3.2.65 \(\int (2+3 x^2) (3+5 x^2+x^4)^{3/2} \, dx\) [165]

3.2.65.1 Optimal result

Integrand size = 22, antiderivative size = 308 \[ \int \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=\frac {203 x \left (5+\sqrt {13}+2 x^2\right )}{30 \sqrt {3+5 x^2+x^4}}-\frac {1}{15} x \left (5+12 x^2\right ) \sqrt {3+5 x^2+x^4}+\frac {1}{3} x \left (3+x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}-\frac {203 \sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{30 \sqrt {3+5 x^2+x^4}}+\frac {5 \sqrt {\frac {2}{3 \left (5+\sqrt {13}\right )}} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {3+5 x^2+x^4}} \]

output

1/3*x*(x^2+3)*(x^4+5*x^2+3)^(3/2)+203/30*x*(5+2*x^2+13^(1/2))/(x^4+5*x^2+3 
)^(1/2)-1/15*x*(12*x^2+5)*(x^4+5*x^2+3)^(1/2)+5/3*(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^(1/2)/(5+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)-203/180*(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)
 

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

Result contains complex when optimal does not.

Time = 6.66 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.78 \[ \int \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=\frac {4 x \left (120+434 x^2+550 x^4+293 x^6+65 x^8+5 x^{10}\right )+203 i \sqrt {2} \left (-5+\sqrt {13}\right ) \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} E\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right )|\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )-i \sqrt {2} \left (-715+203 \sqrt {13}\right ) \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right ),\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )}{60 \sqrt {3+5 x^2+x^4}} \]

input

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

output

(4*x*(120 + 434*x^2 + 550*x^4 + 293*x^6 + 65*x^8 + 5*x^10) + (203*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] - I*Sqrt[2]*(-715 + 203*Sqrt[13])*Sqrt[(-5 + Sqrt[13] - 2*x^ 
2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2/ 
(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6])/(60*Sqrt[3 + 5*x^2 + x^4])
 

3.2.65.3 Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1490, 27, 1490, 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.

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

\(\Big \downarrow \) 1490

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1490

\(\displaystyle \frac {1}{15} \int \frac {203 x^2+150}{\sqrt {x^4+5 x^2+3}}dx+\frac {1}{3} x \left (x^2+3\right ) \left (x^4+5 x^2+3\right )^{3/2}-\frac {1}{15} x \left (12 x^2+5\right ) \sqrt {x^4+5 x^2+3}\)

\(\Big \downarrow \) 1503

\(\displaystyle \frac {1}{15} \left (150 \int \frac {1}{\sqrt {x^4+5 x^2+3}}dx+203 \int \frac {x^2}{\sqrt {x^4+5 x^2+3}}dx\right )+\frac {1}{3} x \left (x^2+3\right ) \left (x^4+5 x^2+3\right )^{3/2}-\frac {1}{15} x \left (12 x^2+5\right ) \sqrt {x^4+5 x^2+3}\)

\(\Big \downarrow \) 1412

\(\displaystyle \frac {1}{15} \left (203 \int \frac {x^2}{\sqrt {x^4+5 x^2+3}}dx+\frac {25 \sqrt {\frac {6}{5+\sqrt {13}}} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {x^4+5 x^2+3}}\right )+\frac {1}{3} x \left (x^2+3\right ) \left (x^4+5 x^2+3\right )^{3/2}-\frac {1}{15} x \left (12 x^2+5\right ) \sqrt {x^4+5 x^2+3}\)

\(\Big \downarrow \) 1455

\(\displaystyle \frac {1}{15} \left (\frac {25 \sqrt {\frac {6}{5+\sqrt {13}}} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {x^4+5 x^2+3}}+203 \left (\frac {x \left (2 x^2+\sqrt {13}+5\right )}{2 \sqrt {x^4+5 x^2+3}}-\frac {\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{2 \sqrt {x^4+5 x^2+3}}\right )\right )+\frac {1}{3} x \left (x^2+3\right ) \left (x^4+5 x^2+3\right )^{3/2}-\frac {1}{15} x \left (12 x^2+5\right ) \sqrt {x^4+5 x^2+3}\)

