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

Optimal. Leaf size=98 \[ -\frac{89}{48} \sqrt{x^4+5 x^2+3} x^4-\frac{1}{384} \left (24243-3802 x^2\right ) \sqrt{x^4+5 x^2+3}+\frac{32801}{256} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )+\frac{3}{8} \sqrt{x^4+5 x^2+3} x^6 \]

[Out]

(-89*x^4*Sqrt[3 + 5*x^2 + x^4])/48 + (3*x^6*Sqrt[3 + 5*x^2 + x^4])/8 - ((24243 -
3802*x^2)*Sqrt[3 + 5*x^2 + x^4])/384 + (32801*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5
*x^2 + x^4])])/256

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Rubi [A]  time = 0.247569, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{89}{48} \sqrt{x^4+5 x^2+3} x^4-\frac{1}{384} \left (24243-3802 x^2\right ) \sqrt{x^4+5 x^2+3}+\frac{32801}{256} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )+\frac{3}{8} \sqrt{x^4+5 x^2+3} x^6 \]

Antiderivative was successfully verified.

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

[Out]

(-89*x^4*Sqrt[3 + 5*x^2 + x^4])/48 + (3*x^6*Sqrt[3 + 5*x^2 + x^4])/8 - ((24243 -
3802*x^2)*Sqrt[3 + 5*x^2 + x^4])/384 + (32801*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5
*x^2 + x^4])])/256

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Rubi in Sympy [A]  time = 26.0606, size = 92, normalized size = 0.94 \[ \frac{3 x^{6} \sqrt{x^{4} + 5 x^{2} + 3}}{8} - \frac{89 x^{4} \sqrt{x^{4} + 5 x^{2} + 3}}{48} - \frac{\left (- \frac{1901 x^{2}}{4} + \frac{24243}{8}\right ) \sqrt{x^{4} + 5 x^{2} + 3}}{48} + \frac{32801 \operatorname{atanh}{\left (\frac{2 x^{2} + 5}{2 \sqrt{x^{4} + 5 x^{2} + 3}} \right )}}{256} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7*(3*x**2+2)/(x**4+5*x**2+3)**(1/2),x)

[Out]

3*x**6*sqrt(x**4 + 5*x**2 + 3)/8 - 89*x**4*sqrt(x**4 + 5*x**2 + 3)/48 - (-1901*x
**2/4 + 24243/8)*sqrt(x**4 + 5*x**2 + 3)/48 + 32801*atanh((2*x**2 + 5)/(2*sqrt(x
**4 + 5*x**2 + 3)))/256

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Mathematica [A]  time = 0.0527585, size = 64, normalized size = 0.65 \[ \frac{1}{768} \left (98403 \log \left (2 x^2+2 \sqrt{x^4+5 x^2+3}+5\right )+2 \sqrt{x^4+5 x^2+3} \left (144 x^6-712 x^4+3802 x^2-24243\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[3 + 5*x^2 + x^4]*(-24243 + 3802*x^2 - 712*x^4 + 144*x^6) + 98403*Log[5 +
2*x^2 + 2*Sqrt[3 + 5*x^2 + x^4]])/768

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Maple [A]  time = 0.027, size = 87, normalized size = 0.9 \[ -{\frac{89\,{x}^{4}}{48}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{1901\,{x}^{2}}{192}\sqrt{{x}^{4}+5\,{x}^{2}+3}}-{\frac{8081}{128}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{32801}{256}\ln \left ({x}^{2}+{\frac{5}{2}}+\sqrt{{x}^{4}+5\,{x}^{2}+3} \right ) }+{\frac{3\,{x}^{6}}{8}\sqrt{{x}^{4}+5\,{x}^{2}+3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7*(3*x^2+2)/(x^4+5*x^2+3)^(1/2),x)

[Out]

-89/48*x^4*(x^4+5*x^2+3)^(1/2)+1901/192*x^2*(x^4+5*x^2+3)^(1/2)-8081/128*(x^4+5*
x^2+3)^(1/2)+32801/256*ln(x^2+5/2+(x^4+5*x^2+3)^(1/2))+3/8*x^6*(x^4+5*x^2+3)^(1/
2)

