3.2.51 \(\int x^4 (2+3 x^2) \sqrt {3+5 x^2+x^4} \, dx\) [151]

3.2.51.1 Optimal result

Integrand size = 25, antiderivative size = 322 \[ \int x^4 \left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx=-\frac {1924 x \left (5+\sqrt {13}+2 x^2\right )}{105 \sqrt {3+5 x^2+x^4}}+\frac {13}{3} x \sqrt {3+5 x^2+x^4}-\frac {26}{35} x^3 \sqrt {3+5 x^2+x^4}+\frac {1}{21} x^5 \left (11+7 x^2\right ) \sqrt {3+5 x^2+x^4}+\frac {962 \sqrt {\frac {2}{3} \left (5+\sqrt {13}\right )} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{105 \sqrt {3+5 x^2+x^4}}-\frac {13 \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {6 \left (5+\sqrt {13}\right )} \sqrt {3+5 x^2+x^4}} \]

output

-1924/105*x*(5+2*x^2+13^(1/2))/(x^4+5*x^2+3)^(1/2)+13/3*x*(x^4+5*x^2+3)^(1 
/2)-26/35*x^3*(x^4+5*x^2+3)^(1/2)+1/21*x^5*(7*x^2+11)*(x^4+5*x^2+3)^(1/2)+ 
962/315*(1/(36+x^2*(30+6*13^(1/2))))^(1/2)*(36+x^2*(30+6*13^(1/2)))^(1/2)* 
EllipticE(x*(30+6*13^(1/2))^(1/2)/(36+x^2*(30+6*13^(1/2)))^(1/2),1/6*(-78+ 
30*13^(1/2))^(1/2))*(6+x^2*(5+13^(1/2)))*(30+6*13^(1/2))^(1/2)*((6+x^2*(5- 
13^(1/2)))/(6+x^2*(5+13^(1/2))))^(1/2)/(x^4+5*x^2+3)^(1/2)-13*(1/(36+x^2*( 
30+6*13^(1/2))))^(1/2)*(36+x^2*(30+6*13^(1/2)))^(1/2)*EllipticF(x*(30+6*13 
^(1/2))^(1/2)/(36+x^2*(30+6*13^(1/2)))^(1/2),1/6*(-78+30*13^(1/2))^(1/2))* 
(6+x^2*(5+13^(1/2)))*((6+x^2*(5-13^(1/2)))/(6+x^2*(5+13^(1/2))))^(1/2)/(x^ 
4+5*x^2+3)^(1/2)/(30+6*13^(1/2))^(1/2)
 

3.2.51.2 Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 6.06 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.74 \[ \int x^4 \left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx=\frac {2730 x+4082 x^3+460 x^5+604 x^7+460 x^9+70 x^{11}-1924 i \sqrt {2} \left (-5+\sqrt {13}\right ) \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} E\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right )|\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )+13 i \sqrt {2} \left (-635+148 \sqrt {13}\right ) \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right ),\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )}{210 \sqrt {3+5 x^2+x^4}} \]

input

Integrate[x^4*(2 + 3*x^2)*Sqrt[3 + 5*x^2 + x^4],x]
 

output

(2730*x + 4082*x^3 + 460*x^5 + 604*x^7 + 460*x^9 + 70*x^11 - (1924*I)*Sqrt 
[2]*(-5 + Sqrt[13])*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + 
 Sqrt[13] + 2*x^2]*EllipticE[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + ( 
5*Sqrt[13])/6] + (13*I)*Sqrt[2]*(-635 + 148*Sqrt[13])*Sqrt[(-5 + Sqrt[13] 
- 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticF[I*ArcSinh[S 
qrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6])/(210*Sqrt[3 + 5*x^2 + x^ 
4])
 

3.2.51.3 Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {1596, 27, 1602, 1602, 1503, 1412, 1455}

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.

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

\(\Big \downarrow \) 1596

\(\displaystyle \frac {1}{63} \int -\frac {117 x^4 \left (2 x^2+1\right )}{\sqrt {x^4+5 x^2+3}}dx+\frac {1}{21} \left (7 x^2+11\right ) \sqrt {x^4+5 x^2+3} x^5\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{21} x^5 \left (7 x^2+11\right ) \sqrt {x^4+5 x^2+3}-\frac {13}{7} \int \frac {x^4 \left (2 x^2+1\right )}{\sqrt {x^4+5 x^2+3}}dx\)

\(\Big \downarrow \) 1602

\(\displaystyle \frac {1}{21} x^5 \left (7 x^2+11\right ) \sqrt {x^4+5 x^2+3}-\frac {13}{7} \left (\frac {2}{5} x^3 \sqrt {x^4+5 x^2+3}-\frac {1}{5} \int \frac {x^2 \left (35 x^2+18\right )}{\sqrt {x^4+5 x^2+3}}dx\right )\)

