\(\int \frac {1}{(3+7 x^2+2 x^4)^{3/2}} \, dx\) [247]

Optimal result

Integrand size = 16, antiderivative size = 122 \[ \int \frac {1}{\left (3+7 x^2+2 x^4\right )^{3/2}} \, dx=\frac {2 x}{5 \sqrt {3+7 x^2+2 x^4}}+\frac {7 \sqrt {3+7 x^2+2 x^4} E\left (\left .\arctan \left (\frac {x}{\sqrt {3}}\right )\right |-5\right )}{75 \sqrt {3+x^2} \sqrt {1+2 x^2}}-\frac {4 \sqrt {3+x^2} \sqrt {1+2 x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {x}{\sqrt {3}}\right ),-5\right )}{25 \sqrt {3+7 x^2+2 x^4}} \] Output:

2/5*x/(2*x^4+7*x^2+3)^(1/2)+7/75*(2*x^4+7*x^2+3)^(1/2)*EllipticE(x*3^(1/2) 
/(3*x^2+9)^(1/2),I*5^(1/2))/(x^2+3)^(1/2)/(2*x^2+1)^(1/2)-4/25*(x^2+3)^(1/ 
2)*(2*x^2+1)^(1/2)*InverseJacobiAM(arctan(1/3*x*3^(1/2)),I*5^(1/2))/(2*x^4 
+7*x^2+3)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 5.86 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\left (3+7 x^2+2 x^4\right )^{3/2}} \, dx=\frac {37 x+14 x^3+7 i \sqrt {6} \sqrt {3+x^2} \sqrt {1+2 x^2} E\left (i \text {arcsinh}\left (\sqrt {2} x\right )|\frac {1}{6}\right )-5 i \sqrt {6} \sqrt {3+x^2} \sqrt {1+2 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} x\right ),\frac {1}{6}\right )}{75 \sqrt {3+7 x^2+2 x^4}} \] Input:

Integrate[(3 + 7*x^2 + 2*x^4)^(-3/2),x]
 

Output:

(37*x + 14*x^3 + (7*I)*Sqrt[6]*Sqrt[3 + x^2]*Sqrt[1 + 2*x^2]*EllipticE[I*A 
rcSinh[Sqrt[2]*x], 1/6] - (5*I)*Sqrt[6]*Sqrt[3 + x^2]*Sqrt[1 + 2*x^2]*Elli 
pticF[I*ArcSinh[Sqrt[2]*x], 1/6])/(75*Sqrt[3 + 7*x^2 + 2*x^4])
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.50, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {1405, 27, 1503, 1412, 1455}

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.

\(\displaystyle \int \frac {1}{\left (2 x^4+7 x^2+3\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 1405

\(\displaystyle \frac {x \left (14 x^2+37\right )}{75 \sqrt {2 x^4+7 x^2+3}}-\frac {1}{75} \int \frac {2 \left (7 x^2+6\right )}{\sqrt {2 x^4+7 x^2+3}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x \left (14 x^2+37\right )}{75 \sqrt {2 x^4+7 x^2+3}}-\frac {2}{75} \int \frac {7 x^2+6}{\sqrt {2 x^4+7 x^2+3}}dx\)

\(\Big \downarrow \) 1503

\(\displaystyle \frac {x \left (14 x^2+37\right )}{75 \sqrt {2 x^4+7 x^2+3}}-\frac {2}{75} \left (6 \int \frac {1}{\sqrt {2 x^4+7 x^2+3}}dx+7 \int \frac {x^2}{\sqrt {2 x^4+7 x^2+3}}dx\right )\)

\(\Big \downarrow \) 1412

\(\displaystyle \frac {x \left (14 x^2+37\right )}{75 \sqrt {2 x^4+7 x^2+3}}-\frac {2}{75} \left (7 \int \frac {x^2}{\sqrt {2 x^4+7 x^2+3}}dx+\frac {\sqrt {6} \sqrt {\frac {x^2+3}{2 x^2+1}} \left (2 x^2+1\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {2} x\right ),\frac {5}{6}\right )}{\sqrt {2 x^4+7 x^2+3}}\right )\)

