3.2.67 \(\int \frac {(2+3 x^2) (3+5 x^2+x^4)^{3/2}}{x^4} \, dx\) [167]

3.2.67.1 Optimal result

Integrand size = 25, antiderivative size = 314 \[ \int \frac {\left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{x^4} \, dx=\frac {949 x \left (5+\sqrt {13}+2 x^2\right )}{30 \sqrt {3+5 x^2+x^4}}-\frac {13 \left (24-5 x^2\right ) \sqrt {3+5 x^2+x^4}}{15 x}-\frac {\left (10-9 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{15 x^3}-\frac {949 \sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{30 \sqrt {3+5 x^2+x^4}}+\frac {65 \sqrt {\frac {2}{3 \left (5+\sqrt {13}\right )}} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {3+5 x^2+x^4}} \]

output

-1/15*(-9*x^2+10)*(x^4+5*x^2+3)^(3/2)/x^3+949/30*x*(5+2*x^2+13^(1/2))/(x^4 
+5*x^2+3)^(1/2)-13/15*(-5*x^2+24)*(x^4+5*x^2+3)^(1/2)/x+65/3*(1/(36+x^2*(3 
0+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)-949/180*(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+1 
3^(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)
 

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

Result contains complex when optimal does not.

Time = 10.25 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.79 \[ \int \frac {\left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{x^4} \, dx=\frac {4 \left (-90-1155 x^2-1405 x^4+192 x^6+145 x^8+9 x^{10}\right )+949 i \sqrt {2} \left (-5+\sqrt {13}\right ) x^3 \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} E\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right )|\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )-13 i \sqrt {2} \left (-65+73 \sqrt {13}\right ) x^3 \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right ),\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )}{60 x^3 \sqrt {3+5 x^2+x^4}} \]

input

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

output

(4*(-90 - 1155*x^2 - 1405*x^4 + 192*x^6 + 145*x^8 + 9*x^10) + (949*I)*Sqrt 
[2]*(-5 + Sqrt[13])*x^3*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]*(-65 + 73*Sqrt[13])*x^3*Sqrt[(-5 + Sqr 
t[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticF[I*Arc 
Sinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6])/(60*x^3*Sqrt[3 + 5 
*x^2 + x^4])
 

3.2.67.3 Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {1594, 27, 1594, 25, 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.

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

\(\Big \downarrow \) 1594

\(\displaystyle -\frac {1}{5} \int -\frac {13 \left (5 x^2+8\right ) \sqrt {x^4+5 x^2+3}}{x^2}dx-\frac {\left (10-9 x^2\right ) \left (x^4+5 x^2+3\right )^{3/2}}{15 x^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {13}{5} \int \frac {\left (5 x^2+8\right ) \sqrt {x^4+5 x^2+3}}{x^2}dx-\frac {\left (10-9 x^2\right ) \left (x^4+5 x^2+3\right )^{3/2}}{15 x^3}\)

\(\Big \downarrow \) 1594

\(\displaystyle \frac {13}{5} \left (-\frac {1}{3} \int -\frac {73 x^2+150}{\sqrt {x^4+5 x^2+3}}dx-\frac {\sqrt {x^4+5 x^2+3} \left (24-5 x^2\right )}{3 x}\right )-\frac {\left (10-9 x^2\right ) \left (x^4+5 x^2+3\right )^{3/2}}{15 x^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {13}{5} \left (\frac {1}{3} \int \frac {73 x^2+150}{\sqrt {x^4+5 x^2+3}}dx-\frac {\left (24-5 x^2\right ) \sqrt {x^4+5 x^2+3}}{3 x}\right )-\frac {\left (10-9 x^2\right ) \left (x^4+5 x^2+3\right )^{3/2}}{15 x^3}\)

\(\Big \downarrow \) 1503

\(\displaystyle \frac {13}{5} \left (\frac {1}{3} \left (150 \int \frac {1}{\sqrt {x^4+5 x^2+3}}dx+73 \int \frac {x^2}{\sqrt {x^4+5 x^2+3}}dx\right )-\frac {\left (24-5 x^2\right ) \sqrt {x^4+5 x^2+3}}{3 x}\right )-\frac {\left (10-9 x^2\right ) \left (x^4+5 x^2+3\right )^{3/2}}{15 x^3}\)

\(\Big \downarrow \) 1412

\(\displaystyle \frac {13}{5} \left (\frac {1}{3} \left (73 \int \frac {x^2}{\sqrt {x^4+5 x^2+3}}dx+\frac {25 \sqrt {\frac {6}{5+\sqrt {13}}} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {x^4+5 x^2+3}}\right )-\frac {\left (24-5 x^2\right ) \sqrt {x^4+5 x^2+3}}{3 x}\right )-\frac {\left (10-9 x^2\right ) \left (x^4+5 x^2+3\right )^{3/2}}{15 x^3}\)

