\right )}-\frac{13}{108} \log \left (x^4+2 x^2+3\right )+\frac{13 \log (x)}{27} \]

Antiderivative was successfully verified.

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

[Out]

-1/(9*x^4) + 13/(54*x^2) + (25*(7 + 5*x^2))/(216*(3 + 2*x^2 + x^4)) + (125*ArcTa
n[(1 + x^2)/Sqrt[2]])/(216*Sqrt[2]) + (13*Log[x])/27 - (13*Log[3 + 2*x^2 + x^4])
/108

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Rubi in Sympy [A]  time = 27.0219, size = 95, normalized size = 1.19 \[ \frac{5 \left (26 x^{2} + 22\right )}{432 \left (x^{4} + 2 x^{2} + 3\right )} + \frac{13 \log{\left (x^{2} \right )}}{54} - \frac{13 \log{\left (x^{4} + 2 x^{2} + 3 \right )}}{108} + \frac{125 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \left (\frac{x^{2}}{2} + \frac{1}{2}\right ) \right )}}{432} + \frac{14}{27 x^{2}} - \frac{5}{6 x^{2} \left (x^{4} + 2 x^{2} + 3\right )} - \frac{1}{9 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*(26*x**2 + 22)/(432*(x**4 + 2*x**2 + 3)) + 13*log(x**2)/54 - 13*log(x**4 + 2*x
**2 + 3)/108 + 125*sqrt(2)*atan(sqrt(2)*(x**2/2 + 1/2))/432 + 14/(27*x**2) - 5/(
6*x**2*(x**4 + 2*x**2 + 3)) - 1/(9*x**4)

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Mathematica [C]  time = 0.10529, size = 105, normalized size = 1.31 \[ \frac{1}{864} \left (-\frac{96}{x^4}+\frac{208}{x^2}-\sqrt{2} \left (52 \sqrt{2}+125 i\right ) \log \left (x^2-i \sqrt{2}+1\right )+\sqrt{2} \left (-52 \sqrt{2}+125 i\right ) \log \left (x^2+i \sqrt{2}+1\right )+\frac{100 \left (5 x^2+7\right )}{x^4+2 x^2+3}+416 \log (x)\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-96/x^4 + 208/x^2 + (100*(7 + 5*x^2))/(3 + 2*x^2 + x^4) + 416*Log[x] - Sqrt[2]*
(125*I + 52*Sqrt[2])*Log[1 - I*Sqrt[2] + x^2] + Sqrt[2]*(125*I - 52*Sqrt[2])*Log
[1 + I*Sqrt[2] + x^2])/864

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Maple [A]  time = 0.02, size = 68, normalized size = 0.9 \[ -{\frac{1}{9\,{x}^{4}}}+{\frac{13}{54\,{x}^{2}}}+{\frac{13\,\ln \left ( x \right ) }{27}}-{\frac{1}{54\,{x}^{4}+108\,{x}^{2}+162} \left ( -{\frac{125\,{x}^{2}}{4}}-{\frac{175}{4}} \right ) }-{\frac{13\,\ln \left ({x}^{4}+2\,{x}^{2}+3 \right ) }{108}}+{\frac{125\,\sqrt{2}}{432}\arctan \left ({\frac{ \left ( 2\,{x}^{2}+2 \right ) \sqrt{2}}{4}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9/x^4+13/54/x^2+13/27*ln(x)-1/54*(-125/4*x^2-175/4)/(x^4+2*x^2+3)-13/108*ln(x
^4+2*x^2+3)+125/432*2^(1/2)*arctan(1/4*(2*x^2+2)*2^(1/2))

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Maxima [A]  time = 0.804584, size = 96, normalized size = 1.2 \[ \frac{125}{432} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) + \frac{59 \, x^{6} + 85 \, x^{4} + 36 \, x^{2} - 24}{72 \,{\left (x^{8} + 2 \, x^{6} + 3 \, x^{4}\right )}} - \frac{13}{108} \, \log \left (x^{4} + 2 \, x^{2} + 3\right ) + \frac{13}{54} \, \log \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

125/432*sqrt(2)*arctan(1/2*sqrt(2)*(x^2 + 1)) + 1/72*(59*x^6 + 85*x^4 + 36*x^2 -
24)/(x^8 + 2*x^6 + 3*x^4) - 13/108*log(x^4 + 2*x^2 + 3) + 13/54*log(x^2)

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Fricas [A]  time = 0.288041, size = 165, normalized size = 2.06 \[ -\frac{\sqrt{2}{\left (26 \, \sqrt{2}{\left (x^{8} + 2 \, x^{6} + 3 \, x^{4}\right )} \log \left (x^{4} + 2 \, x^{2} + 3\right ) - 104 \, \sqrt{2}{\left (x^{8} + 2 \, x^{6} + 3 \, x^{4}\right )} \log \left (x\right ) - 125 \,{\left (x^{8} + 2 \, x^{6} + 3 \, x^{4}\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) - 3 \, \sqrt{2}{\left (59 \, x^{6} + 85 \, x^{4} + 36 \, x^{2} - 24\right )}\right )}}{432 \,{\left (x^{8} + 2 \, x^{6} + 3 \, x^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/432*sqrt(2)*(26*sqrt(2)*(x^8 + 2*x^6 + 3*x^4)*log(x^4 + 2*x^2 + 3) - 104*sqrt
(2)*(x^8 + 2*x^6 + 3*x^4)*log(x) - 125*(x^8 + 2*x^6 + 3*x^4)*arctan(1/2*sqrt(2)*
(x^2 + 1)) - 3*sqrt(2)*(59*x^6 + 85*x^4 + 36*x^2 - 24))/(x^8 + 2*x^6 + 3*x^4)

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Sympy [A]  time = 0.641964, size = 80, normalized size = 1. \[ \frac{13 \log{\left (x \right )}}{27} - \frac{13 \log{\left (x^{4} + 2 x^{2} + 3 \right )}}{108} + \frac{125 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x^{2}}{2} + \frac{\sqrt{2}}{2} \right )}}{432} + \frac{59 x^{6} + 85 x^{4} + 36 x^{2} - 24}{72 x^{8} + 144 x^{6} + 216 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

13*log(x)/27 - 13*log(x**4 + 2*x**2 + 3)/108 + 125*sqrt(2)*atan(sqrt(2)*x**2/2 +
sqrt(2)/2)/432 + (59*x**6 + 85*x**4 + 36*x**2 - 24)/(72*x**8 + 144*x**6 + 216*x
**4)

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GIAC/XCAS [A]  time = 0.274018, size = 107, normalized size = 1.34 \[ \frac{125}{432} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) + \frac{26 \, x^{4} + 177 \, x^{2} + 253}{216 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} - \frac{39 \, x^{4} - 26 \, x^{2} + 12}{108 \, x^{4}} - \frac{13}{108} \,{\rm ln}\left (x^{4} + 2 \, x^{2} + 3\right ) + \frac{13}{54} \,{\rm ln}\left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

125/432*sqrt(2)*arctan(1/2*sqrt(2)*(x^2 + 1)) + 1/216*(26*x^4 + 177*x^2 + 253)/(
x^4 + 2*x^2 + 3) - 1/108*(39*x^4 - 26*x^2 + 12)/x^4 - 13/108*ln(x^4 + 2*x^2 + 3)
+ 13/54*ln(x^2)