Integrand size = 40, antiderivative size = 695 \[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}} \, dx=\frac {\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) x}{3 a b^2 (b c-a d) \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}+\frac {\left (2 A b^3 (b c-4 a d)+a \left (b^3 B c-a b^2 (4 c C-5 B d)-a^3 d D-a^2 b (2 C d-7 c D)\right )\right ) x}{3 a^2 b^2 (b c-a d)^2 \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}+\frac {\left (A b d \left (2 b^2 c^2-9 a b c d-a^2 d^2\right )+a c \left (b^2 B c d-a^2 d (3 C d-7 c D)-a b \left (5 c C d-7 B d^2-c^2 D\right )\right )\right ) x \sqrt {a+b x^2}}{3 a^2 b c (b c-a d)^3 \left (c+d x^2\right )^{3/2}}+\frac {\left (2 A b d \left (b^3 c^3-5 a b^2 c^2 d-5 a^2 b c d^2+a^3 d^3\right )+a c \left (b^3 B c^2 d+a^3 c d^2 D-a^2 b d \left (8 c C d-B d^2-14 c^2 D\right )-a b^2 c \left (8 c C d-14 B d^2-c^2 D\right )\right )\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a^2 b c^{3/2} \sqrt {d} (b c-a d)^4 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {\left (A b^3 c^2 d-a b^2 c \left (3 c^2 C-8 B c d+18 A d^2\right )-a^3 c d (3 C d-8 c D)-a^2 b \left (10 c^2 C d-8 B c d^2-A d^3-8 c^3 D\right )\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a^2 \sqrt {c} \sqrt {d} (b c-a d)^4 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:
1/3*(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)^(3/2)/(d*x^ 2+c)^(3/2)+1/3*(2*A*b^3*(-4*a*d+b*c)+a*(b^3*B*c-a*b^2*(-5*B*d+4*C*c)-a^3*d *D-a^2*b*(2*C*d-7*D*c)))*x/a^2/b^2/(-a*d+b*c)^2/(b*x^2+a)^(1/2)/(d*x^2+c)^ (3/2)+1/3*(A*b*d*(-a^2*d^2-9*a*b*c*d+2*b^2*c^2)+a*c*(b^2*B*c*d-a^2*d*(3*C* d-7*D*c)-a*b*(-7*B*d^2+5*C*c*d-D*c^2)))*x*(b*x^2+a)^(1/2)/a^2/b/c/(-a*d+b* c)^3/(d*x^2+c)^(3/2)+1/3*(2*A*b*d*(a^3*d^3-5*a^2*b*c*d^2-5*a*b^2*c^2*d+b^3 *c^3)+a*c*(b^3*B*c^2*d+a^3*c*d^2*D-a^2*b*d*(-B*d^2+8*C*c*d-14*D*c^2)-a*b^2 *c*(-14*B*d^2+8*C*c*d-D*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^2/b/c^(3/2)/d^(1/2)/(-a*d+b*c)^4/( c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)-1/3*(A*b^3*c^2*d-a*b^2*c*(1 8*A*d^2-8*B*c*d+3*C*c^2)-a^3*c*d*(3*C*d-8*D*c)-a^2*b*(-A*d^3-8*B*c*d^2+10* C*c^2*d-8*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^2/c^(1/2)/d^(1/2)/(-a*d+b*c)^4/(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 = 12.71 (sec) , antiderivative size = 616, normalized size of antiderivative = 0.89 \[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}} \, dx=\frac {-\sqrt {\frac {b}{a}} d x \left (a^2 c (-b c+a d) \left (-c^2 C d+B c d^2-A d^3+c^3 