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

Optimal. Leaf size=42 \[ 4 \log \left (x^2+1\right )-\frac{3}{2} \log \left (x^2+2\right )+\frac{25 x^2+24}{2 \left (x^4+3 x^2+2\right )} \]

[Out]

(24 + 25*x^2)/(2*(2 + 3*x^2 + x^4)) + 4*Log[1 + x^2] - (3*Log[2 + x^2])/2

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Rubi [A]  time = 0.084288, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ 4 \log \left (x^2+1\right )-\frac{3}{2} \log \left (x^2+2\right )+\frac{25 x^2+24}{2 \left (x^4+3 x^2+2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(24 + 25*x^2)/(2*(2 + 3*x^2 + x^4)) + 4*Log[1 + x^2] - (3*Log[2 + x^2])/2

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Rubi in Sympy [A]  time = 17.3863, size = 48, normalized size = 1.14 \[ \frac{0.25 \left (50 x^{2} + 48\right )}{x^{4} + 3 x^{2} + 2} + 2.75 \log{\left (x^{2} + 1 \right )} - 2.75 \log{\left (x^{2} + 2 \right )} + \frac{5 \log{\left (x^{4} + 3 x^{2} + 2 \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

0.25*(50*x**2 + 48)/(x**4 + 3*x**2 + 2) + 2.75*log(x**2 + 1) - 2.75*log(x**2 + 2
) + 5*log(x**4 + 3*x**2 + 2)/4

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Mathematica [A]  time = 0.0312979, size = 42, normalized size = 1. \[ 4 \log \left (x^2+1\right )-\frac{3}{2} \log \left (x^2+2\right )+\frac{25 x^2+24}{2 \left (x^4+3 x^2+2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(24 + 25*x^2)/(2*(2 + 3*x^2 + x^4)) + 4*Log[1 + x^2] - (3*Log[2 + x^2])/2

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Maple [A]  time = 0.022, size = 36, normalized size = 0.9 \[ -{\frac{3\,\ln \left ({x}^{2}+2 \right ) }{2}}+13\, \left ({x}^{2}+2 \right ) ^{-1}-{\frac{1}{2\,{x}^{2}+2}}+4\,\ln \left ({x}^{2}+1 \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/2*ln(x^2+2)+13/(x^2+2)-1/2/(x^2+1)+4*ln(x^2+1)

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Maxima [A]  time = 0.725112, size = 51, normalized size = 1.21 \[ \frac{25 \, x^{2} + 24}{2 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} - \frac{3}{2} \, \log \left (x^{2} + 2\right ) + 4 \, \log \left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(25*x^2 + 24)/(x^4 + 3*x^2 + 2) - 3/2*log(x^2 + 2) + 4*log(x^2 + 1)

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Fricas [A]  time = 0.252317, size = 77, normalized size = 1.83 \[ \frac{25 \, x^{2} - 3 \,{\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 2\right ) + 8 \,{\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 1\right ) + 24}{2 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(25*x^2 - 3*(x^4 + 3*x^2 + 2)*log(x^2 + 2) + 8*(x^4 + 3*x^2 + 2)*log(x^2 + 1
) + 24)/(x^4 + 3*x^2 + 2)

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Sympy [A]  time = 0.394701, size = 36, normalized size = 0.86 \[ \frac{25 x^{2} + 24}{2 x^{4} + 6 x^{2} + 4} + 4 \log{\left (x^{2} + 1 \right )} - \frac{3 \log{\left (x^{2} + 2 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(25*x**2 + 24)/(2*x**4 + 6*x**2 + 4) + 4*log(x**2 + 1) - 3*log(x**2 + 2)/2

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GIAC/XCAS [A]  time = 0.274234, size = 54, normalized size = 1.29 \[ \frac{25 \, x^{2} + 24}{2 \,{\left (x^{2} + 2\right )}{\left (x^{2} + 1\right )}} - \frac{3}{2} \,{\rm ln}\left (x^{2} + 2\right ) + 4 \,{\rm ln}\left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(25*x^2 + 24)/((x^2 + 2)*(x^2 + 1)) - 3/2*ln(x^2 + 2) + 4*ln(x^2 + 1)