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

Optimal. Leaf size=54 \[ \frac{5 x^4}{4}-\frac{27 x^2}{2}+3 \log \left (x^2+1\right )+46 \log \left (x^2+2\right )+\frac{103 x^2+102}{2 \left (x^4+3 x^2+2\right )} \]

[Out]

(-27*x^2)/2 + (5*x^4)/4 + (102 + 103*x^2)/(2*(2 + 3*x^2 + x^4)) + 3*Log[1 + x^2]
+ 46*Log[2 + x^2]

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Rubi [A]  time = 0.173221, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161 \[ \frac{5 x^4}{4}-\frac{27 x^2}{2}+3 \log \left (x^2+1\right )+46 \log \left (x^2+2\right )+\frac{103 x^2+102}{2 \left (x^4+3 x^2+2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(-27*x^2)/2 + (5*x^4)/4 + (102 + 103*x^2)/(2*(2 + 3*x^2 + x^4)) + 3*Log[1 + x^2]
+ 46*Log[2 + x^2]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{5 x^{8}}{4 \left (x^{4} + 3 x^{2} + 2\right )} + 3 \log{\left (x^{2} + 1 \right )} + 46 \log{\left (x^{2} + 2 \right )} + \frac{\int ^{x^{2}} \left (- \frac{39}{2}\right )\, dx}{2} + \frac{72}{x^{2} + 2} - \frac{7}{4 \left (x^{2} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*x**8/(4*(x**4 + 3*x**2 + 2)) + 3*log(x**2 + 1) + 46*log(x**2 + 2) + Integral(-
39/2, (x, x**2))/2 + 72/(x**2 + 2) - 7/(4*(x**2 + 1))

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Mathematica [A]  time = 0.0428921, size = 54, normalized size = 1. \[ \frac{5 x^4}{4}-\frac{27 x^2}{2}+3 \log \left (x^2+1\right )+46 \log \left (x^2+2\right )+\frac{103 x^2+102}{2 \left (x^4+3 x^2+2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(-27*x^2)/2 + (5*x^4)/4 + (102 + 103*x^2)/(2*(2 + 3*x^2 + x^4)) + 3*Log[1 + x^2]
+ 46*Log[2 + x^2]

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Maple [A]  time = 0.022, size = 46, normalized size = 0.9 \[{\frac{5\,{x}^{4}}{4}}-{\frac{27\,{x}^{2}}{2}}+46\,\ln \left ({x}^{2}+2 \right ) +52\, \left ({x}^{2}+2 \right ) ^{-1}-{\frac{1}{2\,{x}^{2}+2}}+3\,\ln \left ({x}^{2}+1 \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/4*x^4-27/2*x^2+46*ln(x^2+2)+52/(x^2+2)-1/2/(x^2+1)+3*ln(x^2+1)

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Maxima [A]  time = 0.722565, size = 65, normalized size = 1.2 \[ \frac{5}{4} \, x^{4} - \frac{27}{2} \, x^{2} + \frac{103 \, x^{2} + 102}{2 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} + 46 \, \log \left (x^{2} + 2\right ) + 3 \, \log \left (x^{2} + 1\right ) \]

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 + 3*x^2 + 2)^2,x, algorithm="maxima")

[Out]

5/4*x^4 - 27/2*x^2 + 1/2*(103*x^2 + 102)/(x^4 + 3*x^2 + 2) + 46*log(x^2 + 2) + 3
*log(x^2 + 1)

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Fricas [A]  time = 0.255956, size = 97, normalized size = 1.8 \[ \frac{5 \, x^{8} - 39 \, x^{6} - 152 \, x^{4} + 98 \, x^{2} + 184 \,{\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 2\right ) + 12 \,{\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 1\right ) + 204}{4 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} \]

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 + 3*x^2 + 2)^2,x, algorithm="fricas")

[Out]

1/4*(5*x^8 - 39*x^6 - 152*x^4 + 98*x^2 + 184*(x^4 + 3*x^2 + 2)*log(x^2 + 2) + 12
*(x^4 + 3*x^2 + 2)*log(x^2 + 1) + 204)/(x^4 + 3*x^2 + 2)

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Sympy [A]  time = 0.407439, size = 48, normalized size = 0.89 \[ \frac{5 x^{4}}{4} - \frac{27 x^{2}}{2} + \frac{103 x^{2} + 102}{2 x^{4} + 6 x^{2} + 4} + 3 \log{\left (x^{2} + 1 \right )} + 46 \log{\left (x^{2} + 2 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*x**4/4 - 27*x**2/2 + (103*x**2 + 102)/(2*x**4 + 6*x**2 + 4) + 3*log(x**2 + 1)
+ 46*log(x**2 + 2)

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GIAC/XCAS [A]  time = 0.271441, size = 72, normalized size = 1.33 \[ \frac{5}{4} \, x^{4} - \frac{27}{2} \, x^{2} - \frac{49 \, x^{4} + 44 \, x^{2} - 4}{2 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} + 46 \,{\rm ln}\left (x^{2} + 2\right ) + 3 \,{\rm ln}\left (x^{2} + 1\right ) \]

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 + 3*x^2 + 2)^2,x, algorithm="giac")

[Out]

5/4*x^4 - 27/2*x^2 - 1/2*(49*x^4 + 44*x^2 - 4)/(x^4 + 3*x^2 + 2) + 46*ln(x^2 + 2
) + 3*ln(x^2 + 1)