D\right ) \left (a+b x^2\right )^2-a^2 \left (b c \left (-4 c^2 C d+7 B c d^2-10 A d^3+c^3 D\right )+a d \left (-4 c^2 C d+B c d^2+2 A d^3+7 c^3 D\right )\right ) \left (a+b x^2\right )^2 \left (c+d x^2\right )-a c^2 (-b c+a d) \left (-A b^3+a \left (b^2 B-a b C+a^2 D\right )\right ) \left (c+d x^2\right )^2-c^2 \left (2 A b^3 (b c-5 a d)+a \left (b^3 B c+a b^2 (-4 c C+7 B d)+a^3 d D+a^2 b (-4 C d+7 c D)\right )\right ) \left (a+b x^2\right ) \left (c+d x^2\right )^2\right )+i c \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \left (c+d x^2\right ) \sqrt {1+\frac {d x^2}{c}} \left (\left (2 A b d \left (b^3 c^3-5 a b^2 c^2 d-5 a^2 b c d^2+a^3 d^3\right )+a c \left (b^3 B c^2 d+a^3 c d^2 D+a b^2 c \left (-8 c C d+14 B d^2+c^2 D\right )+a^2 b d \left (-8 c C d+B d^2+14 c^2 D\right )\right )\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-(b c-a d) \left (A b d \left (2 b^2 c^2-9 a b c d-a^2 d^2\right )+a c \left (b^2 B c d+a^2 d (-3 C d+7 c D)+a b \left (-5 c C d+7 B d^2+c^2 D\right )\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{3 a^2 \sqrt {\frac {b}{a}} c^2 d (b c-a d)^4 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \] Input:
Integrate[(A + B*x^2 + C*x^4 + D*x^6)/((a + b*x^2)^(5/2)*(c + d*x^2)^(5/2) ),x]
Output:
(-(Sqrt[b/a]*d*x*(a^2*c*(-(b*c) + a*d)*(-(c^2*C*d) + B*c*d^2 - A*d^3 + c^3 *D)*(a + b*x^2)^2 - a^2*(b*c*(-4*c^2*C*d + 7*B*c*d^2 - 10*A*d^3 + c^3*D) + a*d*(-4*c^2*C*d + B*c*d^2 + 2*A*d^3 + 7*c^3*D))*(a + b*x^2)^2*(c + d*x^2) - a*c^2*(-(b*c) + a*d)*(-(A*b^3) + a*(b^2*B - a*b*C + a^2*D))*(c + d*x^2) ^2 - c^2*(2*A*b^3*(b*c - 5*a*d) + a*(b^3*B*c + a*b^2*(-4*c*C + 7*B*d) + a^ 3*d*D + a^2*b*(-4*C*d + 7*c*D)))*(a + b*x^2)*(c + d*x^2)^2)) + I*c*(a + b* x^2)*Sqrt[1 + (b*x^2)/a]*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*((2*A*b*d*(b^3*c^ 3 - 5*a*b^2*c^2*d - 5*a^2*b*c*d^2 + a^3*d^3) + a*c*(b^3*B*c^2*d + a^3*c*d^ 2*D + a*b^2*c*(-8*c*C*d + 14*B*d^2 + c^2*D) + a^2*b*d*(-8*c*C*d + B*d^2 + 14*c^2*D)))*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - (b*c - a*d)*( A*b*d*(2*b^2*c^2 - 9*a*b*c*d - a^2*d^2) + a*c*(b^2*B*c*d + a^2*d*(-3*C*d + 7*c*D) + a*b*(-5*c*C*d + 7*B*d^2 + c^2*D)))*EllipticF[I*ArcSinh[Sqrt[b/a] *x], (a*d)/(b*c)]))/(3*a^2*Sqrt[b/a]*c^2*d*(b*c - a*d)^4*(a + b*x^2)^(3/2) *(c + d*x^2)^(3/2))
Leaf count is larger than twice the leaf count of optimal. \(1507\) vs. \(2(695)=1390\).
Time = 2.62 (sec) , antiderivative size = 1507, normalized size of antiderivative = 2.17, 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.
