\(\int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} (c-d x^2)^{3/2} (e+f x^2)} \, dx\) [27]

Optimal result

Integrand size = 45, antiderivative size = 426 \[ \int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} \left (c-d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\frac {\left (c^2 C+B c d+A d^2\right ) x \sqrt {a+b x^2}}{c (b c+a d) (d e+c f) \sqrt {c-d x^2}}-\frac {\left (c^2 C+B c d+A d^2\right ) \sqrt {a+b x^2} \sqrt {1-\frac {d x^2}{c}} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c+a d) (d e+c f) \sqrt {1+\frac {b x^2}{a}} \sqrt {c-d x^2}}-\frac {(c C e-B c f-A d f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} f (d e+c f) \sqrt {a+b x^2} \sqrt {c-d x^2}}+\frac {\sqrt {c} \left (C e^2-B e f+A f^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {c f}{d e},\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{\sqrt {d} e f (d e+c f) \sqrt {a+b x^2} \sqrt {c-d x^2}} \] 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*d^2+B*c*d+C*c^2)*(b*x^2+a)^(1/2)*(1-d*x^2/c)^(1/2)*EllipticE(d^(1/2) 
*x/c^(1/2),(-b*c/a/d)^(1/2))/c^(1/2)/d^(1/2)/(a*d+b*c)/(c*f+d*e)/(1+b*x^2/ 
a)^(1/2)/(-d*x^2+c)^(1/2)-(-A*d*f-B*c*f+C*c*e)*(1+b*x^2/a)^(1/2)*(1-d*x^2/ 
c)^(1/2)*EllipticF(d^(1/2)*x/c^(1/2),(-b*c/a/d)^(1/2))/c^(1/2)/d^(1/2)/f/( 
c*f+d*e)/(b*x^2+a)^(1/2)/(-d*x^2+c)^(1/2)+c^(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(d^(1/2)*x/c^(1/2),-c*f/d/e,(-b* 
c/a/d)^(1/2))/d^(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 = 10.95 (sec) , antiderivative size = 350, normalized size of antiderivative = 0.82 \[ \int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} \left (c-d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\frac {\sqrt {\frac {b}{a}} d \left (c^2 C+B c d+A d^2\right ) e f x \left (a+b x^2\right )-i b c \left (c^2 C+B c d+A d^2\right ) e f \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )+i c C (b c+a d) e (d e+c f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )-i c d (b c+a d) \left (C e^2+f (-B e+A f)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )}{\sqrt {\frac {b}{a}} c d (b c+a d) e f (d e+c f) \sqrt {a+b x^2} \sqrt {c-d x^2}} \] 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*ArcSi 
nh[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))*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]*Sq 
rt[c - d*x^2])
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(956\) vs. \(2(426)=852\).

Time = 2.06 (sec) , antiderivative size = 956, normalized size of antiderivative = 2.24, 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 {b x^2+a} d^2}{c (b c+a d) f^2 (d e+c f) \sqrt {c-d x^2}}-\frac {\left (C e^2-B f e+A f^2\right ) \sqrt {b x^2+a} \sqrt {1-\frac {d x^2}{c}} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right ) d^{3/2}}{\sqrt {c} (b c+a d) f^2 (d e+c f) \sqrt {\frac {b x^2}{a}+1} \sqrt {c-d x^2}}-\frac {(C e-B f) x \sqrt {b x^2+a} d}{c (b c+a d) f^2 \sqrt {c-d x^2}}+\frac {(C e-B f) \sqrt {b x^2+a} \sqrt {1-\frac {d x^2}{c}} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right ) \sqrt {d}}{\sqrt {c} (b c+a d) f^2 \sqrt {\frac {b x^2}{a}+1} \sqrt {c-d x^2}}+\frac {(d e+2 c f) \left (C e^2-B f e+A f^2\right ) \sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right ) \sqrt {d}}{\sqrt {c} f^2 (d e+c f)^2 \sqrt {b x^2+a} \sqrt {c-d x^2}}-\frac {\sqrt {c} \left (C e^2-B f e+A f^2\right ) \sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right ) \sqrt {d}}{f (d e+c f)^2 \sqrt {b x^2+a} \sqrt {c-d x^2}}+\frac {C x \sqrt {b x^2+a}}{(b c+a d) f \sqrt {c-d x^2}}-\frac {\sqrt {c} C \sqrt {b x^2+a} \sqrt {1-\frac {d x^2}{c}} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{(b c+a d) f \sqrt {\frac {b x^2}{a}+1} \sqrt {c-d x^2} \sqrt {d}}-\frac {(C e-B f) \sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{\sqrt {c} f^2 \sqrt {b x^2+a} \sqrt {c-d x^2} \sqrt {d}}+\frac {\sqrt {c} \left (C e^2-B f e+A f^2\right ) \sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {c f}{d e},\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{e f (d e+c f) \sqrt {b x^2+a} \sqrt {c-d x^2} \sqrt {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*S 
qrt[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[c]*C*Sqrt[a + b*x^2]*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(Sqrt[ 
d]*x)/Sqrt[c]], -((b*c)/(a*d))])/(Sqrt[d]*(b*c + a*d)*f*Sqrt[1 + (b*x^2)/a 
]*Sqrt[c - d*x^2]) + (Sqrt[d]*(C*e - B*f)*Sqrt[a + b*x^2]*Sqrt[1 - (d*x^2) 
/c]*EllipticE[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((b*c)/(a*d))])/(Sqrt[c]*(b*c 
+ a*d)*f^2*Sqrt[1 + (b*x^2)/a]*Sqrt[c - d*x^2]) - (d^(3/2)*(C*e^2 - B*e*f 
+ A*f^2)*Sqrt[a + b*x^2]*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(Sqrt[d]*x)/ 
Sqrt[c]], -((b*c)/(a*d))])/(Sqrt[c]*(b*c + a*d)*f^2*(d*e + c*f)*Sqrt[1 + ( 
b*x^2)/a]*Sqrt[c - d*x^2]) - ((C*e - B*f)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 - (d* 
x^2)/c]*EllipticF[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((b*c)/(a*d))])/(Sqrt[c]*S 
qrt[d]*f^2*Sqrt[a + b*x^2]*Sqrt[c - d*x^2]) - (Sqrt[c]*Sqrt[d]*(C*e^2 - B* 
e*f + A*f^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(Sqr 
t[d]*x)/Sqrt[c]], -((b*c)/(a*d))])/(f*(d*e + c*f)^2*Sqrt[a + b*x^2]*Sqrt[c 
 - d*x^2]) + (Sqrt[d]*(d*e + 2*c*f)*(C*e^2 - B*e*f + A*f^2)*Sqrt[1 + (b*x^ 
2)/a]*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((b*c)/( 
a*d))])/(Sqrt[c]*f^2*(d*e + c*f)^2*Sqrt[a + b*x^2]*Sqrt[c - d*x^2]) + (Sqr 
t[c]*(C*e^2 - B*e*f + A*f^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*Ellip 
ticPi[-((c*f)/(d*e)), ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((b*c)/(a*d))])/(Sq...
 

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. \(1126\) vs. \(2(377)=754\).

Time = 8.45 (sec) , antiderivative size = 1127, normalized size of antiderivative = 2.65

method	result	size
default	\(\text {Expression too large to display}\)	\(1127\)
elliptic	\(\text {Expression too large to display}\)	\(1183\)

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

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