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

Optimal. Leaf size=86 \[ \frac{5 x^8}{8}-\frac{17 x^6}{6}+\frac{19 x^4}{4}+19 x^2+\frac{201 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{8 \sqrt{2}}-\frac{25 \left (7 x^2+15\right )}{8 \left (x^4+2 x^2+3\right )}-\frac{183}{4} \log \left (x^4+2 x^2+3\right ) \]

[Out]

19*x^2 + (19*x^4)/4 - (17*x^6)/6 + (5*x^8)/8 - (25*(15 + 7*x^2))/(8*(3 + 2*x^2 +
x^4)) + (201*ArcTan[(1 + x^2)/Sqrt[2]])/(8*Sqrt[2]) - (183*Log[3 + 2*x^2 + x^4]
)/4

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Rubi [A]  time = 0.239025, antiderivative size = 86, 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 \[ \frac{5 x^8}{8}-\frac{17 x^6}{6}+\frac{19 x^4}{4}+19 x^2+\frac{201 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{8 \sqrt{2}}-\frac{25 \left (7 x^2+15\right )}{8 \left (x^4+2 x^2+3\right )}-\frac{183}{4} \log \left (x^4+2 x^2+3\right ) \]

Antiderivative was successfully verified.

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

[Out]

19*x^2 + (19*x^4)/4 - (17*x^6)/6 + (5*x^8)/8 - (25*(15 + 7*x^2))/(8*(3 + 2*x^2 +
x^4)) + (201*ArcTan[(1 + x^2)/Sqrt[2]])/(8*Sqrt[2]) - (183*Log[3 + 2*x^2 + x^4]
)/4

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{5 x^{12}}{8 \left (x^{4} + 2 x^{2} + 3\right )} - \frac{19 x^{6}}{12} + \frac{33 x^{2}}{2} - \frac{45 \left (20 x^{2} + 48\right )}{32 \left (x^{4} + 2 x^{2} + 3\right )} - \frac{183 \log{\left (x^{4} + 2 x^{2} + 3 \right )}}{4} + \frac{201 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \left (\frac{x^{2}}{2} + \frac{1}{2}\right ) \right )}}{16} + \frac{33 \int ^{x^{2}} x\, dx}{4} \]

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

[Out]

5*x**12/(8*(x**4 + 2*x**2 + 3)) - 19*x**6/12 + 33*x**2/2 - 45*(20*x**2 + 48)/(32
*(x**4 + 2*x**2 + 3)) - 183*log(x**4 + 2*x**2 + 3)/4 + 201*sqrt(2)*atan(sqrt(2)*
(x**2/2 + 1/2))/16 + 33*Integral(x, (x, x**2))/4

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Mathematica [A]  time = 0.0793939, size = 78, normalized size = 0.91 \[ \frac{1}{48} \left (30 x^8-136 x^6+228 x^4+912 x^2+603 \sqrt{2} \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )-\frac{150 \left (7 x^2+15\right )}{x^4+2 x^2+3}-2196 \log \left (x^4+2 x^2+3\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(912*x^2 + 228*x^4 - 136*x^6 + 30*x^8 - (150*(15 + 7*x^2))/(3 + 2*x^2 + x^4) + 6
03*Sqrt[2]*ArcTan[(1 + x^2)/Sqrt[2]] - 2196*Log[3 + 2*x^2 + x^4])/48

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Maple [A]  time = 0.014, size = 74, normalized size = 0.9 \[{\frac{5\,{x}^{8}}{8}}-{\frac{17\,{x}^{6}}{6}}+{\frac{19\,{x}^{4}}{4}}+19\,{x}^{2}-{\frac{1}{2\,{x}^{4}+4\,{x}^{2}+6} \left ({\frac{175\,{x}^{2}}{4}}+{\frac{375}{4}} \right ) }-{\frac{183\,\ln \left ({x}^{4}+2\,{x}^{2}+3 \right ) }{4}}+{\frac{201\,\sqrt{2}}{16}\arctan \left ({\frac{ \left ( 2\,{x}^{2}+2 \right ) \sqrt{2}}{4}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/8*x^8-17/6*x^6+19/4*x^4+19*x^2-1/2*(175/4*x^2+375/4)/(x^4+2*x^2+3)-183/4*ln(x^
4+2*x^2+3)+201/16*2^(1/2)*arctan(1/4*(2*x^2+2)*2^(1/2))