| \(\displaystyle \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {A}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}}+\frac {B x^2}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}}+\frac {C x^4}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}}+\frac {D x^6}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a D x^3}{3 b (b c-a d) \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}-\frac {C d (5 b c+3 a d) \sqrt {b x^2+a} x}{3 b (b c-a d)^3 \left (d x^2+c\right )^{3/2}}+\frac {B d (b c+7 a d) \sqrt {b x^2+a} x}{3 a (b c-a d)^3 \left (d x^2+c\right )^{3/2}}+\frac {A d \left (2 b^2 c^2-9 a b d c-a^2 d^2\right ) \sqrt {b x^2+a} x}{3 a^2 c (b c-a d)^3 \left (d x^2+c\right )^{3/2}}+\frac {c (b c+7 a d) D \sqrt {b x^2+a} x}{3 b (b c-a d)^3 \left (d x^2+c\right )^{3/2}}-\frac {2 C (2 b c+a d) x}{3 b (b c-a d)^2 \sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}+\frac {B (b c+5 a d) x}{3 a (b c-a d)^2 \sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}+\frac {2 a c D x}{b (b c-a d)^2 \sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}+\frac {2 A b (b c-4 a d) x}{3 a^2 (b c-a d)^2 \sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}+\frac {A b x}{3 a (b c-a d) \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}-\frac {B x}{3 (b c-a d) \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}+\frac {a C x}{3 b (b c-a d) \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}-\frac {8 \sqrt {c} C \sqrt {d} (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 c-a d)^4 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {2 A \sqrt {d} (b c+a d) \left (b^2 c^2-6 a b d c+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^2 c^{3/2} (b c-a d)^4 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {B \sqrt {d} \left (b^2 c^2+14 a b d c+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 \sqrt {c} (b c-a d)^4 \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 (b^2 c^2+14 a b d c+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 \sqrt {d} (b c-a d)^4 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {8 b B \sqrt {c} \sqrt {d} (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 (b c-a d)^4 \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 (3 b c+a d) (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 a \sqrt {d} (b c-a d)^4 \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} \left (b^2 c^2-18 a b d c+a^2 d^2\right ) \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^2 \sqrt {c} (b c-a d)^4 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {8 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 )}{3 \sqrt {d} (b c-a d)^4 \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)^(5/2)*(c + d*x^2)^(5/2)),x]
Output:
(A*b*x)/(3*a*(b*c - a*d)*(a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)) - (B*x)/(3*( b*c - a*d)*(a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)) + (a*C*x)/(3*b*(b*c - a*d) *(a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)) + (a*D*x^3)/(3*b*(b*c - a*d)*(a + b* x^2)^(3/2)*(c + d*x^2)^(3/2)) + (2*A*b*(b*c - 4*a*d)*x)/(3*a^2*(b*c - a*d) ^2*Sqrt[a + b*x^2]*(c + d*x^2)^(3/2)) - (2*C*(2*b*c + a*d)*x)/(3*b*(b*c - a*d)^2*Sqrt[a + b*x^2]*(c + d*x^2)^(3/2)) + (B*(b*c + 5*a*d)*x)/(3*a*(b*c - a*d)^2*Sqrt[a + b*x^2]*(c + d*x^2)^(3/2)) + (2*a*c*D*x)/(b*(b*c - a*d)^2 *Sqrt[a + b*x^2]*(c + d*x^2)^(3/2)) - (C*d*(5*b*c + 3*a*d)*x*Sqrt[a + b*x^ 2])/(3*b*(b*c - a*d)^3*(c + d*x^2)^(3/2)) + (B*d*(b*c + 7*a*d)*x*Sqrt[a + b*x^2])/(3*a*(b*c - a*d)^3*(c + d*x^2)^(3/2)) + (A*d*(2*b^2*c^2 - 9*a*b*c* d - a^2*d^2)*x*Sqrt[a + b*x^2])/(3*a^2*c*(b*c - a*d)^3*(c + d*x^2)^(3/2)) + (c*(b*c + 7*a*d)*D*x*Sqrt[a + b*x^2])/(3*b*(b*c - a*d)^3*(c + d*x^2)^(3/ 2)) - (8*Sqrt[c]*C*Sqrt[d]*(b*c + a*d)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(S qrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*(b*c - a*d)^4*Sqrt[(c*(a + b*x^2) )/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (2*A*Sqrt[d]*(b*c + a*d)*(b^2*c^2 - 6*a*b*c*d + a^2*d^2)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]] , 1 - (b*c)/(a*d)])/(3*a^2*c^(3/2)*(b*c - a*d)^4*Sqrt[(c*(a + b*x^2))/(a*( c + d*x^2))]*Sqrt[c + d*x^2]) + (B*Sqrt[d]*(b^2*c^2 + 14*a*b*c*d + a^2*d^2 )*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)]) /(3*a*Sqrt[c]*(b*c - a*d)^4*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[...