\(\Big \downarrow \) 1455

\(\displaystyle \frac {13}{5} \left (\frac {1}{3} \left (\frac {25 \sqrt {\frac {6}{5+\sqrt {13}}} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {x^4+5 x^2+3}}+73 \left (\frac {x \left (2 x^2+\sqrt {13}+5\right )}{2 \sqrt {x^4+5 x^2+3}}-\frac {\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{2 \sqrt {x^4+5 x^2+3}}\right )\right )-\frac {\left (24-5 x^2\right ) \sqrt {x^4+5 x^2+3}}{3 x}\right )-\frac {\left (10-9 x^2\right ) \left (x^4+5 x^2+3\right )^{3/2}}{15 x^3}\)

input

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

output

-1/15*((10 - 9*x^2)*(3 + 5*x^2 + x^4)^(3/2))/x^3 + (13*(-1/3*((24 - 5*x^2) 
*Sqrt[3 + 5*x^2 + x^4])/x + (73*((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 + Sq 
rt[13])/6]*x], (-13 + 5*Sqrt[13])/6])/(2*Sqrt[3 + 5*x^2 + x^4])) + (25*Sqr 
t[6/(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], (-1 
3 + 5*Sqrt[13])/6])/Sqrt[3 + 5*x^2 + x^4])/3))/5
 

3.2.67.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 1594

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*((d*(m 
+ 4*p + 3) + e*(m + 1)*x^2)/(f*(m + 1)*(m + 4*p + 3))), x] + Simp[2*(p/(f^2 
*(m + 1)*(m + 4*p + 3)))   Int[(f*x)^(m + 2)*(a + b*x^2 + c*x^4)^(p - 1)*Si 
mp[2*a*e*(m + 1) - b*d*(m + 4*p + 3) + (b*e*(m + 1) - 2*c*d*(m + 4*p + 3))* 
x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && G 
tQ[p, 0] && LtQ[m, -1] && m + 4*p + 3 != 0 && IntegerQ[2*p] && (IntegerQ[p] 
 || IntegerQ[m])
 

3.2.67.4 Maple [A] (verified)

Time = 2.13 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.76

method	result	size
risch	\(\frac {9 x^{10}+145 x^{8}+192 x^{6}-1405 x^{4}-1155 x^{2}-90}{15 x^{3} \sqrt {x^{4}+5 x^{2}+3}}+\frac {780 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {11388 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{5 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\)	\(238\)
default	\(-\frac {67 \sqrt {x^{4}+5 x^{2}+3}}{3 x}+\frac {3 x^{3} \sqrt {x^{4}+5 x^{2}+3}}{5}+\frac {20 x \sqrt {x^{4}+5 x^{2}+3}}{3}+\frac {780 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {11388 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{5 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}-\frac {2 \sqrt {x^{4}+5 x^{2}+3}}{x^{3}}\)	\(260\)
elliptic	\(-\frac {67 \sqrt {x^{4}+5 x^{2}+3}}{3 x}+\frac {3 x^{3} \sqrt {x^{4}+5 x^{2}+3}}{5}+\frac {20 x \sqrt {x^{4}+5 x^{2}+3}}{3}+\frac {780 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {11388 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{5 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}-\frac {2 \sqrt {x^{4}+5 x^{2}+3}}{x^{3}}\)	\(260\)

input

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

output

1/15*(9*x^10+145*x^8+192*x^6-1405*x^4-1155*x^2-90)/x^3/(x^4+5*x^2+3)^(1/2) 
+780/(-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))-11388/5/(-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))*(EllipticF(1/6*x*(-30+6*13^(1/2))^(1/2),5/6*3^(1/2)+1/6*3 
9^(1/2))-EllipticE(1/6*x*(-30+6*13^(1/2))^(1/2),5/6*3^(1/2)+1/6*39^(1/2)))
 

3.2.67.5 Fricas [F]

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

input

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

output

integral((3*x^6 + 17*x^4 + 19*x^2 + 6)*sqrt(x^4 + 5*x^2 + 3)/x^4, x)
 

3.2.67.6 Sympy [F]

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

input

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

output

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

3.2.67.7 Maxima [F]

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

input

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

output

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

3.2.67.8 Giac [F]

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

input

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

output

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

3.2.67.9 Mupad [F(-1)]

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

input

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

output

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