\(\int \frac {4-7 x^2+x^4}{(2+5 x^2+3 x^4)^{3/2}} \, dx\) [214]

Optimal result

Integrand size = 27, antiderivative size = 130 \[ \int \frac {4-7 x^2+x^4}{\left (2+5 x^2+3 x^4\right )^{3/2}} \, dx=\frac {41 x}{3 \sqrt {2+5 x^2+3 x^4}}+\frac {77 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{3 \sqrt {2+5 x^2+3 x^4}}-\frac {63 \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {2+5 x^2+3 x^4}} \] Output:

41/3*x/(3*x^4+5*x^2+2)^(1/2)+77/3*2^(1/2)*(x^2+1)*((3*x^2+2)/(x^2+1))^(1/2 
)*EllipticE(x/(x^2+1)^(1/2),1/2*I*2^(1/2))/(3*x^4+5*x^2+2)^(1/2)-63/2*2^(1 
/2)*(x^2+1)*((3*x^2+2)/(x^2+1))^(1/2)*InverseJacobiAM(arctan(x),1/2*I*2^(1 
/2))/(3*x^4+5*x^2+2)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.11 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.96 \[ \int \frac {4-7 x^2+x^4}{\left (2+5 x^2+3 x^4\right )^{3/2}} \, dx=\frac {3 x \left (65+77 x^2\right )+77 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} E\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )|\frac {2}{3}\right )-14 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right ),\frac {2}{3}\right )}{3 \sqrt {2+5 x^2+3 x^4}} \] Input:

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

Output:

(3*x*(65 + 77*x^2) + (77*I)*Sqrt[3]*Sqrt[1 + x^2]*Sqrt[2 + 3*x^2]*Elliptic 
E[I*ArcSinh[Sqrt[3/2]*x], 2/3] - (14*I)*Sqrt[3]*Sqrt[1 + x^2]*Sqrt[2 + 3*x 
^2]*EllipticF[I*ArcSinh[Sqrt[3/2]*x], 2/3])/(3*Sqrt[2 + 5*x^2 + 3*x^4])
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.29, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2206, 27, 1503, 1413, 1456}

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.

\(\displaystyle \int \frac {x^4-7 x^2+4}{\left (3 x^4+5 x^2+2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 2206

\(\displaystyle \frac {x \left (77 x^2+65\right )}{\sqrt {3 x^4+5 x^2+2}}-\frac {1}{2} \int \frac {14 \left (11 x^2+9\right )}{\sqrt {3 x^4+5 x^2+2}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x \left (77 x^2+65\right )}{\sqrt {3 x^4+5 x^2+2}}-7 \int \frac {11 x^2+9}{\sqrt {3 x^4+5 x^2+2}}dx\)

\(\Big \downarrow \) 1503

\(\displaystyle \frac {x \left (77 x^2+65\right )}{\sqrt {3 x^4+5 x^2+2}}-7 \left (9 \int \frac {1}{\sqrt {3 x^4+5 x^2+2}}dx+11 \int \frac {x^2}{\sqrt {3 x^4+5 x^2+2}}dx\right )\)

\(\Big \downarrow \) 1413

\(\displaystyle \frac {x \left (77 x^2+65\right )}{\sqrt {3 x^4+5 x^2+2}}-7 \left (11 \int \frac {x^2}{\sqrt {3 x^4+5 x^2+2}}dx+\frac {9 \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {3 x^4+5 x^2+2}}\right )\)

\(\Big \downarrow \) 1456

\(\displaystyle \frac {x \left (77 x^2+65\right )}{\sqrt {3 x^4+5 x^2+2}}-7 \left (\frac {9 \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {3 x^4+5 x^2+2}}+11 \left (\frac {x \left (3 x^2+2\right )}{3 \sqrt {3 x^4+5 x^2+2}}-\frac {\sqrt {2} \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{3 \sqrt {3 x^4+5 x^2+2}}\right )\right )\)

Input:

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

Output:

