3.2.62 \(\int \frac {(2+3 x^2) (3+5 x^2+x^4)^{3/2}}{x^7} \, dx\) [162]

3.2.62.1 Optimal result

Integrand size = 25, antiderivative size = 127 \[ \int \frac {\left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{x^7} \, dx=-\frac {\left (67-32 x^2\right ) \sqrt {3+5 x^2+x^4}}{12 x^2}-\frac {\left (2+7 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{6 x^6}+\frac {49}{4} \text {arctanh}\left (\frac {5+2 x^2}{2 \sqrt {3+5 x^2+x^4}}\right )-\frac {527 \text {arctanh}\left (\frac {6+5 x^2}{2 \sqrt {3} \sqrt {3+5 x^2+x^4}}\right )}{24 \sqrt {3}} \]

output

-1/6*(7*x^2+2)*(x^4+5*x^2+3)^(3/2)/x^6+49/4*arctanh(1/2*(2*x^2+5)/(x^4+5*x 
^2+3)^(1/2))-527/72*arctanh(1/6*(5*x^2+6)*3^(1/2)/(x^4+5*x^2+3)^(1/2))*3^( 
1/2)-1/12*(-32*x^2+67)*(x^4+5*x^2+3)^(1/2)/x^2
 

3.2.62.2 Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.80 \[ \int \frac {\left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{x^7} \, dx=\frac {1}{36} \left (\frac {3 \sqrt {3+5 x^2+x^4} \left (-12-62 x^2-141 x^4+18 x^6\right )}{x^6}+527 \sqrt {3} \text {arctanh}\left (\frac {x^2-\sqrt {3+5 x^2+x^4}}{\sqrt {3}}\right )-441 \log \left (-5-2 x^2+2 \sqrt {3+5 x^2+x^4}\right )\right ) \]

input

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

output

((3*Sqrt[3 + 5*x^2 + x^4]*(-12 - 62*x^2 - 141*x^4 + 18*x^6))/x^6 + 527*Sqr 
t[3]*ArcTanh[(x^2 - Sqrt[3 + 5*x^2 + x^4])/Sqrt[3]] - 441*Log[-5 - 2*x^2 + 
 2*Sqrt[3 + 5*x^2 + x^4]])/36
 

3.2.62.3 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, 1229, 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.

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

\(\Big \downarrow \) 1578

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

\(\Big \downarrow \) 1229

\(\displaystyle \frac {1}{2} \left (-\frac {1}{12} \int -\frac {2 \left (32 x^2+67\right ) \sqrt {x^4+5 x^2+3}}{x^4}dx^2-\frac {\left (7 x^2+2\right ) \left (x^4+5 x^2+3\right )^{3/2}}{3 x^6}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (\frac {1}{6} \int \frac {\left (32 x^2+67\right ) \sqrt {x^4+5 x^2+3}}{x^4}dx^2-\frac {\left (7 x^2+2\right ) \left (x^4+5 x^2+3\right )^{3/2}}{3 x^6}\right )\)

