\(\int (1+2 x^2)^2 (4-7 x^2+x^4) (2+5 x^2+3 x^4)^{3/2} \, dx\) [198]

Optimal result

Integrand size = 36, antiderivative size = 262 \[ \int \left (1+2 x^2\right )^2 \left (4-7 x^2+x^4\right ) \left (2+5 x^2+3 x^4\right )^{3/2} \, dx=\frac {15107402 x \left (2+3 x^2\right )}{6567561 \sqrt {2+5 x^2+3 x^4}}+\frac {2 x \left (2823722+1514979 x^2\right ) \sqrt {2+5 x^2+3 x^4}}{2189187}+\frac {x \left (57469-63175 x^2\right ) \left (2+5 x^2+3 x^4\right )^{3/2}}{243243}+\frac {6067 x \left (2+5 x^2+3 x^4\right )^{5/2}}{11583}-\frac {256}{351} x^3 \left (2+5 x^2+3 x^4\right )^{5/2}+\frac {4}{45} x^5 \left (2+5 x^2+3 x^4\right )^{5/2}-\frac {15107402 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{6567561 \sqrt {2+5 x^2+3 x^4}}+\frac {6244958 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{2189187 \sqrt {2+5 x^2+3 x^4}} \] Output:

15107402/6567561*x*(3*x^2+2)/(3*x^4+5*x^2+2)^(1/2)+2/2189187*x*(1514979*x^ 
2+2823722)*(3*x^4+5*x^2+2)^(1/2)+1/243243*x*(-63175*x^2+57469)*(3*x^4+5*x^ 
2+2)^(3/2)+6067/11583*x*(3*x^4+5*x^2+2)^(5/2)-256/351*x^3*(3*x^4+5*x^2+2)^ 
(5/2)+4/45*x^5*(3*x^4+5*x^2+2)^(5/2)-15107402/6567561*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)+6244958/2189187*2^(1/2)*(x^2+1)*((3*x^2+2)/(x^2+1))^(1/2)*Inver 
seJacobiAM(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 = 10.17 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.61 \[ \int \left (1+2 x^2\right )^2 \left (4-7 x^2+x^4\right ) \left (2+5 x^2+3 x^4\right )^{3/2} \, dx=\frac {3 x \left (112685380+491968300 x^2+794467761 x^4+339980580 x^6-716368428 x^8-1182189330 x^{10}-651494907 x^{12}-84199500 x^{14}[phone] x^{16}\right )-75537010 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} E\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )|\frac {2}{3}\right )[phone] i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right ),\frac {2}{3}\right )}{32837805 \sqrt {2+5 x^2+3 x^4}} \] Input:

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

Output:

(3*x*(112685380 + 491968300*x^2 + 794467761*x^4 + 339980580*x^6 - 71636842 
8*x^8 - 1182189330*x^10 - 651494907*x^12 - 84199500*x^14 + 26270244*x^16) 
- (75537010*I)*Sqrt[3]*Sqrt[1 + x^2]*Sqrt[2 + 3*x^2]*EllipticE[I*ArcSinh[S 
qrt[3/2]*x], 2/3] + (13087430*I)*Sqrt[3]*Sqrt[1 + x^2]*Sqrt[2 + 3*x^2]*Ell 
ipticF[I*ArcSinh[Sqrt[3/2]*x], 2/3])/(32837805*Sqrt[2 + 5*x^2 + 3*x^4])
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.306, Rules used = {2207, 27, 2207, 2207, 1490, 27, 1490, 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.

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

\(\Big \downarrow \) 2207

\(\displaystyle \frac {1}{45} \int 5 \left (3 x^4+5 x^2+2\right )^{3/2} \left (-256 x^6-107 x^4+81 x^2+36\right )dx+\frac {4}{45} \left (3 x^4+5 x^2+2\right )^{5/2} x^5\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{9} \int \left (3 x^4+5 x^2+2\right )^{3/2} \left (-256 x^6-107 x^4+81 x^2+36\right )dx+\frac {4}{45} \left (3 x^4+5 x^2+2\right )^{5/2} x^5\)

\(\Big \downarrow \) 2207

\(\displaystyle \frac {1}{9} \left (\frac {1}{39} \int \left (3 x^4+5 x^2+2\right )^{3/2} \left (6067 x^4+4695 x^2+1404\right )dx-\frac {256}{39} x^3 \left (3 x^4+5 x^2+2\right )^{5/2}\right )+\frac {4}{45} \left (3 x^4+5 x^2+2\right )^{5/2} x^5\)

