3.2.61 \(\int \frac {(2+3 x^2) (3+5 x^2+x^4)^{3/2}}{x^5} \, dx\) [161]

3.2.61.1 Optimal result
3.2.61.2 Mathematica [A] (verified)
3.2.61.3 Rubi [A] (verified)
3.2.61.4 Maple [A] (verified)
3.2.61.5 Fricas [A] (verification not implemented)
3.2.61.6 Sympy [F]
3.2.61.7 Maxima [A] (verification not implemented)
3.2.61.8 Giac [A] (verification not implemented)
3.2.61.9 Mupad [F(-1)]

3.2.61.1 Optimal result

Integrand size = 25, antiderivative size = 127 \[ \int \frac {\left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{x^5} \, dx=-\frac {3 \left (28-19 x^2\right ) \sqrt {3+5 x^2+x^4}}{8 x^2}-\frac {\left (2-3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{4 x^4}+\frac {453}{16} \text {arctanh}\left (\frac {5+2 x^2}{2 \sqrt {3+5 x^2+x^4}}\right )-\frac {127}{8} \sqrt {3} \text {arctanh}\left (\frac {6+5 x^2}{2 \sqrt {3} \sqrt {3+5 x^2+x^4}}\right ) \]

output
-1/4*(-3*x^2+2)*(x^4+5*x^2+3)^(3/2)/x^4+453/16*arctanh(1/2*(2*x^2+5)/(x^4+ 
5*x^2+3)^(1/2))-127/8*arctanh(1/6*(5*x^2+6)*3^(1/2)/(x^4+5*x^2+3)^(1/2))*3 
^(1/2)-3/8*(-19*x^2+28)*(x^4+5*x^2+3)^(1/2)/x^2
 
3.2.61.2 Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.80 \[ \int \frac {\left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{x^5} \, dx=\frac {1}{16} \left (\frac {2 \sqrt {3+5 x^2+x^4} \left (-12-86 x^2+83 x^4+6 x^6\right )}{x^4}+508 \sqrt {3} \text {arctanh}\left (\frac {x^2-\sqrt {3+5 x^2+x^4}}{\sqrt {3}}\right )-453 \log \left (-5-2 x^2+2 \sqrt {3+5 x^2+x^4}\right )\right ) \]

input
Integrate[((2 + 3*x^2)*(3 + 5*x^2 + x^4)^(3/2))/x^5,x]
 
output
((2*Sqrt[3 + 5*x^2 + x^4]*(-12 - 86*x^2 + 83*x^4 + 6*x^6))/x^4 + 508*Sqrt[ 
3]*ArcTanh[(x^2 - Sqrt[3 + 5*x^2 + x^4])/Sqrt[3]] - 453*Log[-5 - 2*x^2 + 2 
*Sqrt[3 + 5*x^2 + x^4]])/16
 
3.2.61.3 Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.06, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1578, 1230, 27, 1230, 25, 1269, 1092, 219, 1154, 219}

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.

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

\(\Big \downarrow \) 1578

\(\displaystyle \frac {1}{2} \int \frac {\left (3 x^2+2\right ) \left (x^4+5 x^2+3\right )^{3/2}}{x^6}dx^2\)

\(\Big \downarrow \) 1230

\(\displaystyle \frac {1}{2} \left (-\frac {3}{8} \int -\frac {2 \left (19 x^2+28\right ) \sqrt {x^4+5 x^2+3}}{x^4}dx^2-\frac {\left (2-3 x^2\right ) \left (x^4+5 x^2+3\right )^{3/2}}{2 x^4}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (\frac {3}{4} \int \frac {\left (19 x^2+28\right ) \sqrt {x^4+5 x^2+3}}{x^4}dx^2-\frac {\left (2-3 x^2\right ) \left (x^4+5 x^2+3\right )^{3/2}}{2 x^4}\right )\)

