\(\int \frac {A+B x^2+C x^4+D x^6}{(a+b x^2)^{3/2} (c+d x^2)^{5/2}} \, dx\) [47]

Optimal result

Integrand size = 40, antiderivative size = 531 \[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2}} \, dx=\frac {\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) x}{a b^2 (b c-a d) \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}+\frac {\left (3 A b^3 c d^2+3 a^2 b c C d^2-3 a^3 c d^2 D+a b^2 \left (c^2 C d-4 B c d^2+A d^3-c^3 D\right )\right ) x \sqrt {a+b x^2}}{3 a b^2 c d (b c-a d)^2 \left (c+d x^2\right )^{3/2}}+\frac {\left (3 A b^3 c^2 d^2-3 a^3 c^2 d^2 D+a^2 b d \left (7 c^2 C d-B c d^2-2 A d^3-7 c^3 D\right )+a b^2 c \left (c^2 C d-7 B c d^2+7 A d^3+2 c^3 D\right )\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a b c^{3/2} d^{3/2} (b c-a d)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {\left (3 b^2 c d (B c-3 A d)-3 a^2 c d (C d-3 c D)-a b \left (5 c^2 C d-5 B c d^2-A d^3+c^3 D\right )\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a \sqrt {c} d^{3/2} (b c-a d)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:

(A*b^3-a*(B*b^2-C*a*b+D*a^2))*x/a/b^2/(-a*d+b*c)/(b*x^2+a)^(1/2)/(d*x^2+c) 
^(3/2)+1/3*(3*A*b^3*c*d^2+3*a^2*b*c*C*d^2-3*a^3*c*d^2*D+a*b^2*(A*d^3-4*B*c 
*d^2+C*c^2*d-D*c^3))*x*(b*x^2+a)^(1/2)/a/b^2/c/d/(-a*d+b*c)^2/(d*x^2+c)^(3 
/2)+1/3*(3*A*b^3*c^2*d^2-3*a^3*c^2*d^2*D+a^2*b*d*(-2*A*d^3-B*c*d^2+7*C*c^2 
*d-7*D*c^3)+a*b^2*c*(7*A*d^3-7*B*c*d^2+C*c^2*d+2*D*c^3))*(b*x^2+a)^(1/2)*E 
llipticE(d^(1/2)*x/c^(1/2)/(1+d*x^2/c)^(1/2),(1-b*c/a/d)^(1/2))/a/b/c^(3/2 
)/d^(3/2)/(-a*d+b*c)^3/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)+1/3 
*(3*b^2*c*d*(-3*A*d+B*c)-3*a^2*c*d*(C*d-3*D*c)-a*b*(-A*d^3-5*B*c*d^2+5*C*c 
^2*d+D*c^3))*(b*x^2+a)^(1/2)*InverseJacobiAM(arctan(d^(1/2)*x/c^(1/2)),(1- 
b*c/a/d)^(1/2))/a/c^(1/2)/d^(3/2)/(-a*d+b*c)^3/(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 = 13.29 (sec) , antiderivative size = 640, normalized size of antiderivative = 1.21 \[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2}} \, dx=\frac {\sqrt {\frac {b}{a}} \left (\sqrt {\frac {b}{a}} d x \left (-A d \left (3 b^3 c^2 \left (c+d x^2\right )^2-a^3 d^3 \left (3 c+2 d x^2\right )+a b^2 c d^2 x^2 \left (8 c+7 d x^2\right )+2 a^2 b d^2 \left (4 c^2+2 c d x^2-d^2 x^4\right )\right )+a c \left (a^2 d \left (9 c^3 D+B d^3 x^2+c d^2 x^2 \left (-4 C+3 D x^2\right )+c^2 \left (-3 C d+13 d D x^2\right )\right )+a b \left (-c^4 D+B d^4 x^4+c d^3 x^2 \left (4 B-7 C x^2\right )+c^3 \left (-5 C d+4 d D x^2\right )+c^2 d^2 \left (5 B-10 C x^2+7 D x^4\right )\right )+b^2 c \left (B d \left (3 c^2+11 c d x^2+7 d^2 x^4\right )-c x^2 \left (c^2 D+C d^2 x^2+2 c d \left (C+D x^2\right )\right )\right )\right )\right )+i c \left (-3 A b^3 c^2 d^2+3 a^3 c^2 d^2 D-a b^2 c \left (c^2 C d-7 B c d^2+7 A d^3+2 c^3 D\right )+a^2 b d \left (-7 c^2 C d+B c d^2+2 A d^3+7 c^3 D\right )\right ) \sqrt {1+\frac {b x^2}{a}} \left (c+d x^2\right ) \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i c (-b c+a d) \left (-3 A b^2 c d^2+3 a^2 c d (-C d+2 c D)-a b \left (c^2 C d-4 B c d^2+A d^3+2 c^3 D\right )\right ) \sqrt {1+\frac {b x^2}{a}} \left (c+d x^2\right ) \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{3 b c^2 d^2 (-b c+a d)^3 \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}} \] Input:

