\(\int \frac {\sqrt {a x+b x^3+c x^5}}{x^{3/2}} \, dx\) [55]

Optimal result

Integrand size = 24, antiderivative size = 194 \[ \int \frac {\sqrt {a x+b x^3+c x^5}}{x^{3/2}} \, dx=\frac {\sqrt {a x+b x^3+c x^5}}{2 \sqrt {x}}-\frac {\sqrt {a} \sqrt {x} \sqrt {a+b x^2+c x^4} \text {arctanh}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{2 \sqrt {a x+b x^3+c x^5}}+\frac {b \sqrt {x} \sqrt {a+b x^2+c x^4} \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{4 \sqrt {c} \sqrt {a x+b x^3+c x^5}} \] Output:

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

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt {a x+b x^3+c x^5}}{x^{3/2}} \, dx=\frac {\sqrt {x} \sqrt {a+b x^2+c x^4} \left (2 \sqrt {c} \sqrt {a+b x^2+c x^4}+4 \sqrt {a} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x^2-\sqrt {a+b x^2+c x^4}}{\sqrt {a}}\right )-b \log \left (b+2 c x^2-2 \sqrt {c} \sqrt {a+b x^2+c x^4}\right )\right )}{4 \sqrt {c} \sqrt {x \left (a+b x^2+c x^4\right )}} \] Input:

Integrate[Sqrt[a*x + b*x^3 + c*x^5]/x^(3/2),x]
 

Output:

(Sqrt[x]*Sqrt[a + b*x^2 + c*x^4]*(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4] + 4*Sq 
rt[a]*Sqrt[c]*ArcTanh[(Sqrt[c]*x^2 - Sqrt[a + b*x^2 + c*x^4])/Sqrt[a]] - b 
*Log[b + 2*c*x^2 - 2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4]]))/(4*Sqrt[c]*Sqrt[x* 
(a + b*x^2 + c*x^4)])
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.80, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1968, 2000, 1578, 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.

\(\displaystyle \int \frac {\sqrt {a x+b x^3+c x^5}}{x^{3/2}} \, dx\)

\(\Big \downarrow \) 1968

\(\displaystyle \frac {1}{2} \int \frac {b x^2+2 a}{\sqrt {x} \sqrt {c x^5+b x^3+a x}}dx+\frac {\sqrt {a x+b x^3+c x^5}}{2 \sqrt {x}}\)

\(\Big \downarrow \) 2000

\(\displaystyle \frac {\sqrt {x} \sqrt {a+b x^2+c x^4} \int \frac {b x^2+2 a}{x \sqrt {c x^4+b x^2+a}}dx}{2 \sqrt {a x+b x^3+c x^5}}+\frac {\sqrt {a x+b x^3+c x^5}}{2 \sqrt {x}}\)

\(\Big \downarrow \) 1578

\(\displaystyle \frac {\sqrt {x} \sqrt {a+b x^2+c x^4} \int \frac {b x^2+2 a}{x^2 \sqrt {c x^4+b x^2+a}}dx^2}{4 \sqrt {a x+b x^3+c x^5}}+\frac {\sqrt {a x+b x^3+c x^5}}{2 \sqrt {x}}\)

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

Int[Sqrt[a*x + b*x^3 + c*x^5]/x^(3/2),x]
 

Output:

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

Defintions of rubi rules used

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 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]
 

rule 1968

Int[(x_)^(m_.)*((b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(p_ 
), x_Symbol] :> Simp[x^(m + 1)*((a*x^q + b*x^n + c*x^(2*n - q))^p/(m + p*(2 
*n - q) + 1)), x] + Simp[(n - q)*(p/(m + p*(2*n - q) + 1))   Int[x^(m + q)* 
(2*a + b*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^(p - 1), x], x] /; Free 
Q[{a, b, c}, x] && EqQ[r, 2*n - q] && PosQ[n - q] &&  !IntegerQ[p] && NeQ[b 
^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[p, 0] && RationalQ[m, q] && GtQ[m + p*q 
 + 1, -(n - q)] && NeQ[m + p*(2*n - q) + 1, 0]
 