\(\Big \downarrow \) 1602

\(\displaystyle \frac {1}{21} x^5 \left (7 x^2+11\right ) \sqrt {x^4+5 x^2+3}-\frac {13}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {296 x^2+105}{\sqrt {x^4+5 x^2+3}}dx-\frac {35}{3} x \sqrt {x^4+5 x^2+3}\right )+\frac {2}{5} \sqrt {x^4+5 x^2+3} x^3\right )\)

\(\Big \downarrow \) 1503

\(\displaystyle \frac {1}{21} x^5 \left (7 x^2+11\right ) \sqrt {x^4+5 x^2+3}-\frac {13}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (105 \int \frac {1}{\sqrt {x^4+5 x^2+3}}dx+296 \int \frac {x^2}{\sqrt {x^4+5 x^2+3}}dx\right )-\frac {35}{3} x \sqrt {x^4+5 x^2+3}\right )+\frac {2}{5} \sqrt {x^4+5 x^2+3} x^3\right )\)

\(\Big \downarrow \) 1412

\(\displaystyle \frac {1}{21} x^5 \left (7 x^2+11\right ) \sqrt {x^4+5 x^2+3}-\frac {13}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (296 \int \frac {x^2}{\sqrt {x^4+5 x^2+3}}dx+\frac {35 \sqrt {\frac {3}{2 \left (5+\sqrt {13}\right )}} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {x^4+5 x^2+3}}\right )-\frac {35}{3} x \sqrt {x^4+5 x^2+3}\right )+\frac {2}{5} \sqrt {x^4+5 x^2+3} x^3\right )\)

\(\Big \downarrow \) 1455

\(\displaystyle \frac {1}{21} x^5 \left (7 x^2+11\right ) \sqrt {x^4+5 x^2+3}-\frac {13}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {35 \sqrt {\frac {3}{2 \left (5+\sqrt {13}\right )}} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {x^4+5 x^2+3}}+296 \left (\frac {x \left (2 x^2+\sqrt {13}+5\right )}{2 \sqrt {x^4+5 x^2+3}}-\frac {\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{2 \sqrt {x^4+5 x^2+3}}\right )\right )-\frac {35}{3} x \sqrt {x^4+5 x^2+3}\right )+\frac {2}{5} \sqrt {x^4+5 x^2+3} x^3\right )\)

input

Int[x^4*(2 + 3*x^2)*Sqrt[3 + 5*x^2 + x^4],x]
 

output

(x^5*(11 + 7*x^2)*Sqrt[3 + 5*x^2 + x^4])/21 - (13*((2*x^3*Sqrt[3 + 5*x^2 + 
 x^4])/5 + ((-35*x*Sqrt[3 + 5*x^2 + x^4])/3 + (296*((x*(5 + Sqrt[13] + 2*x 
^2))/(2*Sqrt[3 + 5*x^2 + x^4]) - (Sqrt[(5 + Sqrt[13])/6]*Sqrt[(6 + (5 - Sq 
rt[13])*x^2)/(6 + (5 + Sqrt[13])*x^2)]*(6 + (5 + Sqrt[13])*x^2)*EllipticE[ 
ArcTan[Sqrt[(5 + Sqrt[13])/6]*x], (-13 + 5*Sqrt[13])/6])/(2*Sqrt[3 + 5*x^2 
 + x^4])) + (35*Sqrt[3/(2*(5 + Sqrt[13]))]*Sqrt[(6 + (5 - Sqrt[13])*x^2)/( 
6 + (5 + Sqrt[13])*x^2)]*(6 + (5 + Sqrt[13])*x^2)*EllipticF[ArcTan[Sqrt[(5 
 + Sqrt[13])/6]*x], (-13 + 5*Sqrt[13])/6])/Sqrt[3 + 5*x^2 + x^4])/3)/5))/7
 

3.2.51.3.1 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 1412

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] && 
!(PosQ[(b - q)/a] && SimplerSqrtQ[(b - q)/(2*a), (b + q)/(2*a)])] /; FreeQ[ 
{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]
 

rule 1455

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] &&  !(PosQ[ 
(b - q)/a] && SimplerSqrtQ[(b - q)/(2*a), (b + q)/(2*a)])] /; FreeQ[{a, b, 
c}, x] && GtQ[b^2 - 4*a*c, 0]
 