\(\Big \downarrow \) 1230

\(\displaystyle \frac {1}{2} \left (\frac {1}{6} \left (-\frac {1}{2} \int -\frac {294 x^2+527}{x^2 \sqrt {x^4+5 x^2+3}}dx^2-\frac {\sqrt {x^4+5 x^2+3} \left (67-32 x^2\right )}{x^2}\right )-\frac {\left (7 x^2+2\right ) \left (x^4+5 x^2+3\right )^{3/2}}{3 x^6}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {294 x^2+527}{x^2 \sqrt {x^4+5 x^2+3}}dx^2-\frac {\left (67-32 x^2\right ) \sqrt {x^4+5 x^2+3}}{x^2}\right )-\frac {\left (7 x^2+2\right ) \left (x^4+5 x^2+3\right )^{3/2}}{3 x^6}\right )\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {1}{2} \left (\frac {1}{6} \left (\frac {1}{2} \left (294 \int \frac {1}{\sqrt {x^4+5 x^2+3}}dx^2+527 \int \frac {1}{x^2 \sqrt {x^4+5 x^2+3}}dx^2\right )-\frac {\left (67-32 x^2\right ) \sqrt {x^4+5 x^2+3}}{x^2}\right )-\frac {\left (7 x^2+2\right ) \left (x^4+5 x^2+3\right )^{3/2}}{3 x^6}\right )\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {1}{2} \left (\frac {1}{6} \left (\frac {1}{2} \left (588 \int \frac {1}{4-x^4}d\frac {2 x^2+5}{\sqrt {x^4+5 x^2+3}}+527 \int \frac {1}{x^2 \sqrt {x^4+5 x^2+3}}dx^2\right )-\frac {\left (67-32 x^2\right ) \sqrt {x^4+5 x^2+3}}{x^2}\right )-\frac {\left (7 x^2+2\right ) \left (x^4+5 x^2+3\right )^{3/2}}{3 x^6}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{2} \left (\frac {1}{6} \left (\frac {1}{2} \left (527 \int \frac {1}{x^2 \sqrt {x^4+5 x^2+3}}dx^2+294 \text {arctanh}\left (\frac {2 x^2+5}{2 \sqrt {x^4+5 x^2+3}}\right )\right )-\frac {\left (67-32 x^2\right ) \sqrt {x^4+5 x^2+3}}{x^2}\right )-\frac {\left (7 x^2+2\right ) \left (x^4+5 x^2+3\right )^{3/2}}{3 x^6}\right )\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {1}{2} \left (\frac {1}{6} \left (\frac {1}{2} \left (294 \text {arctanh}\left (\frac {2 x^2+5}{2 \sqrt {x^4+5 x^2+3}}\right )-1054 \int \frac {1}{12-x^4}d\frac {5 x^2+6}{\sqrt {x^4+5 x^2+3}}\right )-\frac {\left (67-32 x^2\right ) \sqrt {x^4+5 x^2+3}}{x^2}\right )-\frac {\left (7 x^2+2\right ) \left (x^4+5 x^2+3\right )^{3/2}}{3 x^6}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{2} \left (\frac {1}{6} \left (\frac {1}{2} \left (294 \text {arctanh}\left (\frac {2 x^2+5}{2 \sqrt {x^4+5 x^2+3}}\right )-\frac {527 \text {arctanh}\left (\frac {5 x^2+6}{2 \sqrt {3} \sqrt {x^4+5 x^2+3}}\right )}{\sqrt {3}}\right )-\frac {\left (67-32 x^2\right ) \sqrt {x^4+5 x^2+3}}{x^2}\right )-\frac {\left (7 x^2+2\right ) \left (x^4+5 x^2+3\right )^{3/2}}{3 x^6}\right )\)

input

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

output

(-1/3*((2 + 7*x^2)*(3 + 5*x^2 + x^4)^(3/2))/x^6 + (-(((67 - 32*x^2)*Sqrt[3 
 + 5*x^2 + x^4])/x^2) + (294*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4]) 
] - (527*ArcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])])/Sqrt[3])/ 
2)/6)/2
 

3.2.62.3.1 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 1229

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2 
)^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2))*(c* 
d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 
- b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x), x] - Simp[p/(e^2*(m + 1 
)*(m + 2)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2 
)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + 
p + 2)) + b*(a*e^2*g*(m + 1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c 
*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1) - b*(d*g*( 
m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g 
}, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 
0]
 

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]
 

