3.101 \(\int \frac{x^7 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^2} \, dx\)

Optimal. Leaf size=81 \[ \frac{5 x^6}{6}-\frac{17 x^4}{4}+\frac{19 x^2}{2}-\frac{455 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{8 \sqrt{2}}+\frac{25 \left (5 x^2+3\right )}{8 \left (x^4+2 x^2+3\right )}+\frac{19}{2} \log \left (x^4+2 x^2+3\right ) \]

[Out]

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

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Rubi [A]  time = 0.218502, antiderivative size = 81, 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 \[ \frac{5 x^6}{6}-\frac{17 x^4}{4}+\frac{19 x^2}{2}-\frac{455 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{8 \sqrt{2}}+\frac{25 \left (5 x^2+3\right )}{8 \left (x^4+2 x^2+3\right )}+\frac{19}{2} \log \left (x^4+2 x^2+3\right ) \]

Antiderivative was successfully verified.

[In]  Int[(x^7*(4 + x^2 + 3*x^4 + 5*x^6))/(3 + 2*x^2 + x^4)^2,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{5 x^{10}}{6 \left (x^{4} + 2 x^{2} + 3\right )} + \frac{5 \left (238 x^{2} + 186\right )}{48 \left (x^{4} + 2 x^{2} + 3\right )} + \frac{19 \log{\left (x^{4} + 2 x^{2} + 3 \right )}}{2} - \frac{455 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \left (\frac{x^{2}}{2} + \frac{1}{2}\right ) \right )}}{16} + \frac{\int ^{x^{2}} \frac{52}{3}\, dx}{2} - \frac{31 \int ^{x^{2}} x\, dx}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7*(5*x**6+3*x**4+x**2+4)/(x**4+2*x**2+3)**2,x)

[Out]

5*x**10/(6*(x**4 + 2*x**2 + 3)) + 5*(238*x**2 + 186)/(48*(x**4 + 2*x**2 + 3)) +
19*log(x**4 + 2*x**2 + 3)/2 - 455*sqrt(2)*atan(sqrt(2)*(x**2/2 + 1/2))/16 + Inte
gral(52/3, (x, x**2))/2 - 31*Integral(x, (x, x**2))/6

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Mathematica [A]  time = 0.0544429, size = 73, normalized size = 0.9 \[ \frac{1}{48} \left (40 x^6-204 x^4+456 x^2-1365 \sqrt{2} \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )+\frac{150 \left (5 x^2+3\right )}{x^4+2 x^2+3}+456 \log \left (x^4+2 x^2+3\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(x^7*(4 + x^2 + 3*x^4 + 5*x^6))/(3 + 2*x^2 + x^4)^2,x]

[Out]

(456*x^2 - 204*x^4 + 40*x^6 + (150*(3 + 5*x^2))/(3 + 2*x^2 + x^4) - 1365*Sqrt[2]
*ArcTan[(1 + x^2)/Sqrt[2]] + 456*Log[3 + 2*x^2 + x^4])/48

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Maple [A]  time = 0.013, size = 69, normalized size = 0.9 \[{\frac{5\,{x}^{6}}{6}}-{\frac{17\,{x}^{4}}{4}}+{\frac{19\,{x}^{2}}{2}}+{\frac{1}{2\,{x}^{4}+4\,{x}^{2}+6} \left ({\frac{125\,{x}^{2}}{4}}+{\frac{75}{4}} \right ) }+{\frac{19\,\ln \left ({x}^{4}+2\,{x}^{2}+3 \right ) }{2}}-{\frac{455\,\sqrt{2}}{16}\arctan \left ({\frac{ \left ( 2\,{x}^{2}+2 \right ) \sqrt{2}}{4}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7*(5*x^6+3*x^4+x^2+4)/(x^4+2*x^2+3)^2,x)

[Out]

5/6*x^6-17/4*x^4+19/2*x^2+1/2*(125/4*x^2+75/4)/(x^4+2*x^2+3)+19/2*ln(x^4+2*x^2+3
)-455/16*2^(1/2)*arctan(1/4*(2*x^2+2)*2^(1/2))

