\(\int (1+2 x^2) (4-7 x^2+x^4) (2+5 x^2+3 x^4)^{3/2} \, dx\) [199]

Optimal result

Integrand size = 34, antiderivative size = 239 \[ \int \left (1+2 x^2\right ) \left (4-7 x^2+x^4\right ) \left (2+5 x^2+3 x^4\right )^{3/2} \, dx=\frac {9842698 x \left (2+3 x^2\right )}{3648645 \sqrt {2+5 x^2+3 x^4}}+\frac {2 x \left (1708960+836181 x^2\right ) \sqrt {2+5 x^2+3 x^4}}{1216215}+\frac {x \left (49801+43169 x^2\right ) \left (2+5 x^2+3 x^4\right )^{3/2}}{27027}-\frac {587 x \left (2+5 x^2+3 x^4\right )^{5/2}}{1287}+\frac {2}{39} x^3 \left (2+5 x^2+3 x^4\right )^{5/2}-\frac {9842698 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{3648645 \sqrt {2+5 x^2+3 x^4}}+\frac {809714 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{243243 \sqrt {2+5 x^2+3 x^4}} \] Output:

9842698/3648645*x*(3*x^2+2)/(3*x^4+5*x^2+2)^(1/2)+2/1216215*x*(836181*x^2+ 
1708960)*(3*x^4+5*x^2+2)^(1/2)+1/27027*x*(43169*x^2+49801)*(3*x^4+5*x^2+2) 
^(3/2)-587/1287*x*(3*x^4+5*x^2+2)^(5/2)+2/39*x^3*(3*x^4+5*x^2+2)^(5/2)-984 
2698/3648645*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)+809714/243243*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 = 10.13 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.65 \[ \int \left (1+2 x^2\right ) \left (4-7 x^2+x^4\right ) \left (2+5 x^2+3 x^4\right )^{3/2} \, dx=\frac {3 x \left (11362300+40241704 x^2+40951545 x^4-13462434 x^6-57295080 x^8-40001850 x^{10}-6557355 x^{12}+1683990 x^{14}\right )-9842698 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} E\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )|\frac {2}{3}\right )+1745558 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right ),\frac {2}{3}\right )}{3648645 \sqrt {2+5 x^2+3 x^4}} \] Input:

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

Output:

(3*x*(11362300 + 40241704*x^2 + 40951545*x^4 - 13462434*x^6 - 57295080*x^8 
 - 40001850*x^10 - 6557355*x^12 + 1683990*x^14) - (9842698*I)*Sqrt[3]*Sqrt 
[1 + x^2]*Sqrt[2 + 3*x^2]*EllipticE[I*ArcSinh[Sqrt[3/2]*x], 2/3] + (174555 
8*I)*Sqrt[3]*Sqrt[1 + x^2]*Sqrt[2 + 3*x^2]*EllipticF[I*ArcSinh[Sqrt[3/2]*x 
], 2/3])/(3648645*Sqrt[2 + 5*x^2 + 3*x^4])
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2207, 2207, 1490, 27, 1490, 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.

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

\(\Big \downarrow \) 2207

\(\displaystyle \frac {1}{39} \int \left (-587 x^4+27 x^2+156\right ) \left (3 x^4+5 x^2+2\right )^{3/2}dx+\frac {2}{39} \left (3 x^4+5 x^2+2\right )^{5/2} x^3\)

\(\Big \downarrow \) 2207

\(\displaystyle \frac {1}{39} \left (\frac {1}{33} \int \left (18501 x^2+6322\right ) \left (3 x^4+5 x^2+2\right )^{3/2}dx-\frac {587}{33} x \left (3 x^4+5 x^2+2\right )^{5/2}\right )+\frac {2}{39} \left (3 x^4+5 x^2+2\right )^{5/2} x^3\)

\(\Big \downarrow \) 1490

\(\displaystyle \frac {1}{39} \left (\frac {1}{33} \left (\frac {1}{63} \int 6 \left (92909 x^2+82961\right ) \sqrt {3 x^4+5 x^2+2}dx+\frac {1}{21} x \left (43169 x^2+49801\right ) \left (3 x^4+5 x^2+2\right )^{3/2}\right )-\frac {587}{33} x \left (3 x^4+5 x^2+2\right )^{5/2}\right )+\frac {2}{39} \left (3 x^4+5 x^2+2\right )^{5/2} x^3\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{39} \left (\frac {1}{33} \left (\frac {2}{21} \int \left (92909 x^2+82961\right ) \sqrt {3 x^4+5 x^2+2}dx+\frac {1}{21} x \left (43169 x^2+49801\right ) \left (3 x^4+5 x^2+2\right )^{3/2}\right )-\frac {587}{33} x \left (3 x^4+5 x^2+2\right )^{5/2}\right )+\frac {2}{39} \left (3 x^4+5 x^2+2\right )^{5/2} x^3\)

