\(\int \frac {x^7}{(a+b x^2+c x^4)^{3/2}} \, dx\) [1009]

Optimal result

Integrand size = 20, antiderivative size = 134 \[ \int \frac {x^7}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {x^4 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}+\frac {\left (3 b^2-8 a c-2 b c x^2\right ) \sqrt {a+b x^2+c x^4}}{2 c^2 \left (b^2-4 a c\right )}-\frac {3 b \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{4 c^{5/2}} \] Output:

x^4*(b*x^2+2*a)/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^(1/2)+1/2*(-2*b*c*x^2-8*a*c+3 
*b^2)*(c*x^4+b*x^2+a)^(1/2)/c^2/(-4*a*c+b^2)-3/4*b*arctanh(1/2*(2*c*x^2+b) 
/c^(1/2)/(c*x^4+b*x^2+a)^(1/2))/c^(5/2)
 

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.98 \[ \int \frac {x^7}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {-3 a b^2+8 a^2 c-3 b^3 x^2+10 a b c x^2-b^2 c x^4+4 a c^2 x^4}{2 c^2 \left (-b^2+4 a c\right ) \sqrt {a+b x^2+c x^4}}+\frac {3 b \log \left (b c^2+2 c^3 x^2-2 c^{5/2} \sqrt {a+b x^2+c x^4}\right )}{4 c^{5/2}} \] Input:

Integrate[x^7/(a + b*x^2 + c*x^4)^(3/2),x]
 

Output:

(-3*a*b^2 + 8*a^2*c - 3*b^3*x^2 + 10*a*b*c*x^2 - b^2*c*x^4 + 4*a*c^2*x^4)/ 
(2*c^2*(-b^2 + 4*a*c)*Sqrt[a + b*x^2 + c*x^4]) + (3*b*Log[b*c^2 + 2*c^3*x^ 
2 - 2*c^(5/2)*Sqrt[a + b*x^2 + c*x^4]])/(4*c^(5/2))
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.12, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1434, 1164, 27, 1225, 1092, 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.

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

\(\Big \downarrow \) 1434

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

\(\Big \downarrow \) 1164

\(\displaystyle \frac {1}{2} \left (\frac {2 x^4 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {2 \int \frac {2 x^2 \left (b x^2+2 a\right )}{\sqrt {c x^4+b x^2+a}}dx^2}{b^2-4 a c}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (\frac {2 x^4 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {4 \int \frac {x^2 \left (b x^2+2 a\right )}{\sqrt {c x^4+b x^2+a}}dx^2}{b^2-4 a c}\right )\)

\(\Big \downarrow \) 1225

\(\displaystyle \frac {1}{2} \left (\frac {2 x^4 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {4 \left (\frac {3 b \left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx^2}{8 c^2}-\frac {\left (-8 a c+3 b^2-2 b c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c^2}\right )}{b^2-4 a c}\right )\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {1}{2} \left (\frac {2 x^4 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {4 \left (\frac {3 b \left (b^2-4 a c\right ) \int \frac {1}{4 c-x^4}d\frac {2 c x^2+b}{\sqrt {c x^4+b x^2+a}}}{4 c^2}-\frac {\left (-8 a c+3 b^2-2 b c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c^2}\right )}{b^2-4 a c}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{2} \left (\frac {2 x^4 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {4 \left (\frac {3 b \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{8 c^{5/2}}-\frac {\left (-8 a c+3 b^2-2 b c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c^2}\right )}{b^2-4 a c}\right )\)

Input:

Int[x^7/(a + b*x^2 + c*x^4)^(3/2),x]
 

Output:

((2*x^4*(2*a + b*x^2))/((b^2 - 4*a*c)*Sqrt[a + b*x^2 + c*x^4]) - (4*(-1/4* 
((3*b^2 - 8*a*c - 2*b*c*x^2)*Sqrt[a + b*x^2 + c*x^4])/c^2 + (3*b*(b^2 - 4* 
a*c)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(8*c^(5/2 
))))/(b^2 - 4*a*c))/2
 

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 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 1164

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m - 1)*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x 
+ c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a* 
c))   Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2* 
c*d^2*(2*p + 3) + e*(b*e - 2*d*c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p 
+ 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[m, 1] && Int 
QuadraticQ[a, b, c, d, e, m, p, x]
 

rule 1225

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

rule 1434

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp 
[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x + c*x^2)^p, x], x, x^2], x] /; Free 
Q[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]
 