Leaf count of result is larger than twice the leaf count of optimal. \(1510\) vs. \(2(660)=1320\).
Time = 22.03 (sec) , antiderivative size = 1511, normalized size of antiderivative = 2.17
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1511\) |
| default | \(\text {Expression too large to display}\) | \(6642\) |
Input:
int((D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(5/2)/(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)*((1/3/b^3/d^3* (A*a*b^2*d^3+A*b^3*c*d^2-2*B*a*b^2*c*d^2+C*a^2*b*c*d^2+C*a*b^2*c^2*d-D*a^3 *c*d^2-D*a*b^2*c^3)/a/c/(a^2*d^2-2*a*b*c*d+b^2*c^2)*x^3+1/3/b^3/d^3*(A*a^2 *b*d^3+A*b^3*c^2*d-B*a^2*b*c*d^2-B*a*b^2*c^2*d+2*C*a^2*b*c^2*d-D*a^3*c^2*d -D*a^2*b*c^3)/a/c/(a^2*d^2-2*a*b*c*d+b^2*c^2)*x)*(b*d*x^4+a*d*x^2+b*c*x^2+ a*c)^(1/2)/(x^4+(a*d+b*c)/d/b*x^2+a*c/d/b)^2-2*b*d*(-1/6/b/d*(2*A*a^3*b*d^ 4-10*A*a^2*b^2*c*d^3-10*A*a*b^3*c^2*d^2+2*A*b^4*c^3*d+B*a^3*b*c*d^3+14*B*a ^2*b^2*c^2*d^2+B*a*b^3*c^3*d-8*C*a^3*b*c^2*d^2-8*C*a^2*b^2*c^3*d+D*a^4*c^2 *d^2+14*D*a^3*b*c^3*d+D*a^2*b^2*c^4)/a^2/c^2/(a^2*d^2-2*a*b*c*d+b^2*c^2)^2 *x^3-1/6*(2*A*a^4*b*d^5-9*A*a^3*b^2*c*d^4-2*A*a^2*b^3*c^2*d^3-9*A*a*b^4*c^ 3*d^2+2*A*b^5*c^4*d+B*a^4*b*c*d^4+7*B*a^3*b^2*c^2*d^3+7*B*a^2*b^3*c^3*d^2+ B*a*b^4*c^4*d-5*C*a^4*b*c^2*d^3-6*C*a^3*b^2*c^3*d^2-5*C*a^2*b^3*c^4*d+D*a^ 5*c^2*d^3+7*D*a^4*b*c^3*d^2+7*D*a^3*b^2*c^4*d+D*a^2*b^3*c^5)/a^2/c^2/(a^2* d^2-2*a*b*c*d+b^2*c^2)^2/b^2/d^2*x)/((x^4+(a*d+b*c)/d/b*x^2+a*c/d/b)*b*d)^ (1/2)+(1/3/b/d/(a^2*d^2-2*a*b*c*d+b^2*c^2)*(2*A*a^2*b*d^3-6*A*a*b^2*c*d^2+ 2*A*b^3*c^2*d+B*a^2*b*c*d^2+B*a*b^2*c^2*d-2*C*a^2*b*c^2*d+D*a^3*c^2*d+D*a^ 2*b*c^3)/a^2/c^2-1/3/b/d*(2*A*a^4*b*d^5-9*A*a^3*b^2*c*d^4-2*A*a^2*b^3*c^2* d^3-9*A*a*b^4*c^3*d^2+2*A*b^5*c^4*d+B*a^4*b*c*d^4+7*B*a^3*b^2*c^2*d^3+7*B* a^2*b^3*c^3*d^2+B*a*b^4*c^4*d-5*C*a^4*b*c^2*d^3-6*C*a^3*b^2*c^3*d^2-5*C*a^ 2*b^3*c^4*d+D*a^5*c^2*d^3+7*D*a^4*b*c^3*d^2+7*D*a^3*b^2*c^4*d+D*a^2*b^3...