input

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

output

-1/15*(x*(5 + 12*x^2)*Sqrt[3 + 5*x^2 + x^4]) + (x*(3 + x^2)*(3 + 5*x^2 + x 
^4)^(3/2))/3 + (203*((x*(5 + Sqrt[13] + 2*x^2))/(2*Sqrt[3 + 5*x^2 + x^4]) 
- (Sqrt[(5 + Sqrt[13])/6]*Sqrt[(6 + (5 - Sqrt[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])) + (25*Sqrt[6/(5 + Sqr 
t[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[1 
3])/6])/Sqrt[3 + 5*x^2 + x^4])/15
 

3.2.65.3.1 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 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]
 

3.2.65.4 Maple [A] (verified)

Time = 1.40 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.73

method	result	size
risch	\(\frac {x \left (5 x^{6}+40 x^{4}+78 x^{2}+40\right ) \sqrt {x^{4}+5 x^{2}+3}}{15}+\frac {60 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {2436 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{5 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\)	\(226\)
default	\(\frac {8 x^{5} \sqrt {x^{4}+5 x^{2}+3}}{3}+\frac {26 x^{3} \sqrt {x^{4}+5 x^{2}+3}}{5}+\frac {8 x \sqrt {x^{4}+5 x^{2}+3}}{3}+\frac {60 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {2436 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{5 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}+\frac {x^{7} \sqrt {x^{4}+5 x^{2}+3}}{3}\)	\(260\)
elliptic	\(\frac {8 x^{5} \sqrt {x^{4}+5 x^{2}+3}}{3}+\frac {26 x^{3} \sqrt {x^{4}+5 x^{2}+3}}{5}+\frac {8 x \sqrt {x^{4}+5 x^{2}+3}}{3}+\frac {60 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {2436 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{5 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}+\frac {x^{7} \sqrt {x^{4}+5 x^{2}+3}}{3}\)	\(260\)

input

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

output

1/15*x*(5*x^6+40*x^4+78*x^2+40)*(x^4+5*x^2+3)^(1/2)+60/(-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*3 
9^(1/2))-2436/5/(-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))*(Ellipti 
cF(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)))
 

3.2.65.5 Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.45 \[ \int \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=\frac {203 \, {\left (\sqrt {13} \sqrt {2} x - 5 \, \sqrt {2} x\right )} \sqrt {\sqrt {13} - 5} E(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {13} - 5}}{2 \, x}\right )\,|\,\frac {5}{6} \, \sqrt {13} + \frac {19}{6}) - {\left (153 \, \sqrt {13} \sqrt {2} x - 1265 \, \sqrt {2} x\right )} \sqrt {\sqrt {13} - 5} F(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {13} - 5}}{2 \, x}\right )\,|\,\frac {5}{6} \, \sqrt {13} + \frac {19}{6}) + 4 \, {\left (5 \, x^{8} + 40 \, x^{6} + 78 \, x^{4} + 40 \, x^{2} + 203\right )} \sqrt {x^{4} + 5 \, x^{2} + 3}}{60 \, x} \]

input

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

output

1/60*(203*(sqrt(13)*sqrt(2)*x - 5*sqrt(2)*x)*sqrt(sqrt(13) - 5)*elliptic_e 
(arcsin(1/2*sqrt(2)*sqrt(sqrt(13) - 5)/x), 5/6*sqrt(13) + 19/6) - (153*sqr 
t(13)*sqrt(2)*x - 1265*sqrt(2)*x)*sqrt(sqrt(13) - 5)*elliptic_f(arcsin(1/2 
*sqrt(2)*sqrt(sqrt(13) - 5)/x), 5/6*sqrt(13) + 19/6) + 4*(5*x^8 + 40*x^6 + 
 78*x^4 + 40*x^2 + 203)*sqrt(x^4 + 5*x^2 + 3))/x
 

3.2.65.6 Sympy [F]

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

input

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

output

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

3.2.65.7 Maxima [F]

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

input

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

output

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

3.2.65.8 Giac [F]

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

input

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

output

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

3.2.65.9 Mupad [F(-1)]

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

input

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

output

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