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.27

method	result	size
pseudoelliptic	\(\frac {\frac {x^{4}}{2 \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {3 b \left (-\frac {x^{2}}{c \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {b \left (b \,x^{2}+2 a \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {-\ln \left (2\right )+\ln \left (\frac {2 c \,x^{2}+2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {c}+b}{\sqrt {c}}\right )}{c^{\frac {3}{2}}}\right )}{4}+\frac {2 a \left (b \,x^{2}+2 a \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}}{c}\)	\(170\)
default	\(\frac {x^{4}}{2 c \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {3 b \left (-\frac {x^{2}}{c \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b \left (2 c \,x^{2}+b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{c^{\frac {3}{2}}}\right )}{4 c}-\frac {a \left (-\frac {1}{c \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b \left (2 c \,x^{2}+b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{c}\)	\(218\)
elliptic	\(\frac {x^{4}}{2 c \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {3 b \left (-\frac {x^{2}}{c \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b \left (2 c \,x^{2}+b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{c^{\frac {3}{2}}}\right )}{4 c}-\frac {a \left (-\frac {1}{c \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b \left (2 c \,x^{2}+b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{c}\)	\(218\)
risch	\(\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{2 c^{2}}-\frac {b^{3} x^{2}}{4 c^{2} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {2 a^{2}}{c \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (4 a c -b^{2}\right )}+\frac {b^{2} a}{2 c^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (4 a c -b^{2}\right )}+\frac {3 b \,x^{2}}{4 c^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {3 b^{2}}{8 c^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {3 b^{4}}{8 c^{3} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {3 b \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{4 c^{\frac {5}{2}}}\)	\(241\)

Input:

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

Output:

1/c*(1/2*x^4/(c*x^4+b*x^2+a)^(1/2)-3/4*b*(-x^2/c/(c*x^4+b*x^2+a)^(1/2)+b/c 
*(b*x^2+2*a)/(4*a*c-b^2)/(c*x^4+b*x^2+a)^(1/2)+1/c^(3/2)*(-ln(2)+ln((2*c*x 
^2+2*(c*x^4+b*x^2+a)^(1/2)*c^(1/2)+b)/c^(1/2))))+2*a*(b*x^2+2*a)/(4*a*c-b^ 
2)/(c*x^4+b*x^2+a)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 459, normalized size of antiderivative = 3.43 \[ \int \frac {x^7}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\left [\frac {3 \, {\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} x^{4} + a b^{3} - 4 \, a^{2} b c + {\left (b^{4} - 4 \, a b^{2} c\right )} x^{2}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{4} - 8 \, b c x^{2} - b^{2} + 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{4} + 3 \, a b^{2} c - 8 \, a^{2} c^{2} + {\left (3 \, b^{3} c - 10 \, a b c^{2}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{8 \, {\left (a b^{2} c^{3} - 4 \, a^{2} c^{4} + {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{4} + {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{2}\right )}}, \frac {3 \, {\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} x^{4} + a b^{3} - 4 \, a^{2} b c + {\left (b^{4} - 4 \, a b^{2} c\right )} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{4} + b c x^{2} + a c\right )}}\right ) + 2 \, {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{4} + 3 \, a b^{2} c - 8 \, a^{2} c^{2} + {\left (3 \, b^{3} c - 10 \, a b c^{2}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{4 \, {\left (a b^{2} c^{3} - 4 \, a^{2} c^{4} + {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{4} + {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{2}\right )}}\right ] \] Input:

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

Output:

[1/8*(3*((b^3*c - 4*a*b*c^2)*x^4 + a*b^3 - 4*a^2*b*c + (b^4 - 4*a*b^2*c)*x 
^2)*sqrt(c)*log(-8*c^2*x^4 - 8*b*c*x^2 - b^2 + 4*sqrt(c*x^4 + b*x^2 + a)*( 
2*c*x^2 + b)*sqrt(c) - 4*a*c) + 4*((b^2*c^2 - 4*a*c^3)*x^4 + 3*a*b^2*c - 8 
*a^2*c^2 + (3*b^3*c - 10*a*b*c^2)*x^2)*sqrt(c*x^4 + b*x^2 + a))/(a*b^2*c^3 
 - 4*a^2*c^4 + (b^2*c^4 - 4*a*c^5)*x^4 + (b^3*c^3 - 4*a*b*c^4)*x^2), 1/4*( 
3*((b^3*c - 4*a*b*c^2)*x^4 + a*b^3 - 4*a^2*b*c + (b^4 - 4*a*b^2*c)*x^2)*sq 
rt(-c)*arctan(1/2*sqrt(c*x^4 + b*x^2 + a)*(2*c*x^2 + b)*sqrt(-c)/(c^2*x^4 
+ b*c*x^2 + a*c)) + 2*((b^2*c^2 - 4*a*c^3)*x^4 + 3*a*b^2*c - 8*a^2*c^2 + ( 
3*b^3*c - 10*a*b*c^2)*x^2)*sqrt(c*x^4 + b*x^2 + a))/(a*b^2*c^3 - 4*a^2*c^4 
 + (b^2*c^4 - 4*a*c^5)*x^4 + (b^3*c^3 - 4*a*b*c^4)*x^2)]
 

