\(\int \frac {4-7 x^2+x^4}{(1+2 x^2)^3 (2+5 x^2+3 x^4)^{3/2}} \, dx\) [217]

Optimal result

Integrand size = 36, antiderivative size = 293 \[ \int \frac {4-7 x^2+x^4}{\left (1+2 x^2\right )^3 \left (2+5 x^2+3 x^4\right )^{3/2}} \, dx=\frac {31 x}{4 \left (1+2 x^2\right )^2 \sqrt {2+5 x^2+3 x^4}}-\frac {591 x}{8 \left (1+2 x^2\right ) \sqrt {2+5 x^2+3 x^4}}+\frac {2363 x \left (2+3 x^2\right )}{4 \sqrt {2+5 x^2+3 x^4}}-\frac {x \left (14735+14178 x^2\right )}{8 \sqrt {2+5 x^2+3 x^4}}-\frac {2363 \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{2 \sqrt {2} \sqrt {2+5 x^2+3 x^4}}-\frac {2565 \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {2+5 x^2+3 x^4}}+\frac {6385 \left (1+x^2\right ) \operatorname {EllipticPi}\left (-\frac {1}{3},\arctan \left (\sqrt {\frac {3}{2}} x\right ),\frac {1}{3}\right )}{2 \sqrt {3} \sqrt {\frac {1+x^2}{2+3 x^2}} \sqrt {2+5 x^2+3 x^4}} \] Output:

31/4*x/(2*x^2+1)^2/(3*x^4+5*x^2+2)^(1/2)-591/8*x/(2*x^2+1)/(3*x^4+5*x^2+2) 
^(1/2)+2363/4*x*(3*x^2+2)/(3*x^4+5*x^2+2)^(1/2)-1/8*x*(14178*x^2+14735)/(3 
*x^4+5*x^2+2)^(1/2)-2363/4*2^(1/2)*(x^2+1)*((3*x^2+2)/(x^2+1))^(1/2)*Ellip 
ticE(x/(x^2+1)^(1/2),1/2*I*2^(1/2))/(3*x^4+5*x^2+2)^(1/2)-2565/4*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)+6385/6*(x^2+1)*EllipticPi(x*6^(1/2)/(6*x^2+4)^(1/2), 
-1/3,1/3*3^(1/2))*3^(1/2)/((x^2+1)/(3*x^2+2))^(1/2)/(3*x^4+5*x^2+2)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.88 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.66 \[ \int \frac {4-7 x^2+x^4}{\left (1+2 x^2\right )^3 \left (2+5 x^2+3 x^4\right )^{3/2}} \, dx=\frac {-\frac {12 x \left (3816+18575 x^2+28913 x^4+14178 x^6\right )}{\left (1+2 x^2\right )^2}-14178 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} E\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )|\frac {2}{3}\right )+5283 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right ),\frac {2}{3}\right )-6385 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} \operatorname {EllipticPi}\left (\frac {4}{3},i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right ),\frac {2}{3}\right )}{24 \sqrt {2+5 x^2+3 x^4}} \] Input:

Integrate[(4 - 7*x^2 + x^4)/((1 + 2*x^2)^3*(2 + 5*x^2 + 3*x^4)^(3/2)),x]
 

Output:

((-12*x*(3816 + 18575*x^2 + 28913*x^4 + 14178*x^6))/(1 + 2*x^2)^2 - (14178 
*I)*Sqrt[3]*Sqrt[1 + x^2]*Sqrt[2 + 3*x^2]*EllipticE[I*ArcSinh[Sqrt[3/2]*x] 
, 2/3] + (5283*I)*Sqrt[3]*Sqrt[1 + x^2]*Sqrt[2 + 3*x^2]*EllipticF[I*ArcSin 
h[Sqrt[3/2]*x], 2/3] - (6385*I)*Sqrt[3]*Sqrt[1 + x^2]*Sqrt[2 + 3*x^2]*Elli 
pticPi[4/3, I*ArcSinh[Sqrt[3/2]*x], 2/3])/(24*Sqrt[2 + 5*x^2 + 3*x^4])
 

Rubi [A] (verified)

Time = 1.14 (sec) , antiderivative size = 567, normalized size of antiderivative = 1.94, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2258, 2009}

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.

