\(\int \frac {1}{(2+x^2+3 x^4)^{3/2}} \, dx\) [237]

Optimal result

Integrand size = 14, antiderivative size = 254 \[ \int \frac {1}{\left (2+x^2+3 x^4\right )^{3/2}} \, dx=\frac {x \left (11-3 x^2\right )}{46 \sqrt {2+x^2+3 x^4}}+\frac {3 x \sqrt {2+x^2+3 x^4}}{46 \left (\sqrt {6}+3 x^2\right )}-\frac {\sqrt [4]{3} \left (2+\sqrt {6} x^2\right ) \sqrt {\frac {2+x^2+3 x^4}{\left (2+\sqrt {6} x^2\right )^2}} E\left (2 \arctan \left (\sqrt [4]{\frac {3}{2}} x\right )|\frac {1}{24} \left (12-\sqrt {6}\right )\right )}{23\ 2^{3/4} \sqrt {2+x^2+3 x^4}}+\frac {\sqrt [4]{3} \left (1+2 \sqrt {6}\right ) \left (2+\sqrt {6} x^2\right ) \sqrt {\frac {2+x^2+3 x^4}{\left (2+\sqrt {6} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{\frac {3}{2}} x\right ),\frac {1}{24} \left (12-\sqrt {6}\right )\right )}{46\ 2^{3/4} \sqrt {2+x^2+3 x^4}} \] Output:

1/46*x*(-3*x^2+11)/(3*x^4+x^2+2)^(1/2)+3*x*(3*x^4+x^2+2)^(1/2)/(46*6^(1/2) 
+138*x^2)-1/46*3^(1/4)*(2+6^(1/2)*x^2)*((3*x^4+x^2+2)/(2+6^(1/2)*x^2)^2)^( 
1/2)*EllipticE(sin(2*arctan(1/2*3^(1/4)*2^(3/4)*x)),1/12*(72-6*6^(1/2))^(1 
/2))*2^(1/4)/(3*x^4+x^2+2)^(1/2)+1/92*3^(1/4)*(1+2*6^(1/2))*(2+6^(1/2)*x^2 
)*((3*x^4+x^2+2)/(2+6^(1/2)*x^2)^2)^(1/2)*InverseJacobiAM(2*arctan(1/2*3^( 
1/4)*2^(3/4)*x),1/12*(72-6*6^(1/2))^(1/2))*2^(1/4)/(3*x^4+x^2+2)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.10 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.28 \[ \int \frac {1}{\left (2+x^2+3 x^4\right )^{3/2}} \, dx=\frac {12 \sqrt {-\frac {i}{-i+\sqrt {23}}} x \left (11-3 x^2\right )-\sqrt {6} \left (i+\sqrt {23}\right ) \sqrt {\frac {-i+\sqrt {23}-6 i x^2}{-i+\sqrt {23}}} \sqrt {\frac {i+\sqrt {23}+6 i x^2}{i+\sqrt {23}}} E\left (i \text {arcsinh}\left (\sqrt {-\frac {6 i}{-i+\sqrt {23}}} x\right )|\frac {i-\sqrt {23}}{i+\sqrt {23}}\right )+\sqrt {6} \left (-23 i+\sqrt {23}\right ) \sqrt {\frac {-i+\sqrt {23}-6 i x^2}{-i+\sqrt {23}}} \sqrt {\frac {i+\sqrt {23}+6 i x^2}{i+\sqrt {23}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {6 i}{-i+\sqrt {23}}} x\right ),\frac {i-\sqrt {23}}{i+\sqrt {23}}\right )}{552 \sqrt {-\frac {i}{-i+\sqrt {23}}} \sqrt {2+x^2+3 x^4}} \] Input:

Integrate[(2 + x^2 + 3*x^4)^(-3/2),x]
 

Output:

(12*Sqrt[(-I)/(-I + Sqrt[23])]*x*(11 - 3*x^2) - Sqrt[6]*(I + Sqrt[23])*Sqr 
t[(-I + Sqrt[23] - (6*I)*x^2)/(-I + Sqrt[23])]*Sqrt[(I + Sqrt[23] + (6*I)* 
x^2)/(I + Sqrt[23])]*EllipticE[I*ArcSinh[Sqrt[(-6*I)/(-I + Sqrt[23])]*x], 
(I - Sqrt[23])/(I + Sqrt[23])] + Sqrt[6]*(-23*I + Sqrt[23])*Sqrt[(-I + Sqr 
t[23] - (6*I)*x^2)/(-I + Sqrt[23])]*Sqrt[(I + Sqrt[23] + (6*I)*x^2)/(I + S 
qrt[23])]*EllipticF[I*ArcSinh[Sqrt[(-6*I)/(-I + Sqrt[23])]*x], (I - Sqrt[2 
3])/(I + Sqrt[23])])/(552*Sqrt[(-I)/(-I + Sqrt[23])]*Sqrt[2 + x^2 + 3*x^4] 
)
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {1405, 27, 1511, 27, 1416, 1509}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {1}{\left (3 x^4+x^2+2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 1405