\(\Big \downarrow \) 1230

\(\displaystyle \frac {1}{2} \left (\frac {3}{4} \left (-\frac {1}{2} \int -\frac {151 x^2+254}{x^2 \sqrt {x^4+5 x^2+3}}dx^2-\frac {\sqrt {x^4+5 x^2+3} \left (28-19 x^2\right )}{x^2}\right )-\frac {\left (2-3 x^2\right ) \left (x^4+5 x^2+3\right )^{3/2}}{2 x^4}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {151 x^2+254}{x^2 \sqrt {x^4+5 x^2+3}}dx^2-\frac {\left (28-19 x^2\right ) \sqrt {x^4+5 x^2+3}}{x^2}\right )-\frac {\left (2-3 x^2\right ) \left (x^4+5 x^2+3\right )^{3/2}}{2 x^4}\right )\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {1}{2} \left (\frac {3}{4} \left (\frac {1}{2} \left (151 \int \frac {1}{\sqrt {x^4+5 x^2+3}}dx^2+254 \int \frac {1}{x^2 \sqrt {x^4+5 x^2+3}}dx^2\right )-\frac {\left (28-19 x^2\right ) \sqrt {x^4+5 x^2+3}}{x^2}\right )-\frac {\left (2-3 x^2\right ) \left (x^4+5 x^2+3\right )^{3/2}}{2 x^4}\right )\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {1}{2} \left (\frac {3}{4} \left (\frac {1}{2} \left (302 \int \frac {1}{4-x^4}d\frac {2 x^2+5}{\sqrt {x^4+5 x^2+3}}+254 \int \frac {1}{x^2 \sqrt {x^4+5 x^2+3}}dx^2\right )-\frac {\left (28-19 x^2\right ) \sqrt {x^4+5 x^2+3}}{x^2}\right )-\frac {\left (2-3 x^2\right ) \left (x^4+5 x^2+3\right )^{3/2}}{2 x^4}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{2} \left (\frac {3}{4} \left (\frac {1}{2} \left (254 \int \frac {1}{x^2 \sqrt {x^4+5 x^2+3}}dx^2+151 \text {arctanh}\left (\frac {2 x^2+5}{2 \sqrt {x^4+5 x^2+3}}\right )\right )-\frac {\left (28-19 x^2\right ) \sqrt {x^4+5 x^2+3}}{x^2}\right )-\frac {\left (2-3 x^2\right ) \left (x^4+5 x^2+3\right )^{3/2}}{2 x^4}\right )\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {1}{2} \left (\frac {3}{4} \left (\frac {1}{2} \left (151 \text {arctanh}\left (\frac {2 x^2+5}{2 \sqrt {x^4+5 x^2+3}}\right )-508 \int \frac {1}{12-x^4}d\frac {5 x^2+6}{\sqrt {x^4+5 x^2+3}}\right )-\frac {\left (28-19 x^2\right ) \sqrt {x^4+5 x^2+3}}{x^2}\right )-\frac {\left (2-3 x^2\right ) \left (x^4+5 x^2+3\right )^{3/2}}{2 x^4}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{2} \left (\frac {3}{4} \left (\frac {1}{2} \left (151 \text {arctanh}\left (\frac {2 x^2+5}{2 \sqrt {x^4+5 x^2+3}}\right )-\frac {254 \text {arctanh}\left (\frac {5 x^2+6}{2 \sqrt {3} \sqrt {x^4+5 x^2+3}}\right )}{\sqrt {3}}\right )-\frac {\left (28-19 x^2\right ) \sqrt {x^4+5 x^2+3}}{x^2}\right )-\frac {\left (2-3 x^2\right ) \left (x^4+5 x^2+3\right )^{3/2}}{2 x^4}\right )\)

input
Int[((2 + 3*x^2)*(3 + 5*x^2 + x^4)^(3/2))/x^5,x]
 
output
(-1/2*((2 - 3*x^2)*(3 + 5*x^2 + x^4)^(3/2))/x^4 + (3*(-(((28 - 19*x^2)*Sqr 
t[3 + 5*x^2 + x^4])/x^2) + (151*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^ 
4])] - (254*ArcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])])/Sqrt[3 
])/2))/4)/2
 

3.2.61.3.1 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 219
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1092
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1154
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1230
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - 
 d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2*p 
+ 2))), x] + Simp[p/(e^2*(m + 1)*(m + 2*p + 2))   Int[(d + e*x)^(m + 1)*(a 
+ b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m 
+ 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, 
 x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, - 
1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&  !ILtQ 
[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1269
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

rule 1578
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ 
)^4)^(p_.), x_Symbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a 
+ b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int 
egerQ[(m - 1)/2]
 