\(\displaystyle \int \frac {x^4-7 x^2+4}{\left (2 x^2+1\right )^3 \left (3 x^4+5 x^2+2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 2258

\(\displaystyle \int \left (\frac {12}{\left (x^2+1\right ) \sqrt {3 x^4+5 x^2+2}}+\frac {468}{\left (2 x^2+1\right ) \sqrt {3 x^4+5 x^2+2}}-\frac {738}{\left (3 x^2+2\right ) \sqrt {3 x^4+5 x^2+2}}-\frac {140}{\left (2 x^2+1\right )^2 \sqrt {3 x^4+5 x^2+2}}+\frac {31}{\left (2 x^2+1\right )^3 \sqrt {3 x^4+5 x^2+2}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {912 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{\sqrt {3 x^4+5 x^2+2}}-\frac {465 \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {3 x^4+5 x^2+2}}+\frac {387 \sqrt {2} \left (3 x^2+2\right ) \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{\sqrt {\frac {3 x^2+2}{x^2+1}} \sqrt {3 x^4+5 x^2+2}}-\frac {140 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{\sqrt {3 x^4+5 x^2+2}}-\frac {279 \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{2 \sqrt {2} \sqrt {3 x^4+5 x^2+2}}-\frac {381 \sqrt {2} \left (3 x^2+2\right ) E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{\sqrt {\frac {3 x^2+2}{x^2+1}} \sqrt {3 x^4+5 x^2+2}}+\frac {904 \sqrt {3} \left (x^2+1\right ) \operatorname {EllipticPi}\left (-\frac {1}{3},\arctan \left (\sqrt {\frac {3}{2}} x\right ),\frac {1}{3}\right )}{\sqrt {\frac {x^2+1}{3 x^2+2}} \sqrt {3 x^4+5 x^2+2}}+\frac {961 \left (x^2+1\right ) \operatorname {EllipticPi}\left (-\frac {1}{3},\arctan \left (\sqrt {\frac {3}{2}} x\right ),\frac {1}{3}\right )}{2 \sqrt {3} \sqrt {\frac {x^2+1}{3 x^2+2}} \sqrt {3 x^4+5 x^2+2}}-\frac {839 \sqrt {3 x^4+5 x^2+2} x}{2 \left (2 x^2+1\right )}+\frac {31 \sqrt {3 x^4+5 x^2+2} x}{\left (2 x^2+1\right )^2}-\frac {1107 \left (x^2+1\right ) x}{\sqrt {3 x^4+5 x^2+2}}+\frac {2315 \left (3 x^2+2\right ) x}{4 \sqrt {3 x^4+5 x^2+2}}\)

Input:

Int[(4 - 7*x^2 + x^4)/((1 + 2*x^2)^3*(2 + 5*x^2 + 3*x^4)^(3/2)),x]
 

Output:

