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

Optimal. Leaf size=77 \[ \frac{1}{2} \sqrt{x^4+5 x^2+3} x^4+\frac{3}{16} \left (89-14 x^2\right ) \sqrt{x^4+5 x^2+3}-\frac{1083}{32} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right ) \]

[Out]

(x^4*Sqrt[3 + 5*x^2 + x^4])/2 + (3*(89 - 14*x^2)*Sqrt[3 + 5*x^2 + x^4])/16 - (10
83*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4])])/32

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Rubi [A]  time = 0.183952, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{1}{2} \sqrt{x^4+5 x^2+3} x^4+\frac{3}{16} \left (89-14 x^2\right ) \sqrt{x^4+5 x^2+3}-\frac{1083}{32} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x^4*Sqrt[3 + 5*x^2 + x^4])/2 + (3*(89 - 14*x^2)*Sqrt[3 + 5*x^2 + x^4])/16 - (10
83*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4])])/32

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Rubi in Sympy [A]  time = 19.4908, size = 70, normalized size = 0.91 \[ \frac{x^{4} \sqrt{x^{4} + 5 x^{2} + 3}}{2} + \frac{\left (- \frac{63 x^{2}}{2} + \frac{801}{4}\right ) \sqrt{x^{4} + 5 x^{2} + 3}}{12} - \frac{1083 \operatorname{atanh}{\left (\frac{2 x^{2} + 5}{2 \sqrt{x^{4} + 5 x^{2} + 3}} \right )}}{32} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**4*sqrt(x**4 + 5*x**2 + 3)/2 + (-63*x**2/2 + 801/4)*sqrt(x**4 + 5*x**2 + 3)/12
- 1083*atanh((2*x**2 + 5)/(2*sqrt(x**4 + 5*x**2 + 3)))/32

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Mathematica [A]  time = 0.0409706, size = 61, normalized size = 0.79 \[ \frac{1}{2} \left (x^4-\frac{21 x^2}{4}+\frac{267}{8}\right ) \sqrt{x^4+5 x^2+3}-\frac{1083}{32} \log \left (2 x^2+2 \sqrt{x^4+5 x^2+3}+5\right ) \]

Antiderivative was successfully verified.

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

[Out]

((267/8 - (21*x^2)/4 + x^4)*Sqrt[3 + 5*x^2 + x^4])/2 - (1083*Log[5 + 2*x^2 + 2*S
qrt[3 + 5*x^2 + x^4]])/32

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Maple [A]  time = 0.017, size = 70, normalized size = 0.9 \[ -{\frac{21\,{x}^{2}}{8}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{267}{16}\sqrt{{x}^{4}+5\,{x}^{2}+3}}-{\frac{1083}{32}\ln \left ({x}^{2}+{\frac{5}{2}}+\sqrt{{x}^{4}+5\,{x}^{2}+3} \right ) }+{\frac{{x}^{4}}{2}\sqrt{{x}^{4}+5\,{x}^{2}+3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-21/8*x^2*(x^4+5*x^2+3)^(1/2)+267/16*(x^4+5*x^2+3)^(1/2)-1083/32*ln(x^2+5/2+(x^4
+5*x^2+3)^(1/2))+1/2*x^4*(x^4+5*x^2+3)^(1/2)

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Maxima [A]  time = 0.702596, size = 99, normalized size = 1.29 \[ \frac{1}{2} \, \sqrt{x^{4} + 5 \, x^{2} + 3} x^{4} - \frac{21}{8} \, \sqrt{x^{4} + 5 \, x^{2} + 3} x^{2} + \frac{267}{16} \, \sqrt{x^{4} + 5 \, x^{2} + 3} - \frac{1083}{32} \, \log \left (2 \, x^{2} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} + 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(x^4 + 5*x^2 + 3)*x^4 - 21/8*sqrt(x^4 + 5*x^2 + 3)*x^2 + 267/16*sqrt(x^4
+ 5*x^2 + 3) - 1083/32*log(2*x^2 + 2*sqrt(x^4 + 5*x^2 + 3) + 5)

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Fricas [A]  time = 0.266837, size = 252, normalized size = 3.27 \[ -\frac{2048 \, x^{12} + 9728 \, x^{10} + 29312 \, x^{8} + 302080 \, x^{6} + 1052928 \, x^{4} + 941238 \, x^{2} - 4332 \,{\left (32 \, x^{6} + 240 \, x^{4} + 522 \, x^{2} - 2 \,{\left (16 \, x^{4} + 80 \, x^{2} + 87\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} + 305\right )} \log \left (-2 \, x^{2} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} - 5\right ) - 2 \,{\left (1024 \, x^{10} + 2304 \, x^{8} + 10560 \, x^{6} + 124224 \, x^{4} + 235456 \, x^{2} + 31245\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} + 82567}{128 \,{\left (32 \, x^{6} + 240 \, x^{4} + 522 \, x^{2} - 2 \,{\left (16 \, x^{4} + 80 \, x^{2} + 87\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} + 305\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/128*(2048*x^12 + 9728*x^10 + 29312*x^8 + 302080*x^6 + 1052928*x^4 + 941238*x^
2 - 4332*(32*x^6 + 240*x^4 + 522*x^2 - 2*(16*x^4 + 80*x^2 + 87)*sqrt(x^4 + 5*x^2
+ 3) + 305)*log(-2*x^2 + 2*sqrt(x^4 + 5*x^2 + 3) - 5) - 2*(1024*x^10 + 2304*x^8
+ 10560*x^6 + 124224*x^4 + 235456*x^2 + 31245)*sqrt(x^4 + 5*x^2 + 3) + 82567)/(
32*x^6 + 240*x^4 + 522*x^2 - 2*(16*x^4 + 80*x^2 + 87)*sqrt(x^4 + 5*x^2 + 3) + 30
5)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.283432, size = 72, normalized size = 0.94 \[ \frac{1}{16} \, \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (2 \,{\left (4 \, x^{2} - 21\right )} x^{2} + 267\right )} + \frac{1083}{32} \,{\rm ln}\left (2 \, x^{2} - 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} + 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*sqrt(x^4 + 5*x^2 + 3)*(2*(4*x^2 - 21)*x^2 + 267) + 1083/32*ln(2*x^2 - 2*sqr
t(x^4 + 5*x^2 + 3) + 5)