\(\int \frac {A+B x^2+C x^4+D x^6}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx\) [39]

Optimal result

Integrand size = 40, antiderivative size = 428 \[ \int \frac {A+B x^2+C x^4+D x^6}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\frac {\left (8 a^2 d^2 D-a b d (10 C d-7 c D)-b^2 \left (10 c C d-15 B d^2-8 c^2 D\right )\right ) x \sqrt {c+d x^2}}{15 b^2 d^3 \sqrt {a+b x^2}}+\frac {(5 b C d-4 b c D-4 a d D) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 b^2 d^2}+\frac {D x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 b d}-\frac {\sqrt {a} \left (8 a^2 d^2 D-a b d (10 C d-7 c D)-b^2 \left (10 c C d-15 B d^2-8 c^2 D\right )\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{15 b^{5/2} d^3 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {\sqrt {a} \left (15 A b^2 d^2+4 a^2 c d D-a b c (5 C d-4 c D)\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{15 b^{5/2} c d^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

1/15*(8*a^2*d^2*D-a*b*d*(10*C*d-7*D*c)-b^2*(-15*B*d^2+10*C*c*d-8*D*c^2))*x 
*(d*x^2+c)^(1/2)/b^2/d^3/(b*x^2+a)^(1/2)+1/15*(5*C*b*d-4*D*a*d-4*D*b*c)*x* 
(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b^2/d^2+1/5*D*x^3*(b*x^2+a)^(1/2)*(d*x^2+c 
)^(1/2)/b/d-1/15*a^(1/2)*(8*a^2*d^2*D-a*b*d*(10*C*d-7*D*c)-b^2*(-15*B*d^2+ 
10*C*c*d-8*D*c^2))*(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^(5/2)/d^3/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^ 
2+a))^(1/2)+1/15*a^(1/2)*(15*A*b^2*d^2+4*a^2*c*d*D-a*b*c*(5*C*d-4*D*c))*(d 
*x^2+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2)) 
/b^(5/2)/c/d^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 = 6.36 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.73 \[ \int \frac {A+B x^2+C x^4+D x^6}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\frac {-\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (4 a d D+b \left (-5 C d+4 c D-3 d D x^2\right )\right )-i c \left (8 a^2 d^2 D+a b d (-10 C d+7 c D)+b^2 \left (-10 c C d+15 B d^2+8 c^2 D\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i \left (4 a^2 c d^2 D+a b c d (-5 C d+3 c D)+b^2 \left (-10 c^2 C d+15 B c d^2-15 A d^3+8 c^3 D\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{15 a^2 \left (\frac {b}{a}\right )^{5/2} d^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

Integrate[(A + B*x^2 + C*x^4 + D*x^6)/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]
 

Output:

(-(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(4*a*d*D + b*(-5*C*d + 4*c*D - 3* 
d*D*x^2))) - I*c*(8*a^2*d^2*D + a*b*d*(-10*C*d + 7*c*D) + b^2*(-10*c*C*d + 
 15*B*d^2 + 8*c^2*D))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I* 
ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + I*(4*a^2*c*d^2*D + a*b*c*d*(-5*C*d + 
3*c*D) + b^2*(-10*c^2*C*d + 15*B*c*d^2 - 15*A*d^3 + 8*c^3*D))*Sqrt[1 + (b* 
x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] 
)/(15*a^2*(b/a)^(5/2)*d^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
 

Rubi [A] (verified)

Time = 1.40 (sec) , antiderivative size = 816, normalized size of antiderivative = 1.91, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {7293, 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.

\(\displaystyle \int \frac {A+B x^2+C x^4+D x^6}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {A}{\sqrt {a+b x^2} \sqrt {c+d x^2}}+\frac {B x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}}+\frac {C x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2}}+\frac {D x^6}{\sqrt {a+b x^2} \sqrt {c+d x^2}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {D \sqrt {b x^2+a} \sqrt {d x^2+c} x^3}{5 b d}-\frac {4 (b c+a d) D \sqrt {b x^2+a} \sqrt {d x^2+c} x}{15 b^2 d^2}+\frac {C \sqrt {b x^2+a} \sqrt {d x^2+c} x}{3 b d}+\frac {B \sqrt {b x^2+a} x}{b \sqrt {d x^2+c}}-\frac {2 C (b c+a d) \sqrt {b x^2+a} x}{3 b^2 d \sqrt {d x^2+c}}+\frac {\left (8 b^2 c^2+7 a b d c+8 a^2 d^2\right ) D \sqrt {b x^2+a} x}{15 b^3 d^2 \sqrt {d x^2+c}}+\frac {2 \sqrt {c} C (b c+a d) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 b^2 d^{3/2} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\sqrt {c} \left (8 b^2 c^2+7 a b d c+8 a^2 d^2\right ) D \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 b^3 d^{5/2} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {B \sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {4 c^{3/2} (b c+a d) D \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{15 b^2 d^{5/2} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {A \sqrt {c} \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {c^{3/2} C \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 b d^{3/2} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\)