\(\displaystyle \frac {1}{46} \int \frac {3 \left (x^2+4\right )}{\sqrt {3 x^4+x^2+2}}dx+\frac {x \left (11-3 x^2\right )}{46 \sqrt {3 x^4+x^2+2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{46} \int \frac {x^2+4}{\sqrt {3 x^4+x^2+2}}dx+\frac {x \left (11-3 x^2\right )}{46 \sqrt {3 x^4+x^2+2}}\)

\(\Big \downarrow \) 1511

\(\displaystyle \frac {3}{46} \left (\frac {1}{3} \left (12+\sqrt {6}\right ) \int \frac {1}{\sqrt {3 x^4+x^2+2}}dx-\sqrt {\frac {2}{3}} \int \frac {2-\sqrt {6} x^2}{2 \sqrt {3 x^4+x^2+2}}dx\right )+\frac {x \left (11-3 x^2\right )}{46 \sqrt {3 x^4+x^2+2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{46} \left (\frac {1}{3} \left (12+\sqrt {6}\right ) \int \frac {1}{\sqrt {3 x^4+x^2+2}}dx-\frac {\int \frac {2-\sqrt {6} x^2}{\sqrt {3 x^4+x^2+2}}dx}{\sqrt {6}}\right )+\frac {x \left (11-3 x^2\right )}{46 \sqrt {3 x^4+x^2+2}}\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {3}{46} \left (\frac {\left (12+\sqrt {6}\right ) \left (\sqrt {6} x^2+2\right ) \sqrt {\frac {3 x^4+x^2+2}{\left (\sqrt {6} x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{\frac {3}{2}} x\right ),\frac {1}{24} \left (12-\sqrt {6}\right )\right )}{6 \sqrt [4]{6} \sqrt {3 x^4+x^2+2}}-\frac {\int \frac {2-\sqrt {6} x^2}{\sqrt {3 x^4+x^2+2}}dx}{\sqrt {6}}\right )+\frac {x \left (11-3 x^2\right )}{46 \sqrt {3 x^4+x^2+2}}\)

\(\Big \downarrow \) 1509

\(\displaystyle \frac {3}{46} \left (\frac {\left (12+\sqrt {6}\right ) \left (\sqrt {6} x^2+2\right ) \sqrt {\frac {3 x^4+x^2+2}{\left (\sqrt {6} x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{\frac {3}{2}} x\right ),\frac {1}{24} \left (12-\sqrt {6}\right )\right )}{6 \sqrt [4]{6} \sqrt {3 x^4+x^2+2}}-\frac {\frac {2^{3/4} \left (\sqrt {6} x^2+2\right ) \sqrt {\frac {3 x^4+x^2+2}{\left (\sqrt {6} x^2+2\right )^2}} E\left (2 \arctan \left (\sqrt [4]{\frac {3}{2}} x\right )|\frac {1}{24} \left (12-\sqrt {6}\right )\right )}{\sqrt [4]{3} \sqrt {3 x^4+x^2+2}}-\frac {2 x \sqrt {3 x^4+x^2+2}}{\sqrt {6} x^2+2}}{\sqrt {6}}\right )+\frac {x \left (11-3 x^2\right )}{46 \sqrt {3 x^4+x^2+2}}\)

Input:

Int[(2 + x^2 + 3*x^4)^(-3/2),x]
 

Output:

(x*(11 - 3*x^2))/(46*Sqrt[2 + x^2 + 3*x^4]) + (3*(-(((-2*x*Sqrt[2 + x^2 + 
3*x^4])/(2 + Sqrt[6]*x^2) + (2^(3/4)*(2 + Sqrt[6]*x^2)*Sqrt[(2 + x^2 + 3*x 
^4)/(2 + Sqrt[6]*x^2)^2]*EllipticE[2*ArcTan[(3/2)^(1/4)*x], (12 - Sqrt[6]) 
/24])/(3^(1/4)*Sqrt[2 + x^2 + 3*x^4]))/Sqrt[6]) + ((12 + Sqrt[6])*(2 + Sqr 
t[6]*x^2)*Sqrt[(2 + x^2 + 3*x^4)/(2 + Sqrt[6]*x^2)^2]*EllipticF[2*ArcTan[( 
3/2)^(1/4)*x], (12 - Sqrt[6])/24])/(6*6^(1/4)*Sqrt[2 + x^2 + 3*x^4])))/46
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 1405