rule 2000

Int[((x_)^(m_.)*((A_) + (B_.)*(x_)^(j_.)))/Sqrt[(b_.)*(x_)^(n_.) + (a_.)*(x 
_)^(q_.) + (c_.)*(x_)^(r_.)], x_Symbol] :> Simp[x^(q/2)*(Sqrt[a + b*x^(n - 
q) + c*x^(2*(n - q))]/Sqrt[a*x^q + b*x^n + c*x^(2*n - q)])   Int[x^(m - q/2 
)*((A + B*x^(n - q))/Sqrt[a + b*x^(n - q) + c*x^(2*(n - q))]), x], x] /; Fr 
eeQ[{a, b, c, A, B, m, n, q}, x] && EqQ[j, n - q] && EqQ[r, 2*n - q] && Pos 
Q[n - q] && (EqQ[m, 1/2] || EqQ[m, -2^(-1)]) && EqQ[n, 3] && EqQ[q, 1]
 

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.70

method	result	size
default	\(-\frac {\sqrt {x \left (c \,x^{4}+b \,x^{2}+a \right )}\, \left (2 \sqrt {a}\, \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right ) \sqrt {c}-b \ln \left (\frac {2 c \,x^{2}+2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {c}+b}{2 \sqrt {c}}\right )-2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {c}\right )}{4 \sqrt {x}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {c}}\)	\(136\)

Input:

int((c*x^5+b*x^3+a*x)^(1/2)/x^(3/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 666, normalized size of antiderivative = 3.43 \[ \int \frac {\sqrt {a x+b x^3+c x^5}}{x^{3/2}} \, dx=\left [\frac {b \sqrt {c} x \log \left (-\frac {8 \, c^{2} x^{5} + 8 \, b c x^{3} + 4 \, \sqrt {c x^{5} + b x^{3} + a x} {\left (2 \, c x^{2} + b\right )} \sqrt {c} \sqrt {x} + {\left (b^{2} + 4 \, a c\right )} x}{x}\right ) + 2 \, \sqrt {a} c x \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{5} + 8 \, a b x^{3} + 8 \, a^{2} x - 4 \, \sqrt {c x^{5} + b x^{3} + a x} {\left (b x^{2} + 2 \, a\right )} \sqrt {a} \sqrt {x}}{x^{5}}\right ) + 4 \, \sqrt {c x^{5} + b x^{3} + a x} c \sqrt {x}}{8 \, c x}, -\frac {b \sqrt {-c} x \arctan \left (\frac {\sqrt {c x^{5} + b x^{3} + a x} {\left (2 \, c x^{2} + b\right )} \sqrt {-c} \sqrt {x}}{2 \, {\left (c^{2} x^{5} + b c x^{3} + a c x\right )}}\right ) - \sqrt {a} c x \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{5} + 8 \, a b x^{3} + 8 \, a^{2} x - 4 \, \sqrt {c x^{5} + b x^{3} + a x} {\left (b x^{2} + 2 \, a\right )} \sqrt {a} \sqrt {x}}{x^{5}}\right ) - 2 \, \sqrt {c x^{5} + b x^{3} + a x} c \sqrt {x}}{4 \, c x}, \frac {4 \, \sqrt {-a} c x \arctan \left (\frac {\sqrt {c x^{5} + b x^{3} + a x} {\left (b x^{2} + 2 \, a\right )} \sqrt {-a} \sqrt {x}}{2 \, {\left (a c x^{5} + a b x^{3} + a^{2} x\right )}}\right ) + b \sqrt {c} x \log \left (-\frac {8 \, c^{2} x^{5} + 8 \, b c x^{3} + 4 \, \sqrt {c x^{5} + b x^{3} + a x} {\left (2 \, c x^{2} + b\right )} \sqrt {c} \sqrt {x} + {\left (b^{2} + 4 \, a c\right )} x}{x}\right ) + 4 \, \sqrt {c x^{5} + b x^{3} + a x} c \sqrt {x}}{8 \, c x}, \frac {2 \, \sqrt {-a} c x \arctan \left (\frac {\sqrt {c x^{5} + b x^{3} + a x} {\left (b x^{2} + 2 \, a\right )} \sqrt {-a} \sqrt {x}}{2 \, {\left (a c x^{5} + a b x^{3} + a^{2} x\right )}}\right ) - b \sqrt {-c} x \arctan \left (\frac {\sqrt {c x^{5} + b x^{3} + a x} {\left (2 \, c x^{2} + b\right )} \sqrt {-c} \sqrt {x}}{2 \, {\left (c^{2} x^{5} + b c x^{3} + a c x\right )}}\right ) + 2 \, \sqrt {c x^{5} + b x^{3} + a x} c \sqrt {x}}{4 \, c x}\right ] \] Input:

integrate((c*x^5+b*x^3+a*x)^(1/2)/x^(3/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {\sqrt {a x+b x^3+c x^5}}{x^{3/2}} \, dx=\int \frac {\sqrt {x \left (a + b x^{2} + c x^{4}\right )}}{x^{\frac {3}{2}}}\, dx \] Input:

integrate((c*x**5+b*x**3+a*x)**(1/2)/x**(3/2),x)
 