(x*(65 + 77*x^2))/Sqrt[2 + 5*x^2 + 3*x^4] - 7*(11*((x*(2 + 3*x^2))/(3*Sqrt 
[2 + 5*x^2 + 3*x^4]) - (Sqrt[2]*(1 + x^2)*Sqrt[(2 + 3*x^2)/(1 + x^2)]*Elli 
pticE[ArcTan[x], -1/2])/(3*Sqrt[2 + 5*x^2 + 3*x^4])) + (9*(1 + x^2)*Sqrt[( 
2 + 3*x^2)/(1 + x^2)]*EllipticF[ArcTan[x], -1/2])/(Sqrt[2]*Sqrt[2 + 5*x^2 
+ 3*x^4]))
 

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 1413

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]] /; 
 FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]
 

rule 1456

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]] /; 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]
 

rule 2206

Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = 
 Coeff[PolynomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Poly 
nomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^ 
4)^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b 
^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c))   Int[(a + b*x^2 + c 
*x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[Px, 
a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4* 
p + 7)*(b*d - 2*a*e)*x^2, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x 
^2] && Expon[Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]
 

Maple [A] (verified)

Time = 4.27 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.95

method	result	size
risch	\(\frac {x \left (77 x^{2}+65\right )}{\sqrt {3 x^{4}+5 x^{2}+2}}+\frac {63 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{2 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {77 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{3 \sqrt {3 x^{4}+5 x^{2}+2}}\)	\(124\)
elliptic	\(-\frac {6 \left (-\frac {77}{6} x^{3}-\frac {65}{6} x \right )}{\sqrt {3 x^{4}+5 x^{2}+2}}+\frac {63 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{2 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {77 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{3 \sqrt {3 x^{4}+5 x^{2}+2}}\)	\(126\)
default	\(-\frac {6 \left (-\frac {5}{6} x^{3}-\frac {2}{3} x \right )}{\sqrt {3 x^{4}+5 x^{2}+2}}+\frac {63 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{2 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {77 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{3 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {24 \left (-\frac {5}{4} x^{3}-\frac {13}{12} x \right )}{\sqrt {3 x^{4}+5 x^{2}+2}}+\frac {42 x^{3}+35 x}{\sqrt {3 x^{4}+5 x^{2}+2}}\)	\(174\)

Input:

int((x^4-7*x^2+4)/(3*x^4+5*x^2+2)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

x*(77*x^2+65)/(3*x^4+5*x^2+2)^(1/2)+63/2*I*(x^2+1)^(1/2)*(6*x^2+4)^(1/2)/( 
3*x^4+5*x^2+2)^(1/2)*EllipticF(I*x,1/2*6^(1/2))-77/3*I*(x^2+1)^(1/2)*(6*x^ 
2+4)^(1/2)/(3*x^4+5*x^2+2)^(1/2)*(EllipticF(I*x,1/2*6^(1/2))-EllipticE(I*x 
,1/2*6^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.68 \[ \int \frac {4-7 x^2+x^4}{\left (2+5 x^2+3 x^4\right )^{3/2}} \, dx=-\frac {154 \, \sqrt {2} {\left (3 i \, x^{4} + 5 i \, x^{2} + 2 i\right )} E(\arcsin \left (i \, x\right )\,|\,\frac {3}{2}) + 343 \, \sqrt {2} {\left (-3 i \, x^{4} - 5 i \, x^{2} - 2 i\right )} F(\arcsin \left (i \, x\right )\,|\,\frac {3}{2}) - 6 \, \sqrt {3 \, x^{4} + 5 \, x^{2} + 2} {\left (77 \, x^{3} + 65 \, x\right )}}{6 \, {\left (3 \, x^{4} + 5 \, x^{2} + 2\right )}} \] Input:

integrate((x^4-7*x^2+4)/(3*x^4+5*x^2+2)^(3/2),x, algorithm="fricas")
 

Output:

-1/6*(154*sqrt(2)*(3*I*x^4 + 5*I*x^2 + 2*I)*elliptic_e(arcsin(I*x), 3/2) + 
 343*sqrt(2)*(-3*I*x^4 - 5*I*x^2 - 2*I)*elliptic_f(arcsin(I*x), 3/2) - 6*s 
qrt(3*x^4 + 5*x^2 + 2)*(77*x^3 + 65*x))/(3*x^4 + 5*x^2 + 2)
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {4-7 x^2+x^4}{\left (2+5 x^2+3 x^4\right )^{3/2}} \, dx=\int \frac {x^{4} - 7 x^{2} + 4}{\left (\left (x^{2} + 1\right ) \left (3 x^{2} + 2\right )\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((x**4-7*x**2+4)/(3*x**4+5*x**2+2)**(3/2),x)
 