Integrate[(A + B*x^2 + C*x^4 + D*x^6)/((a + b*x^2)^(3/2)*(c + d*x^2)^(5/2) 
),x]
 

Output:

(Sqrt[b/a]*(Sqrt[b/a]*d*x*(-(A*d*(3*b^3*c^2*(c + d*x^2)^2 - a^3*d^3*(3*c + 
 2*d*x^2) + a*b^2*c*d^2*x^2*(8*c + 7*d*x^2) + 2*a^2*b*d^2*(4*c^2 + 2*c*d*x 
^2 - d^2*x^4))) + a*c*(a^2*d*(9*c^3*D + B*d^3*x^2 + c*d^2*x^2*(-4*C + 3*D* 
x^2) + c^2*(-3*C*d + 13*d*D*x^2)) + a*b*(-(c^4*D) + B*d^4*x^4 + c*d^3*x^2* 
(4*B - 7*C*x^2) + c^3*(-5*C*d + 4*d*D*x^2) + c^2*d^2*(5*B - 10*C*x^2 + 7*D 
*x^4)) + b^2*c*(B*d*(3*c^2 + 11*c*d*x^2 + 7*d^2*x^4) - c*x^2*(c^2*D + C*d^ 
2*x^2 + 2*c*d*(C + D*x^2))))) + I*c*(-3*A*b^3*c^2*d^2 + 3*a^3*c^2*d^2*D - 
a*b^2*c*(c^2*C*d - 7*B*c*d^2 + 7*A*d^3 + 2*c^3*D) + a^2*b*d*(-7*c^2*C*d + 
B*c*d^2 + 2*A*d^3 + 7*c^3*D))*Sqrt[1 + (b*x^2)/a]*(c + d*x^2)*Sqrt[1 + (d* 
x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + I*c*(-(b*c) + a*d 
)*(-3*A*b^2*c*d^2 + 3*a^2*c*d*(-(C*d) + 2*c*D) - a*b*(c^2*C*d - 4*B*c*d^2 
+ A*d^3 + 2*c^3*D))*Sqrt[1 + (b*x^2)/a]*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*El 
lipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(3*b*c^2*d^2*(-(b*c) + a*d) 
^3*Sqrt[a + b*x^2]*(c + d*x^2)^(3/2))
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1245\) vs. \(2(531)=1062\).

Time = 2.08 (sec) , antiderivative size = 1245, normalized size of antiderivative = 2.34, 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.

