\(\int \frac {(4-7 x^2+x^4) \sqrt {2+5 x^2+3 x^4}}{1+2 x^2} \, dx\) [194]

Optimal result

Integrand size = 36, antiderivative size = 249 \[ \int \frac {\left (4-7 x^2+x^4\right ) \sqrt {2+5 x^2+3 x^4}}{1+2 x^2} \, dx=\frac {1879 x \left (2+3 x^2\right )}{1080 \sqrt {2+5 x^2+3 x^4}}-\frac {43}{36} x \sqrt {2+5 x^2+3 x^4}+\frac {1}{10} x^3 \sqrt {2+5 x^2+3 x^4}-\frac {1879 \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{540 \sqrt {2} \sqrt {2+5 x^2+3 x^4}}+\frac {95 \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{36 \sqrt {2} \sqrt {2+5 x^2+3 x^4}}+\frac {31 \left (1+x^2\right ) \operatorname {EllipticPi}\left (-\frac {1}{3},\arctan \left (\sqrt {\frac {3}{2}} x\right ),\frac {1}{3}\right )}{4 \sqrt {3} \sqrt {\frac {1+x^2}{2+3 x^2}} \sqrt {2+5 x^2+3 x^4}} \] Output:

1879/1080*x*(3*x^2+2)/(3*x^4+5*x^2+2)^(1/2)-43/36*x*(3*x^4+5*x^2+2)^(1/2)+ 
1/10*x^3*(3*x^4+5*x^2+2)^(1/2)-1879/1080*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)+95 
/72*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)+31/12*(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 = 10.36 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.73 \[ \int \frac {\left (4-7 x^2+x^4\right ) \sqrt {2+5 x^2+3 x^4}}{1+2 x^2} \, dx=\frac {-5160 x-12468 x^3-6660 x^5+648 x^7-3758 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} E\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )|\frac {2}{3}\right )-2327 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right ),\frac {2}{3}\right )-1395 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} \operatorname {EllipticPi}\left (\frac {4}{3},i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right ),\frac {2}{3}\right )}{2160 \sqrt {2+5 x^2+3 x^4}} \] Input:

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

Output:

(-5160*x - 12468*x^3 - 6660*x^5 + 648*x^7 - (3758*I)*Sqrt[3]*Sqrt[1 + x^2] 
*Sqrt[2 + 3*x^2]*EllipticE[I*ArcSinh[Sqrt[3/2]*x], 2/3] - (2327*I)*Sqrt[3] 
*Sqrt[1 + x^2]*Sqrt[2 + 3*x^2]*EllipticF[I*ArcSinh[Sqrt[3/2]*x], 2/3] - (1 
395*I)*Sqrt[3]*Sqrt[1 + x^2]*Sqrt[2 + 3*x^2]*EllipticPi[4/3, I*ArcSinh[Sqr 
t[3/2]*x], 2/3])/(2160*Sqrt[2 + 5*x^2 + 3*x^4])
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.44, 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.

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

\(\Big \downarrow \) 2258

\(\displaystyle \int \left (-\frac {35 x^4}{4 \sqrt {3 x^4+5 x^2+2}}-\frac {49 x^2}{8 \sqrt {3 x^4+5 x^2+2}}+\frac {31}{16 \left (2 x^2+1\right ) \sqrt {3 x^4+5 x^2+2}}+\frac {97}{16 \sqrt {3 x^4+5 x^2+2}}+\frac {3 x^6}{2 \sqrt {3 x^4+5 x^2+2}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{9 \sqrt {3 x^4+5 x^2+2}}+\frac {79 \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{36 \sqrt {2} \sqrt {3 x^4+5 x^2+2}}-\frac {73 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{135 \sqrt {3 x^4+5 x^2+2}}-\frac {259 \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{108 \sqrt {2} \sqrt {3 x^4+5 x^2+2}}+\frac {31 \left (x^2+1\right ) \operatorname {EllipticPi}\left (-\frac {1}{3},\arctan \left (\sqrt {\frac {3}{2}} x\right ),\frac {1}{3}\right )}{4 \sqrt {3} \sqrt {\frac {x^2+1}{3 x^2+2}} \sqrt {3 x^4+5 x^2+2}}-\frac {43}{36} \sqrt {3 x^4+5 x^2+2} x+\frac {1879 \left (3 x^2+2\right ) x}{1080 \sqrt {3 x^4+5 x^2+2}}+\frac {1}{10} \sqrt {3 x^4+5 x^2+2} x^3\)

Input:

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

Output:

(1879*x*(2 + 3*x^2))/(1080*Sqrt[2 + 5*x^2 + 3*x^4]) - (43*x*Sqrt[2 + 5*x^2 
 + 3*x^4])/36 + (x^3*Sqrt[2 + 5*x^2 + 3*x^4])/10 - (259*(1 + x^2)*Sqrt[(2 
+ 3*x^2)/(1 + x^2)]*EllipticE[ArcTan[x], -1/2])/(108*Sqrt[2]*Sqrt[2 + 5*x^ 
2 + 3*x^4]) - (73*Sqrt[2]*(1 + x^2)*Sqrt[(2 + 3*x^2)/(1 + x^2)]*EllipticE[ 
ArcTan[x], -1/2])/(135*Sqrt[2 + 5*x^2 + 3*x^4]) + (79*(1 + x^2)*Sqrt[(2 + 
3*x^2)/(1 + x^2)]*EllipticF[ArcTan[x], -1/2])/(36*Sqrt[2]*Sqrt[2 + 5*x^2 + 
 3*x^4]) + (2*Sqrt[2]*(1 + x^2)*Sqrt[(2 + 3*x^2)/(1 + x^2)]*EllipticF[ArcT 
an[x], -1/2])/(9*Sqrt[2 + 5*x^2 + 3*x^4]) + (31*(1 + x^2)*EllipticPi[-1/3, 
 ArcTan[Sqrt[3/2]*x], 1/3])/(4*Sqrt[3]*Sqrt[(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 = 5.14 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.69

method	result	size
elliptic	\(\frac {x^{3} \sqrt {3 x^{4}+5 x^{2}+2}}{10}-\frac {43 x \sqrt {3 x^{4}+5 x^{2}+2}}{36}-\frac {10739 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{4320 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {1879 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )}{1080 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {31 i \sqrt {x^{2}+1}\, \sqrt {1+\frac {3 x^{2}}{2}}\, \operatorname {EllipticPi}\left (i x , 2, \frac {i \sqrt {-3}\, \sqrt {2}}{2}\right )}{16 \sqrt {3 x^{4}+5 x^{2}+2}}\)	\(172\)
risch	\(\frac {x \left (18 x^{2}-215\right ) \sqrt {3 x^{4}+5 x^{2}+2}}{180}-\frac {1217 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{288 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {1879 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{1080 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {31 i \sqrt {x^{2}+1}\, \sqrt {1+\frac {3 x^{2}}{2}}\, \operatorname {EllipticPi}\left (i x , 2, \frac {i \sqrt {-3}\, \sqrt {2}}{2}\right )}{16 \sqrt {3 x^{4}+5 x^{2}+2}}\)	\(173\)
default	\(-\frac {43 x \sqrt {3 x^{4}+5 x^{2}+2}}{36}-\frac {101 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{288 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {1153 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{540 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {x^{3} \sqrt {3 x^{4}+5 x^{2}+2}}{10}-\frac {31 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )}{8 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {31 i \sqrt {x^{2}+1}\, \sqrt {1+\frac {3 x^{2}}{2}}\, \operatorname {EllipticPi}\left (i x , 2, \frac {i \sqrt {-3}\, \sqrt {2}}{2}\right )}{16 \sqrt {3 x^{4}+5 x^{2}+2}}\)	\(228\)

Input:

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

Output:

1/10*x^3*(3*x^4+5*x^2+2)^(1/2)-43/36*x*(3*x^4+5*x^2+2)^(1/2)-10739/4320*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))-1879/1080*I*(x^2+1)^(1/2)*(6*x^2+4)^(1/2)/(3*x^4+5*x^2+2)^(1/2)*Ellip 
ticE(I*x,1/2*6^(1/2))-31/16*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]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (4-7 x^2+x^4\right ) \sqrt {2+5 x^2+3 x^4}}{1+2 x^2} \, dx=\frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{3}}{10}-\frac {43 \sqrt {3 x^{4}+5 x^{2}+2}\, x}{36}+\frac {187 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{6 x^{6}+13 x^{4}+9 x^{2}+2}d x \right )}{18}+\frac {1879 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{4}}{6 x^{6}+13 x^{4}+9 x^{2}+2}d x \right )}{180}+\frac {1991 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{2}}{6 x^{6}+13 x^{4}+9 x^{2}+2}d x \right )}{90} \] Input:

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

Output:

(18*sqrt(3*x**4 + 5*x**2 + 2)*x**3 - 215*sqrt(3*x**4 + 5*x**2 + 2)*x + 187 
0*int(sqrt(3*x**4 + 5*x**2 + 2)/(6*x**6 + 13*x**4 + 9*x**2 + 2),x) + 1879* 
int((sqrt(3*x**4 + 5*x**2 + 2)*x**4)/(6*x**6 + 13*x**4 + 9*x**2 + 2),x) + 
3982*int((sqrt(3*x**4 + 5*x**2 + 2)*x**2)/(6*x**6 + 13*x**4 + 9*x**2 + 2), 
x))/180