(-1107*x*(1 + x^2))/Sqrt[2 + 5*x^2 + 3*x^4] + (2315*x*(2 + 3*x^2))/(4*Sqrt 
[2 + 5*x^2 + 3*x^4]) + (31*x*Sqrt[2 + 5*x^2 + 3*x^4])/(1 + 2*x^2)^2 - (839 
*x*Sqrt[2 + 5*x^2 + 3*x^4])/(2*(1 + 2*x^2)) - (381*Sqrt[2]*(2 + 3*x^2)*Ell 
ipticE[ArcTan[x], -1/2])/(Sqrt[(2 + 3*x^2)/(1 + x^2)]*Sqrt[2 + 5*x^2 + 3*x 
^4]) - (279*(1 + x^2)*Sqrt[(2 + 3*x^2)/(1 + x^2)]*EllipticE[ArcTan[x], -1/ 
2])/(2*Sqrt[2]*Sqrt[2 + 5*x^2 + 3*x^4]) - (140*Sqrt[2]*(1 + x^2)*Sqrt[(2 + 
 3*x^2)/(1 + x^2)]*EllipticE[ArcTan[x], -1/2])/Sqrt[2 + 5*x^2 + 3*x^4] + ( 
387*Sqrt[2]*(2 + 3*x^2)*EllipticF[ArcTan[x], -1/2])/(Sqrt[(2 + 3*x^2)/(1 + 
 x^2)]*Sqrt[2 + 5*x^2 + 3*x^4]) - (465*(1 + x^2)*Sqrt[(2 + 3*x^2)/(1 + x^2 
)]*EllipticF[ArcTan[x], -1/2])/(2*Sqrt[2]*Sqrt[2 + 5*x^2 + 3*x^4]) - (912* 
Sqrt[2]*(1 + x^2)*Sqrt[(2 + 3*x^2)/(1 + x^2)]*EllipticF[ArcTan[x], -1/2])/ 
Sqrt[2 + 5*x^2 + 3*x^4] + (961*(1 + x^2)*EllipticPi[-1/3, ArcTan[Sqrt[3/2] 
*x], 1/3])/(2*Sqrt[3]*Sqrt[(1 + x^2)/(2 + 3*x^2)]*Sqrt[2 + 5*x^2 + 3*x^4]) 
 + (904*Sqrt[3]*(1 + x^2)*EllipticPi[-1/3, ArcTan[Sqrt[3/2]*x], 1/3])/(Sqr 
t[(1 + x^2)/(2 + 3*x^2)]*Sqrt[2 + 5*x^2 + 3*x^4])
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2258

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^ 
(p_), x_Symbol] :> Int[ExpandIntegrand[1/Sqrt[a + b*x^2 + c*x^4], Px*(d + e 
*x^2)^q*(a + b*x^2 + c*x^4)^(p + 1/2), x], x] /; FreeQ[{a, b, c, d, e}, x] 
&& PolyQ[Px, x] && IntegerQ[p + 1/2] && IntegerQ[q]
 

Maple [A] (verified)

Time = 10.62 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.66

method	result	size
risch	\(-\frac {x \left (14178 x^{6}+28913 x^{4}+18575 x^{2}+3816\right )}{2 \left (2 x^{2}+1\right )^{2} \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {8895 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{16 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {2363 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{4 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {6385 i \sqrt {x^{2}+1}\, \sqrt {1+\frac {3 x^{2}}{2}}\, \operatorname {EllipticPi}\left (i x , 2, \frac {i \sqrt {-3}\, \sqrt {2}}{2}\right )}{8 \sqrt {3 x^{4}+5 x^{2}+2}}\)	\(192\)
elliptic	\(\frac {31 x \sqrt {3 x^{4}+5 x^{2}+2}}{\left (2 x^{2}+1\right )^{2}}-\frac {839 x \sqrt {3 x^{4}+5 x^{2}+2}}{2 \left (2 x^{2}+1\right )}-\frac {6 \left (\frac {381}{2} x^{3}+\frac {377}{2} x \right )}{\sqrt {3 x^{4}+5 x^{2}+2}}+\frac {557 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{16 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {2363 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )}{4 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {6385 i \sqrt {x^{2}+1}\, \sqrt {1+\frac {3 x^{2}}{2}}\, \operatorname {EllipticPi}\left (i x , 2, \frac {i \sqrt {-3}\, \sqrt {2}}{2}\right )}{8 \sqrt {3 x^{4}+5 x^{2}+2}}\)	\(213\)
default	\(\frac {31 x \sqrt {3 x^{4}+5 x^{2}+2}}{\left (2 x^{2}+1\right )^{2}}-\frac {839 x \sqrt {3 x^{4}+5 x^{2}+2}}{2 \left (2 x^{2}+1\right )}-\frac {93 \left (\frac {83}{4} x^{3}+\frac {247}{12} x \right )}{2 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {557 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{16 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {2363 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )}{4 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {6385 i \sqrt {x^{2}+1}\, \sqrt {1+\frac {3 x^{2}}{2}}\, \operatorname {EllipticPi}\left (i x , 2, \frac {i \sqrt {-3}\, \sqrt {2}}{2}\right )}{8 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {-174 x^{3}-170 x}{\sqrt {3 x^{4}+5 x^{2}+2}}-\frac {3 \left (\frac {11}{4} x^{3}+\frac {31}{12} x \right )}{2 \sqrt {3 x^{4}+5 x^{2}+2}}\)	\(263\)