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-x)*(b^2 
- 2*a*c + b*c*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c)) 
), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c))   Int[(b^2 - 2*a*c + 2*(p + 1)*( 
b^2 - 4*a*c) + b*c*(4*p + 7)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; Fr 
eeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p]
 

rule 1416

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c 
/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ 
(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) 
], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

rule 1509

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q 
^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* 
x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 
/(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 
- 4*a*c, 0] && PosQ[c/a]
 

rule 1511

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q   Int[1/Sqrt[a + b*x^2 + c*x^ 
4], x], x] - Simp[e/q   Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; 
NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos 
Q[c/a]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.85 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.91

method	result	size
risch	\(-\frac {x \left (3 x^{2}-11\right )}{46 \sqrt {3 x^{4}+x^{2}+2}}+\frac {12 \sqrt {1-\left (-\frac {1}{4}+\frac {i \sqrt {23}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{4}-\frac {i \sqrt {23}}{4}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-1+i \sqrt {23}}}{2}, \frac {\sqrt {-33+3 i \sqrt {23}}}{6}\right )}{23 \sqrt {-1+i \sqrt {23}}\, \sqrt {3 x^{4}+x^{2}+2}}-\frac {12 \sqrt {1-\left (-\frac {1}{4}+\frac {i \sqrt {23}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{4}-\frac {i \sqrt {23}}{4}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {-1+i \sqrt {23}}}{2}, \frac {\sqrt {-33+3 i \sqrt {23}}}{6}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {-1+i \sqrt {23}}}{2}, \frac {\sqrt {-33+3 i \sqrt {23}}}{6}\right )\right )}{23 \sqrt {-1+i \sqrt {23}}\, \sqrt {3 x^{4}+x^{2}+2}\, \left (1+i \sqrt {23}\right )}\)	\(231\)
default	\(-\frac {6 \left (-\frac {11}{276} x +\frac {1}{92} x^{3}\right )}{\sqrt {3 x^{4}+x^{2}+2}}+\frac {12 \sqrt {1-\left (-\frac {1}{4}+\frac {i \sqrt {23}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{4}-\frac {i \sqrt {23}}{4}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-1+i \sqrt {23}}}{2}, \frac {\sqrt {-33+3 i \sqrt {23}}}{6}\right )}{23 \sqrt {-1+i \sqrt {23}}\, \sqrt {3 x^{4}+x^{2}+2}}-\frac {12 \sqrt {1-\left (-\frac {1}{4}+\frac {i \sqrt {23}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{4}-\frac {i \sqrt {23}}{4}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {-1+i \sqrt {23}}}{2}, \frac {\sqrt {-33+3 i \sqrt {23}}}{6}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {-1+i \sqrt {23}}}{2}, \frac {\sqrt {-33+3 i \sqrt {23}}}{6}\right )\right )}{23 \sqrt {-1+i \sqrt {23}}\, \sqrt {3 x^{4}+x^{2}+2}\, \left (1+i \sqrt {23}\right )}\)	\(232\)
elliptic	\(-\frac {6 \left (-\frac {11}{276} x +\frac {1}{92} x^{3}\right )}{\sqrt {3 x^{4}+x^{2}+2}}+\frac {12 \sqrt {1-\left (-\frac {1}{4}+\frac {i \sqrt {23}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{4}-\frac {i \sqrt {23}}{4}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-1+i \sqrt {23}}}{2}, \frac {\sqrt {-33+3 i \sqrt {23}}}{6}\right )}{23 \sqrt {-1+i \sqrt {23}}\, \sqrt {3 x^{4}+x^{2}+2}}-\frac {12 \sqrt {1-\left (-\frac {1}{4}+\frac {i \sqrt {23}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{4}-\frac {i \sqrt {23}}{4}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {-1+i \sqrt {23}}}{2}, \frac {\sqrt {-33+3 i \sqrt {23}}}{6}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {-1+i \sqrt {23}}}{2}, \frac {\sqrt {-33+3 i \sqrt {23}}}{6}\right )\right )}{23 \sqrt {-1+i \sqrt {23}}\, \sqrt {3 x^{4}+x^{2}+2}\, \left (1+i \sqrt {23}\right )}\)	\(232\)

