3.74 \(\int \frac{x^9 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^2} \, dx\)

Optimal. Leaf size=68 \[ \frac{5 x^8}{8}-\frac{9 x^6}{2}+\frac{49 x^4}{2}-\frac{293 x^2}{2}+2 \log \left (x^2+1\right )+392 \log \left (x^2+2\right )+\frac{415 x^2+414}{2 \left (x^4+3 x^2+2\right )} \]

[Out]

(-293*x^2)/2 + (49*x^4)/2 - (9*x^6)/2 + (5*x^8)/8 + (414 + 415*x^2)/(2*(2 + 3*x^
2 + x^4)) + 2*Log[1 + x^2] + 392*Log[2 + x^2]

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Rubi [A]  time = 0.198708, antiderivative size = 68, 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 \[ \frac{5 x^8}{8}-\frac{9 x^6}{2}+\frac{49 x^4}{2}-\frac{293 x^2}{2}+2 \log \left (x^2+1\right )+392 \log \left (x^2+2\right )+\frac{415 x^2+414}{2 \left (x^4+3 x^2+2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(x^9*(4 + x^2 + 3*x^4 + 5*x^6))/(2 + 3*x^2 + x^4)^2,x]

[Out]

(-293*x^2)/2 + (49*x^4)/2 - (9*x^6)/2 + (5*x^8)/8 + (414 + 415*x^2)/(2*(2 + 3*x^
2 + x^4)) + 2*Log[1 + x^2] + 392*Log[2 + x^2]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{5 x^{12}}{8 \left (x^{4} + 3 x^{2} + 2\right )} - \frac{21 x^{6}}{8} + 2 \log{\left (x^{2} + 1 \right )} + 392 \log{\left (x^{2} + 2 \right )} + \frac{\int ^{x^{2}} \left (- \frac{1097}{4}\right )\, dx}{2} + \frac{161 \int ^{x^{2}} x\, dx}{4} + \frac{248}{x^{2} + 2} - \frac{9}{8 \left (x^{2} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9*(5*x**6+3*x**4+x**2+4)/(x**4+3*x**2+2)**2,x)

[Out]

5*x**12/(8*(x**4 + 3*x**2 + 2)) - 21*x**6/8 + 2*log(x**2 + 1) + 392*log(x**2 + 2
) + Integral(-1097/4, (x, x**2))/2 + 161*Integral(x, (x, x**2))/4 + 248/(x**2 +
2) - 9/(8*(x**2 + 1))

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Mathematica [A]  time = 0.0478874, size = 62, normalized size = 0.91 \[ \frac{1}{8} \left (5 x^8-36 x^6+196 x^4-1172 x^2+16 \log \left (x^2+1\right )+3136 \log \left (x^2+2\right )+\frac{4 \left (415 x^2+414\right )}{x^4+3 x^2+2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(x^9*(4 + x^2 + 3*x^4 + 5*x^6))/(2 + 3*x^2 + x^4)^2,x]

[Out]

(-1172*x^2 + 196*x^4 - 36*x^6 + 5*x^8 + (4*(414 + 415*x^2))/(2 + 3*x^2 + x^4) +
16*Log[1 + x^2] + 3136*Log[2 + x^2])/8

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Maple [A]  time = 0.024, size = 56, normalized size = 0.8 \[{\frac{5\,{x}^{8}}{8}}-{\frac{9\,{x}^{6}}{2}}+{\frac{49\,{x}^{4}}{2}}-{\frac{293\,{x}^{2}}{2}}+392\,\ln \left ({x}^{2}+2 \right ) +208\, \left ({x}^{2}+2 \right ) ^{-1}-{\frac{1}{2\,{x}^{2}+2}}+2\,\ln \left ({x}^{2}+1 \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9*(5*x^6+3*x^4+x^2+4)/(x^4+3*x^2+2)^2,x)

[Out]

5/8*x^8-9/2*x^6+49/2*x^4-293/2*x^2+392*ln(x^2+2)+208/(x^2+2)-1/2/(x^2+1)+2*ln(x^
2+1)

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Maxima [A]  time = 0.731391, size = 78, normalized size = 1.15 \[ \frac{5}{8} \, x^{8} - \frac{9}{2} \, x^{6} + \frac{49}{2} \, x^{4} - \frac{293}{2} \, x^{2} + \frac{415 \, x^{2} + 414}{2 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} + 392 \, \log \left (x^{2} + 2\right ) + 2 \, \log \left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^6 + 3*x^4 + x^2 + 4)*x^9/(x^4 + 3*x^2 + 2)^2,x, algorithm="maxima")

[Out]

5/8*x^8 - 9/2*x^6 + 49/2*x^4 - 293/2*x^2 + 1/2*(415*x^2 + 414)/(x^4 + 3*x^2 + 2)
+ 392*log(x^2 + 2) + 2*log(x^2 + 1)

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Fricas [A]  time = 0.259687, size = 111, normalized size = 1.63 \[ \frac{5 \, x^{12} - 21 \, x^{10} + 98 \, x^{8} - 656 \, x^{6} - 3124 \, x^{4} - 684 \, x^{2} + 3136 \,{\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 2\right ) + 16 \,{\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 1\right ) + 1656}{8 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^6 + 3*x^4 + x^2 + 4)*x^9/(x^4 + 3*x^2 + 2)^2,x, algorithm="fricas")

[Out]

1/8*(5*x^12 - 21*x^10 + 98*x^8 - 656*x^6 - 3124*x^4 - 684*x^2 + 3136*(x^4 + 3*x^
2 + 2)*log(x^2 + 2) + 16*(x^4 + 3*x^2 + 2)*log(x^2 + 1) + 1656)/(x^4 + 3*x^2 + 2
)

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Sympy [A]  time = 0.411799, size = 61, normalized size = 0.9 \[ \frac{5 x^{8}}{8} - \frac{9 x^{6}}{2} + \frac{49 x^{4}}{2} - \frac{293 x^{2}}{2} + \frac{415 x^{2} + 414}{2 x^{4} + 6 x^{2} + 4} + 2 \log{\left (x^{2} + 1 \right )} + 392 \log{\left (x^{2} + 2 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**9*(5*x**6+3*x**4+x**2+4)/(x**4+3*x**2+2)**2,x)

[Out]

5*x**8/8 - 9*x**6/2 + 49*x**4/2 - 293*x**2/2 + (415*x**2 + 414)/(2*x**4 + 6*x**2
+ 4) + 2*log(x**2 + 1) + 392*log(x**2 + 2)

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GIAC/XCAS [A]  time = 0.285976, size = 85, normalized size = 1.25 \[ \frac{5}{8} \, x^{8} - \frac{9}{2} \, x^{6} + \frac{49}{2} \, x^{4} - \frac{293}{2} \, x^{2} - \frac{394 \, x^{4} + 767 \, x^{2} + 374}{2 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} + 392 \,{\rm ln}\left (x^{2} + 2\right ) + 2 \,{\rm ln}\left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^6 + 3*x^4 + x^2 + 4)*x^9/(x^4 + 3*x^2 + 2)^2,x, algorithm="giac")

[Out]

5/8*x^8 - 9/2*x^6 + 49/2*x^4 - 293/2*x^2 - 1/2*(394*x^4 + 767*x^2 + 374)/(x^4 +
3*x^2 + 2) + 392*ln(x^2 + 2) + 2*ln(x^2 + 1)