\(\int (1+2 x^2)^3 (4-7 x^2+x^4) (2+5 x^2+3 x^4)^{3/2} \, dx\) [197]

Optimal result

Integrand size = 36, antiderivative size = 285 \[ \int \left (1+2 x^2\right )^3 \left (4-7 x^2+x^4\right ) \left (2+5 x^2+3 x^4\right )^{3/2} \, dx=\frac {1152397882 x \left (2+3 x^2\right )}{558242685 \sqrt {2+5 x^2+3 x^4}}+\frac {2 x \left (202977940+101236329 x^2\right ) \sqrt {2+5 x^2+3 x^4}}{186080895}+\frac {x \left (4895329+3917921 x^2\right ) \left (2+5 x^2+3 x^4\right )^{3/2}}{4135131}+\frac {2609 x \left (2+5 x^2+3 x^4\right )^{5/2}}{11583}+\frac {1706 x^3 \left (2+5 x^2+3 x^4\right )^{5/2}}{5967}-\frac {908}{765} x^5 \left (2+5 x^2+3 x^4\right )^{5/2}+\frac {8}{51} x^7 \left (2+5 x^2+3 x^4\right )^{5/2}-\frac {1152397882 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{558242685 \sqrt {2+5 x^2+3 x^4}}+\frac {94891466 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{37216179 \sqrt {2+5 x^2+3 x^4}} \] Output:

1152397882/558242685*x*(3*x^2+2)/(3*x^4+5*x^2+2)^(1/2)+2/186080895*x*(1012 
36329*x^2+202977940)*(3*x^4+5*x^2+2)^(1/2)+1/4135131*x*(3917921*x^2+489532 
9)*(3*x^4+5*x^2+2)^(3/2)+2609/11583*x*(3*x^4+5*x^2+2)^(5/2)+1706/5967*x^3* 
(3*x^4+5*x^2+2)^(5/2)-908/765*x^5*(3*x^4+5*x^2+2)^(5/2)+8/51*x^7*(3*x^4+5* 
x^2+2)^(5/2)-1152397882/558242685*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)[phone]/ 
37216179*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 = 10.19 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.58 \[ \int \left (1+2 x^2\right )^3 \left (4-7 x^2+x^4\right ) \left (2+5 x^2+3 x^4\right )^{3/2} \, dx=\frac {3 x \left (2028379660+10486174576 x^2+23128188873 x^4+23400228114 x^6-4137291324 x^8-39846191490 x^{10}-44404657731 x^{12}-20236507830 x^{14}-2022808788 x^{16}[phone] x^{18}\right )-1152397882 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} E\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )|\frac {2}{3}\right )[phone] i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right ),\frac {2}{3}\right )}{558242685 \sqrt {2+5 x^2+3 x^4}} \] Input:

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

Output:

(3*x*(2028379660 + 10486174576*x^2 + 23128188873*x^4 + 23400228114*x^6 - 4 
137291324*x^8 - 39846191490*x^10 - 44404657731*x^12 - 20236507830*x^14 - 2 
022808788*x^16 + 788107320*x^18) - (1152397882*I)*Sqrt[3]*Sqrt[1 + x^2]*Sq 
rt[2 + 3*x^2]*EllipticE[I*ArcSinh[Sqrt[3/2]*x], 2/3] + (203483222*I)*Sqrt[ 
3]*Sqrt[1 + x^2]*Sqrt[2 + 3*x^2]*EllipticF[I*ArcSinh[Sqrt[3/2]*x], 2/3])/( 
558242685*Sqrt[2 + 5*x^2 + 3*x^4])
 

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.11, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.306, Rules used = {2207, 2207, 27, 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.