\(\Big \downarrow \) 1490

\(\displaystyle \frac {1}{39} \left (\frac {1}{33} \left (\frac {2}{21} \left (\frac {1}{45} \int \frac {4921349 x^2+4048570}{\sqrt {3 x^4+5 x^2+2}}dx+\frac {1}{45} x \sqrt {3 x^4+5 x^2+2} \left (836181 x^2+1708960\right )\right )+\frac {1}{21} x \left (43169 x^2+49801\right ) \left (3 x^4+5 x^2+2\right )^{3/2}\right )-\frac {587}{33} x \left (3 x^4+5 x^2+2\right )^{5/2}\right )+\frac {2}{39} \left (3 x^4+5 x^2+2\right )^{5/2} x^3\)

\(\Big \downarrow \) 1503

\(\displaystyle \frac {1}{39} \left (\frac {1}{33} \left (\frac {2}{21} \left (\frac {1}{45} \left (4048570 \int \frac {1}{\sqrt {3 x^4+5 x^2+2}}dx+4921349 \int \frac {x^2}{\sqrt {3 x^4+5 x^2+2}}dx\right )+\frac {1}{45} x \sqrt {3 x^4+5 x^2+2} \left (836181 x^2+1708960\right )\right )+\frac {1}{21} x \left (43169 x^2+49801\right ) \left (3 x^4+5 x^2+2\right )^{3/2}\right )-\frac {587}{33} x \left (3 x^4+5 x^2+2\right )^{5/2}\right )+\frac {2}{39} \left (3 x^4+5 x^2+2\right )^{5/2} x^3\)

\(\Big \downarrow \) 1413

\(\displaystyle \frac {1}{39} \left (\frac {1}{33} \left (\frac {2}{21} \left (\frac {1}{45} \left (4921349 \int \frac {x^2}{\sqrt {3 x^4+5 x^2+2}}dx+\frac {2024285 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{\sqrt {3 x^4+5 x^2+2}}\right )+\frac {1}{45} x \sqrt {3 x^4+5 x^2+2} \left (836181 x^2+1708960\right )\right )+\frac {1}{21} x \left (43169 x^2+49801\right ) \left (3 x^4+5 x^2+2\right )^{3/2}\right )-\frac {587}{33} x \left (3 x^4+5 x^2+2\right )^{5/2}\right )+\frac {2}{39} \left (3 x^4+5 x^2+2\right )^{5/2} x^3\)

\(\Big \downarrow \) 1456

\(\displaystyle \frac {1}{39} \left (\frac {1}{33} \left (\frac {2}{21} \left (\frac {1}{45} \left (\frac {2024285 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{\sqrt {3 x^4+5 x^2+2}}+4921349 \left (\frac {x \left (3 x^2+2\right )}{3 \sqrt {3 x^4+5 x^2+2}}-\frac {\sqrt {2} \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{3 \sqrt {3 x^4+5 x^2+2}}\right )\right )+\frac {1}{45} x \sqrt {3 x^4+5 x^2+2} \left (836181 x^2+1708960\right )\right )+\frac {1}{21} x \left (43169 x^2+49801\right ) \left (3 x^4+5 x^2+2\right )^{3/2}\right )-\frac {587}{33} x \left (3 x^4+5 x^2+2\right )^{5/2}\right )+\frac {2}{39} \left (3 x^4+5 x^2+2\right )^{5/2} x^3\)

Input:

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

Output:

(2*x^3*(2 + 5*x^2 + 3*x^4)^(5/2))/39 + ((-587*x*(2 + 5*x^2 + 3*x^4)^(5/2)) 
/33 + ((x*(49801 + 43169*x^2)*(2 + 5*x^2 + 3*x^4)^(3/2))/21 + (2*((x*(1708 
960 + 836181*x^2)*Sqrt[2 + 5*x^2 + 3*x^4])/45 + (4921349*((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])) + (2024285*Sqr 
t[2]*(1 + x^2)*Sqrt[(2 + 3*x^2)/(1 + x^2)]*EllipticF[ArcTan[x], -1/2])/Sqr 
t[2 + 5*x^2 + 3*x^4])/45))/21)/33)/39
 