\(\Big \downarrow \) 2207

\(\displaystyle \frac {1}{9} \left (\frac {1}{39} \left (\frac {1}{33} \int \left (34198-27075 x^2\right ) \left (3 x^4+5 x^2+2\right )^{3/2}dx+\frac {6067}{33} x \left (3 x^4+5 x^2+2\right )^{5/2}\right )-\frac {256}{39} x^3 \left (3 x^4+5 x^2+2\right )^{5/2}\right )+\frac {4}{45} \left (3 x^4+5 x^2+2\right )^{5/2} x^5\)

\(\Big \downarrow \) 1490

\(\displaystyle \frac {1}{9} \left (\frac {1}{39} \left (\frac {1}{33} \left (\frac {1}{63} \int 6 \left (841655 x^2+660689\right ) \sqrt {3 x^4+5 x^2+2}dx+\frac {1}{21} x \left (57469-63175 x^2\right ) \left (3 x^4+5 x^2+2\right )^{3/2}\right )+\frac {6067}{33} x \left (3 x^4+5 x^2+2\right )^{5/2}\right )-\frac {256}{39} x^3 \left (3 x^4+5 x^2+2\right )^{5/2}\right )+\frac {4}{45} \left (3 x^4+5 x^2+2\right )^{5/2} x^5\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{9} \left (\frac {1}{39} \left (\frac {1}{33} \left (\frac {2}{21} \int \left (841655 x^2+660689\right ) \sqrt {3 x^4+5 x^2+2}dx+\frac {1}{21} x \left (57469-63175 x^2\right ) \left (3 x^4+5 x^2+2\right )^{3/2}\right )+\frac {6067}{33} x \left (3 x^4+5 x^2+2\right )^{5/2}\right )-\frac {256}{39} x^3 \left (3 x^4+5 x^2+2\right )^{5/2}\right )+\frac {4}{45} \left (3 x^4+5 x^2+2\right )^{5/2} x^5\)

\(\Big \downarrow \) 1490

\(\displaystyle \frac {1}{9} \left (\frac {1}{39} \left (\frac {1}{33} \left (\frac {2}{21} \left (\frac {1}{45} \int \frac {5 \left (7553701 x^2+6244958\right )}{\sqrt {3 x^4+5 x^2+2}}dx+\frac {1}{9} x \sqrt {3 x^4+5 x^2+2} \left (1514979 x^2+2823722\right )\right )+\frac {1}{21} x \left (57469-63175 x^2\right ) \left (3 x^4+5 x^2+2\right )^{3/2}\right )+\frac {6067}{33} x \left (3 x^4+5 x^2+2\right )^{5/2}\right )-\frac {256}{39} x^3 \left (3 x^4+5 x^2+2\right )^{5/2}\right )+\frac {4}{45} \left (3 x^4+5 x^2+2\right )^{5/2} x^5\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{9} \left (\frac {1}{39} \left (\frac {1}{33} \left (\frac {2}{21} \left (\frac {1}{9} \int \frac {7553701 x^2+6244958}{\sqrt {3 x^4+5 x^2+2}}dx+\frac {1}{9} x \sqrt {3 x^4+5 x^2+2} \left (1514979 x^2+2823722\right )\right )+\frac {1}{21} x \left (57469-63175 x^2\right ) \left (3 x^4+5 x^2+2\right )^{3/2}\right )+\frac {6067}{33} x \left (3 x^4+5 x^2+2\right )^{5/2}\right )-\frac {256}{39} x^3 \left (3 x^4+5 x^2+2\right )^{5/2}\right )+\frac {4}{45} \left (3 x^4+5 x^2+2\right )^{5/2} x^5\)

\(\Big \downarrow \) 1503

\(\displaystyle \frac {1}{9} \left (\frac {1}{39} \left (\frac {1}{33} \left (\frac {2}{21} \left (\frac {1}{9} \left (6244958 \int \frac {1}{\sqrt {3 x^4+5 x^2+2}}dx+7553701 \int \frac {x^2}{\sqrt {3 x^4+5 x^2+2}}dx\right )+\frac {1}{9} x \sqrt {3 x^4+5 x^2+2} \left (1514979 x^2+2823722\right )\right )+\frac {1}{21} x \left (57469-63175 x^2\right ) \left (3 x^4+5 x^2+2\right )^{3/2}\right )+\frac {6067}{33} x \left (3 x^4+5 x^2+2\right )^{5/2}\right )-\frac {256}{39} x^3 \left (3 x^4+5 x^2+2\right )^{5/2}\right )+\frac {4}{45} \left (3 x^4+5 x^2+2\right )^{5/2} x^5\)