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

\(\Big \downarrow \) 2207

\(\displaystyle \frac {1}{51} \int \left (3 x^4+5 x^2+2\right )^{3/2} \left (-2724 x^8-2458 x^6+357 x^4+867 x^2+204\right )dx+\frac {8}{51} \left (3 x^4+5 x^2+2\right )^{5/2} x^7\)

\(\Big \downarrow \) 2207

\(\displaystyle \frac {1}{51} \left (\frac {1}{45} \int 15 \left (3 x^4+5 x^2+2\right )^{3/2} \left (1706 x^6+2887 x^4+2601 x^2+612\right )dx-\frac {908}{15} x^5 \left (3 x^4+5 x^2+2\right )^{5/2}\right )+\frac {8}{51} \left (3 x^4+5 x^2+2\right )^{5/2} x^7\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{51} \left (\frac {1}{3} \int \left (3 x^4+5 x^2+2\right )^{3/2} \left (1706 x^6+2887 x^4+2601 x^2+612\right )dx-\frac {908}{15} x^5 \left (3 x^4+5 x^2+2\right )^{5/2}\right )+\frac {8}{51} \left (3 x^4+5 x^2+2\right )^{5/2} x^7\)

\(\Big \downarrow \) 2207

\(\displaystyle \frac {1}{51} \left (\frac {1}{3} \left (\frac {1}{39} \int \left (3 x^4+5 x^2+2\right )^{3/2} \left (44353 x^4+91203 x^2+23868\right )dx+\frac {1706}{39} \left (3 x^4+5 x^2+2\right )^{5/2} x^3\right )-\frac {908}{15} x^5 \left (3 x^4+5 x^2+2\right )^{5/2}\right )+\frac {8}{51} \left (3 x^4+5 x^2+2\right )^{5/2} x^7\)

\(\Big \downarrow \) 2207

\(\displaystyle \frac {1}{51} \left (\frac {1}{3} \left (\frac {1}{39} \left (\frac {1}{33} \int \left (1679109 x^2+698938\right ) \left (3 x^4+5 x^2+2\right )^{3/2}dx+\frac {44353}{33} x \left (3 x^4+5 x^2+2\right )^{5/2}\right )+\frac {1706}{39} \left (3 x^4+5 x^2+2\right )^{5/2} x^3\right )-\frac {908}{15} x^5 \left (3 x^4+5 x^2+2\right )^{5/2}\right )+\frac {8}{51} \left (3 x^4+5 x^2+2\right )^{5/2} x^7\)

\(\Big \downarrow \) 1490

\(\displaystyle \frac {1}{51} \left (\frac {1}{3} \left (\frac {1}{39} \left (\frac {1}{33} \left (\frac {1}{63} \int 6 \left (11248481 x^2+9782369\right ) \sqrt {3 x^4+5 x^2+2}dx+\frac {1}{21} x \left (3917921 x^2+4895329\right ) \left (3 x^4+5 x^2+2\right )^{3/2}\right )+\frac {44353}{33} x \left (3 x^4+5 x^2+2\right )^{5/2}\right )+\frac {1706}{39} \left (3 x^4+5 x^2+2\right )^{5/2} x^3\right )-\frac {908}{15} x^5 \left (3 x^4+5 x^2+2\right )^{5/2}\right )+\frac {8}{51} \left (3 x^4+5 x^2+2\right )^{5/2} x^7\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{51} \left (\frac {1}{3} \left (\frac {1}{39} \left (\frac {1}{33} \left (\frac {2}{21} \int \left (11248481 x^2+9782369\right ) \sqrt {3 x^4+5 x^2+2}dx+\frac {1}{21} x \left (3917921 x^2+4895329\right ) \left (3 x^4+5 x^2+2\right )^{3/2}\right )+\frac {44353}{33} x \left (3 x^4+5 x^2+2\right )^{5/2}\right )+\frac {1706}{39} \left (3 x^4+5 x^2+2\right )^{5/2} x^3\right )-\frac {908}{15} x^5 \left (3 x^4+5 x^2+2\right )^{5/2}\right )+\frac {8}{51} \left (3 x^4+5 x^2+2\right )^{5/2} x^7\)

