\(\int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} (c+d x^2)^{3/2} (e+f x^2)} \, dx\) [26]

Optimal result

Integrand size = 44, antiderivative size = 415 \[ \int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=-\frac {\left (c^2 C-B c d+A d^2\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c-a d) (d e-c f) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {\sqrt {c} \left (a \left (2 c C d e-c^2 C f-d^2 (B e-A f)\right )+b \left (A d^2 e-2 A c d f-c^2 (C e-B f)\right )\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} (b c-a d) (d e-c f)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {c^{3/2} \left (C e^2-B e f+A f^2\right ) \sqrt {a+b x^2} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e (d e-c f)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:

-(A*d^2-B*c*d+C*c^2)*(b*x^2+a)^(1/2)*EllipticE(d^(1/2)*x/c^(1/2)/(1+d*x^2/ 
c)^(1/2),(1-b*c/a/d)^(1/2))/c^(1/2)/d^(1/2)/(-a*d+b*c)/(-c*f+d*e)/(c*(b*x^ 
2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)+c^(1/2)*(a*(2*c*C*d*e-c^2*C*f-d^2* 
(-A*f+B*e))+b*(A*d^2*e-2*A*c*d*f-c^2*(-B*f+C*e)))*(b*x^2+a)^(1/2)*InverseJ 
acobiAM(arctan(d^(1/2)*x/c^(1/2)),(1-b*c/a/d)^(1/2))/a/d^(1/2)/(-a*d+b*c)/ 
(-c*f+d*e)^2/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)+c^(3/2)*(A*f^ 
2-B*e*f+C*e^2)*(b*x^2+a)^(1/2)*EllipticPi(d^(1/2)*x/c^(1/2)/(1+d*x^2/c)^(1 
/2),1-c*f/d/e,(1-b*c/a/d)^(1/2))/a/d^(1/2)/e/(-c*f+d*e)^2/(c*(b*x^2+a)/a/( 
d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.89 (sec) , antiderivative size = 352, normalized size of antiderivative = 0.85 \[ \int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\frac {-i b c \left (c^2 C-B c d+A d^2\right ) e f \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c C (-b c+a d) e (-d e+c f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )-d \left (\sqrt {\frac {b}{a}} \left (c^2 C-B c d+A d^2\right ) e f x \left (a+b x^2\right )+i c (-b c+a d) \left (C e^2+f (-B e+A f)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{\sqrt {\frac {b}{a}} c d (-b c+a d) e f (-d e+c f) \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

Integrate[(A + B*x^2 + C*x^4)/(Sqrt[a + b*x^2]*(c + d*x^2)^(3/2)*(e + f*x^ 
2)),x]
 

Output:

((-I)*b*c*(c^2*C - B*c*d + A*d^2)*e*f*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2) 
/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*c*C*(-(b*c) + a*d)* 
e*(-(d*e) + c*f)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSi 
nh[Sqrt[b/a]*x], (a*d)/(b*c)] - d*(Sqrt[b/a]*(c^2*C - B*c*d + A*d^2)*e*f*x 
*(a + b*x^2) + I*c*(-(b*c) + a*d)*(C*e^2 + f*(-(B*e) + A*f))*Sqrt[1 + (b*x 
^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticPi[(a*f)/(b*e), I*ArcSinh[Sqrt[b/a]*x], 
 (a*d)/(b*c)]))/(Sqrt[b/a]*c*d*(-(b*c) + a*d)*e*f*(-(d*e) + c*f)*Sqrt[a + 
b*x^2]*Sqrt[c + d*x^2])
 

Rubi [A] (verified)

Time = 1.72 (sec) , antiderivative size = 805, normalized size of antiderivative = 1.94, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {7276, 2009}

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.

\(\displaystyle \int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx\)

\(\Big \downarrow \) 7276

\(\displaystyle \int \left (\frac {A f^2-B e f+C e^2}{f^2 \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )}-\frac {C e-B f}{f^2 \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}+\frac {C x^2}{f \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\left (C e^2-B f e+A f^2\right ) \sqrt {b x^2+a} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) c^{3/2}}{a \sqrt {d} e (d e-c f)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {C \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right ) \sqrt {c}}{\sqrt {d} (b c-a d) f \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {b (C e-B f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) \sqrt {c}}{a \sqrt {d} (b c-a d) f^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {\sqrt {d} (b d e-2 b c f+a d f) \left (C e^2-B f e+A f^2\right ) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) \sqrt {c}}{a (b c-a d) f^2 (d e-c f)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {C \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) \sqrt {c}}{\sqrt {d} (b c-a d) f \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {\sqrt {d} (C e-B f) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{(b c-a d) f^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c} \sqrt {c}}-\frac {d^{3/2} \left (C e^2-B f e+A f^2\right ) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{(b c-a d) f^2 (d e-c f) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c} \sqrt {c}}\)