3.2.61.4 Maple [A] (verified)

Time = 0.53 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.76

method result size
pseudoelliptic \(\frac {-254 \,\operatorname {arctanh}\left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right ) \sqrt {3}\, x^{4}+453 \ln \left (2 x^{2}+5+2 \sqrt {x^{4}+5 x^{2}+3}\right ) x^{4}+12 \sqrt {x^{4}+5 x^{2}+3}\, \left (x^{6}+\frac {83}{6} x^{4}-\frac {43}{3} x^{2}-2\right )}{16 x^{4}}\) \(96\)
trager \(\frac {\left (6 x^{6}+83 x^{4}-86 x^{2}-12\right ) \sqrt {x^{4}+5 x^{2}+3}}{8 x^{4}}+\frac {453 \ln \left (-2 x^{2}-2 \sqrt {x^{4}+5 x^{2}+3}-5\right )}{16}+\frac {127 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{2}+6 \sqrt {x^{4}+5 x^{2}+3}-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )}{x^{2}}\right )}{8}\) \(108\)
default \(\frac {453 \ln \left (\frac {5}{2}+x^{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{16}+\frac {3 x^{2} \sqrt {x^{4}+5 x^{2}+3}}{4}+\frac {83 \sqrt {x^{4}+5 x^{2}+3}}{8}-\frac {43 \sqrt {x^{4}+5 x^{2}+3}}{4 x^{2}}-\frac {127 \,\operatorname {arctanh}\left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right ) \sqrt {3}}{8}-\frac {3 \sqrt {x^{4}+5 x^{2}+3}}{2 x^{4}}\) \(117\)
risch \(-\frac {43 x^{6}+221 x^{4}+159 x^{2}+18}{4 x^{4} \sqrt {x^{4}+5 x^{2}+3}}+\frac {453 \ln \left (\frac {5}{2}+x^{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{16}+\frac {3 x^{2} \sqrt {x^{4}+5 x^{2}+3}}{4}+\frac {83 \sqrt {x^{4}+5 x^{2}+3}}{8}-\frac {127 \,\operatorname {arctanh}\left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right ) \sqrt {3}}{8}\) \(117\)
elliptic \(\frac {453 \ln \left (\frac {5}{2}+x^{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{16}+\frac {3 x^{2} \sqrt {x^{4}+5 x^{2}+3}}{4}+\frac {83 \sqrt {x^{4}+5 x^{2}+3}}{8}-\frac {43 \sqrt {x^{4}+5 x^{2}+3}}{4 x^{2}}-\frac {127 \,\operatorname {arctanh}\left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right ) \sqrt {3}}{8}-\frac {3 \sqrt {x^{4}+5 x^{2}+3}}{2 x^{4}}\) \(117\)

input
int((3*x^2+2)*(x^4+5*x^2+3)^(3/2)/x^5,x,method=_RETURNVERBOSE)
 
output
1/16*(-254*arctanh(1/6*(5*x^2+6)*3^(1/2)/(x^4+5*x^2+3)^(1/2))*3^(1/2)*x^4+ 
453*ln(2*x^2+5+2*(x^4+5*x^2+3)^(1/2))*x^4+12*(x^4+5*x^2+3)^(1/2)*(x^6+83/6 
*x^4-43/3*x^2-2))/x^4
 
3.2.61.5 Fricas [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.96 \[ \int \frac {\left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{x^5} \, dx=\frac {1016 \, \sqrt {3} x^{4} \log \left (\frac {25 \, x^{2} - 2 \, \sqrt {3} {\left (5 \, x^{2} + 6\right )} - 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (5 \, \sqrt {3} - 6\right )} + 30}{x^{2}}\right ) - 1812 \, x^{4} \log \left (-2 \, x^{2} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} - 5\right ) + 67 \, x^{4} + 8 \, {\left (6 \, x^{6} + 83 \, x^{4} - 86 \, x^{2} - 12\right )} \sqrt {x^{4} + 5 \, x^{2} + 3}}{64 \, x^{4}} \]

input
integrate((3*x^2+2)*(x^4+5*x^2+3)^(3/2)/x^5,x, algorithm="fricas")
 
output
1/64*(1016*sqrt(3)*x^4*log((25*x^2 - 2*sqrt(3)*(5*x^2 + 6) - 2*sqrt(x^4 + 
5*x^2 + 3)*(5*sqrt(3) - 6) + 30)/x^2) - 1812*x^4*log(-2*x^2 + 2*sqrt(x^4 + 
 5*x^2 + 3) - 5) + 67*x^4 + 8*(6*x^6 + 83*x^4 - 86*x^2 - 12)*sqrt(x^4 + 5* 
x^2 + 3))/x^4
 