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 = 7.03 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.61

method	result	size
risch	\(\frac {x \left (561330 x^{10}-3121335 x^{8}-8505945 x^{6}-2840895 x^{4}+5917977 x^{2}+5681150\right ) \sqrt {3 x^{4}+5 x^{2}+2}}{1216215}-\frac {809714 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{243243 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {9842698 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{3648645 \sqrt {3 x^{4}+5 x^{2}+2}}\)	\(145\)
default	\(-\frac {9001 x^{7} \sqrt {3 x^{4}+5 x^{2}+2}}{1287}-\frac {63131 x^{5} \sqrt {3 x^{4}+5 x^{2}+2}}{27027}+\frac {50581 x^{3} \sqrt {3 x^{4}+5 x^{2}+2}}{10395}+\frac {1136230 x \sqrt {3 x^{4}+5 x^{2}+2}}{243243}-\frac {809714 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{243243 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {9842698 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{3648645 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {367 x^{9} \sqrt {3 x^{4}+5 x^{2}+2}}{143}+\frac {6 x^{11} \sqrt {3 x^{4}+5 x^{2}+2}}{13}\)	\(213\)
elliptic	\(-\frac {9001 x^{7} \sqrt {3 x^{4}+5 x^{2}+2}}{1287}-\frac {63131 x^{5} \sqrt {3 x^{4}+5 x^{2}+2}}{27027}+\frac {50581 x^{3} \sqrt {3 x^{4}+5 x^{2}+2}}{10395}+\frac {1136230 x \sqrt {3 x^{4}+5 x^{2}+2}}{243243}-\frac {809714 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{243243 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {9842698 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{3648645 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {367 x^{9} \sqrt {3 x^{4}+5 x^{2}+2}}{143}+\frac {6 x^{11} \sqrt {3 x^{4}+5 x^{2}+2}}{13}\)	\(213\)

Input:

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

Output:

1/1216215*x*(561330*x^10-3121335*x^8-8505945*x^6-2840895*x^4+5917977*x^2+5 
681150)*(3*x^4+5*x^2+2)^(1/2)-809714/243243*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))+9842698/3648645*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 = 0.07 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.38 \[ \int \left (1+2 x^2\right ) \left (4-7 x^2+x^4\right ) \left (2+5 x^2+3 x^4\right )^{3/2} \, dx=-\frac {19685396 \, \sqrt {3} \sqrt {-\frac {2}{3}} x E(\arcsin \left (\frac {\sqrt {-\frac {2}{3}}}{x}\right )\,|\,\frac {3}{2}) - 56122526 \, \sqrt {3} \sqrt {-\frac {2}{3}} x F(\arcsin \left (\frac {\sqrt {-\frac {2}{3}}}{x}\right )\,|\,\frac {3}{2}) - 3 \, {\left (1683990 \, x^{12} - 9364005 \, x^{10} - 25517835 \, x^{8} - 8522685 \, x^{6} + 17753931 \, x^{4} + 17043450 \, x^{2} + 9842698\right )} \sqrt {3 \, x^{4} + 5 \, x^{2} + 2}}{10945935 \, x} \] Input:

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

Output:

-1/10945935*(19685396*sqrt(3)*sqrt(-2/3)*x*elliptic_e(arcsin(sqrt(-2/3)/x) 
, 3/2) - 56122526*sqrt(3)*sqrt(-2/3)*x*elliptic_f(arcsin(sqrt(-2/3)/x), 3/ 
2) - 3*(1683990*x^12 - 9364005*x^10 - 25517835*x^8 - 8522685*x^6 + 1775393 
1*x^4 + 17043450*x^2 + 9842698)*sqrt(3*x^4 + 5*x^2 + 2))/x
 

Sympy [F]

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

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \left (1+2 x^2\right ) \left (4-7 x^2+x^4\right ) \left (2+5 x^2+3 x^4\right )^{3/2} \, dx=\frac {6 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{11}}{13}-\frac {367 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{9}}{143}-\frac {9001 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{7}}{1287}-\frac {63131 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{5}}{27027}+\frac {50581 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{3}}{10395}+\frac {1136230 \sqrt {3 x^{4}+5 x^{2}+2}\, x}{243243}+\frac {1619428 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{3 x^{4}+5 x^{2}+2}d x \right )}{243243}+\frac {9842698 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{2}}{3 x^{4}+5 x^{2}+2}d x \right )}{1216215} \] Input:

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

Output:

(561330*sqrt(3*x**4 + 5*x**2 + 2)*x**11 - 3121335*sqrt(3*x**4 + 5*x**2 + 2 
)*x**9 - 8505945*sqrt(3*x**4 + 5*x**2 + 2)*x**7 - 2840895*sqrt(3*x**4 + 5* 
x**2 + 2)*x**5 + 5917977*sqrt(3*x**4 + 5*x**2 + 2)*x**3 + 5681150*sqrt(3*x 
**4 + 5*x**2 + 2)*x + 8097140*int(sqrt(3*x**4 + 5*x**2 + 2)/(3*x**4 + 5*x* 
*2 + 2),x) + 9842698*int((sqrt(3*x**4 + 5*x**2 + 2)*x**2)/(3*x**4 + 5*x**2 
 + 2),x))/1216215