Input:

Int[(A + B*x^2 + C*x^4)/(Sqrt[a + b*x^2]*(c + d*x^2)^(3/2)*(e + f*x^2)),x]
 

Output:

(Sqrt[c]*C*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c 
)/(a*d)])/(Sqrt[d]*(b*c - a*d)*f*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqr 
t[c + d*x^2]) + (Sqrt[d]*(C*e - B*f)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqr 
t[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[c]*(b*c - a*d)*f^2*Sqrt[(c*(a + 
b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (d^(3/2)*(C*e^2 - B*e*f + A*f^ 
2)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)] 
)/(Sqrt[c]*(b*c - a*d)*f^2*(d*e - c*f)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2) 
)]*Sqrt[c + d*x^2]) - (Sqrt[c]*C*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d] 
*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d]*(b*c - a*d)*f*Sqrt[(c*(a + b*x^2) 
)/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (b*Sqrt[c]*(C*e - B*f)*Sqrt[a + b*x^ 
2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[d]*(b* 
c - a*d)*f^2*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (Sqr 
t[c]*Sqrt[d]*(b*d*e - 2*b*c*f + a*d*f)*(C*e^2 - B*e*f + A*f^2)*Sqrt[a + b* 
x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*(b*c - a* 
d)*f^2*(d*e - c*f)^2*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2] 
) + (c^(3/2)*(C*e^2 - B*e*f + A*f^2)*Sqrt[a + b*x^2]*EllipticPi[1 - (c*f)/ 
(d*e), ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[d]*e*(d*e - 
c*f)^2*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1041\) vs. \(2(401)=802\).

Time = 8.39 (sec) , antiderivative size = 1042, normalized size of antiderivative = 2.51

method	result	size
default	\(\text {Expression too large to display}\)	\(1042\)
elliptic	\(\text {Expression too large to display}\)	\(1833\)

Input:

int((C*x^4+B*x^2+A)/(b*x^2+a)^(1/2)/(d*x^2+c)^(3/2)/(f*x^2+e),x,method=_RE 
TURNVERBOSE)
 

Output:

(-A*(-b/a)^(1/2)*b*d^3*e*f*x^3+B*(-b/a)^(1/2)*b*c*d^2*e*f*x^3-C*(-b/a)^(1/ 
2)*b*c^2*d*e*f*x^3+A*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticE(x*( 
-b/a)^(1/2),(a*d/b/c)^(1/2))*b*c*d^2*e*f+A*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/ 
c)^(1/2)*EllipticPi(x*(-b/a)^(1/2),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2))*a* 
c*d^2*f^2-A*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticPi(x*(-b/a)^(1 
/2),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2))*b*c^2*d*f^2-B*((b*x^2+a)/a)^(1/2) 
*((d*x^2+c)/c)^(1/2)*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*b*c^2*d*e*f 
-B*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticPi(x*(-b/a)^(1/2),a*f/b 
/e,(-1/c*d)^(1/2)/(-b/a)^(1/2))*a*c*d^2*e*f+B*((b*x^2+a)/a)^(1/2)*((d*x^2+ 
c)/c)^(1/2)*EllipticPi(x*(-b/a)^(1/2),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2)) 
*b*c^2*d*e*f+C*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticE(x*(-b/a)^ 
(1/2),(a*d/b/c)^(1/2))*b*c^3*e*f+C*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2) 
*EllipticPi(x*(-b/a)^(1/2),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2))*a*c*d^2*e^ 
2-C*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticPi(x*(-b/a)^(1/2),a*f/ 
b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2))*b*c^2*d*e^2+C*((b*x^2+a)/a)^(1/2)*((d*x^2 
+c)/c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a*c^2*d*e*f-C*((b*x 
^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2 
))*a*c*d^2*e^2-C*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-b/a 
)^(1/2),(a*d/b/c)^(1/2))*b*c^3*e*f+C*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/ 
2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*b*c^2*d*e^2-A*(-b/a)^(1/2)...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:

integrate((C*x^4+B*x^2+A)/(b*x^2+a)^(1/2)/(d*x^2+c)^(3/2)/(f*x^2+e),x, alg 
orithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\int \frac {A + B x^{2} + C x^{4}}{\sqrt {a + b x^{2}} \left (c + d x^{2}\right )^{\frac {3}{2}} \left (e + f x^{2}\right )}\, dx \] Input:

integrate((C*x**4+B*x**2+A)/(b*x**2+a)**(1/2)/(d*x**2+c)**(3/2)/(f*x**2+e) 
,x)
 

