\(\int (A+B x^2+C x^4) (a c+(b c-a d) x^2-b d x^4)^{3/2} \, dx\) [10]

Optimal result

Integrand size = 40, antiderivative size = 821 \[ \int \left (A+B x^2+C x^4\right ) \left (a c+(b c-a d) x^2-b d x^4\right )^{3/2} \, dx=-\frac {x \left (24 a^4 C d^4-11 a^3 b d^3 (3 c C+4 B d)-3 a^2 b^2 d^2 \left (24 c^2 C-11 B c d-33 A d^2\right )-33 a b^3 c d \left (c^2 C+B c d+36 A d^2\right )+b^4 c^2 \left (24 c^2 C+44 B c d+99 A d^2\right )-3 b d \left (9 b d (b c-a d) (a c C+11 A b d)+14 a b c d (6 b c C+11 b B d-6 a C d)+4 (b c-a d)^2 (6 b c C+11 b B d-6 a C d)\right ) x^2\right ) \sqrt {a c+(b c-a d) x^2-b d x^4}}{3465 b^3 d^3}+\frac {x \left (3 (3 b d (a c C+11 A b d)-(b c-a d) (6 b c C+11 b B d-6 a C d))+7 b d (6 b c C+11 b B d-6 a C d) x^2\right ) \left (a c+(b c-a d) x^2-b d x^4\right )^{3/2}}{693 b^2 d^2}-\frac {C x \left (a c+(b c-a d) x^2-b d x^4\right )^{5/2}}{11 b d}-\frac {a \sqrt {c} \left (48 a^5 C d^5+8 a^4 b d^4 (15 c C-11 B d)-6 a^2 b^3 c d^2 \left (8 c^2 C+33 B c d-165 A d^2\right )-2 b^5 c^3 \left (24 c^2 C+44 B c d+99 A d^2\right )+a^3 b^2 d^3 \left (48 c^2 C-275 B c d+198 A d^2\right )-5 a b^4 c^2 d \left (24 c^2 C+55 B c d+198 A d^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{3465 b^4 d^{7/2} \sqrt {a c+(b c-a d) x^2-b d x^4}}+\frac {a \sqrt {c} (b c+a d) \left (48 a^4 C d^4+8 a^3 b d^3 (12 c C-11 B d)-3 a b^3 c d \left (13 c^2 C+33 B c d-297 A d^2\right )+3 a^2 b^2 d^2 \left (3 c^2 C-77 B c d+66 A d^2\right )-b^4 c^2 \left (24 c^2 C+44 B c d+99 A d^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{3465 b^4 d^{7/2} \sqrt {a c+(b c-a d) x^2-b d x^4}} \] Output:

-1/3465*x*(24*a^4*C*d^4-11*a^3*b*d^3*(4*B*d+3*C*c)-3*a^2*b^2*d^2*(-33*A*d^ 
2-11*B*c*d+24*C*c^2)-33*a*b^3*c*d*(36*A*d^2+B*c*d+C*c^2)+b^4*c^2*(99*A*d^2 
+44*B*c*d+24*C*c^2)-3*b*d*(9*b*d*(-a*d+b*c)*(11*A*b*d+C*a*c)+14*a*b*c*d*(1 
1*B*b*d-6*C*a*d+6*C*b*c)+4*(-a*d+b*c)^2*(11*B*b*d-6*C*a*d+6*C*b*c))*x^2)*( 
a*c+(-a*d+b*c)*x^2-b*d*x^4)^(1/2)/b^3/d^3+1/693*x*(9*b*d*(11*A*b*d+C*a*c)- 
3*(-a*d+b*c)*(11*B*b*d-6*C*a*d+6*C*b*c)+7*b*d*(11*B*b*d-6*C*a*d+6*C*b*c)*x 
^2)*(a*c+(-a*d+b*c)*x^2-b*d*x^4)^(3/2)/b^2/d^2-1/11*C*x*(a*c+(-a*d+b*c)*x^ 
2-b*d*x^4)^(5/2)/b/d-1/3465*a*c^(1/2)*(48*a^5*C*d^5+8*a^4*b*d^4*(-11*B*d+1 
5*C*c)-6*a^2*b^3*c*d^2*(-165*A*d^2+33*B*c*d+8*C*c^2)-2*b^5*c^3*(99*A*d^2+4 
4*B*c*d+24*C*c^2)+a^3*b^2*d^3*(198*A*d^2-275*B*c*d+48*C*c^2)-5*a*b^4*c^2*d 
*(198*A*d^2+55*B*c*d+24*C*c^2))*(1+b*x^2/a)^(1/2)*(1-d*x^2/c)^(1/2)*Ellipt 
icE(d^(1/2)*x/c^(1/2),(-b*c/a/d)^(1/2))/b^4/d^(7/2)/(a*c+(-a*d+b*c)*x^2-b* 
d*x^4)^(1/2)+1/3465*a*c^(1/2)*(a*d+b*c)*(48*a^4*C*d^4+8*a^3*b*d^3*(-11*B*d 
+12*C*c)-3*a*b^3*c*d*(-297*A*d^2+33*B*c*d+13*C*c^2)+3*a^2*b^2*d^2*(66*A*d^ 
2-77*B*c*d+3*C*c^2)-b^4*c^2*(99*A*d^2+44*B*c*d+24*C*c^2))*(1+b*x^2/a)^(1/2 
)*(1-d*x^2/c)^(1/2)*EllipticF(d^(1/2)*x/c^(1/2),(-b*c/a/d)^(1/2))/b^4/d^(7 
/2)/(a*c+(-a*d+b*c)*x^2-b*d*x^4)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 12.62 (sec) , antiderivative size = 707, normalized size of antiderivative = 0.86 \[ \int \left (A+B x^2+C x^4\right ) \left (a c+(b c-a d) x^2-b d x^4\right )^{3/2} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (-c+d x^2\right ) \left (24 a^4 C d^4+a^3 b d^3 \left (57 c C-2 d \left (22 B+9 C x^2\right )\right )+3 a^2 b^2 d^2 \left (6 c^2 C-2 c d \left (22 B+7 C x^2\right )+d^2 \left (33 A+11 B x^2+5 C x^4\right )\right )+b^4 \left (24 c^4 C+2 c^3 d \left (22 B+9 C x^2\right )+3 c^2 d^2 \left (33 A+11 B x^2+5 C x^4\right )+5 d^4 x^4 \left (99 A+77 B x^2+63 C x^4\right )-2 c d^3 x^2 \left (396 A+275 B x^2+210 C x^4\right )\right )+a b^3 d \left (57 c^3 C+6 c^2 d \left (22 B+7 C x^2\right )+2 d^3 x^2 \left (396 A+275 B x^2+210 C x^4\right )-c d^2 \left (1683 A+913 B x^2+615 C x^4\right )\right )\right )-i c \left (48 a^5 C d^5+8 a^4 b d^4 (15 c C-11 B d)-6 a^2 b^3 c d^2 \left (8 c^2 C+33 B c d-165 A d^2\right )-2 b^5 c^3 \left (24 c^2 C+44 B c d+99 A d^2\right )+a^3 b^2 d^3 \left (48 c^2 C-275 B c d+198 A d^2\right )-5 a b^4 c^2 d \left (24 c^2 C+55 B c d+198 A d^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )-i c (b c+a d) \left (-24 a^4 C d^4+a^3 b d^3 (-39 c C+44 B d)+9 a^2 b^2 d^2 \left (c^2 C+11 B c d-11 A d^2\right )+2 b^4 c^2 \left (24 c^2 C+44 B c d+99 A d^2\right )+3 a b^3 c d \left (32 c^2 C+77 B c d+297 A d^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )}{3465 b^3 \sqrt {\frac {b}{a}} d^4 \sqrt {\left (a+b x^2\right ) \left (c-d x^2\right )}} \] Input:

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

Output:

(Sqrt[b/a]*d*x*(a + b*x^2)*(-c + d*x^2)*(24*a^4*C*d^4 + a^3*b*d^3*(57*c*C 
- 2*d*(22*B + 9*C*x^2)) + 3*a^2*b^2*d^2*(6*c^2*C - 2*c*d*(22*B + 7*C*x^2) 
+ d^2*(33*A + 11*B*x^2 + 5*C*x^4)) + b^4*(24*c^4*C + 2*c^3*d*(22*B + 9*C*x 
^2) + 3*c^2*d^2*(33*A + 11*B*x^2 + 5*C*x^4) + 5*d^4*x^4*(99*A + 77*B*x^2 + 
 63*C*x^4) - 2*c*d^3*x^2*(396*A + 275*B*x^2 + 210*C*x^4)) + a*b^3*d*(57*c^ 
3*C + 6*c^2*d*(22*B + 7*C*x^2) + 2*d^3*x^2*(396*A + 275*B*x^2 + 210*C*x^4) 
 - c*d^2*(1683*A + 913*B*x^2 + 615*C*x^4))) - I*c*(48*a^5*C*d^5 + 8*a^4*b* 
d^4*(15*c*C - 11*B*d) - 6*a^2*b^3*c*d^2*(8*c^2*C + 33*B*c*d - 165*A*d^2) - 
 2*b^5*c^3*(24*c^2*C + 44*B*c*d + 99*A*d^2) + a^3*b^2*d^3*(48*c^2*C - 275* 
B*c*d + 198*A*d^2) - 5*a*b^4*c^2*d*(24*c^2*C + 55*B*c*d + 198*A*d^2))*Sqrt 
[1 + (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], -((a 
*d)/(b*c))] - I*c*(b*c + a*d)*(-24*a^4*C*d^4 + a^3*b*d^3*(-39*c*C + 44*B*d 
) + 9*a^2*b^2*d^2*(c^2*C + 11*B*c*d - 11*A*d^2) + 2*b^4*c^2*(24*c^2*C + 44 
*B*c*d + 99*A*d^2) + 3*a*b^3*c*d*(32*c^2*C + 77*B*c*d + 297*A*d^2))*Sqrt[1 
 + (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], -((a*d 
)/(b*c))])/(3465*b^3*Sqrt[b/a]*d^4*Sqrt[(a + b*x^2)*(c - d*x^2)])
 

Rubi [A] (verified)

Time = 1.41 (sec) , antiderivative size = 793, normalized size of antiderivative = 0.97, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2207, 25, 1490, 25, 1490, 25, 1514, 399, 321, 327}

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.

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

\(\Big \downarrow \) 2207

\(\displaystyle -\frac {\int -\left (\left ((6 b c C-6 a d C+11 b B d) x^2+a c C+11 A b d\right ) \left (-b d x^4+(b c-a d) x^2+a c\right )^{3/2}\right )dx}{11 b d}-\frac {C x \left (x^2 (b c-a d)+a c-b d x^4\right )^{5/2}}{11 b d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \left ((6 b c C-6 a d C+11 b B d) x^2+a c C+11 A b d\right ) \left (-b d x^4+(b c-a d) x^2+a c\right )^{3/2}dx}{11 b d}-\frac {C x \left (x^2 (b c-a d)+a c-b d x^4\right )^{5/2}}{11 b d}\)