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Maxima [A]  time = 0.791742, size = 89, normalized size = 1.1 \[ \frac{5}{6} \, x^{6} - \frac{17}{4} \, x^{4} + \frac{19}{2} \, x^{2} - \frac{455}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) + \frac{25 \,{\left (5 \, x^{2} + 3\right )}}{8 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} + \frac{19}{2} \, \log \left (x^{4} + 2 \, x^{2} + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^6 + 3*x^4 + x^2 + 4)*x^7/(x^4 + 2*x^2 + 3)^2,x, algorithm="maxima")

[Out]

5/6*x^6 - 17/4*x^4 + 19/2*x^2 - 455/16*sqrt(2)*arctan(1/2*sqrt(2)*(x^2 + 1)) + 2
5/8*(5*x^2 + 3)/(x^4 + 2*x^2 + 3) + 19/2*log(x^4 + 2*x^2 + 3)

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Fricas [A]  time = 0.276443, size = 132, normalized size = 1.63 \[ \frac{\sqrt{2}{\left (228 \, \sqrt{2}{\left (x^{4} + 2 \, x^{2} + 3\right )} \log \left (x^{4} + 2 \, x^{2} + 3\right ) - 1365 \,{\left (x^{4} + 2 \, x^{2} + 3\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) + \sqrt{2}{\left (20 \, x^{10} - 62 \, x^{8} + 84 \, x^{6} + 150 \, x^{4} + 1059 \, x^{2} + 225\right )}\right )}}{48 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^6 + 3*x^4 + x^2 + 4)*x^7/(x^4 + 2*x^2 + 3)^2,x, algorithm="fricas")

[Out]

1/48*sqrt(2)*(228*sqrt(2)*(x^4 + 2*x^2 + 3)*log(x^4 + 2*x^2 + 3) - 1365*(x^4 + 2
*x^2 + 3)*arctan(1/2*sqrt(2)*(x^2 + 1)) + sqrt(2)*(20*x^10 - 62*x^8 + 84*x^6 + 1
50*x^4 + 1059*x^2 + 225))/(x^4 + 2*x^2 + 3)

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Sympy [A]  time = 0.454654, size = 80, normalized size = 0.99 \[ \frac{5 x^{6}}{6} - \frac{17 x^{4}}{4} + \frac{19 x^{2}}{2} + \frac{125 x^{2} + 75}{8 x^{4} + 16 x^{2} + 24} + \frac{19 \log{\left (x^{4} + 2 x^{2} + 3 \right )}}{2} - \frac{455 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x^{2}}{2} + \frac{\sqrt{2}}{2} \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7*(5*x**6+3*x**4+x**2+4)/(x**4+2*x**2+3)**2,x)

[Out]

5*x**6/6 - 17*x**4/4 + 19*x**2/2 + (125*x**2 + 75)/(8*x**4 + 16*x**2 + 24) + 19*
log(x**4 + 2*x**2 + 3)/2 - 455*sqrt(2)*atan(sqrt(2)*x**2/2 + sqrt(2)/2)/16

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GIAC/XCAS [A]  time = 0.270333, size = 96, normalized size = 1.19 \[ \frac{5}{6} \, x^{6} - \frac{17}{4} \, x^{4} + \frac{19}{2} \, x^{2} - \frac{455}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) - \frac{76 \, x^{4} + 27 \, x^{2} + 153}{8 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} + \frac{19}{2} \,{\rm ln}\left (x^{4} + 2 \, x^{2} + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^6 + 3*x^4 + x^2 + 4)*x^7/(x^4 + 2*x^2 + 3)^2,x, algorithm="giac")

[Out]

5/6*x^6 - 17/4*x^4 + 19/2*x^2 - 455/16*sqrt(2)*arctan(1/2*sqrt(2)*(x^2 + 1)) - 1
/8*(76*x^4 + 27*x^2 + 153)/(x^4 + 2*x^2 + 3) + 19/2*ln(x^4 + 2*x^2 + 3)