\(\displaystyle \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {A}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2}}+\frac {B x^2}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2}}+\frac {C x^4}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2}}+\frac {D x^6}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a D x^3}{b (b c-a d) \sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}+\frac {A d (3 b c+a d) \sqrt {b x^2+a} x}{3 a c (b c-a d)^2 \left (d x^2+c\right )^{3/2}}+\frac {C (b c+3 a d) \sqrt {b x^2+a} x}{3 b (b c-a d)^2 \left (d x^2+c\right )^{3/2}}-\frac {c (b c+3 a d) D \sqrt {b x^2+a} x}{3 b d (b c-a d)^2 \left (d x^2+c\right )^{3/2}}-\frac {4 B d \sqrt {b x^2+a} x}{3 (b c-a d)^2 \left (d x^2+c\right )^{3/2}}+\frac {A b x}{a (b c-a d) \sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}-\frac {B x}{(b c-a d) \sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}+\frac {a C x}{b (b c-a d) \sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}-\frac {B \sqrt {d} (7 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 \sqrt {c} (b c-a d)^3 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {\sqrt {c} C (b c+7 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 \sqrt {d} (b c-a d)^3 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {A \sqrt {d} \left (3 b^2 c^2+7 a b d c-2 a^2 d^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 )}{3 a c^{3/2} (b c-a d)^3 \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 (2 b^2 c^2-7 a b d c-3 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 )}{3 b d^{3/2} (b c-a d)^3 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {A b \sqrt {d} (9 b c-a 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 )}{3 a \sqrt {c} (b c-a d)^3 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\sqrt {c} C (5 b c+3 a 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 )}{3 \sqrt {d} (b c-a d)^3 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {b B \sqrt {c} (3 b c+5 a 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 )}{3 a \sqrt {d} (b c-a d)^3 \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} (b c-9 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 )}{3 d^{3/2} (b c-a d)^3 \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)/((a + b*x^2)^(3/2)*(c + d*x^2)^(5/2)),x]
 

Output:

(A*b*x)/(a*(b*c - a*d)*Sqrt[a + b*x^2]*(c + d*x^2)^(3/2)) - (B*x)/((b*c - 
a*d)*Sqrt[a + b*x^2]*(c + d*x^2)^(3/2)) + (a*C*x)/(b*(b*c - a*d)*Sqrt[a + 
b*x^2]*(c + d*x^2)^(3/2)) + (a*D*x^3)/(b*(b*c - a*d)*Sqrt[a + b*x^2]*(c + 
d*x^2)^(3/2)) - (4*B*d*x*Sqrt[a + b*x^2])/(3*(b*c - a*d)^2*(c + d*x^2)^(3/ 
2)) + (A*d*(3*b*c + a*d)*x*Sqrt[a + b*x^2])/(3*a*c*(b*c - a*d)^2*(c + d*x^ 
2)^(3/2)) + (C*(b*c + 3*a*d)*x*Sqrt[a + b*x^2])/(3*b*(b*c - a*d)^2*(c + d* 
x^2)^(3/2)) - (c*(b*c + 3*a*d)*D*x*Sqrt[a + b*x^2])/(3*b*d*(b*c - a*d)^2*( 
c + d*x^2)^(3/2)) - (B*Sqrt[d]*(7*b*c + a*d)*Sqrt[a + b*x^2]*EllipticE[Arc 
Tan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*Sqrt[c]*(b*c - a*d)^3*Sqrt[ 
(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (Sqrt[c]*C*(b*c + 7*a* 
d)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)] 
)/(3*Sqrt[d]*(b*c - a*d)^3*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + 
d*x^2]) + (A*Sqrt[d]*(3*b^2*c^2 + 7*a*b*c*d - 2*a^2*d^2)*Sqrt[a + b*x^2]*E 
llipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*a*c^(3/2)*(b*c 
- a*d)^3*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (Sqrt[c] 
*(2*b^2*c^2 - 7*a*b*c*d - 3*a^2*d^2)*D*Sqrt[a + b*x^2]*EllipticE[ArcTan[(S 
qrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*b*d^(3/2)*(b*c - a*d)^3*Sqrt[(c*( 
a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (A*b*Sqrt[d]*(9*b*c - a*d) 
*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/ 
(3*a*Sqrt[c]*(b*c - a*d)^3*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c...
 

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 = 11.08 (sec) , antiderivative size = 949, normalized size of antiderivative = 1.79