3.2.62.4 Maple [A] (verified)

Time = 0.55 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.76

method	result	size
pseudoelliptic	\(\frac {-527 \,\operatorname {arctanh}\left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right ) \sqrt {3}\, x^{6}+882 \ln \left (2 x^{2}+5+2 \sqrt {x^{4}+5 x^{2}+3}\right ) x^{6}+108 \left (x^{6}-\frac {47}{6} x^{4}-\frac {31}{9} x^{2}-\frac {2}{3}\right ) \sqrt {x^{4}+5 x^{2}+3}}{72 x^{6}}\)	\(96\)
risch	\(-\frac {141 x^{8}+767 x^{6}+745 x^{4}+246 x^{2}+36}{12 x^{6} \sqrt {x^{4}+5 x^{2}+3}}+\frac {49 \ln \left (\frac {5}{2}+x^{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{4}+\frac {3 \sqrt {x^{4}+5 x^{2}+3}}{2}-\frac {527 \,\operatorname {arctanh}\left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right ) \sqrt {3}}{72}\)	\(105\)
trager	\(\frac {\left (18 x^{6}-141 x^{4}-62 x^{2}-12\right ) \sqrt {x^{4}+5 x^{2}+3}}{12 x^{6}}-\frac {49 \ln \left (2 x^{2}-2 \sqrt {x^{4}+5 x^{2}+3}+5\right )}{4}+\frac {527 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{2}+6 \sqrt {x^{4}+5 x^{2}+3}-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )}{x^{2}}\right )}{72}\)	\(108\)
default	\(\frac {49 \ln \left (\frac {5}{2}+x^{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{4}-\frac {\sqrt {x^{4}+5 x^{2}+3}}{x^{6}}-\frac {31 \sqrt {x^{4}+5 x^{2}+3}}{6 x^{4}}-\frac {47 \sqrt {x^{4}+5 x^{2}+3}}{4 x^{2}}-\frac {527 \,\operatorname {arctanh}\left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right ) \sqrt {3}}{72}+\frac {3 \sqrt {x^{4}+5 x^{2}+3}}{2}\)	\(117\)
elliptic	\(\frac {49 \ln \left (\frac {5}{2}+x^{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{4}-\frac {\sqrt {x^{4}+5 x^{2}+3}}{x^{6}}-\frac {31 \sqrt {x^{4}+5 x^{2}+3}}{6 x^{4}}-\frac {47 \sqrt {x^{4}+5 x^{2}+3}}{4 x^{2}}-\frac {527 \,\operatorname {arctanh}\left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right ) \sqrt {3}}{72}+\frac {3 \sqrt {x^{4}+5 x^{2}+3}}{2}\)	\(117\)

input

int((3*x^2+2)*(x^4+5*x^2+3)^(3/2)/x^7,x,method=_RETURNVERBOSE)
 

output

1/72*(-527*arctanh(1/6*(5*x^2+6)*3^(1/2)/(x^4+5*x^2+3)^(1/2))*3^(1/2)*x^6+ 
882*ln(2*x^2+5+2*(x^4+5*x^2+3)^(1/2))*x^6+108*(x^6-47/6*x^4-31/9*x^2-2/3)* 
(x^4+5*x^2+3)^(1/2))/x^6
 

3.2.62.5 Fricas [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.96 \[ \int \frac {\left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{x^7} \, dx=\frac {527 \, \sqrt {3} x^{6} \log \left (\frac {25 \, x^{2} - 2 \, \sqrt {3} {\left (5 \, x^{2} + 6\right )} - 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (5 \, \sqrt {3} - 6\right )} + 30}{x^{2}}\right ) - 882 \, x^{6} \log \left (-2 \, x^{2} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} - 5\right ) - 711 \, x^{6} + 6 \, {\left (18 \, x^{6} - 141 \, x^{4} - 62 \, x^{2} - 12\right )} \sqrt {x^{4} + 5 \, x^{2} + 3}}{72 \, x^{6}} \]

input

integrate((3*x^2+2)*(x^4+5*x^2+3)^(3/2)/x^7,x, algorithm="fricas")
 

output

1/72*(527*sqrt(3)*x^6*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) - 882*x^6*log(-2*x^2 + 2*sqrt(x^4 + 5 
*x^2 + 3) - 5) - 711*x^6 + 6*(18*x^6 - 141*x^4 - 62*x^2 - 12)*sqrt(x^4 + 5 
*x^2 + 3))/x^6
 