Input:

Int[(A + B*x^2 + C*x^4 + D*x^6)/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]
 

Output:

(B*x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (2*C*(b*c + a*d)*x*Sqrt[a + b* 
x^2])/(3*b^2*d*Sqrt[c + d*x^2]) + ((8*b^2*c^2 + 7*a*b*c*d + 8*a^2*d^2)*D*x 
*Sqrt[a + b*x^2])/(15*b^3*d^2*Sqrt[c + d*x^2]) + (C*x*Sqrt[a + b*x^2]*Sqrt 
[c + d*x^2])/(3*b*d) - (4*(b*c + a*d)*D*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]) 
/(15*b^2*d^2) + (D*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(5*b*d) - (B*Sqrt[ 
c]*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)] 
)/(b*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (2*S 
qrt[c]*C*(b*c + a*d)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]] 
, 1 - (b*c)/(a*d)])/(3*b^2*d^(3/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*S 
qrt[c + d*x^2]) - (Sqrt[c]*(8*b^2*c^2 + 7*a*b*c*d + 8*a^2*d^2)*D*Sqrt[a + 
b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*b^3*d^ 
(5/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (c^(3/2)*C* 
Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/( 
3*b*d^(3/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (A*Sq 
rt[c]*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a* 
d)])/(a*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + ( 
4*c^(3/2)*(b*c + a*d)*D*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[ 
c]], 1 - (b*c)/(a*d)])/(15*b^2*d^(5/2)*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 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [A] (verified)

Time = 6.51 (sec) , antiderivative size = 418, normalized size of antiderivative = 0.98

method	result	size
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {D x^{3} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{5 b d}+\frac {\left (C -\frac {D \left (4 a d +4 b c \right )}{5 b d}\right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 b d}+\frac {\left (A -\frac {\left (C -\frac {D \left (4 a d +4 b c \right )}{5 b d}\right ) a c}{3 b d}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (B -\frac {3 D a c}{5 b d}-\frac {\left (C -\frac {D \left (4 a d +4 b c \right )}{5 b d}\right ) \left (2 a d +2 b c \right )}{3 b d}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(418\)
default	\(\text {Expression too large to display}\)	\(1043\)

Input:

int((D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x,method=_RETURN 
VERBOSE)
 

Output:

((b*x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(1/5*D/b/d*x^3 
*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+1/3*(C-1/5*D/b/d*(4*a*d+4*b*c))/b/d*x 
*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+(A-1/3*(C-1/5*D/b/d*(4*a*d+4*b*c))/b/ 
d*a*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)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-(B-3/ 
5*D/b/d*a*c-1/3*(C-1/5*D/b/d*(4*a*d+4*b*c))/b/d*(2*a*d+2*b*c))*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))))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 349, normalized size of antiderivative = 0.82 \[ \int \frac {A+B x^2+C x^4+D x^6}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=-\frac {{\left (8 \, D b^{2} c^{4} + {\left (7 \, D a b - 10 \, C b^{2}\right )} c^{3} d + {\left (8 \, D a^{2} - 10 \, C a b + 15 \, B b^{2}\right )} c^{2} d^{2}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (8 \, D b^{2} c^{4} + 15 \, A b^{2} d^{4} + {\left (7 \, D a b - 10 \, C b^{2}\right )} c^{3} d + {\left (8 \, D a^{2} - 2 \, {\left (5 \, C - 2 \, D\right )} a b + 15 \, B b^{2}\right )} c^{2} d^{2} + {\left (4 \, D a^{2} - 5 \, C a b\right )} c d^{3}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (3 \, D b^{2} c d^{3} x^{4} + 8 \, D b^{2} c^{3} d + {\left (7 \, D a b - 10 \, C b^{2}\right )} c^{2} d^{2} + {\left (8 \, D a^{2} - 10 \, C a b + 15 \, B b^{2}\right )} c d^{3} - {\left (4 \, D b^{2} c^{2} d^{2} + {\left (4 \, D a b - 5 \, C b^{2}\right )} c d^{3}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{15 \, b^{3} c d^{4} x} \] Input:

integrate((D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorit 
hm="fricas")
 