method	result	size
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {\left (b d \,x^{2}+b c \right ) x \left (b^{3} A -a \,b^{2} B +a^{2} b C -a^{3} D\right )}{b a \left (a d -b c \right )^{3} \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}+\frac {x \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right ) \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 c \,d^{3} \left (a d -b c \right )^{2} \left (x^{2}+\frac {c}{d}\right )^{2}}+\frac {\left (b d \,x^{2}+a d \right ) x \left (2 A a \,d^{4}-7 A b c \,d^{3}+B a c \,d^{3}+4 B b \,c^{2} d^{2}-4 C a \,c^{2} d^{2}-c^{3} C b d +7 D a \,c^{3} d -2 D b \,c^{4}\right )}{3 c^{2} d^{2} \left (a d -b c \right )^{3} \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (b d \,x^{2}+a d \right )}}+\frac {\left (\frac {D}{b \,d^{2}}+\frac {b^{3} A -a \,b^{2} B +a^{2} b C -a^{3} D}{b \left (a d -b c \right )^{2} a}+\frac {c \left (b^{3} A -a \,b^{2} B +a^{2} b C -a^{3} D\right )}{a \left (a d -b c \right )^{3}}+\frac {b \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )}{3 d^{2} \left (a d -b c \right )^{2} c}+\frac {2 A a \,d^{4}-7 A b c \,d^{3}+B a c \,d^{3}+4 B b \,c^{2} d^{2}-4 C a \,c^{2} d^{2}-c^{3} C b d +7 D a \,c^{3} d -2 D b \,c^{4}}{3 d^{2} \left (a d -b c \right )^{2} c^{2}}-\frac {a \left (2 A a \,d^{4}-7 A b c \,d^{3}+B a c \,d^{3}+4 B b \,c^{2} d^{2}-4 C a \,c^{2} d^{2}-c^{3} C b d +7 D a \,c^{3} d -2 D b \,c^{4}\right )}{3 d \,c^{2} \left (a d -b c \right )^{3}}\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 (\frac {\left (b^{3} A -a \,b^{2} B +a^{2} b C -a^{3} D\right ) d}{\left (a d -b c \right )^{3} a}-\frac {b \left (2 A a \,d^{4}-7 A b c \,d^{3}+B a c \,d^{3}+4 B b \,c^{2} d^{2}-4 C a \,c^{2} d^{2}-c^{3} C b d +7 D a \,c^{3} d -2 D b \,c^{4}\right )}{3 d \left (a d -b c \right )^{3} c^{2}}\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}}\)	\(949\)
default	\(\text {Expression too large to display}\)	\(3355\)

Input:

int((D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(3/2)/(d*x^2+c)^(5/2),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)*(-(b*d*x^2+b*c 
)/b/a/(a*d-b*c)^3*x*(A*b^3-B*a*b^2+C*a^2*b-D*a^3)/((x^2+a/b)*(b*d*x^2+b*c) 
)^(1/2)+1/3/c/d^3/(a*d-b*c)^2*x*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(b*d*x^4+a*d 
*x^2+b*c*x^2+a*c)^(1/2)/(x^2+c/d)^2+1/3*(b*d*x^2+a*d)/c^2/d^2/(a*d-b*c)^3* 
x*(2*A*a*d^4-7*A*b*c*d^3+B*a*c*d^3+4*B*b*c^2*d^2-4*C*a*c^2*d^2-C*b*c^3*d+7 
*D*a*c^3*d-2*D*b*c^4)/((x^2+c/d)*(b*d*x^2+a*d))^(1/2)+(D/b/d^2+1/b*(A*b^3- 
B*a*b^2+C*a^2*b-D*a^3)/(a*d-b*c)^2/a+c/a/(a*d-b*c)^3*(A*b^3-B*a*b^2+C*a^2* 
b-D*a^3)+1/3/d^2*b*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)/(a*d-b*c)^2/c+1/3/d^2/(a* 
d-b*c)^2*(2*A*a*d^4-7*A*b*c*d^3+B*a*c*d^3+4*B*b*c^2*d^2-4*C*a*c^2*d^2-C*b* 
c^3*d+7*D*a*c^3*d-2*D*b*c^4)/c^2-1/3*a/d/c^2/(a*d-b*c)^3*(2*A*a*d^4-7*A*b* 
c*d^3+B*a*c*d^3+4*B*b*c^2*d^2-4*C*a*c^2*d^2-C*b*c^3*d+7*D*a*c^3*d-2*D*b*c^ 
4))/(-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))-((A*b^3- 
B*a*b^2+C*a^2*b-D*a^3)*d/(a*d-b*c)^3/a-1/3*b/d*(2*A*a*d^4-7*A*b*c*d^3+B*a* 
c*d^3+4*B*b*c^2*d^2-4*C*a*c^2*d^2-C*b*c^3*d+7*D*a*c^3*d-2*D*b*c^4)/(a*d-b* 
c)^3/c^2)*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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2094 vs. \(2 (506) = 1012\).