Leaf count of result is larger than twice the leaf count of optimal. 3222 vs. \(2 (649) = 1298\).
Time = 0.36 (sec) , antiderivative size = 3222, normalized size of antiderivative = 4.64 \[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:
integrate((D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(5/2)/(d*x^2+c)^(5/2),x, algorit hm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((D*x**6+C*x**4+B*x**2+A)/(b*x**2+a)**(5/2)/(d*x**2+c)**(5/2),x)
Output:
Timed out
\[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {D x^{6} + C x^{4} + B x^{2} + A}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(5/2)/(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)^(5/2)*(d*x^2 + c)^(5/2) ), x)
\[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {D x^{6} + C x^{4} + B x^{2} + A}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(5/2)/(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)^(5/2)*(d*x^2 + c)^(5/2) ), x)
Timed out. \[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}} \, dx=\int \frac {A+B\,x^2+C\,x^4+x^6\,D}{{\left (b\,x^2+a\right )}^{5/2}\,{\left (d\,x^2+c\right )}^{5/2}} \,d x \] Input:
int((A + B*x^2 + C*x^4 + x^6*D)/((a + b*x^2)^(5/2)*(c + d*x^2)^(5/2)),x)
Output:
int((A + B*x^2 + C*x^4 + x^6*D)/((a + b*x^2)^(5/2)*(c + d*x^2)^(5/2)), x)
\[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}} \, dx=\text {too large to display} \] Input:
int((D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(5/2)/(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 - 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*x - 2*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*c*x**3 - 9*int((sqrt(c + d*x**2)*sqrt(a + b* x**2)*x**4)/(a**4*c**3*d + 3*a**4*c**2*d**2*x**2 + 3*a**4*c*d**3*x**4 + a* *4*d**4*x**6 + a**3*b*c**4 + 6*a**3*b*c**3*d*x**2 + 12*a**3*b*c**2*d**2*x* *4 + 10*a**3*b*c*d**3*x**6 + 3*a**3*b*d**4*x**8 + 3*a**2*b**2*c**4*x**2 + 12*a**2*b**2*c**3*d*x**4 + 18*a**2*b**2*c**2*d**2*x**6 + 12*a**2*b**2*c*d* *3*x**8 + 3*a**2*b**2*d**4*x**10 + 3*a*b**3*c**4*x**4 + 10*a*b**3*c**3*d*x **6 + 12*a*b**3*c**2*d**2*x**8 + 6*a*b**3*c*d**3*x**10 + a*b**3*d**4*x**12 + b**4*c**4*x**6 + 3*b**4*c**3*d*x**8 + 3*b**4*c**2*d**2*x**10 + b**4*c*d **3*x**12),x)*a**4*b*c**3*d**2 - 18*int((sqrt(c + d*x**2)*sqrt(a + b*x**2) *x**4)/(a**4*c**3*d + 3*a**4*c**2*d**2*x**2 + 3*a**4*c*d**3*x**4 + a**4*d* *4*x**6 + a**3*b*c**4 + 6*a**3*b*c**3*d*x**2 + 12*a**3*b*c**2*d**2*x**4 + 10*a**3*b*c*d**3*x**6 + 3*a**3*b*d**4*x**8 + 3*a**2*b**2*c**4*x**2 + 12*a* *2*b**2*c**3*d*x**4 + 18*a**2*b**2*c**2*d**2*x**6 + 12*a**2*b**2*c*d**3*x* *8 + 3*a**2*b**2*d**4*x**10 + 3*a*b**3*c**4*x**4 + 10*a*b**3*c**3*d*x**6 + 12*a*b**3*c**2*d**2*x**8 + 6*a*b**3*c*d**3*x**10 + a*b**3*d**4*x**12 + b* *4*c**4*x**6 + 3*b**4*c**3*d*x**8 + 3*b**4*c**2*d**2*x**10 + b**4*c*d**3*x **12),x)*a**4*b*c**2*d**3*x**2 - 9*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)* x**4)/(a**4*c**3*d + 3*a**4*c**2*d**2*x**2 + 3*a**4*c*d**3*x**4 + a**4*...