\(\int \frac {A+B x^2+C x^4}{\sqrt {a-b x^2} (c+d x^2)^{3/2} (e+f x^2)} \, dx\) [28]

Optimal result

Integrand size = 45, antiderivative size = 416 \[ \int \frac {A+B x^2+C x^4}{\sqrt {a-b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\frac {\left (c^2 C-B c d+A d^2\right ) x \sqrt {a-b x^2}}{c (b c+a d) (d e-c f) \sqrt {c+d x^2}}+\frac {\sqrt {a} \sqrt {b} \left (c^2 C-B c d+A d^2\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{c d (b c+a d) (d e-c f) \sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}}}+\frac {\sqrt {a} C \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d f \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {\sqrt {a} \left (C e^2-B e f+A f^2\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} e f (d e-c f) \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 13.35 (sec) , antiderivative size = 362, normalized size of antiderivative = 0.87 \[ \int \frac {A+B x^2+C x^4}{\sqrt {a-b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\frac {-\sqrt {-\frac {b}{a}} d \left (c^2 C-B c d+A d^2\right ) e f x \left (a-b x^2\right )+i b c \left (c^2 C-B c d+A d^2\right ) e f \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )-i c C (b c+a d) e (-d e+c f) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )-i c d (b c+a d) \left (C e^2+f (-B e+A f)\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )}{\sqrt {-\frac {b}{a}} c d (b c+a d) e f (-d e+c f) \sqrt {a-b x^2} \sqrt {c+d x^2}} \] 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:

(-(Sqrt[-(b/a)]*d*(c^2*C - B*c*d + A*d^2)*e*f*x*(a - b*x^2)) + 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*ArcSinh[Sqrt[-(b/ 
a)]*x], -((a*d)/(b*c))] - I*c*d*(b*c + a*d)*(C*e^2 + f*(-(B*e) + A*f))*Sqr 
t[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 [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(844\) vs. \(2(416)=832\).

Time = 1.87 (sec) , antiderivative size = 844, normalized size of antiderivative = 2.03, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.044, 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

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

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:

-((C*x*Sqrt[a - b*x^2])/((b*c + a*d)*f*Sqrt[c + d*x^2])) - (d*(C*e - B*f)* 
x*Sqrt[a - b*x^2])/(c*(b*c + a*d)*f^2*Sqrt[c + d*x^2]) + (d^2*(C*e^2 - B*e 
*f + A*f^2)*x*Sqrt[a - b*x^2])/(c*(b*c + a*d)*f^2*(d*e - c*f)*Sqrt[c + d*x 
^2]) - (Sqrt[a]*Sqrt[b]*C*Sqrt[1 - (b*x^2)/a]*Sqrt[c + d*x^2]*EllipticE[Ar 
cSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(d*(b*c + a*d)*f*Sqrt[a - b*x^ 
2]*Sqrt[1 + (d*x^2)/c]) - (Sqrt[a]*Sqrt[b]*(C*e - B*f)*Sqrt[1 - (b*x^2)/a] 
*Sqrt[c + d*x^2]*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/( 
c*(b*c + a*d)*f^2*Sqrt[a - b*x^2]*Sqrt[1 + (d*x^2)/c]) + (Sqrt[a]*Sqrt[b]* 
d*(C*e^2 - B*e*f + A*f^2)*Sqrt[1 - (b*x^2)/a]*Sqrt[c + d*x^2]*EllipticE[Ar 
cSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(c*(b*c + a*d)*f^2*(d*e - c*f) 
*Sqrt[a - b*x^2]*Sqrt[1 + (d*x^2)/c]) + (Sqrt[a]*C*Sqrt[1 - (b*x^2)/a]*Sqr 
t[1 + (d*x^2)/c]*EllipticF[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/( 
Sqrt[b]*d*f*Sqrt[a - b*x^2]*Sqrt[c + d*x^2]) - (Sqrt[a]*d*(C*e^2 - B*e*f + 
 A*f^2)*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[ArcSin[(Sqrt[b]* 
x)/Sqrt[a]], -((a*d)/(b*c))])/(Sqrt[b]*f*(d*e - c*f)^2*Sqrt[a - b*x^2]*Sqr 
t[c + d*x^2]) + (c^(3/2)*(C*e^2 - B*e*f + A*f^2)*Sqrt[a - b*x^2]*EllipticP 
i[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. \(1042\) vs. \(2(367)=734\).

Time = 8.46 (sec) , antiderivative size = 1043, normalized size of antiderivative = 2.51

method	result	size
default	\(\text {Expression too large to display}\)	\(1043\)
elliptic	\(\text {Expression too large to display}\)	\(1869\)

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=_R 
ETURNVERBOSE)
 

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)*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-C*((-b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*Elli 
pticE(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-A*(b/a)^(1/...
 

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