Integrand size = 35, antiderivative size = 439 \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2}} \, dx=\frac {\left (A b^2-a (b B-a C)\right ) x}{a b (b c-a d) \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}+\frac {\left (3 A b^2 c d+3 a^2 c C d+a b \left (c^2 C-4 B c d+A d^2\right )\right ) x \sqrt {a+b x^2}}{3 a b c (b c-a d)^2 \left (c+d x^2\right )^{3/2}}+\frac {\left (3 A b^2 c^2 d+a^2 d \left (7 c^2 C-B c d-2 A d^2\right )+a b c \left (c^2 C-7 B c d+7 A d^2\right )\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a c^{3/2} \sqrt {d} (b c-a d)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {\left (3 a^2 c C d-3 b^2 c (B c-3 A d)+a b \left (5 c^2 C-5 B c d-A d^2\right )\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a \sqrt {c} \sqrt {d} (b c-a d)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:
(A*b^2-a*(B*b-C*a))*x/a/b/(-a*d+b*c)/(b*x^2+a)^(1/2)/(d*x^2+c)^(3/2)+1/3*( 3*A*b^2*c*d+3*a^2*c*C*d+a*b*(A*d^2-4*B*c*d+C*c^2))*x*(b*x^2+a)^(1/2)/a/b/c /(-a*d+b*c)^2/(d*x^2+c)^(3/2)+1/3*(3*A*b^2*c^2*d+a^2*d*(-2*A*d^2-B*c*d+7*C *c^2)+a*b*c*(7*A*d^2-7*B*c*d+C*c^2))*(b*x^2+a)^(1/2)*EllipticE(d^(1/2)*x/c ^(1/2)/(1+d*x^2/c)^(1/2),(1-b*c/a/d)^(1/2))/a/c^(3/2)/d^(1/2)/(-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*a^2*c*C*d-3*b^2*c* (-3*A*d+B*c)+a*b*(-A*d^2-5*B*c*d+5*C*c^2))*(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^(1/2)/(-a*d+b*c) ^3/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)
Result contains complex when optimal does not.
Time = 12.26 (sec) , antiderivative size = 518, normalized size of antiderivative = 1.18 \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2}} \, dx=\frac {\sqrt {\frac {b}{a}} \left (\sqrt {\frac {b}{a}} d x \left (-A \left (3 b^3 c^2 \left (c+d x^2\right )^2-a^3 d^3 \left (3 c+2 d x^2\right )+a b^2 c d^2 x^2 \left (8 c+7 d x^2\right )+2 a^2 b d^2 \left (4 c^2+2 c d x^2-d^2 x^4\right )\right )+a c \left (a^2 d \left (-3 c^2 C-4 c C d x^2+B d^2 x^2\right )+a b \left (-5 c^3 C+B d^3 x^4+c d^2 x^2 \left (4 B-7 C x^2\right )+5 c^2 d \left (B-2 C x^2\right )\right )+b^2 c \left (-c C x^2 \left (2 c+d x^2\right )+B \left (3 c^2+11 c d x^2+7 d^2 x^4\right )\right )\right )\right )-i b c \left (3 A b^2 c^2 d-a^2 d \left (-7 c^2 C+B c d+2 A d^2\right )+a b c \left (c^2 C-7 B c d+7 A d^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \left (c+d x^2\right ) \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c (-b c+a d) \left (3 A b^2 c d+3 a^2 c C d+a b \left (c^2 C-4 B c d+A d^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \left (c+d x^2\right ) \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{3 b c^2 d (-b c+a d)^3 \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}} \] Input:
Integrate[(A + B*x^2 + C*x^4)/((a + b*x^2)^(3/2)*(c + d*x^2)^(5/2)),x]
Output:
(Sqrt[b/a]*(Sqrt[b/a]*d*x*(-(A*(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*(-3*c^2*C - 4*c*C*d*x^2 + B*d^2*x^2) + a*b*(-5* c^3*C + B*d^3*x^4 + c*d^2*x^2*(4*B - 7*C*x^2) + 5*c^2*d*(B - 2*C*x^2)) + b ^2*c*(-(c*C*x^2*(2*c + d*x^2)) + B*(3*c^2 + 11*c*d*x^2 + 7*d^2*x^4)))) - I *b*c*(3*A*b^2*c^2*d - a^2*d*(-7*c^2*C + B*c*d + 2*A*d^2) + a*b*c*(c^2*C - 7*B*c*d + 7*A*d^2))*Sqrt[1 + (b*x^2)/a]*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*El lipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*c*(-(b*c) + a*d)*(3*A*b^2 *c*d + 3*a^2*c*C*d + a*b*(c^2*C - 4*B*c*d + A*d^2))*Sqrt[1 + (b*x^2)/a]*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c )]))/(3*b*c^2*d*(-(b*c) + a*d)^3*Sqrt[a + b*x^2]*(c + d*x^2)^(3/2))
Leaf count is larger than twice the leaf count of optimal. \(921\) vs. \(2(439)=878\).