Input:

int(1/(3*x^4+x^2+2)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/46*x*(3*x^2-11)/(3*x^4+x^2+2)^(1/2)+12/23/(-1+I*23^(1/2))^(1/2)*(1-(-1/ 
4+1/4*I*23^(1/2))*x^2)^(1/2)*(1-(-1/4-1/4*I*23^(1/2))*x^2)^(1/2)/(3*x^4+x^ 
2+2)^(1/2)*EllipticF(1/2*x*(-1+I*23^(1/2))^(1/2),1/6*(-33+3*I*23^(1/2))^(1 
/2))-12/23/(-1+I*23^(1/2))^(1/2)*(1-(-1/4+1/4*I*23^(1/2))*x^2)^(1/2)*(1-(- 
1/4-1/4*I*23^(1/2))*x^2)^(1/2)/(3*x^4+x^2+2)^(1/2)/(1+I*23^(1/2))*(Ellipti 
cF(1/2*x*(-1+I*23^(1/2))^(1/2),1/6*(-33+3*I*23^(1/2))^(1/2))-EllipticE(1/2 
*x*(-1+I*23^(1/2))^(1/2),1/6*(-33+3*I*23^(1/2))^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.59 \[ \int \frac {1}{\left (2+x^2+3 x^4\right )^{3/2}} \, dx=\frac {\sqrt {2} {\left (3 \, x^{4} + x^{2} - \sqrt {-23} {\left (3 \, x^{4} + x^{2} + 2\right )} + 2\right )} \sqrt {\sqrt {-23} - 1} E(\arcsin \left (\frac {1}{2} \, x \sqrt {\sqrt {-23} - 1}\right )\,|\,\frac {1}{12} \, \sqrt {-23} - \frac {11}{12}) - \sqrt {2} {\left (15 \, x^{4} + 5 \, x^{2} + 3 \, \sqrt {-23} {\left (3 \, x^{4} + x^{2} + 2\right )} + 10\right )} \sqrt {\sqrt {-23} - 1} F(\arcsin \left (\frac {1}{2} \, x \sqrt {\sqrt {-23} - 1}\right )\,|\,\frac {1}{12} \, \sqrt {-23} - \frac {11}{12}) - 8 \, \sqrt {3 \, x^{4} + x^{2} + 2} {\left (3 \, x^{3} - 11 \, x\right )}}{368 \, {\left (3 \, x^{4} + x^{2} + 2\right )}} \] Input:

integrate(1/(3*x^4+x^2+2)^(3/2),x, algorithm="fricas")
 

Output:

1/368*(sqrt(2)*(3*x^4 + x^2 - sqrt(-23)*(3*x^4 + x^2 + 2) + 2)*sqrt(sqrt(- 
23) - 1)*elliptic_e(arcsin(1/2*x*sqrt(sqrt(-23) - 1)), 1/12*sqrt(-23) - 11 
/12) - sqrt(2)*(15*x^4 + 5*x^2 + 3*sqrt(-23)*(3*x^4 + x^2 + 2) + 10)*sqrt( 
sqrt(-23) - 1)*elliptic_f(arcsin(1/2*x*sqrt(sqrt(-23) - 1)), 1/12*sqrt(-23 
) - 11/12) - 8*sqrt(3*x^4 + x^2 + 2)*(3*x^3 - 11*x))/(3*x^4 + x^2 + 2)
 

Sympy [F]

\[ \int \frac {1}{\left (2+x^2+3 x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (3 x^{4} + x^{2} + 2\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/(3*x**4+x**2+2)**(3/2),x)
 

Output:

Integral((3*x**4 + x**2 + 2)**(-3/2), x)
 

Maxima [F]

\[ \int \frac {1}{\left (2+x^2+3 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (3 \, x^{4} + x^{2} + 2\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(3*x^4+x^2+2)^(3/2),x, algorithm="maxima")
 

Output:

integrate((3*x^4 + x^2 + 2)^(-3/2), x)
 

Giac [F]

\[ \int \frac {1}{\left (2+x^2+3 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (3 \, x^{4} + x^{2} + 2\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(3*x^4+x^2+2)^(3/2),x, algorithm="giac")
 

Output:

integrate((3*x^4 + x^2 + 2)^(-3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (2+x^2+3 x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (3\,x^4+x^2+2\right )}^{3/2}} \,d x \] Input:

int(1/(x^2 + 3*x^4 + 2)^(3/2),x)
 

Output:

int(1/(x^2 + 3*x^4 + 2)^(3/2), x)
 

Reduce [F]

\[ \int \frac {1}{\left (2+x^2+3 x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {3 x^{4}+x^{2}+2}}{9 x^{8}+6 x^{6}+13 x^{4}+4 x^{2}+4}d x \] Input:

int(1/(3*x^4+x^2+2)^(3/2),x)
 

Output:

int(sqrt(3*x**4 + x**2 + 2)/(9*x**8 + 6*x**6 + 13*x**4 + 4*x**2 + 4),x)