\(\Big \downarrow \) 1490

\(\displaystyle \frac {1}{51} \left (\frac {1}{3} \left (\frac {1}{39} \left (\frac {1}{33} \left (\frac {2}{21} \left (\frac {1}{45} \int \frac {576198941 x^2+474457330}{\sqrt {3 x^4+5 x^2+2}}dx+\frac {1}{45} x \sqrt {3 x^4+5 x^2+2} \left (101236329 x^2+202977940\right )\right )+\frac {1}{21} x \left (3917921 x^2+4895329\right ) \left (3 x^4+5 x^2+2\right )^{3/2}\right )+\frac {44353}{33} x \left (3 x^4+5 x^2+2\right )^{5/2}\right )+\frac {1706}{39} \left (3 x^4+5 x^2+2\right )^{5/2} x^3\right )-\frac {908}{15} x^5 \left (3 x^4+5 x^2+2\right )^{5/2}\right )+\frac {8}{51} \left (3 x^4+5 x^2+2\right )^{5/2} x^7\)

\(\Big \downarrow \) 1503

\(\displaystyle \frac {1}{51} \left (\frac {1}{3} \left (\frac {1}{39} \left (\frac {1}{33} \left (\frac {2}{21} \left (\frac {1}{45} \left (474457330 \int \frac {1}{\sqrt {3 x^4+5 x^2+2}}dx+576198941 \int \frac {x^2}{\sqrt {3 x^4+5 x^2+2}}dx\right )+\frac {1}{45} x \sqrt {3 x^4+5 x^2+2} \left (101236329 x^2+202977940\right )\right )+\frac {1}{21} x \left (3917921 x^2+4895329\right ) \left (3 x^4+5 x^2+2\right )^{3/2}\right )+\frac {44353}{33} x \left (3 x^4+5 x^2+2\right )^{5/2}\right )+\frac {1706}{39} \left (3 x^4+5 x^2+2\right )^{5/2} x^3\right )-\frac {908}{15} x^5 \left (3 x^4+5 x^2+2\right )^{5/2}\right )+\frac {8}{51} \left (3 x^4+5 x^2+2\right )^{5/2} x^7\)

\(\Big \downarrow \) 1413

\(\displaystyle \frac {1}{51} \left (\frac {1}{3} \left (\frac {1}{39} \left (\frac {1}{33} \left (\frac {2}{21} \left (\frac {1}{45} \left (576198941 \int \frac {x^2}{\sqrt {3 x^4+5 x^2+2}}dx+\frac {237228665 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{\sqrt {3 x^4+5 x^2+2}}\right )+\frac {1}{45} x \sqrt {3 x^4+5 x^2+2} \left (101236329 x^2+202977940\right )\right )+\frac {1}{21} x \left (3917921 x^2+4895329\right ) \left (3 x^4+5 x^2+2\right )^{3/2}\right )+\frac {44353}{33} x \left (3 x^4+5 x^2+2\right )^{5/2}\right )+\frac {1706}{39} \left (3 x^4+5 x^2+2\right )^{5/2} x^3\right )-\frac {908}{15} x^5 \left (3 x^4+5 x^2+2\right )^{5/2}\right )+\frac {8}{51} \left (3 x^4+5 x^2+2\right )^{5/2} x^7\)

\(\Big \downarrow \) 1456

\(\displaystyle \frac {1}{51} \left (\frac {1}{3} \left (\frac {1}{39} \left (\frac {1}{33} \left (\frac {2}{21} \left (\frac {1}{45} \left (\frac {237228665 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{\sqrt {3 x^4+5 x^2+2}}[phone] \left (\frac {x \left (3 x^2+2\right )}{3 \sqrt {3 x^4+5 x^2+2}}-\frac {\sqrt {2} \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{3 \sqrt {3 x^4+5 x^2+2}}\right )\right )+\frac {1}{45} x \sqrt {3 x^4+5 x^2+2} \left (101236329 x^2+202977940\right )\right )+\frac {1}{21} x \left (3917921 x^2+4895329\right ) \left (3 x^4+5 x^2+2\right )^{3/2}\right )+\frac {44353}{33} x \left (3 x^4+5 x^2+2\right )^{5/2}\right )+\frac {1706}{39} \left (3 x^4+5 x^2+2\right )^{5/2} x^3\right )-\frac {908}{15} x^5 \left (3 x^4+5 x^2+2\right )^{5/2}\right )+\frac {8}{51} \left (3 x^4+5 x^2+2\right )^{5/2} x^7\)