Input:

int((x^4-7*x^2+4)/(2*x^2+1)^3/(3*x^4+5*x^2+2)^(3/2),x,method=_RETURNVERBOS 
E)
 

Output:

-1/2*x*(14178*x^6+28913*x^4+18575*x^2+3816)/(2*x^2+1)^2/(3*x^4+5*x^2+2)^(1 
/2)-8895/16*I*(x^2+1)^(1/2)*(6*x^2+4)^(1/2)/(3*x^4+5*x^2+2)^(1/2)*Elliptic 
F(I*x,1/2*6^(1/2))+2363/4*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)))-6385/8*I*(x^ 
2+1)^(1/2)*(1+3/2*x^2)^(1/2)/(3*x^4+5*x^2+2)^(1/2)*EllipticPi(I*x,2,1/2*I* 
(-3)^(1/2)*2^(1/2))
 

Fricas [F]

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

integrate((x^4-7*x^2+4)/(2*x^2+1)^3/(3*x^4+5*x^2+2)^(3/2),x, algorithm="fr 
icas")
 

Output:

integral(sqrt(3*x^4 + 5*x^2 + 2)*(x^4 - 7*x^2 + 4)/(72*x^14 + 348*x^12 + 7 
10*x^10 + 793*x^8 + 524*x^6 + 205*x^4 + 44*x^2 + 4), x)
 

Sympy [F]

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

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

Maxima [F]

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

integrate((x^4-7*x^2+4)/(2*x^2+1)^3/(3*x^4+5*x^2+2)^(3/2),x, algorithm="ma 
xima")
 

Output:

integrate((x^4 - 7*x^2 + 4)/((3*x^4 + 5*x^2 + 2)^(3/2)*(2*x^2 + 1)^3), x)
 

Giac [F]

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

integrate((x^4-7*x^2+4)/(2*x^2+1)^3/(3*x^4+5*x^2+2)^(3/2),x, algorithm="gi 
ac")
 

Output:

integrate((x^4 - 7*x^2 + 4)/((3*x^4 + 5*x^2 + 2)^(3/2)*(2*x^2 + 1)^3), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

int((x^4 - 7*x^2 + 4)/((2*x^2 + 1)^3*(5*x^2 + 3*x^4 + 2)^(3/2)),x)
 

Output:

int((x^4 - 7*x^2 + 4)/((2*x^2 + 1)^3*(5*x^2 + 3*x^4 + 2)^(3/2)), x)
 

Reduce [F]

\[ \int \frac {4-7 x^2+x^4}{\left (1+2 x^2\right )^3 \left (2+5 x^2+3 x^4\right )^{3/2}} \, dx=4 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{72 x^{14}+348 x^{12}+710 x^{10}+793 x^{8}+524 x^{6}+205 x^{4}+44 x^{2}+4}d x \right )+\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{4}}{72 x^{14}+348 x^{12}+710 x^{10}+793 x^{8}+524 x^{6}+205 x^{4}+44 x^{2}+4}d x -7 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{2}}{72 x^{14}+348 x^{12}+710 x^{10}+793 x^{8}+524 x^{6}+205 x^{4}+44 x^{2}+4}d x \right ) \] Input:

int((x^4-7*x^2+4)/(2*x^2+1)^3/(3*x^4+5*x^2+2)^(3/2),x)
 

Output:

4*int(sqrt(3*x**4 + 5*x**2 + 2)/(72*x**14 + 348*x**12 + 710*x**10 + 793*x* 
*8 + 524*x**6 + 205*x**4 + 44*x**2 + 4),x) + int((sqrt(3*x**4 + 5*x**2 + 2 
)*x**4)/(72*x**14 + 348*x**12 + 710*x**10 + 793*x**8 + 524*x**6 + 205*x**4 
 + 44*x**2 + 4),x) - 7*int((sqrt(3*x**4 + 5*x**2 + 2)*x**2)/(72*x**14 + 34 
8*x**12 + 710*x**10 + 793*x**8 + 524*x**6 + 205*x**4 + 44*x**2 + 4),x)