Time = 1.65 (sec) , antiderivative size = 921, normalized size of antiderivative = 2.10, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, 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.
| \(\displaystyle \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {A}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2}}+\frac {B x^2}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2}}+\frac {C x^4}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \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 {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 {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}}\) |
Input:
Int[(A + B*x^2 + C*x^4)/((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)) - (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)) - (B*Sqrt[d]*(7*b*c + a*d)*Sqrt[a + b*x^2]*Ellipti cE[ArcTan[(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))]*Sqr t[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]*EllipticE[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]) - (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 + d*x^2]) - (Sqrt[c]*C*(5*b*c + 3*a*d)*Sqrt[a + b*x^ 2]*EllipticF[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]) + (b*B*S qrt[c]*(3*b*c + 5*a*d)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c ]], 1 - (b*c)/(a*d)])/(3*a*Sqrt[d]*(b*c - a*d)^3*Sqrt[(c*(a + b*x^2))/(...
Time = 10.88 (sec) , antiderivative size = 797, normalized size of antiderivative = 1.82
| 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^{2} A -a b B +a^{2} C \right )}{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 (d^{2} A -c d B +C \,c^{2}\right ) \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 c \,d^{2} \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^{3}-7 A b c \,d^{2}+B a c \,d^{2}+4 B b \,c^{2} d -4 C a \,c^{2} d -c^{3} C b \right )}{3 c^{2} d \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 {b^{2} A -a b B +a^{2} C}{\left (a d -b c \right )^{2} a}+\frac {b c \left (b^{2} A -a b B +a^{2} C \right )}{a \left (a d -b c \right )^{3}}+\frac {b \left (d^{2} A -c d B +C \,c^{2}\right )}{3 d \left (a d -b c \right )^{2} c}+\frac {2 A a \,d^{3}-7 A b c \,d^{2}+B a c \,d^{2}+4 B b \,c^{2} d -4 C a \,c^{2} d -c^{3} C b}{3 \left (a d -b c \right )^{2} d \,c^{2}}-\frac {a \left (2 A a \,d^{3}-7 A b c \,d^{2}+B a c \,d^{2}+4 B b \,c^{2} d -4 C a \,c^{2} d -c^{3} C b \right )}{3 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 {b d \left (b^{2} A -a b B +a^{2} C \right )}{a \left (a d -b c \right )^{3}}-\frac {b \left (2 A a \,d^{3}-7 A b c \,d^{2}+B a c \,d^{2}+4 B b \,c^{2} d -4 C a \,c^{2} d -c^{3} C b \right )}{3 \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}}\) | \(797\) |
| default | \(\text {Expression too large to display}\) | \(2417\) |
Input:
int((C*x^4+B*x^2+A)/(b*x^2+a)^(3/2)/(d*x^2+c)^(5/2),x,method=_RETURNVERBOS E)
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 )/a/(a*d-b*c)^3*x*(A*b^2-B*a*b+C*a^2)/((x^2+a/b)*(b*d*x^2+b*c))^(1/2)+1/3/ c/d^2/(a*d-b*c)^2*x*(A*d^2-B*c*d+C*c^2)*(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/(a*d-b*c)^3*x*(2*A*a*d^3-7*A*b*c*d^2 +B*a*c*d^2+4*B*b*c^2*d-4*C*a*c^2*d-C*b*c^3)/((x^2+c/d)*(b*d*x^2+a*d))^(1/2 )+(1/(a*d-b*c)^2*(A*b^2-B*a*b+C*a^2)/a+b*c/a/(a*d-b*c)^3*(A*b^2-B*a*b+C*a^ 2)+1/3*b/d*(A*d^2-B*c*d+C*c^2)/(a*d-b*c)^2/c+1/3/(a*d-b*c)^2/d*(2*A*a*d^3- 7*A*b*c*d^2+B*a*c*d^2+4*B*b*c^2*d-4*C*a*c^2*d-C*b*c^3)/c^2-1/3*a/c^2/(a*d- b*c)^3*(2*A*a*d^3-7*A*b*c*d^2+B*a*c*d^2+4*B*b*c^2*d-4*C*a*c^2*d-C*b*c^3))/ (-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*d*(A*b^2- B*a*b+C*a^2)/a/(a*d-b*c)^3-1/3*b*(2*A*a*d^3-7*A*b*c*d^2+B*a*c*d^2+4*B*b*c^ 2*d-4*C*a*c^2*d-C*b*c^3)/(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))))
Leaf count of result is larger than twice the leaf count of optimal. 1634 vs. \(2 (411) = 822\).
