\(\int \frac {\sqrt {a+b x^2} (A+B x^2+C x^4)}{\sqrt {c+d x^2} (e+f x^2)} \, dx\) [10]

Optimal result

Integrand size = 44, antiderivative size = 443 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4\right )}{\sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\frac {(a C d f-b (3 C d e+2 c C f-3 B d f)) x \sqrt {c+d x^2}}{3 d^2 f^2 \sqrt {a+b x^2}}+\frac {C x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 d f}-\frac {\sqrt {a} (a C d f-b (3 C d e+2 c C f-3 B d f)) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 \sqrt {b} d^2 f^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} (3 C d e+c C f-3 B d f) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{3 \sqrt {b} c d f^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {a^{3/2} \left (C e^2-B e f+A f^2\right ) \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e f^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 5.51 (sec) , antiderivative size = 383, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4\right )}{\sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\frac {-i c e f (a C d f+b (-3 C d e-2 c C f+3 B d f)) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i e \left (a d f (3 C d e+2 c C f-3 B d f)-b \left (C \left (3 d^2 e^2+3 c d e f+2 c^2 f^2\right )+3 d f (A d f-B (d e+c f))\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+d \left (\sqrt {\frac {b}{a}} C e f^2 x \left (a+b x^2\right ) \left (c+d x^2\right )-3 i d (-b e+a f) \left (C e^2+f (-B e+A f)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{3 \sqrt {\frac {b}{a}} d^2 e f^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.27 (sec) , antiderivative size = 623, normalized size of antiderivative = 1.41, 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.

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Time = 8.61 (sec) , antiderivative size = 506, normalized size of antiderivative = 1.14

method	result	size
risch	\(\frac {C x \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{3 d f}+\frac {\left (\frac {\left (3 A b d \,f^{2}+3 B a d \,f^{2}-3 B b d e f -C a c \,f^{2}-3 a C d e f +3 e^{2} C b d \right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{f^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (3 b B d f +a C d f -2 C b f c -3 C b d e \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{f \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}+\frac {3 \left (A a \,f^{3}-A b e \,f^{2}-B a e \,f^{2}+B b \,e^{2} f +C a \,e^{2} f -e^{3} C b \right ) d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right )}{f^{2} e \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{3 d f \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(506\)
default	\(\text {Expression too large to display}\)	\(1344\)
elliptic	\(\text {Expression too large to display}\)	\(1908\)

Input:

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

Output:

1/3*C*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/d/f+1/3/d/f*((3*A*b*d*f^2+3*B*a*d* 
f^2-3*B*b*d*e*f-C*a*c*f^2-3*C*a*d*e*f+3*C*b*d*e^2)/f^2/(-b/a)^(1/2)*(1+b*x 
^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*Elliptic 
F(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-1/f*(3*B*b*d*f+C*a*d*f-2*C*b*c* 
f-3*C*b*d*e)*c/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a 
*d*x^2+b*c*x^2+a*c)^(1/2)/d*(EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^( 
1/2))-EllipticE(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2)))+3*(A*a*f^3-A*b*e 
*f^2-B*a*e*f^2+B*b*e^2*f+C*a*e^2*f-C*b*e^3)*d/f^2/e/(-b/a)^(1/2)*(1+b*x^2/ 
a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticPi( 
x*(-b/a)^(1/2),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2)))*((b*x^2+a)*(d*x^2+c)) 
^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(c + d*x**2)*sqrt(a + b*x**2)*c*x + int((sqrt(c + d*x**2)*sqrt(a + b* 
x**2)*x**4)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b*c*e*x**2 + b 
*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*a*c*d*f + 3*int((sqrt(c + d*x**2)* 
sqrt(a + b*x**2)*x**4)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b*c 
*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*b**2*d*f - 2*int((sqrt( 
c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d* 
f*x**4 + b*c*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*b*c**2*f - 
3*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c*e + a*c*f*x**2 + a*d*e 
*x**2 + a*d*f*x**4 + b*c*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x) 
*b*c*d*e + 6*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c*e + a*c*f*x 
**2 + a*d*e*x**2 + a*d*f*x**4 + b*c*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d 
*f*x**6),x)*a*b*d*f - int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c*e 
+ a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b*c*e*x**2 + b*c*f*x**4 + b*d*e*x 
**4 + b*d*f*x**6),x)*a*c**2*f - 2*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x 
**2)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b*c*e*x**2 + b*c*f*x* 
*4 + b*d*e*x**4 + b*d*f*x**6),x)*a*c*d*e - 2*int((sqrt(c + d*x**2)*sqrt(a 
+ b*x**2)*x**2)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b*c*e*x**2 
 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*b*c**2*e + 3*int((sqrt(c + d*x 
**2)*sqrt(a + b*x**2))/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b*c 
*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*a**2*d*f - int((sqrt...