3.2.61.6 Sympy [F]

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

input
integrate((3*x**2+2)*(x**4+5*x**2+3)**(3/2)/x**5,x)
 
output
Integral((3*x**2 + 2)*(x**4 + 5*x**2 + 3)**(3/2)/x**5, x)
 
3.2.61.7 Maxima [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.08 \[ \int \frac {\left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{x^5} \, dx=\frac {7}{2} \, \sqrt {x^{4} + 5 \, x^{2} + 3} x^{2} + \frac {1}{6} \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}} - \frac {127}{8} \, \sqrt {3} \log \left (\frac {2 \, \sqrt {3} \sqrt {x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac {6}{x^{2}} + 5\right ) + \frac {197}{8} \, \sqrt {x^{4} + 5 \, x^{2} + 3} - \frac {23 \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}}}{12 \, x^{2}} - \frac {{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {5}{2}}}{6 \, x^{4}} + \frac {453}{16} \, \log \left (2 \, x^{2} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 5\right ) \]

input
integrate((3*x^2+2)*(x^4+5*x^2+3)^(3/2)/x^5,x, algorithm="maxima")
 
output
7/2*sqrt(x^4 + 5*x^2 + 3)*x^2 + 1/6*(x^4 + 5*x^2 + 3)^(3/2) - 127/8*sqrt(3 
)*log(2*sqrt(3)*sqrt(x^4 + 5*x^2 + 3)/x^2 + 6/x^2 + 5) + 197/8*sqrt(x^4 + 
5*x^2 + 3) - 23/12*(x^4 + 5*x^2 + 3)^(3/2)/x^2 - 1/6*(x^4 + 5*x^2 + 3)^(5/ 
2)/x^4 + 453/16*log(2*x^2 + 2*sqrt(x^4 + 5*x^2 + 3) + 5)
 
3.2.61.8 Giac [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.50 \[ \int \frac {\left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{x^5} \, dx=\frac {1}{8} \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (6 \, x^{2} + 83\right )} + \frac {127}{8} \, \sqrt {3} \log \left (\frac {x^{2} + \sqrt {3} - \sqrt {x^{4} + 5 \, x^{2} + 3}}{x^{2} - \sqrt {3} - \sqrt {x^{4} + 5 \, x^{2} + 3}}\right ) + \frac {227 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{3} + 348 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{2} - 459 \, x^{2} + 459 \, \sqrt {x^{4} + 5 \, x^{2} + 3} - 684}{4 \, {\left ({\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{2} - 3\right )}^{2}} - \frac {453}{16} \, \log \left (2 \, x^{2} - 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 5\right ) \]

input
integrate((3*x^2+2)*(x^4+5*x^2+3)^(3/2)/x^5,x, algorithm="giac")
 
output
1/8*sqrt(x^4 + 5*x^2 + 3)*(6*x^2 + 83) + 127/8*sqrt(3)*log((x^2 + sqrt(3) 
- sqrt(x^4 + 5*x^2 + 3))/(x^2 - sqrt(3) - sqrt(x^4 + 5*x^2 + 3))) + 1/4*(2 
27*(x^2 - sqrt(x^4 + 5*x^2 + 3))^3 + 348*(x^2 - sqrt(x^4 + 5*x^2 + 3))^2 - 
 459*x^2 + 459*sqrt(x^4 + 5*x^2 + 3) - 684)/((x^2 - sqrt(x^4 + 5*x^2 + 3)) 
^2 - 3)^2 - 453/16*log(2*x^2 - 2*sqrt(x^4 + 5*x^2 + 3) + 5)
 
3.2.61.9 Mupad [F(-1)]

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

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