Time = 0.15 (sec) , antiderivative size = 1634, normalized size of antiderivative = 3.72 \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:
integrate((C*x^4+B*x^2+A)/(b*x^2+a)^(3/2)/(d*x^2+c)^(5/2),x, algorithm="fr icas")
Output:
-1/3*((C*a^2*b^3*c^5 - 2*A*a^3*b^2*c^2*d^3 + (C*a*b^4*c^3*d^2 - 2*A*a^2*b^ 3*d^5 + (7*C*a^2*b^3 - 7*B*a*b^4 + 3*A*b^5)*c^2*d^3 - (B*a^2*b^3 - 7*A*a*b ^4)*c*d^4)*x^6 + (7*C*a^3*b^2 - 7*B*a^2*b^3 + 3*A*a*b^4)*c^4*d - (B*a^3*b^ 2 - 7*A*a^2*b^3)*c^3*d^2 + (2*C*a*b^4*c^4*d - 2*A*a^3*b^2*d^5 + (15*C*a^2* b^3 - 14*B*a*b^4 + 6*A*b^5)*c^3*d^2 + (7*C*a^3*b^2 - 9*B*a^2*b^3 + 17*A*a* b^4)*c^2*d^3 - (B*a^3*b^2 - 3*A*a^2*b^3)*c*d^4)*x^4 + (C*a*b^4*c^5 - 4*A*a ^3*b^2*c*d^4 + (9*C*a^2*b^3 - 7*B*a*b^4 + 3*A*b^5)*c^4*d + (14*C*a^3*b^2 - 15*B*a^2*b^3 + 13*A*a*b^4)*c^3*d^2 - 2*(B*a^3*b^2 - 6*A*a^2*b^3)*c^2*d^3) *x^2)*sqrt(a*c)*sqrt(-b/a)*elliptic_e(arcsin(x*sqrt(-b/a)), a*d/(b*c)) - ( C*a^2*b^3*c^5 + (C*a*b^4*c^3*d^2 + (5*C*a^3*b^2 - (3*B - 7*C)*a^2*b^3 - 7* B*a*b^4 + 3*A*b^5)*c^2*d^3 + (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^4 - (A*a^3*b^2 + 2*A*a^2*b^3)*d^5)*x^6 + (5*C*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 + (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^2 - (A*a^4*b + 2*A*a^3*b^2)*c^2*d^ 3 + (2*C*a*b^4*c^4*d + (10*C*a^3*b^2 - 3*(2*B - 5*C)*a^2*b^3 - 14*B*a*b^4 + 6*A*b^5)*c^3*d^2 + (11*C*a^4*b - (13*B - 7*C)*a^3*b^2 + 9*(2*A - B)*a^2* b^3 + 17*A*a*b^4)*c^2*d^3 + (3*C*a^5 - 5*B*a^4*b + (7*A - B)*a^3*b^2 + 3*A *a^2*b^3)*c*d^4 - (A*a^4*b + 2*A*a^3*b^2)*d^5)*x^4 + (C*a*b^4*c^5 + (5*C*a ^3*b^2 - 3*(B - 3*C)*a^2*b^3 - 7*B*a*b^4 + 3*A*b^5)*c^4*d + (13*C*a^4*b - (11*B - 14*C)*a^3*b^2 + 3*(3*A - 5*B)*a^2*b^3 + 13*A*a*b^4)*c^3*d^2 + (...
Timed out. \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((C*x**4+B*x**2+A)/(b*x**2+a)**(3/2)/(d*x**2+c)**(5/2),x)
Output:
Timed out
\[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((C*x^4+B*x^2+A)/(b*x^2+a)^(3/2)/(d*x^2+c)^(5/2),x, algorithm="ma xima")
Output:
integrate((C*x^4 + B*x^2 + A)/((b*x^2 + a)^(3/2)*(d*x^2 + c)^(5/2)), x)
\[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((C*x^4+B*x^2+A)/(b*x^2+a)^(3/2)/(d*x^2+c)^(5/2),x, algorithm="gi ac")
Output:
integrate((C*x^4 + B*x^2 + A)/((b*x^2 + a)^(3/2)*(d*x^2 + c)^(5/2)), x)
Timed out. \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2}} \, dx=\int \frac {C\,x^4+B\,x^2+A}{{\left (b\,x^2+a\right )}^{3/2}\,{\left (d\,x^2+c\right )}^{5/2}} \,d x \] Input:
int((A + B*x^2 + C*x^4)/((a + b*x^2)^(3/2)*(c + d*x^2)^(5/2)),x)
Output:
int((A + B*x^2 + C*x^4)/((a + b*x^2)^(3/2)*(c + d*x^2)^(5/2)), x)
\[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2}} \, dx=\text {too large to display} \] Input:
int((C*x^4+B*x^2+A)/(b*x^2+a)^(3/2)/(d*x^2+c)^(5/2),x)
Output:
( - sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*x + 2*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*c**3*d + 4*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*c**2*d**2*x**2 + 2*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*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**2*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**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*...