\(\Big \downarrow \) 1413

\(\displaystyle \frac {1}{9} \left (\frac {1}{39} \left (\frac {1}{33} \left (\frac {2}{21} \left (\frac {1}{9} \left (7553701 \int \frac {x^2}{\sqrt {3 x^4+5 x^2+2}}dx+\frac {3122479 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{\sqrt {3 x^4+5 x^2+2}}\right )+\frac {1}{9} x \sqrt {3 x^4+5 x^2+2} \left (1514979 x^2+2823722\right )\right )+\frac {1}{21} x \left (57469-63175 x^2\right ) \left (3 x^4+5 x^2+2\right )^{3/2}\right )+\frac {6067}{33} x \left (3 x^4+5 x^2+2\right )^{5/2}\right )-\frac {256}{39} x^3 \left (3 x^4+5 x^2+2\right )^{5/2}\right )+\frac {4}{45} \left (3 x^4+5 x^2+2\right )^{5/2} x^5\)

\(\Big \downarrow \) 1456

\(\displaystyle \frac {1}{9} \left (\frac {1}{39} \left (\frac {1}{33} \left (\frac {2}{21} \left (\frac {1}{9} \left (\frac {3122479 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{\sqrt {3 x^4+5 x^2+2}}+7553701 \left (\frac {x \left (3 x^2+2\right )}{3 \sqrt {3 x^4+5 x^2+2}}-\frac {\sqrt {2} \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{3 \sqrt {3 x^4+5 x^2+2}}\right )\right )+\frac {1}{9} x \sqrt {3 x^4+5 x^2+2} \left (1514979 x^2+2823722\right )\right )+\frac {1}{21} x \left (57469-63175 x^2\right ) \left (3 x^4+5 x^2+2\right )^{3/2}\right )+\frac {6067}{33} x \left (3 x^4+5 x^2+2\right )^{5/2}\right )-\frac {256}{39} x^3 \left (3 x^4+5 x^2+2\right )^{5/2}\right )+\frac {4}{45} \left (3 x^4+5 x^2+2\right )^{5/2} x^5\)

Input:

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

Output:

(4*x^5*(2 + 5*x^2 + 3*x^4)^(5/2))/45 + ((-256*x^3*(2 + 5*x^2 + 3*x^4)^(5/2 
))/39 + ((6067*x*(2 + 5*x^2 + 3*x^4)^(5/2))/33 + ((x*(57469 - 63175*x^2)*( 
2 + 5*x^2 + 3*x^4)^(3/2))/21 + (2*((x*(2823722 + 1514979*x^2)*Sqrt[2 + 5*x 
^2 + 3*x^4])/9 + (7553701*((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)]*EllipticE[ArcTan[x], -1/2])/ 
(3*Sqrt[2 + 5*x^2 + 3*x^4])) + (3122479*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])/9))/21)/3 
3)/39)/9
 

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 1490

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb 
ol] :> Simp[x*(2*b*e*p + c*d*(4*p + 3) + c*e*(4*p + 1)*x^2)*((a + b*x^2 + c 
*x^4)^p/(c*(4*p + 1)*(4*p + 3))), x] + Simp[2*(p/(c*(4*p + 1)*(4*p + 3))) 
 Int[Simp[2*a*c*d*(4*p + 3) - a*b*e + (2*a*c*e*(4*p + 1) + b*c*d*(4*p + 3) 
- b^2*e*(2*p + 1))*x^2, x]*(a + b*x^2 + c*x^4)^(p - 1), x], x] /; FreeQ[{a, 
 b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && 
 GtQ[p, 0] && FractionQ[p] && IntegerQ[2*p]
 

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 2207

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

Maple [A] (verified)

Time = 10.86 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.57

method	result	size
risch	\(\frac {x \left (8756748 x^{12}-42661080 x^{10}-151901001 x^{8}-112454055 x^{6}[phone] x^{4}[phone] x^{2}[phone]\right ) \sqrt {3 x^{4}+5 x^{2}+2}}{10945935}-\frac {6244958 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{2189187 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {15107402 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{6567561 \sqrt {3 x^{4}+5 x^{2}+2}}\)	\(150\)
default	\(\frac {5544587 x^{5} \sqrt {3 x^{4}+5 x^{2}+2}}{1216215}+\frac {179705 x^{3} \sqrt {3 x^{4}+5 x^{2}+2}}{18711}+\frac {11268538 x \sqrt {3 x^{4}+5 x^{2}+2}}{2189187}-\frac {6244958 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{2189187 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {15107402 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{6567561 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {118999 x^{7} \sqrt {3 x^{4}+5 x^{2}+2}}{11583}-\frac {29767 x^{9} \sqrt {3 x^{4}+5 x^{2}+2}}{2145}-\frac {152 x^{11} \sqrt {3 x^{4}+5 x^{2}+2}}{39}+\frac {4 x^{13} \sqrt {3 x^{4}+5 x^{2}+2}}{5}\)	\(232\)
elliptic	\(\frac {5544587 x^{5} \sqrt {3 x^{4}+5 x^{2}+2}}{1216215}+\frac {179705 x^{3} \sqrt {3 x^{4}+5 x^{2}+2}}{18711}+\frac {11268538 x \sqrt {3 x^{4}+5 x^{2}+2}}{2189187}-\frac {6244958 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{2189187 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {15107402 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{6567561 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {118999 x^{7} \sqrt {3 x^{4}+5 x^{2}+2}}{11583}-\frac {29767 x^{9} \sqrt {3 x^{4}+5 x^{2}+2}}{2145}-\frac {152 x^{11} \sqrt {3 x^{4}+5 x^{2}+2}}{39}+\frac {4 x^{13} \sqrt {3 x^{4}+5 x^{2}+2}}{5}\)	\(232\)