Time = 0.17 (sec) , antiderivative size = 2094, normalized size of antiderivative = 3.94 \[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate((D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(3/2)/(d*x^2+c)^(5/2),x, algorit 
hm="fricas")
 

Output:

-1/3*((2*D*a^2*b^3*c^6 - 2*A*a^3*b^2*c^2*d^4 - (7*D*a^3*b^2 - C*a^2*b^3)*c 
^5*d - (3*D*a^4*b - 7*C*a^3*b^2 + 7*B*a^2*b^3 - 3*A*a*b^4)*c^4*d^2 - (B*a^ 
3*b^2 - 7*A*a^2*b^3)*c^3*d^3 + (2*D*a*b^4*c^4*d^2 - 2*A*a^2*b^3*d^6 - (7*D 
*a^2*b^3 - C*a*b^4)*c^3*d^3 - (3*D*a^3*b^2 - 7*C*a^2*b^3 + 7*B*a*b^4 - 3*A 
*b^5)*c^2*d^4 - (B*a^2*b^3 - 7*A*a*b^4)*c*d^5)*x^6 + (4*D*a*b^4*c^5*d - 2* 
A*a^3*b^2*d^6 - 2*(6*D*a^2*b^3 - C*a*b^4)*c^4*d^2 - (13*D*a^3*b^2 - 15*C*a 
^2*b^3 + 14*B*a*b^4 - 6*A*b^5)*c^3*d^3 - (3*D*a^4*b - 7*C*a^3*b^2 + 9*B*a^ 
2*b^3 - 17*A*a*b^4)*c^2*d^4 - (B*a^3*b^2 - 3*A*a^2*b^3)*c*d^5)*x^4 + (2*D* 
a*b^4*c^6 - 4*A*a^3*b^2*c*d^5 - (3*D*a^2*b^3 - C*a*b^4)*c^5*d - (17*D*a^3* 
b^2 - 9*C*a^2*b^3 + 7*B*a*b^4 - 3*A*b^5)*c^4*d^2 - (6*D*a^4*b - 14*C*a^3*b 
^2 + 15*B*a^2*b^3 - 13*A*a*b^4)*c^3*d^3 - 2*(B*a^3*b^2 - 6*A*a^2*b^3)*c^2* 
d^4)*x^2)*sqrt(a*c)*sqrt(-b/a)*elliptic_e(arcsin(x*sqrt(-b/a)), a*d/(b*c)) 
 - (2*D*a^2*b^3*c^6 + (D*a^4*b - 7*D*a^3*b^2 + C*a^2*b^3)*c^5*d - (9*D*a^5 
 - (5*C - 3*D)*a^4*b + (3*B - 7*C)*a^3*b^2 + 7*B*a^2*b^3 - 3*A*a*b^4)*c^4* 
d^2 + (3*C*a^5 - 5*B*a^4*b + (9*A - B)*a^3*b^2 + 7*A*a^2*b^3)*c^3*d^3 - (A 
*a^4*b + 2*A*a^3*b^2)*c^2*d^4 + (2*D*a*b^4*c^4*d^2 + (D*a^3*b^2 - 7*D*a^2* 
b^3 + C*a*b^4)*c^3*d^3 - (9*D*a^4*b - (5*C - 3*D)*a^3*b^2 + (3*B - 7*C)*a^ 
2*b^3 + 7*B*a*b^4 - 3*A*b^5)*c^2*d^4 + (3*C*a^4*b - 5*B*a^3*b^2 + (9*A - B 
)*a^2*b^3 + 7*A*a*b^4)*c*d^5 - (A*a^3*b^2 + 2*A*a^2*b^3)*d^6)*x^6 + (4*D*a 
*b^4*c^5*d + 2*(D*a^3*b^2 - 6*D*a^2*b^3 + C*a*b^4)*c^4*d^2 - (17*D*a^4*...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((D*x**6+C*x**4+B*x**2+A)/(b*x**2+a)**(3/2)/(d*x**2+c)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {D x^{6} + C x^{4} + B x^{2} + A}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(3/2)/(d*x^2+c)^(5/2),x, algorit 
hm="maxima")
 