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Maxima [A]  time = 0.790036, size = 96, normalized size = 1.12 \[ \frac{5}{8} \, x^{8} - \frac{17}{6} \, x^{6} + \frac{19}{4} \, x^{4} + 19 \, x^{2} + \frac{201}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) - \frac{25 \,{\left (7 \, x^{2} + 15\right )}}{8 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} - \frac{183}{4} \, \log \left (x^{4} + 2 \, x^{2} + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/8*x^8 - 17/6*x^6 + 19/4*x^4 + 19*x^2 + 201/16*sqrt(2)*arctan(1/2*sqrt(2)*(x^2
+ 1)) - 25/8*(7*x^2 + 15)/(x^4 + 2*x^2 + 3) - 183/4*log(x^4 + 2*x^2 + 3)

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Fricas [A]  time = 0.270653, size = 140, normalized size = 1.63 \[ -\frac{\sqrt{2}{\left (1098 \, \sqrt{2}{\left (x^{4} + 2 \, x^{2} + 3\right )} \log \left (x^{4} + 2 \, x^{2} + 3\right ) - 603 \,{\left (x^{4} + 2 \, x^{2} + 3\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) - \sqrt{2}{\left (15 \, x^{12} - 38 \, x^{10} + 23 \, x^{8} + 480 \, x^{6} + 1254 \, x^{4} + 843 \, x^{2} - 1125\right )}\right )}}{48 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/48*sqrt(2)*(1098*sqrt(2)*(x^4 + 2*x^2 + 3)*log(x^4 + 2*x^2 + 3) - 603*(x^4 +
2*x^2 + 3)*arctan(1/2*sqrt(2)*(x^2 + 1)) - sqrt(2)*(15*x^12 - 38*x^10 + 23*x^8 +
480*x^6 + 1254*x^4 + 843*x^2 - 1125))/(x^4 + 2*x^2 + 3)

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Sympy [A]  time = 0.452715, size = 85, normalized size = 0.99 \[ \frac{5 x^{8}}{8} - \frac{17 x^{6}}{6} + \frac{19 x^{4}}{4} + 19 x^{2} - \frac{175 x^{2} + 375}{8 x^{4} + 16 x^{2} + 24} - \frac{183 \log{\left (x^{4} + 2 x^{2} + 3 \right )}}{4} + \frac{201 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x^{2}}{2} + \frac{\sqrt{2}}{2} \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*x**8/8 - 17*x**6/6 + 19*x**4/4 + 19*x**2 - (175*x**2 + 375)/(8*x**4 + 16*x**2
+ 24) - 183*log(x**4 + 2*x**2 + 3)/4 + 201*sqrt(2)*atan(sqrt(2)*x**2/2 + sqrt(2)
/2)/16

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GIAC/XCAS [A]  time = 0.271471, size = 103, normalized size = 1.2 \[ \frac{5}{8} \, x^{8} - \frac{17}{6} \, x^{6} + \frac{19}{4} \, x^{4} + 19 \, x^{2} + \frac{201}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) + \frac{366 \, x^{4} + 557 \, x^{2} + 723}{8 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} - \frac{183}{4} \,{\rm ln}\left (x^{4} + 2 \, x^{2} + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/8*x^8 - 17/6*x^6 + 19/4*x^4 + 19*x^2 + 201/16*sqrt(2)*arctan(1/2*sqrt(2)*(x^2
+ 1)) + 1/8*(366*x^4 + 557*x^2 + 723)/(x^4 + 2*x^2 + 3) - 183/4*ln(x^4 + 2*x^2 +
3)