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

Optimal. Leaf size=74 \[ \frac{1}{2} \left (x^4+5 x^2+3\right )^{3/2}-\frac{11}{16} \left (2 x^2+5\right ) \sqrt{x^4+5 x^2+3}+\frac{143}{32} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right ) \]

[Out]

(-11*(5 + 2*x^2)*Sqrt[3 + 5*x^2 + x^4])/16 + (3 + 5*x^2 + x^4)^(3/2)/2 + (143*Ar
cTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4])])/32

_______________________________________________________________________________________

Rubi [A]  time = 0.10209, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ \frac{1}{2} \left (x^4+5 x^2+3\right )^{3/2}-\frac{11}{16} \left (2 x^2+5\right ) \sqrt{x^4+5 x^2+3}+\frac{143}{32} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-11*(5 + 2*x^2)*Sqrt[3 + 5*x^2 + x^4])/16 + (3 + 5*x^2 + x^4)^(3/2)/2 + (143*Ar
cTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4])])/32

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 12.0704, size = 65, normalized size = 0.88 \[ - \frac{11 \left (2 x^{2} + 5\right ) \sqrt{x^{4} + 5 x^{2} + 3}}{16} + \frac{\left (x^{4} + 5 x^{2} + 3\right )^{\frac{3}{2}}}{2} + \frac{143 \operatorname{atanh}{\left (\frac{2 x^{2} + 5}{2 \sqrt{x^{4} + 5 x^{2} + 3}} \right )}}{32} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-11*(2*x**2 + 5)*sqrt(x**4 + 5*x**2 + 3)/16 + (x**4 + 5*x**2 + 3)**(3/2)/2 + 143
*atanh((2*x**2 + 5)/(2*sqrt(x**4 + 5*x**2 + 3)))/32

_______________________________________________________________________________________

Mathematica [A]  time = 0.0378313, size = 61, normalized size = 0.82 \[ \frac{1}{2} \sqrt{x^4+5 x^2+3} \left (x^4+\frac{9 x^2}{4}-\frac{31}{8}\right )+\frac{143}{32} \log \left (2 x^2+2 \sqrt{x^4+5 x^2+3}+5\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-31/8 + (9*x^2)/4 + x^4)*Sqrt[3 + 5*x^2 + x^4])/2 + (143*Log[5 + 2*x^2 + 2*Sqr
t[3 + 5*x^2 + x^4]])/32

_______________________________________________________________________________________

Maple [A]  time = 0.016, size = 57, normalized size = 0.8 \[ -{\frac{22\,{x}^{2}+55}{16}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{143}{32}\ln \left ({x}^{2}+{\frac{5}{2}}+\sqrt{{x}^{4}+5\,{x}^{2}+3} \right ) }+{\frac{1}{2} \left ({x}^{4}+5\,{x}^{2}+3 \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-11/16*(2*x^2+5)*(x^4+5*x^2+3)^(1/2)+143/32*ln(x^2+5/2+(x^4+5*x^2+3)^(1/2))+1/2*
(x^4+5*x^2+3)^(3/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.730575, size = 95, normalized size = 1.28 \[ -\frac{11}{8} \, \sqrt{x^{4} + 5 \, x^{2} + 3} x^{2} + \frac{1}{2} \,{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}} - \frac{55}{16} \, \sqrt{x^{4} + 5 \, x^{2} + 3} + \frac{143}{32} \, \log \left (2 \, x^{2} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} + 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-11/8*sqrt(x^4 + 5*x^2 + 3)*x^2 + 1/2*(x^4 + 5*x^2 + 3)^(3/2) - 55/16*sqrt(x^4 +
5*x^2 + 3) + 143/32*log(2*x^2 + 2*sqrt(x^4 + 5*x^2 + 3) + 5)

_______________________________________________________________________________________

Fricas [A]  time = 0.262023, size = 252, normalized size = 3.41 \[ -\frac{2048 \, x^{12} + 25088 \, x^{10} + 106624 \, x^{8} + 172160 \, x^{6} + 45248 \, x^{4} - 79542 \, x^{2} + 572 \,{\left (32 \, x^{6} + 240 \, x^{4} + 522 \, x^{2} - 2 \,{\left (16 \, x^{4} + 80 \, x^{2} + 87\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} + 305\right )} \log \left (-2 \, x^{2} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} - 5\right ) - 2 \,{\left (1024 \, x^{10} + 9984 \, x^{8} + 30016 \, x^{6} + 23104 \, x^{4} - 15168 \, x^{2} - 7805\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} - 24231}{128 \,{\left (32 \, x^{6} + 240 \, x^{4} + 522 \, x^{2} - 2 \,{\left (16 \, x^{4} + 80 \, x^{2} + 87\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} + 305\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/128*(2048*x^12 + 25088*x^10 + 106624*x^8 + 172160*x^6 + 45248*x^4 - 79542*x^2
+ 572*(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 + 9984*x^8 +
30016*x^6 + 23104*x^4 - 15168*x^2 - 7805)*sqrt(x^4 + 5*x^2 + 3) - 24231)/(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)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.27468, size = 72, normalized size = 0.97 \[ \frac{1}{16} \, \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (2 \,{\left (4 \, x^{2} + 9\right )} x^{2} - 31\right )} - \frac{143}{32} \,{\rm ln}\left (2 \, x^{2} - 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} + 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*sqrt(x^4 + 5*x^2 + 3)*(2*(4*x^2 + 9)*x^2 - 31) - 143/32*ln(2*x^2 - 2*sqrt(x
^4 + 5*x^2 + 3) + 5)