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 1596

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

rule 1602

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

3.2.51.4 Maple [A] (verified)

Time = 5.58 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.70

method	result	size
risch	\(\frac {x \left (35 x^{6}+55 x^{4}-78 x^{2}+455\right ) \sqrt {x^{4}+5 x^{2}+3}}{105}-\frac {78 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}+\frac {46176 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{35 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\)	\(226\)
default	\(\frac {x^{7} \sqrt {x^{4}+5 x^{2}+3}}{3}+\frac {11 x^{5} \sqrt {x^{4}+5 x^{2}+3}}{21}-\frac {26 x^{3} \sqrt {x^{4}+5 x^{2}+3}}{35}+\frac {13 x \sqrt {x^{4}+5 x^{2}+3}}{3}-\frac {78 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}+\frac {46176 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{35 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\)	\(260\)
elliptic	\(\frac {x^{7} \sqrt {x^{4}+5 x^{2}+3}}{3}+\frac {11 x^{5} \sqrt {x^{4}+5 x^{2}+3}}{21}-\frac {26 x^{3} \sqrt {x^{4}+5 x^{2}+3}}{35}+\frac {13 x \sqrt {x^{4}+5 x^{2}+3}}{3}-\frac {78 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}+\frac {46176 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{35 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\)	\(260\)

input

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

output

1/105*x*(35*x^6+55*x^4-78*x^2+455)*(x^4+5*x^2+3)^(1/2)-78/(-30+6*13^(1/2)) 
^(1/2)*(1-(-5/6+1/6*13^(1/2))*x^2)^(1/2)*(1-(-5/6-1/6*13^(1/2))*x^2)^(1/2) 
/(x^4+5*x^2+3)^(1/2)*EllipticF(1/6*x*(-30+6*13^(1/2))^(1/2),5/6*3^(1/2)+1/ 
6*39^(1/2))+46176/35/(-30+6*13^(1/2))^(1/2)*(1-(-5/6+1/6*13^(1/2))*x^2)^(1 
/2)*(1-(-5/6-1/6*13^(1/2))*x^2)^(1/2)/(x^4+5*x^2+3)^(1/2)/(5+13^(1/2))*(El 
lipticF(1/6*x*(-30+6*13^(1/2))^(1/2),5/6*3^(1/2)+1/6*39^(1/2))-EllipticE(1 
/6*x*(-30+6*13^(1/2))^(1/2),5/6*3^(1/2)+1/6*39^(1/2)))
 

3.2.51.5 Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.43 \[ \int x^4 \left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx=-\frac {3848 \, {\left (\sqrt {13} \sqrt {2} x - 5 \, \sqrt {2} x\right )} \sqrt {\sqrt {13} - 5} E(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {13} - 5}}{2 \, x}\right )\,|\,\frac {5}{6} \, \sqrt {13} + \frac {19}{6}) - 13 \, {\left (261 \, \sqrt {13} \sqrt {2} x - 1655 \, \sqrt {2} x\right )} \sqrt {\sqrt {13} - 5} F(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {13} - 5}}{2 \, x}\right )\,|\,\frac {5}{6} \, \sqrt {13} + \frac {19}{6}) - 4 \, {\left (35 \, x^{8} + 55 \, x^{6} - 78 \, x^{4} + 455 \, x^{2} - 3848\right )} \sqrt {x^{4} + 5 \, x^{2} + 3}}{420 \, x} \]

input

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

output

-1/420*(3848*(sqrt(13)*sqrt(2)*x - 5*sqrt(2)*x)*sqrt(sqrt(13) - 5)*ellipti 
c_e(arcsin(1/2*sqrt(2)*sqrt(sqrt(13) - 5)/x), 5/6*sqrt(13) + 19/6) - 13*(2 
61*sqrt(13)*sqrt(2)*x - 1655*sqrt(2)*x)*sqrt(sqrt(13) - 5)*elliptic_f(arcs 
in(1/2*sqrt(2)*sqrt(sqrt(13) - 5)/x), 5/6*sqrt(13) + 19/6) - 4*(35*x^8 + 5 
5*x^6 - 78*x^4 + 455*x^2 - 3848)*sqrt(x^4 + 5*x^2 + 3))/x
 

3.2.51.6 Sympy [F]

\[ \int x^4 \left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx=\int x^{4} \cdot \left (3 x^{2} + 2\right ) \sqrt {x^{4} + 5 x^{2} + 3}\, dx \]

input

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

output

Integral(x**4*(3*x**2 + 2)*sqrt(x**4 + 5*x**2 + 3), x)
 

3.2.51.7 Maxima [F]

\[ \int x^4 \left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx=\int { \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (3 \, x^{2} + 2\right )} x^{4} \,d x } \]

input

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

output

integrate(sqrt(x^4 + 5*x^2 + 3)*(3*x^2 + 2)*x^4, x)
 

3.2.51.8 Giac [F]

\[ \int x^4 \left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx=\int { \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (3 \, x^{2} + 2\right )} x^{4} \,d x } \]

input

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

output

integrate(sqrt(x^4 + 5*x^2 + 3)*(3*x^2 + 2)*x^4, x)
 

3.2.51.9 Mupad [F(-1)]

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

input

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

output

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