\(\Big \downarrow \) 1455

\(\displaystyle \frac {x \left (14 x^2+37\right )}{75 \sqrt {2 x^4+7 x^2+3}}-\frac {2}{75} \left (\frac {\sqrt {6} \sqrt {\frac {x^2+3}{2 x^2+1}} \left (2 x^2+1\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {2} x\right ),\frac {5}{6}\right )}{\sqrt {2 x^4+7 x^2+3}}+7 \left (\frac {x \left (x^2+3\right )}{\sqrt {2 x^4+7 x^2+3}}-\frac {\sqrt {\frac {3}{2}} \sqrt {\frac {x^2+3}{2 x^2+1}} \left (2 x^2+1\right ) E\left (\arctan \left (\sqrt {2} x\right )|\frac {5}{6}\right )}{\sqrt {2 x^4+7 x^2+3}}\right )\right )\)

Input:

Int[(3 + 7*x^2 + 2*x^4)^(-3/2),x]
 

Output:

(x*(37 + 14*x^2))/(75*Sqrt[3 + 7*x^2 + 2*x^4]) - (2*(7*((x*(3 + x^2))/Sqrt 
[3 + 7*x^2 + 2*x^4] - (Sqrt[3/2]*Sqrt[(3 + x^2)/(1 + 2*x^2)]*(1 + 2*x^2)*E 
llipticE[ArcTan[Sqrt[2]*x], 5/6])/Sqrt[3 + 7*x^2 + 2*x^4]) + (Sqrt[6]*Sqrt 
[(3 + x^2)/(1 + 2*x^2)]*(1 + 2*x^2)*EllipticF[ArcTan[Sqrt[2]*x], 5/6])/Sqr 
t[3 + 7*x^2 + 2*x^4]))/75
 

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 1412

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

rule 1455

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

rule 1503

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

Maple [A] (verified)

Time = 2.65 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.13

method	result	size
risch	\(\frac {x \left (14 x^{2}+37\right )}{75 \sqrt {2 x^{4}+7 x^{2}+3}}+\frac {4 i \sqrt {3}\, \sqrt {3 x^{2}+9}\, \sqrt {2 x^{2}+1}\, \operatorname {EllipticF}\left (\frac {i x \sqrt {3}}{3}, \sqrt {6}\right )}{75 \sqrt {2 x^{4}+7 x^{2}+3}}-\frac {7 i \sqrt {3}\, \sqrt {3 x^{2}+9}\, \sqrt {2 x^{2}+1}\, \left (\operatorname {EllipticF}\left (\frac {i x \sqrt {3}}{3}, \sqrt {6}\right )-\operatorname {EllipticE}\left (\frac {i x \sqrt {3}}{3}, \sqrt {6}\right )\right )}{225 \sqrt {2 x^{4}+7 x^{2}+3}}\)	\(138\)
default	\(-\frac {4 \left (-\frac {37}{300} x -\frac {7}{150} x^{3}\right )}{\sqrt {2 x^{4}+7 x^{2}+3}}+\frac {4 i \sqrt {3}\, \sqrt {3 x^{2}+9}\, \sqrt {2 x^{2}+1}\, \operatorname {EllipticF}\left (\frac {i x \sqrt {3}}{3}, \sqrt {6}\right )}{75 \sqrt {2 x^{4}+7 x^{2}+3}}-\frac {7 i \sqrt {3}\, \sqrt {3 x^{2}+9}\, \sqrt {2 x^{2}+1}\, \left (\operatorname {EllipticF}\left (\frac {i x \sqrt {3}}{3}, \sqrt {6}\right )-\operatorname {EllipticE}\left (\frac {i x \sqrt {3}}{3}, \sqrt {6}\right )\right )}{225 \sqrt {2 x^{4}+7 x^{2}+3}}\)	\(139\)
elliptic	\(-\frac {4 \left (-\frac {37}{300} x -\frac {7}{150} x^{3}\right )}{\sqrt {2 x^{4}+7 x^{2}+3}}+\frac {4 i \sqrt {3}\, \sqrt {3 x^{2}+9}\, \sqrt {2 x^{2}+1}\, \operatorname {EllipticF}\left (\frac {i x \sqrt {3}}{3}, \sqrt {6}\right )}{75 \sqrt {2 x^{4}+7 x^{2}+3}}-\frac {7 i \sqrt {3}\, \sqrt {3 x^{2}+9}\, \sqrt {2 x^{2}+1}\, \left (\operatorname {EllipticF}\left (\frac {i x \sqrt {3}}{3}, \sqrt {6}\right )-\operatorname {EllipticE}\left (\frac {i x \sqrt {3}}{3}, \sqrt {6}\right )\right )}{225 \sqrt {2 x^{4}+7 x^{2}+3}}\)	\(139\)