Output:

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

Maxima [F]

\[ \int \frac {\sqrt {a x+b x^3+c x^5}}{x^{3/2}} \, dx=\int { \frac {\sqrt {c x^{5} + b x^{3} + a x}}{x^{\frac {3}{2}}} \,d x } \] Input:

integrate((c*x^5+b*x^3+a*x)^(1/2)/x^(3/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(c*x^5 + b*x^3 + a*x)/x^(3/2), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {a x+b x^3+c x^5}}{x^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((c*x^5+b*x^3+a*x)^(1/2)/x^(3/2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Degree mismatch inside factorisatio 
n over extensionNot implemented, e.g. for multivariate mod/approx polynomi 
alsError:
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a x+b x^3+c x^5}}{x^{3/2}} \, dx=\int \frac {\sqrt {c\,x^5+b\,x^3+a\,x}}{x^{3/2}} \,d x \] Input:

int((a*x + b*x^3 + c*x^5)^(1/2)/x^(3/2),x)
 

Output:

int((a*x + b*x^3 + c*x^5)^(1/2)/x^(3/2), x)
 

Reduce [F]

\[ \int \frac {\sqrt {a x+b x^3+c x^5}}{x^{3/2}} \, dx=\frac {2 \sqrt {c}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{4}+b \,x^{2}+a}+b +2 c \,x^{2}}{\sqrt {4 a c -b^{2}}}\right ) b +2 \sqrt {c}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, b +4 \sqrt {c}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, c \,x^{2}+8 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{c \,x^{5}+b \,x^{3}+a x}d x \right ) a c +4 \sqrt {c}\, \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{c \,x^{5}+b \,x^{3}+a x}d x \right ) a b +8 \sqrt {c}\, \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{c \,x^{5}+b \,x^{3}+a x}d x \right ) a c \,x^{2}+\mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{4}+b \,x^{2}+a}+b +2 c \,x^{2}}{\sqrt {4 a c -b^{2}}}\right ) b^{2}+2 \,\mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{4}+b \,x^{2}+a}+b +2 c \,x^{2}}{\sqrt {4 a c -b^{2}}}\right ) b c \,x^{2}+4 a c +4 b c \,x^{2}+4 c^{2} x^{4}}{8 \sqrt {c \,x^{4}+b \,x^{2}+a}\, c +4 \sqrt {c}\, b +8 \sqrt {c}\, c \,x^{2}} \] Input:

int((c*x^5+b*x^3+a*x)^(1/2)/x^(3/2),x)
 

Output:

(2*sqrt(c)*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))*b + 2*sqrt(c)*sqrt(a + b*x**2 + c* 
x**4)*b + 4*sqrt(c)*sqrt(a + b*x**2 + c*x**4)*c*x**2 + 8*sqrt(a + b*x**2 + 
 c*x**4)*int(sqrt(a + b*x**2 + c*x**4)/(a*x + b*x**3 + c*x**5),x)*a*c + 4* 
sqrt(c)*int(sqrt(a + b*x**2 + c*x**4)/(a*x + b*x**3 + c*x**5),x)*a*b + 8*s 
qrt(c)*int(sqrt(a + b*x**2 + c*x**4)/(a*x + b*x**3 + c*x**5),x)*a*c*x**2 + 
 log((2*sqrt(c)*sqrt(a + b*x**2 + c*x**4) + b + 2*c*x**2)/sqrt(4*a*c - b** 
2))*b**2 + 2*log((2*sqrt(c)*sqrt(a + b*x**2 + c*x**4) + b + 2*c*x**2)/sqrt 
(4*a*c - b**2))*b*c*x**2 + 4*a*c + 4*b*c*x**2 + 4*c**2*x**4)/(4*(2*sqrt(a 
+ b*x**2 + c*x**4)*c + sqrt(c)*b + 2*sqrt(c)*c*x**2))