Output:

integrate((D*x^6 + C*x^4 + B*x^2 + A)/((b*x^2 + a)^(3/2)*(d*x^2 + c)^(5/2) 
), x)
 

Giac [F]

\[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {D x^{6} + C x^{4} + B x^{2} + A}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(3/2)/(d*x^2+c)^(5/2),x, algorit 
hm="giac")
 

Output:

integrate((D*x^6 + C*x^4 + B*x^2 + A)/((b*x^2 + a)^(3/2)*(d*x^2 + c)^(5/2) 
), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2}} \, dx=\int \frac {A+B\,x^2+C\,x^4+x^6\,D}{{\left (b\,x^2+a\right )}^{3/2}\,{\left (d\,x^2+c\right )}^{5/2}} \,d x \] Input:

int((A + B*x^2 + C*x^4 + x^6*D)/((a + b*x^2)^(3/2)*(c + d*x^2)^(5/2)),x)
 

Output:

int((A + B*x^2 + C*x^4 + x^6*D)/((a + b*x^2)^(3/2)*(c + d*x^2)^(5/2)), x)
 

Reduce [F]

\[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2}} \, dx=\text {too large to display} \] Input:

int((D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(3/2)/(d*x^2+c)^(5/2),x)
 

Output:

( - 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*c*x - 2*sqrt(c + d*x**2)*sqrt(a 
+ b*x**2)*a*d*x**3 - sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*x - 3*int((sqr 
t(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**2*c**3 + 3*a**2*c**2*d*x**2 + 3*a 
**2*c*d**2*x**4 + a**2*d**3*x**6 + 2*a*b*c**3*x**2 + 6*a*b*c**2*d*x**4 + 6 
*a*b*c*d**2*x**6 + 2*a*b*d**3*x**8 + b**2*c**3*x**4 + 3*b**2*c**2*d*x**6 + 
 3*b**2*c*d**2*x**8 + b**2*d**3*x**10),x)*a**2*b*c**3*d - 6*int((sqrt(c + 
d*x**2)*sqrt(a + b*x**2)*x**4)/(a**2*c**3 + 3*a**2*c**2*d*x**2 + 3*a**2*c* 
d**2*x**4 + a**2*d**3*x**6 + 2*a*b*c**3*x**2 + 6*a*b*c**2*d*x**4 + 6*a*b*c 
*d**2*x**6 + 2*a*b*d**3*x**8 + b**2*c**3*x**4 + 3*b**2*c**2*d*x**6 + 3*b** 
2*c*d**2*x**8 + b**2*d**3*x**10),x)*a**2*b*c**2*d**2*x**2 - 3*int((sqrt(c 
+ d*x**2)*sqrt(a + b*x**2)*x**4)/(a**2*c**3 + 3*a**2*c**2*d*x**2 + 3*a**2* 
c*d**2*x**4 + a**2*d**3*x**6 + 2*a*b*c**3*x**2 + 6*a*b*c**2*d*x**4 + 6*a*b 
*c*d**2*x**6 + 2*a*b*d**3*x**8 + b**2*c**3*x**4 + 3*b**2*c**2*d*x**6 + 3*b 
**2*c*d**2*x**8 + b**2*d**3*x**10),x)*a**2*b*c*d**3*x**4 - 3*int((sqrt(c + 
 d*x**2)*sqrt(a + b*x**2)*x**4)/(a**2*c**3 + 3*a**2*c**2*d*x**2 + 3*a**2*c 
*d**2*x**4 + a**2*d**3*x**6 + 2*a*b*c**3*x**2 + 6*a*b*c**2*d*x**4 + 6*a*b* 
c*d**2*x**6 + 2*a*b*d**3*x**8 + b**2*c**3*x**4 + 3*b**2*c**2*d*x**6 + 3*b* 
*2*c*d**2*x**8 + b**2*d**3*x**10),x)*a*b**3*c**2*d - 6*int((sqrt(c + d*x** 
2)*sqrt(a + b*x**2)*x**4)/(a**2*c**3 + 3*a**2*c**2*d*x**2 + 3*a**2*c*d**2* 
x**4 + a**2*d**3*x**6 + 2*a*b*c**3*x**2 + 6*a*b*c**2*d*x**4 + 6*a*b*c*d...