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

Optimal. Leaf size=65 \[ \frac{5 x^2}{2}-\frac{17 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{8 \sqrt{2}}-\frac{25 \left (x^2+3\right )}{8 \left (x^4+2 x^2+3\right )}-\frac{17}{4} \log \left (x^4+2 x^2+3\right ) \]

[Out]

(5*x^2)/2 - (25*(3 + x^2))/(8*(3 + 2*x^2 + x^4)) - (17*ArcTan[(1 + x^2)/Sqrt[2]]
)/(8*Sqrt[2]) - (17*Log[3 + 2*x^2 + x^4])/4

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Rubi [A]  time = 0.179814, antiderivative size = 65, 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 \[ \frac{5 x^2}{2}-\frac{17 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{8 \sqrt{2}}-\frac{25 \left (x^2+3\right )}{8 \left (x^4+2 x^2+3\right )}-\frac{17}{4} \log \left (x^4+2 x^2+3\right ) \]

Antiderivative was successfully verified.

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

[Out]

(5*x^2)/2 - (25*(3 + x^2))/(8*(3 + 2*x^2 + x^4)) - (17*ArcTan[(1 + x^2)/Sqrt[2]]
)/(8*Sqrt[2]) - (17*Log[3 + 2*x^2 + x^4])/4

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Rubi in Sympy [A]  time = 26.3388, size = 73, normalized size = 1.12 \[ \frac{5 x^{6}}{2 \left (x^{4} + 2 x^{2} + 3\right )} - \frac{15 \left (6 x^{2} + 26\right )}{16 \left (x^{4} + 2 x^{2} + 3\right )} - \frac{17 \log{\left (x^{4} + 2 x^{2} + 3 \right )}}{4} - \frac{17 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \left (\frac{x^{2}}{2} + \frac{1}{2}\right ) \right )}}{16} \]

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+2*x**2+3)**2,x)

[Out]

5*x**6/(2*(x**4 + 2*x**2 + 3)) - 15*(6*x**2 + 26)/(16*(x**4 + 2*x**2 + 3)) - 17*
log(x**4 + 2*x**2 + 3)/4 - 17*sqrt(2)*atan(sqrt(2)*(x**2/2 + 1/2))/16

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Mathematica [A]  time = 0.0466926, size = 61, normalized size = 0.94 \[ \frac{1}{16} \left (40 x^2-17 \sqrt{2} \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )-\frac{50 \left (x^2+3\right )}{x^4+2 x^2+3}-68 \log \left (x^4+2 x^2+3\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(40*x^2 - (50*(3 + x^2))/(3 + 2*x^2 + x^4) - 17*Sqrt[2]*ArcTan[(1 + x^2)/Sqrt[2]
] - 68*Log[3 + 2*x^2 + x^4])/16

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Maple [A]  time = 0.012, size = 59, normalized size = 0.9 \[{\frac{5\,{x}^{2}}{2}}-{\frac{1}{2\,{x}^{4}+4\,{x}^{2}+6} \left ({\frac{25\,{x}^{2}}{4}}+{\frac{75}{4}} \right ) }-{\frac{17\,\ln \left ({x}^{4}+2\,{x}^{2}+3 \right ) }{4}}-{\frac{17\,\sqrt{2}}{16}\arctan \left ({\frac{ \left ( 2\,{x}^{2}+2 \right ) \sqrt{2}}{4}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/2*x^2-1/2*(25/4*x^2+75/4)/(x^4+2*x^2+3)-17/4*ln(x^4+2*x^2+3)-17/16*2^(1/2)*arc
tan(1/4*(2*x^2+2)*2^(1/2))

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Maxima [A]  time = 0.793378, size = 73, normalized size = 1.12 \[ \frac{5}{2} \, x^{2} - \frac{17}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) - \frac{25 \,{\left (x^{2} + 3\right )}}{8 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} - \frac{17}{4} \, \log \left (x^{4} + 2 \, x^{2} + 3\right ) \]

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

[Out]

5/2*x^2 - 17/16*sqrt(2)*arctan(1/2*sqrt(2)*(x^2 + 1)) - 25/8*(x^2 + 3)/(x^4 + 2*
x^2 + 3) - 17/4*log(x^4 + 2*x^2 + 3)

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Fricas [A]  time = 0.271286, size = 120, normalized size = 1.85 \[ -\frac{\sqrt{2}{\left (34 \, \sqrt{2}{\left (x^{4} + 2 \, x^{2} + 3\right )} \log \left (x^{4} + 2 \, x^{2} + 3\right ) + 17 \,{\left (x^{4} + 2 \, x^{2} + 3\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) - 5 \, \sqrt{2}{\left (4 \, x^{6} + 8 \, x^{4} + 7 \, x^{2} - 15\right )}\right )}}{16 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} \]

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

[Out]

-1/16*sqrt(2)*(34*sqrt(2)*(x^4 + 2*x^2 + 3)*log(x^4 + 2*x^2 + 3) + 17*(x^4 + 2*x
^2 + 3)*arctan(1/2*sqrt(2)*(x^2 + 1)) - 5*sqrt(2)*(4*x^6 + 8*x^4 + 7*x^2 - 15))/
(x^4 + 2*x^2 + 3)

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Sympy [A]  time = 0.45492, size = 66, normalized size = 1.02 \[ \frac{5 x^{2}}{2} - \frac{25 x^{2} + 75}{8 x^{4} + 16 x^{2} + 24} - \frac{17 \log{\left (x^{4} + 2 x^{2} + 3 \right )}}{4} - \frac{17 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x^{2}}{2} + \frac{\sqrt{2}}{2} \right )}}{16} \]

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+2*x**2+3)**2,x)

[Out]

5*x**2/2 - (25*x**2 + 75)/(8*x**4 + 16*x**2 + 24) - 17*log(x**4 + 2*x**2 + 3)/4
- 17*sqrt(2)*atan(sqrt(2)*x**2/2 + sqrt(2)/2)/16

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GIAC/XCAS [A]  time = 0.274086, size = 73, normalized size = 1.12 \[ \frac{5}{2} \, x^{2} - \frac{17}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) - \frac{25 \,{\left (x^{2} + 3\right )}}{8 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} - \frac{17}{4} \,{\rm ln}\left (x^{4} + 2 \, x^{2} + 3\right ) \]

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 + 2*x^2 + 3)^2,x, algorithm="giac")

[Out]

5/2*x^2 - 17/16*sqrt(2)*arctan(1/2*sqrt(2)*(x^2 + 1)) - 25/8*(x^2 + 3)/(x^4 + 2*
x^2 + 3) - 17/4*ln(x^4 + 2*x^2 + 3)