Input:

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

Output:

1/10945935*x*(8756748*x^12-42661080*x^10-151901001*x^8-112454055*x^6+49901 
283*x^4+105127425*x^2+56342690)*(3*x^4+5*x^2+2)^(1/2)-6244958/2189187*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 
))[phone]/6567561*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 = 97, normalized size of antiderivative = 0.37 \[ \int \left (1+2 x^2\right )^2 \left (4-7 x^2+x^4\right ) \left (2+5 x^2+3 x^4\right )^{3/2} \, dx=-\frac {151074020 \, \sqrt {3} \sqrt {-\frac {2}{3}} x E(\arcsin \left (\frac {\sqrt {-\frac {2}{3}}}{x}\right )\,|\,\frac {3}{2}) - 432097130 \, \sqrt {3} \sqrt {-\frac {2}{3}} x F(\arcsin \left (\frac {\sqrt {-\frac {2}{3}}}{x}\right )\,|\,\frac {3}{2}) - 3 \, {\left (26270244 \, x^{14} - 127983240 \, x^{12} - 455703003 \, x^{10} - 337362165 \, x^{8} + 149703849 \, x^{6} + 315382275 \, x^{4} + 169028070 \, x^{2} + 75537010\right )} \sqrt {3 \, x^{4} + 5 \, x^{2} + 2}}{98513415 \, x} \] Input:

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

Output:

-1/98513415*(151074020*sqrt(3)*sqrt(-2/3)*x*elliptic_e(arcsin(sqrt(-2/3)/x 
), 3/2) - 432097130*sqrt(3)*sqrt(-2/3)*x*elliptic_f(arcsin(sqrt(-2/3)/x), 
3/2) - 3*(26270244*x^14 - 127983240*x^12 - 455703003*x^10 - 337362165*x^8 
+ 149703849*x^6 + 315382275*x^4 + 169028070*x^2 + 75537010)*sqrt(3*x^4 + 5 
*x^2 + 2))/x
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \left (1+2 x^2\right )^2 \left (4-7 x^2+x^4\right ) \left (2+5 x^2+3 x^4\right )^{3/2} \, dx=\frac {4 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{13}}{5}-\frac {152 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{11}}{39}-\frac {29767 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{9}}{2145}-\frac {118999 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{7}}{11583}+\frac {5544587 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{5}}{1216215}+\frac {179705 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{3}}{18711}+\frac {11268538 \sqrt {3 x^{4}+5 x^{2}+2}\, x}{2189187}+\frac {12489916 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{3 x^{4}+5 x^{2}+2}d x \right )}{2189187}+\frac {15107402 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{2}}{3 x^{4}+5 x^{2}+2}d x \right )}{2189187} \] Input:

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

Output:

(8756748*sqrt(3*x**4 + 5*x**2 + 2)*x**13 - 42661080*sqrt(3*x**4 + 5*x**2 + 
 2)*x**11 - 151901001*sqrt(3*x**4 + 5*x**2 + 2)*x**9 - 112454055*sqrt(3*x* 
*4 + 5*x**2 + 2)*x**7 + 49901283*sqrt(3*x**4 + 5*x**2 + 2)*x**5 + 10512742 
5*sqrt(3*x**4 + 5*x**2 + 2)*x**3 + 56342690*sqrt(3*x**4 + 5*x**2 + 2)*x + 
62449580*int(sqrt(3*x**4 + 5*x**2 + 2)/(3*x**4 + 5*x**2 + 2),x) + 75537010 
*int((sqrt(3*x**4 + 5*x**2 + 2)*x**2)/(3*x**4 + 5*x**2 + 2),x))/10945935