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

Optimal. Leaf size=49 \[ \frac{5 x^2}{2}-\frac{7}{2} \log \left (x^2+1\right )-10 \log \left (x^2+2\right )-\frac{51 x^2+50}{2 \left (x^4+3 x^2+2\right )} \]

[Out]

(5*x^2)/2 - (50 + 51*x^2)/(2*(2 + 3*x^2 + x^4)) - (7*Log[1 + x^2])/2 - 10*Log[2
+ x^2]

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Rubi [A]  time = 0.144343, antiderivative size = 49, 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 \[ \frac{5 x^2}{2}-\frac{7}{2} \log \left (x^2+1\right )-10 \log \left (x^2+2\right )-\frac{51 x^2+50}{2 \left (x^4+3 x^2+2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(5*x^2)/2 - (50 + 51*x^2)/(2*(2 + 3*x^2 + x^4)) - (7*Log[1 + x^2])/2 - 10*Log[2
+ x^2]

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Rubi in Sympy [A]  time = 22.557, size = 48, normalized size = 0.98 \[ \frac{5 x^{6}}{2 \left (x^{4} + 3 x^{2} + 2\right )} - \frac{7 \log{\left (x^{2} + 1 \right )}}{2} - 10 \log{\left (x^{2} + 2 \right )} - \frac{46}{x^{2} + 2} + \frac{3}{x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*x**6/(2*(x**4 + 3*x**2 + 2)) - 7*log(x**2 + 1)/2 - 10*log(x**2 + 2) - 46/(x**2
+ 2) + 3/(x**2 + 1)

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Mathematica [A]  time = 0.0404558, size = 49, normalized size = 1. \[ \frac{5 x^2}{2}-\frac{7}{2} \log \left (x^2+1\right )-10 \log \left (x^2+2\right )+\frac{-51 x^2-50}{2 \left (x^4+3 x^2+2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(5*x^2)/2 + (-50 - 51*x^2)/(2*(2 + 3*x^2 + x^4)) - (7*Log[1 + x^2])/2 - 10*Log[2
+ x^2]

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Maple [A]  time = 0.025, size = 41, normalized size = 0.8 \[{\frac{5\,{x}^{2}}{2}}-10\,\ln \left ({x}^{2}+2 \right ) -26\, \left ({x}^{2}+2 \right ) ^{-1}+{\frac{1}{2\,{x}^{2}+2}}-{\frac{7\,\ln \left ({x}^{2}+1 \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/2*x^2-10*ln(x^2+2)-26/(x^2+2)+1/2/(x^2+1)-7/2*ln(x^2+1)

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Maxima [A]  time = 0.738623, size = 58, normalized size = 1.18 \[ \frac{5}{2} \, x^{2} - \frac{51 \, x^{2} + 50}{2 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} - 10 \, \log \left (x^{2} + 2\right ) - \frac{7}{2} \, \log \left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/2*x^2 - 1/2*(51*x^2 + 50)/(x^4 + 3*x^2 + 2) - 10*log(x^2 + 2) - 7/2*log(x^2 +
1)

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Fricas [A]  time = 0.282631, size = 90, normalized size = 1.84 \[ \frac{5 \, x^{6} + 15 \, x^{4} - 41 \, x^{2} - 20 \,{\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 2\right ) - 7 \,{\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 1\right ) - 50}{2 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(5*x^6 + 15*x^4 - 41*x^2 - 20*(x^4 + 3*x^2 + 2)*log(x^2 + 2) - 7*(x^4 + 3*x^
2 + 2)*log(x^2 + 1) - 50)/(x^4 + 3*x^2 + 2)

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Sympy [A]  time = 0.413091, size = 42, normalized size = 0.86 \[ \frac{5 x^{2}}{2} - \frac{51 x^{2} + 50}{2 x^{4} + 6 x^{2} + 4} - \frac{7 \log{\left (x^{2} + 1 \right )}}{2} - 10 \log{\left (x^{2} + 2 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*x**2/2 - (51*x**2 + 50)/(2*x**4 + 6*x**2 + 4) - 7*log(x**2 + 1)/2 - 10*log(x**
2 + 2)

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GIAC/XCAS [A]  time = 0.273641, size = 61, normalized size = 1.24 \[ \frac{5}{2} \, x^{2} - \frac{51 \, x^{2} + 50}{2 \,{\left (x^{2} + 2\right )}{\left (x^{2} + 1\right )}} - 10 \,{\rm ln}\left (x^{2} + 2\right ) - \frac{7}{2} \,{\rm ln}\left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/2*x^2 - 1/2*(51*x^2 + 50)/((x^2 + 2)*(x^2 + 1)) - 10*ln(x^2 + 2) - 7/2*ln(x^2
+ 1)