Output:

-1/15*((8*D*b^2*c^4 + (7*D*a*b - 10*C*b^2)*c^3*d + (8*D*a^2 - 10*C*a*b + 1 
5*B*b^2)*c^2*d^2)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), 
a*d/(b*c)) - (8*D*b^2*c^4 + 15*A*b^2*d^4 + (7*D*a*b - 10*C*b^2)*c^3*d + (8 
*D*a^2 - 2*(5*C - 2*D)*a*b + 15*B*b^2)*c^2*d^2 + (4*D*a^2 - 5*C*a*b)*c*d^3 
)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_f(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - (3* 
D*b^2*c*d^3*x^4 + 8*D*b^2*c^3*d + (7*D*a*b - 10*C*b^2)*c^2*d^2 + (8*D*a^2 
- 10*C*a*b + 15*B*b^2)*c*d^3 - (4*D*b^2*c^2*d^2 + (4*D*a*b - 5*C*b^2)*c*d^ 
3)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(b^3*c*d^4*x)
 

Sympy [F]

\[ \int \frac {A+B x^2+C x^4+D x^6}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int \frac {A + B x^{2} + C x^{4} + D x^{6}}{\sqrt {a + b x^{2}} \sqrt {c + d x^{2}}}\, dx \] Input:

integrate((D*x**6+C*x**4+B*x**2+A)/(b*x**2+a)**(1/2)/(d*x**2+c)**(1/2),x)
 

Output:

Integral((A + B*x**2 + C*x**4 + D*x**6)/(sqrt(a + b*x**2)*sqrt(c + d*x**2) 
), x)
 

Maxima [F]

\[ \int \frac {A+B x^2+C x^4+D x^6}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int { \frac {D x^{6} + C x^{4} + B x^{2} + A}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c}} \,d x } \] Input:

integrate((D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorit 
hm="maxima")
 

Output:

integrate((D*x^6 + C*x^4 + B*x^2 + A)/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)), x 
)
 

Giac [F]

\[ \int \frac {A+B x^2+C x^4+D x^6}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int { \frac {D x^{6} + C x^{4} + B x^{2} + A}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c}} \,d x } \] Input:

integrate((D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorit 
hm="giac")
 

Output:

integrate((D*x^6 + C*x^4 + B*x^2 + A)/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)), x 
)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x^2+C x^4+D x^6}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int \frac {A+B\,x^2+C\,x^4+x^6\,D}{\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}} \,d x \] Input:

int((A + B*x^2 + C*x^4 + x^6*D)/((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)),x)
 

Output:

int((A + B*x^2 + C*x^4 + x^6*D)/((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {A+B x^2+C x^4+D x^6}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\frac {-4 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a d x +\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b c x +3 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b d \,x^{3}+8 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{2} d^{2}-3 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a b c d +15 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b^{3} d -2 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b^{2} c^{2}+4 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{2} c d +15 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a \,b^{2} d -\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a b \,c^{2}}{15 b^{2} d} \] Input:

int((D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x)
 

Output:

( - 4*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*d*x + sqrt(c + d*x**2)*sqrt(a + 
b*x**2)*b*c*x + 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*d*x**3 + 8*int((sqrt 
(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4) 
,x)*a**2*d**2 - 3*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d* 
x**2 + b*c*x**2 + b*d*x**4),x)*a*b*c*d + 15*int((sqrt(c + d*x**2)*sqrt(a + 
 b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*b**3*d - 2*int((s 
qrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x* 
*4),x)*b**2*c**2 + 4*int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a*c + a*d*x* 
*2 + b*c*x**2 + b*d*x**4),x)*a**2*c*d + 15*int((sqrt(c + d*x**2)*sqrt(a + 
b*x**2))/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a*b**2*d - int((sqrt(c 
+ d*x**2)*sqrt(a + b*x**2))/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a*b* 
c**2)/(15*b**2*d)