3.2.62.6 Sympy [F]

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

input

integrate((3*x**2+2)*(x**4+5*x**2+3)**(3/2)/x**7,x)
 

output

Integral((3*x**2 + 2)*(x**4 + 5*x**2 + 3)**(3/2)/x**7, x)
 

3.2.62.7 Maxima [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.21 \[ \int \frac {\left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{x^7} \, dx=\frac {67}{36} \, \sqrt {x^{4} + 5 \, x^{2} + 3} x^{2} + \frac {11}{54} \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}} - \frac {527}{72} \, \sqrt {3} \log \left (\frac {2 \, \sqrt {3} \sqrt {x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac {6}{x^{2}} + 5\right ) + \frac {431}{36} \, \sqrt {x^{4} + 5 \, x^{2} + 3} - \frac {79 \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}}}{108 \, x^{2}} - \frac {11 \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {5}{2}}}{54 \, x^{4}} - \frac {{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {5}{2}}}{9 \, x^{6}} + \frac {49}{4} \, \log \left (2 \, x^{2} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 5\right ) \]

input

integrate((3*x^2+2)*(x^4+5*x^2+3)^(3/2)/x^7,x, algorithm="maxima")
 

output

67/36*sqrt(x^4 + 5*x^2 + 3)*x^2 + 11/54*(x^4 + 5*x^2 + 3)^(3/2) - 527/72*s 
qrt(3)*log(2*sqrt(3)*sqrt(x^4 + 5*x^2 + 3)/x^2 + 6/x^2 + 5) + 431/36*sqrt( 
x^4 + 5*x^2 + 3) - 79/108*(x^4 + 5*x^2 + 3)^(3/2)/x^2 - 11/54*(x^4 + 5*x^2 
 + 3)^(5/2)/x^4 - 1/9*(x^4 + 5*x^2 + 3)^(5/2)/x^6 + 49/4*log(2*x^2 + 2*sqr 
t(x^4 + 5*x^2 + 3) + 5)
 

3.2.62.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 227 vs. \(2 (103) = 206\).

Time = 0.32 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.79 \[ \int \frac {\left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{x^7} \, dx=\frac {527}{72} \, \sqrt {3} \log \left (\frac {x^{2} + \sqrt {3} - \sqrt {x^{4} + 5 \, x^{2} + 3}}{x^{2} - \sqrt {3} - \sqrt {x^{4} + 5 \, x^{2} + 3}}\right ) + \frac {3}{2} \, \sqrt {x^{4} + 5 \, x^{2} + 3} + \frac {829 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{5} + 1824 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{4} - 2200 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{3} - 5292 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{2} + 2799 \, x^{2} - 2799 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 5724}{12 \, {\left ({\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{2} - 3\right )}^{3}} - \frac {49}{4} \, \log \left (2 \, x^{2} - 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 5\right ) \]

input

integrate((3*x^2+2)*(x^4+5*x^2+3)^(3/2)/x^7,x, algorithm="giac")
 

output

527/72*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))) + 3/2*sqrt(x^4 + 5*x^2 + 3) + 1/12*(829*(x^2 - s 
qrt(x^4 + 5*x^2 + 3))^5 + 1824*(x^2 - sqrt(x^4 + 5*x^2 + 3))^4 - 2200*(x^2 
 - sqrt(x^4 + 5*x^2 + 3))^3 - 5292*(x^2 - sqrt(x^4 + 5*x^2 + 3))^2 + 2799* 
x^2 - 2799*sqrt(x^4 + 5*x^2 + 3) + 5724)/((x^2 - sqrt(x^4 + 5*x^2 + 3))^2 
- 3)^3 - 49/4*log(2*x^2 - 2*sqrt(x^4 + 5*x^2 + 3) + 5)
 

3.2.62.9 Mupad [F(-1)]

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

input

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

output

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