Input:

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

Output:

(8*x^7*(2 + 5*x^2 + 3*x^4)^(5/2))/51 + ((-908*x^5*(2 + 5*x^2 + 3*x^4)^(5/2 
))/15 + ((1706*x^3*(2 + 5*x^2 + 3*x^4)^(5/2))/39 + ((44353*x*(2 + 5*x^2 + 
3*x^4)^(5/2))/33 + ((x*(4895329 + 3917921*x^2)*(2 + 5*x^2 + 3*x^4)^(3/2))/ 
21 + (2*((x*(202977940 + 101236329*x^2)*Sqrt[2 + 5*x^2 + 3*x^4])/45 + (576 
198941*((x*(2 + 3*x^2))/(3*Sqrt[2 + 5*x^2 + 3*x^4]) - (Sqrt[2]*(1 + x^2)*S 
qrt[(2 + 3*x^2)/(1 + x^2)]*EllipticE[ArcTan[x], -1/2])/(3*Sqrt[2 + 5*x^2 + 
 3*x^4])) + (237228665*Sqrt[2]*(1 + x^2)*Sqrt[(2 + 3*x^2)/(1 + x^2)]*Ellip 
ticF[ArcTan[x], -1/2])/Sqrt[2 + 5*x^2 + 3*x^4])/45))/21)/33)/39)/3)/51
 

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 = 16.06 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.54

method	result	size
risch	\(\frac {x \left (262702440 x^{14}-1112106996 x^{12}-5067125910 x^{10}-5614938063 x^{8}-545749785 x^{6}[phone] x^{4}[phone] x^{2}[phone]\right ) \sqrt {3 x^{4}+5 x^{2}+2}}{186080895}-\frac {94891466 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{37216179 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {1152397882 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{558242685 \sqrt {3 x^{4}+5 x^{2}+2}}\)	\(155\)
default	\(\frac {363753101 x^{5} \sqrt {3 x^{4}+5 x^{2}+2}}{20675655}+\frac {23141989 x^{3} \sqrt {3 x^{4}+5 x^{2}+2}}{1590435}+\frac {202837966 x \sqrt {3 x^{4}+5 x^{2}+2}}{37216179}+\frac {1152397882 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{558242685 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {94891466 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{37216179 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {577513 x^{7} \sqrt {3 x^{4}+5 x^{2}+2}}{196911}-\frac {1100321 x^{9} \sqrt {3 x^{4}+5 x^{2}+2}}{36465}-\frac {354 x^{11} \sqrt {3 x^{4}+5 x^{2}+2}}{13}-\frac {508 x^{13} \sqrt {3 x^{4}+5 x^{2}+2}}{85}+\frac {24 x^{15} \sqrt {3 x^{4}+5 x^{2}+2}}{17}\)	\(251\)
elliptic	\(\frac {363753101 x^{5} \sqrt {3 x^{4}+5 x^{2}+2}}{20675655}+\frac {23141989 x^{3} \sqrt {3 x^{4}+5 x^{2}+2}}{1590435}+\frac {202837966 x \sqrt {3 x^{4}+5 x^{2}+2}}{37216179}+\frac {1152397882 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{558242685 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {94891466 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{37216179 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {577513 x^{7} \sqrt {3 x^{4}+5 x^{2}+2}}{196911}-\frac {1100321 x^{9} \sqrt {3 x^{4}+5 x^{2}+2}}{36465}-\frac {354 x^{11} \sqrt {3 x^{4}+5 x^{2}+2}}{13}-\frac {508 x^{13} \sqrt {3 x^{4}+5 x^{2}+2}}{85}+\frac {24 x^{15} \sqrt {3 x^{4}+5 x^{2}+2}}{17}\)	\(251\)

Input:

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

Output:

1/186080895*x*(262702440*x^14-1112106996*x^12-5067125910*x^10-5614938063*x 
^8-545749785*x^6+3273777909*x^4+2707612713*x^2+1014189830)*(3*x^4+5*x^2+2) 
^(1/2)-94891466/37216179*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]/558242685*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 = 0.09 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.36 \[ \int \left (1+2 x^2\right )^3 \left (4-7 x^2+x^4\right ) \left (2+5 x^2+3 x^4\right )^{3/2} \, dx=-\frac {2304795764 \, \sqrt {3} \sqrt {-\frac {2}{3}} x E(\arcsin \left (\frac {\sqrt {-\frac {2}{3}}}{x}\right )\,|\,\frac {3}{2}) - 6574911734 \, \sqrt {3} \sqrt {-\frac {2}{3}} x F(\arcsin \left (\frac {\sqrt {-\frac {2}{3}}}{x}\right )\,|\,\frac {3}{2}) - 3 \, {\left (788107320 \, x^{16} - 3336320988 \, x^{14} - 15201377730 \, x^{12} - 16844814189 \, x^{10} - 1637249355 \, x^{8} + 9821333727 \, x^{6} + 8122838139 \, x^{4} + 3042569490 \, x^{2} + 1152397882\right )} \sqrt {3 \, x^{4} + 5 \, x^{2} + 2}}{1674728055 \, x} \] Input:

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

Output:

-1/1674728055*(2304795764*sqrt(3)*sqrt(-2/3)*x*elliptic_e(arcsin(sqrt(-2/3 
)/x), 3/2) - 6574911734*sqrt(3)*sqrt(-2/3)*x*elliptic_f(arcsin(sqrt(-2/3)/ 
x), 3/2) - 3*(788107320*x^16 - 3336320988*x^14 - 15201377730*x^12 - 168448 
14189*x^10 - 1637249355*x^8 + 9821333727*x^6 + 8122838139*x^4 + 3042569490 
*x^2 + 1152397882)*sqrt(3*x^4 + 5*x^2 + 2))/x
 

Sympy [F]

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

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

Maxima [F]

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

integrate((2*x^2+1)^3*(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)^3, x)
 

Giac [F]

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

integrate((2*x^2+1)^3*(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)^3, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \left (1+2 x^2\right )^3 \left (4-7 x^2+x^4\right ) \left (2+5 x^2+3 x^4\right )^{3/2} \, dx=\frac {24 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{15}}{17}-\frac {508 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{13}}{85}-\frac {354 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{11}}{13}-\frac {1100321 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{9}}{36465}-\frac {577513 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{7}}{196911}+\frac {363753101 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{5}}{20675655}+\frac {23141989 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{3}}{1590435}+\frac {202837966 \sqrt {3 x^{4}+5 x^{2}+2}\, x}{37216179}+\frac {189782932 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{3 x^{4}+5 x^{2}+2}d x \right )}{37216179}+\frac {1152397882 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{2}}{3 x^{4}+5 x^{2}+2}d x \right )}{186080895} \] Input:

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

Output:

(262702440*sqrt(3*x**4 + 5*x**2 + 2)*x**15 - 1112106996*sqrt(3*x**4 + 5*x* 
*2 + 2)*x**13 - 5067125910*sqrt(3*x**4 + 5*x**2 + 2)*x**11 - 5614938063*sq 
rt(3*x**4 + 5*x**2 + 2)*x**9 - 545749785*sqrt(3*x**4 + 5*x**2 + 2)*x**7 + 
3273777909*sqrt(3*x**4 + 5*x**2 + 2)*x**5 + 2707612713*sqrt(3*x**4 + 5*x** 
2 + 2)*x**3 + 1014189830*sqrt(3*x**4 + 5*x**2 + 2)*x + 948914660*int(sqrt( 
3*x**4 + 5*x**2 + 2)/(3*x**4 + 5*x**2 + 2),x) + 1152397882*int((sqrt(3*x** 
4 + 5*x**2 + 2)*x**2)/(3*x**4 + 5*x**2 + 2),x))/186080895