Output:

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

Maxima [F]

\[ \int \frac {4-7 x^2+x^4}{\left (2+5 x^2+3 x^4\right )^{3/2}} \, dx=\int { \frac {x^{4} - 7 \, x^{2} + 4}{{\left (3 \, x^{4} + 5 \, x^{2} + 2\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((x^4-7*x^2+4)/(3*x^4+5*x^2+2)^(3/2),x, algorithm="maxima")
 

Output:

integrate((x^4 - 7*x^2 + 4)/(3*x^4 + 5*x^2 + 2)^(3/2), x)
 

Giac [F]

\[ \int \frac {4-7 x^2+x^4}{\left (2+5 x^2+3 x^4\right )^{3/2}} \, dx=\int { \frac {x^{4} - 7 \, x^{2} + 4}{{\left (3 \, x^{4} + 5 \, x^{2} + 2\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((x^4-7*x^2+4)/(3*x^4+5*x^2+2)^(3/2),x, algorithm="giac")
 

Output:

integrate((x^4 - 7*x^2 + 4)/(3*x^4 + 5*x^2 + 2)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {4-7 x^2+x^4}{\left (2+5 x^2+3 x^4\right )^{3/2}} \, dx=\int \frac {x^4-7\,x^2+4}{{\left (3\,x^4+5\,x^2+2\right )}^{3/2}} \,d x \] Input:

int((x^4 - 7*x^2 + 4)/(5*x^2 + 3*x^4 + 2)^(3/2),x)
 

Output:

int((x^4 - 7*x^2 + 4)/(5*x^2 + 3*x^4 + 2)^(3/2), x)
 

Reduce [F]

\[ \int \frac {4-7 x^2+x^4}{\left (2+5 x^2+3 x^4\right )^{3/2}} \, dx=\frac {-\sqrt {3 x^{4}+5 x^{2}+2}\, x +42 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{9 x^{8}+30 x^{6}+37 x^{4}+20 x^{2}+4}d x \right ) x^{4}+70 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{9 x^{8}+30 x^{6}+37 x^{4}+20 x^{2}+4}d x \right ) x^{2}+28 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{9 x^{8}+30 x^{6}+37 x^{4}+20 x^{2}+4}d x \right )-63 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{2}}{9 x^{8}+30 x^{6}+37 x^{4}+20 x^{2}+4}d x \right ) x^{4}-105 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{2}}{9 x^{8}+30 x^{6}+37 x^{4}+20 x^{2}+4}d x \right ) x^{2}-42 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{2}}{9 x^{8}+30 x^{6}+37 x^{4}+20 x^{2}+4}d x \right )}{9 x^{4}+15 x^{2}+6} \] Input:

int((x^4-7*x^2+4)/(3*x^4+5*x^2+2)^(3/2),x)
 

Output:

( - sqrt(3*x**4 + 5*x**2 + 2)*x + 42*int(sqrt(3*x**4 + 5*x**2 + 2)/(9*x**8 
 + 30*x**6 + 37*x**4 + 20*x**2 + 4),x)*x**4 + 70*int(sqrt(3*x**4 + 5*x**2 
+ 2)/(9*x**8 + 30*x**6 + 37*x**4 + 20*x**2 + 4),x)*x**2 + 28*int(sqrt(3*x* 
*4 + 5*x**2 + 2)/(9*x**8 + 30*x**6 + 37*x**4 + 20*x**2 + 4),x) - 63*int((s 
qrt(3*x**4 + 5*x**2 + 2)*x**2)/(9*x**8 + 30*x**6 + 37*x**4 + 20*x**2 + 4), 
x)*x**4 - 105*int((sqrt(3*x**4 + 5*x**2 + 2)*x**2)/(9*x**8 + 30*x**6 + 37* 
x**4 + 20*x**2 + 4),x)*x**2 - 42*int((sqrt(3*x**4 + 5*x**2 + 2)*x**2)/(9*x 
**8 + 30*x**6 + 37*x**4 + 20*x**2 + 4),x))/(3*(3*x**4 + 5*x**2 + 2))