Input:

int(1/(2*x^4+7*x^2+3)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/75*x*(14*x^2+37)/(2*x^4+7*x^2+3)^(1/2)+4/75*I*3^(1/2)*(3*x^2+9)^(1/2)*(2 
*x^2+1)^(1/2)/(2*x^4+7*x^2+3)^(1/2)*EllipticF(1/3*I*x*3^(1/2),6^(1/2))-7/2 
25*I*3^(1/2)*(3*x^2+9)^(1/2)*(2*x^2+1)^(1/2)/(2*x^4+7*x^2+3)^(1/2)*(Ellipt 
icF(1/3*I*x*3^(1/2),6^(1/2))-EllipticE(1/3*I*x*3^(1/2),6^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\left (3+7 x^2+2 x^4\right )^{3/2}} \, dx=-\frac {7 \, \sqrt {3} \sqrt {-\frac {1}{3}} {\left (2 \, x^{4} + 7 \, x^{2} + 3\right )} E(\arcsin \left (\sqrt {-\frac {1}{3}} x\right )\,|\,6) - 43 \, \sqrt {3} \sqrt {-\frac {1}{3}} {\left (2 \, x^{4} + 7 \, x^{2} + 3\right )} F(\arcsin \left (\sqrt {-\frac {1}{3}} x\right )\,|\,6) - 3 \, \sqrt {2 \, x^{4} + 7 \, x^{2} + 3} {\left (14 \, x^{3} + 37 \, x\right )}}{225 \, {\left (2 \, x^{4} + 7 \, x^{2} + 3\right )}} \] Input:

integrate(1/(2*x^4+7*x^2+3)^(3/2),x, algorithm="fricas")
 

Output:

-1/225*(7*sqrt(3)*sqrt(-1/3)*(2*x^4 + 7*x^2 + 3)*elliptic_e(arcsin(sqrt(-1 
/3)*x), 6) - 43*sqrt(3)*sqrt(-1/3)*(2*x^4 + 7*x^2 + 3)*elliptic_f(arcsin(s 
qrt(-1/3)*x), 6) - 3*sqrt(2*x^4 + 7*x^2 + 3)*(14*x^3 + 37*x))/(2*x^4 + 7*x 
^2 + 3)
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {1}{\left (3+7 x^2+2 x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (2 x^{4} + 7 x^{2} + 3\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/(2*x**4+7*x**2+3)**(3/2),x)
 

Output:

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

Maxima [F]

\[ \int \frac {1}{\left (3+7 x^2+2 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (2 \, x^{4} + 7 \, x^{2} + 3\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(2*x^4+7*x^2+3)^(3/2),x, algorithm="maxima")
 

Output:

integrate((2*x^4 + 7*x^2 + 3)^(-3/2), x)
 

Giac [F]

\[ \int \frac {1}{\left (3+7 x^2+2 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (2 \, x^{4} + 7 \, x^{2} + 3\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(2*x^4+7*x^2+3)^(3/2),x, algorithm="giac")
 

Output:

integrate((2*x^4 + 7*x^2 + 3)^(-3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (3+7 x^2+2 x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (2\,x^4+7\,x^2+3\right )}^{3/2}} \,d x \] Input:

int(1/(7*x^2 + 2*x^4 + 3)^(3/2),x)
 

Output:

int(1/(7*x^2 + 2*x^4 + 3)^(3/2), x)
 

Reduce [F]

\[ \int \frac {1}{\left (3+7 x^2+2 x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {2 x^{4}+7 x^{2}+3}}{4 x^{8}+28 x^{6}+61 x^{4}+42 x^{2}+9}d x \] Input:

int(1/(2*x^4+7*x^2+3)^(3/2),x)
 

Output:

int(sqrt(2*x**4 + 7*x**2 + 3)/(4*x**8 + 28*x**6 + 61*x**4 + 42*x**2 + 9),x 
)