Sympy [F]

\[ \int \frac {x^7}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {x^{7}}{\left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(x**7/(c*x**4+b*x**2+a)**(3/2),x)
 

Output:

Integral(x**7/(a + b*x**2 + c*x**4)**(3/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^7}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for 
 more deta
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.13 \[ \int \frac {x^7}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {{\left (\frac {{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2}}{b^{2} c^{2} - 4 \, a c^{3}} + \frac {3 \, b^{3} - 10 \, a b c}{b^{2} c^{2} - 4 \, a c^{3}}\right )} x^{2} + \frac {3 \, a b^{2} - 8 \, a^{2} c}{b^{2} c^{2} - 4 \, a c^{3}}}{2 \, \sqrt {c x^{4} + b x^{2} + a}} + \frac {3 \, b \log \left ({\left | 2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} \sqrt {c} + b \right |}\right )}{4 \, c^{\frac {5}{2}}} \] Input:

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

Output:

1/2*(((b^2*c - 4*a*c^2)*x^2/(b^2*c^2 - 4*a*c^3) + (3*b^3 - 10*a*b*c)/(b^2* 
c^2 - 4*a*c^3))*x^2 + (3*a*b^2 - 8*a^2*c)/(b^2*c^2 - 4*a*c^3))/sqrt(c*x^4 
+ b*x^2 + a) + 3/4*b*log(abs(2*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))*sqr 
t(c) + b))/c^(5/2)
 

Mupad [F(-1)]

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

int(x^7/(a + b*x^2 + c*x^4)^(3/2),x)
 

Output:

int(x^7/(a + b*x^2 + c*x^4)^(3/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 3725, normalized size of antiderivative = 27.80 \[ \int \frac {x^7}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

int(x^7/(c*x^4+b*x^2+a)^(3/2),x)
 

Output:

( - 384*sqrt(a + b*x**2 + c*x**4)*log((2*sqrt(c)*sqrt(a + b*x**2 + c*x**4) 
 + b + 2*c*x**2)/sqrt(4*a*c - b**2))*a**3*b**2*c**3 - 768*sqrt(a + b*x**2 
+ c*x**4)*log((2*sqrt(c)*sqrt(a + b*x**2 + c*x**4) + b + 2*c*x**2)/sqrt(4* 
a*c - b**2))*a**3*b*c**4*x**2 - 1152*sqrt(a + b*x**2 + c*x**4)*log((2*sqrt 
(c)*sqrt(a + b*x**2 + c*x**4) + b + 2*c*x**2)/sqrt(4*a*c - b**2))*a**2*b** 
3*c**3*x**2 - 3456*sqrt(a + b*x**2 + c*x**4)*log((2*sqrt(c)*sqrt(a + b*x** 
2 + c*x**4) + b + 2*c*x**2)/sqrt(4*a*c - b**2))*a**2*b**2*c**4*x**4 - 2304 
*sqrt(a + b*x**2 + c*x**4)*log((2*sqrt(c)*sqrt(a + b*x**2 + c*x**4) + b + 
2*c*x**2)/sqrt(4*a*c - b**2))*a**2*b*c**5*x**6 + 24*sqrt(a + b*x**2 + c*x* 
*4)*log((2*sqrt(c)*sqrt(a + b*x**2 + c*x**4) + b + 2*c*x**2)/sqrt(4*a*c - 
b**2))*a*b**6*c + 240*sqrt(a + b*x**2 + c*x**4)*log((2*sqrt(c)*sqrt(a + b* 
x**2 + c*x**4) + b + 2*c*x**2)/sqrt(4*a*c - b**2))*a*b**5*c**2*x**2 - 192* 
sqrt(a + b*x**2 + c*x**4)*log((2*sqrt(c)*sqrt(a + b*x**2 + c*x**4) + b + 2 
*c*x**2)/sqrt(4*a*c - b**2))*a*b**4*c**3*x**4 - 2688*sqrt(a + b*x**2 + c*x 
**4)*log((2*sqrt(c)*sqrt(a + b*x**2 + c*x**4) + b + 2*c*x**2)/sqrt(4*a*c - 
 b**2))*a*b**3*c**4*x**6 - 3840*sqrt(a + b*x**2 + c*x**4)*log((2*sqrt(c)*s 
qrt(a + b*x**2 + c*x**4) + b + 2*c*x**2)/sqrt(4*a*c - b**2))*a*b**2*c**5*x 
**8 - 1536*sqrt(a + b*x**2 + c*x**4)*log((2*sqrt(c)*sqrt(a + b*x**2 + c*x* 
*4) + b + 2*c*x**2)/sqrt(4*a*c - b**2))*a*b*c**6*x**10 + 24*sqrt(a + b*x** 
2 + c*x**4)*log((2*sqrt(c)*sqrt(a + b*x**2 + c*x**4) + b + 2*c*x**2)/sq...