Output:

Integral((A + B*x**2 + C*x**4)/(sqrt(a + b*x**2)*(c + d*x**2)**(3/2)*(e + 
f*x**2)), x)
 

Maxima [F]

\[ \int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{\sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}} \,d x } \] Input:

integrate((C*x^4+B*x^2+A)/(b*x^2+a)^(1/2)/(d*x^2+c)^(3/2)/(f*x^2+e),x, alg 
orithm="maxima")
 

Output:

integrate((C*x^4 + B*x^2 + A)/(sqrt(b*x^2 + a)*(d*x^2 + c)^(3/2)*(f*x^2 + 
e)), x)
 

Giac [F]

\[ \int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{\sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}} \,d x } \] Input:

integrate((C*x^4+B*x^2+A)/(b*x^2+a)^(1/2)/(d*x^2+c)^(3/2)/(f*x^2+e),x, alg 
orithm="giac")
 

Output:

integrate((C*x^4 + B*x^2 + A)/(sqrt(b*x^2 + a)*(d*x^2 + c)^(3/2)*(f*x^2 + 
e)), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

int((A + B*x^2 + C*x^4)/((a + b*x^2)^(1/2)*(c + d*x^2)^(3/2)*(e + f*x^2)), 
x)
 

Output:

int((A + B*x^2 + C*x^4)/((a + b*x^2)^(1/2)*(c + d*x^2)^(3/2)*(e + f*x^2)), 
 x)
 

Reduce [F]

\[ \int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{4}}{b \,d^{2} f \,x^{8}+a \,d^{2} f \,x^{6}+2 b c d f \,x^{6}+b \,d^{2} e \,x^{6}+2 a c d f \,x^{4}+a \,d^{2} e \,x^{4}+b \,c^{2} f \,x^{4}+2 b c d e \,x^{4}+a \,c^{2} f \,x^{2}+2 a c d e \,x^{2}+b \,c^{2} e \,x^{2}+a \,c^{2} e}d x \right ) c +\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b \,d^{2} f \,x^{8}+a \,d^{2} f \,x^{6}+2 b c d f \,x^{6}+b \,d^{2} e \,x^{6}+2 a c d f \,x^{4}+a \,d^{2} e \,x^{4}+b \,c^{2} f \,x^{4}+2 b c d e \,x^{4}+a \,c^{2} f \,x^{2}+2 a c d e \,x^{2}+b \,c^{2} e \,x^{2}+a \,c^{2} e}d x \right ) b +\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b \,d^{2} f \,x^{8}+a \,d^{2} f \,x^{6}+2 b c d f \,x^{6}+b \,d^{2} e \,x^{6}+2 a c d f \,x^{4}+a \,d^{2} e \,x^{4}+b \,c^{2} f \,x^{4}+2 b c d e \,x^{4}+a \,c^{2} f \,x^{2}+2 a c d e \,x^{2}+b \,c^{2} e \,x^{2}+a \,c^{2} e}d x \right ) a \] Input:

int((C*x^4+B*x^2+A)/(b*x^2+a)^(1/2)/(d*x^2+c)^(3/2)/(f*x^2+e),x)
 

Output:

int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c**2*e + a*c**2*f*x**2 + 2 
*a*c*d*e*x**2 + 2*a*c*d*f*x**4 + a*d**2*e*x**4 + a*d**2*f*x**6 + b*c**2*e* 
x**2 + b*c**2*f*x**4 + 2*b*c*d*e*x**4 + 2*b*c*d*f*x**6 + b*d**2*e*x**6 + b 
*d**2*f*x**8),x)*c + int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c**2* 
e + a*c**2*f*x**2 + 2*a*c*d*e*x**2 + 2*a*c*d*f*x**4 + a*d**2*e*x**4 + a*d* 
*2*f*x**6 + b*c**2*e*x**2 + b*c**2*f*x**4 + 2*b*c*d*e*x**4 + 2*b*c*d*f*x** 
6 + b*d**2*e*x**6 + b*d**2*f*x**8),x)*b + int((sqrt(c + d*x**2)*sqrt(a + b 
*x**2))/(a*c**2*e + a*c**2*f*x**2 + 2*a*c*d*e*x**2 + 2*a*c*d*f*x**4 + a*d* 
*2*e*x**4 + a*d**2*f*x**6 + b*c**2*e*x**2 + b*c**2*f*x**4 + 2*b*c*d*e*x**4 
 + 2*b*c*d*f*x**6 + b*d**2*e*x**6 + b*d**2*f*x**8),x)*a