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Maxima [A]  time = 0.706507, size = 122, normalized size = 1.24 \[ \frac{3}{8} \, \sqrt{x^{4} + 5 \, x^{2} + 3} x^{6} - \frac{89}{48} \, \sqrt{x^{4} + 5 \, x^{2} + 3} x^{4} + \frac{1901}{192} \, \sqrt{x^{4} + 5 \, x^{2} + 3} x^{2} - \frac{8081}{128} \, \sqrt{x^{4} + 5 \, x^{2} + 3} + \frac{32801}{256} \, \log \left (2 \, x^{2} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} + 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/8*sqrt(x^4 + 5*x^2 + 3)*x^6 - 89/48*sqrt(x^4 + 5*x^2 + 3)*x^4 + 1901/192*sqrt(
x^4 + 5*x^2 + 3)*x^2 - 8081/128*sqrt(x^4 + 5*x^2 + 3) + 32801/256*log(2*x^2 + 2*
sqrt(x^4 + 5*x^2 + 3) + 5)

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Fricas [A]  time = 0.27254, size = 306, normalized size = 3.12 \[ -\frac{294912 \, x^{16} + 2228224 \, x^{14} + 6553600 \, x^{12} - 1048576 \, x^{10} - 233473664 \, x^{8} - 1270350080 \, x^{6} - 2376291488 \, x^{4} - 1415183008 \, x^{2} + 787224 \,{\left (128 \, x^{8} + 1280 \, x^{6} + 4384 \, x^{4} + 5920 \, x^{2} - 8 \,{\left (16 \, x^{6} + 120 \, x^{4} + 274 \, x^{2} + 185\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} + 2569\right )} \log \left (-2 \, x^{2} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} - 5\right ) - 8 \,{\left (36864 \, x^{14} + 186368 \, x^{12} + 413184 \, x^{10} - 1010944 \, x^{8} - 26319472 \, x^{6} - 95478600 \, x^{4} - 94593374 \, x^{2} - 7815359\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} - 101042425}{6144 \,{\left (128 \, x^{8} + 1280 \, x^{6} + 4384 \, x^{4} + 5920 \, x^{2} - 8 \,{\left (16 \, x^{6} + 120 \, x^{4} + 274 \, x^{2} + 185\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} + 2569\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6144*(294912*x^16 + 2228224*x^14 + 6553600*x^12 - 1048576*x^10 - 233473664*x^
8 - 1270350080*x^6 - 2376291488*x^4 - 1415183008*x^2 + 787224*(128*x^8 + 1280*x^
6 + 4384*x^4 + 5920*x^2 - 8*(16*x^6 + 120*x^4 + 274*x^2 + 185)*sqrt(x^4 + 5*x^2
+ 3) + 2569)*log(-2*x^2 + 2*sqrt(x^4 + 5*x^2 + 3) - 5) - 8*(36864*x^14 + 186368*
x^12 + 413184*x^10 - 1010944*x^8 - 26319472*x^6 - 95478600*x^4 - 94593374*x^2 -
7815359)*sqrt(x^4 + 5*x^2 + 3) - 101042425)/(128*x^8 + 1280*x^6 + 4384*x^4 + 592
0*x^2 - 8*(16*x^6 + 120*x^4 + 274*x^2 + 185)*sqrt(x^4 + 5*x^2 + 3) + 2569)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{7} \left (3 x^{2} + 2\right )}{\sqrt{x^{4} + 5 x^{2} + 3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7*(3*x**2+2)/(x**4+5*x**2+3)**(1/2),x)

[Out]

Integral(x**7*(3*x**2 + 2)/sqrt(x**4 + 5*x**2 + 3), x)

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GIAC/XCAS [A]  time = 0.282435, size = 81, normalized size = 0.83 \[ \frac{1}{384} \, \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (2 \,{\left (4 \,{\left (18 \, x^{2} - 89\right )} x^{2} + 1901\right )} x^{2} - 24243\right )} - \frac{32801}{256} \,{\rm ln}\left (2 \, x^{2} - 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} + 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/384*sqrt(x^4 + 5*x^2 + 3)*(2*(4*(18*x^2 - 89)*x^2 + 1901)*x^2 - 24243) - 32801
/256*ln(2*x^2 - 2*sqrt(x^4 + 5*x^2 + 3) + 5)