\(\Big \downarrow \) 1490

\(\displaystyle \frac {\frac {x \left (x^2 (b c-a d)+a c-b d x^4\right )^{3/2} \left (3 (3 b d (a c C+11 A b d)-(b c-a d) (-6 a C d+11 b B d+6 b c C))+7 b d x^2 (-6 a C d+11 b B d+6 b c C)\right )}{63 b d}-\frac {\int -\left (\left (\left (4 (6 b c C-6 a d C+11 b B d) (b c-a d)^2+9 b d (a c C+11 A b d) (b c-a d)+14 a b c d (6 b c C-6 a d C+11 b B d)\right ) x^2+a c (18 b d (a c C+11 A b d)+(b c-a d) (6 b c C-6 a d C+11 b B d))\right ) \sqrt {-b d x^4+(b c-a d) x^2+a c}\right )dx}{21 b d}}{11 b d}-\frac {C x \left (x^2 (b c-a d)+a c-b d x^4\right )^{5/2}}{11 b d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \left (\left (4 (6 b c C-6 a d C+11 b B d) (b c-a d)^2+9 b d (a c C+11 A b d) (b c-a d)+14 a b c d (6 b c C-6 a d C+11 b B d)\right ) x^2+a c (18 b d (a c C+11 A b d)+(b c-a d) (6 b c C-6 a d C+11 b B d))\right ) \sqrt {-b d x^4+(b c-a d) x^2+a c}dx}{21 b d}+\frac {x \left (x^2 (b c-a d)+a c-b d x^4\right )^{3/2} \left (3 (3 b d (a c C+11 A b d)-(b c-a d) (-6 a C d+11 b B d+6 b c C))+7 b d x^2 (-6 a C d+11 b B d+6 b c C)\right )}{63 b d}}{11 b d}-\frac {C x \left (x^2 (b c-a d)+a c-b d x^4\right )^{5/2}}{11 b d}\)

\(\Big \downarrow \) 1490

\(\displaystyle \frac {\frac {-\frac {\int -\frac {a c \left (c^2 \left (24 C c^2+44 B d c+99 A d^2\right ) b^4+3 a c d \left (19 C c^2+44 B d c+594 A d^2\right ) b^3+3 a^2 d^2 \left (6 C c^2-44 B d c+33 A d^2\right ) b^2+a^3 d^3 (57 c C-44 B d) b+24 a^4 C d^4\right )-\left (-2 c^3 \left (24 C c^2+44 B d c+99 A d^2\right ) b^5-5 a c^2 d \left (24 C c^2+55 B d c+198 A d^2\right ) b^4-6 a^2 c d^2 \left (8 C c^2+33 B d c-165 A d^2\right ) b^3+a^3 d^3 \left (48 C c^2-275 B d c+198 A d^2\right ) b^2+8 a^4 d^4 (15 c C-11 B d) b+48 a^5 C d^5\right ) x^2}{\sqrt {-b d x^4+(b c-a d) x^2+a c}}dx}{15 b d}-\frac {x \sqrt {x^2 (b c-a d)+a c-b d x^4} \left (24 a^4 C d^4-11 a^3 b d^3 (4 B d+3 c C)-3 a^2 b^2 d^2 \left (-33 A d^2-11 B c d+24 c^2 C\right )-33 a b^3 c d \left (36 A d^2+B c d+c^2 C\right )-3 b d x^2 \left (9 b d (b c-a d) (a c C+11 A b d)+4 (b c-a d)^2 (-6 a C d+11 b B d+6 b c C)+14 a b c d (-6 a C d+11 b B d+6 b c C)\right )+b^4 c^2 \left (99 A d^2+44 B c d+24 c^2 C\right )\right )}{15 b d}}{21 b d}+\frac {x \left (x^2 (b c-a d)+a c-b d x^4\right )^{3/2} \left (3 (3 b d (a c C+11 A b d)-(b c-a d) (-6 a C d+11 b B d+6 b c C))+7 b d x^2 (-6 a C d+11 b B d+6 b c C)\right )}{63 b d}}{11 b d}-\frac {C x \left (x^2 (b c-a d)+a c-b d x^4\right )^{5/2}}{11 b d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {\int \frac {a c \left (c^2 \left (24 C c^2+44 B d c+99 A d^2\right ) b^4+3 a c d \left (19 C c^2+44 B d c+594 A d^2\right ) b^3+3 a^2 d^2 \left (6 C c^2-44 B d c+33 A d^2\right ) b^2+a^3 d^3 (57 c C-44 B d) b+24 a^4 C d^4\right )-\left (-2 c^3 \left (24 C c^2+44 B d c+99 A d^2\right ) b^5-5 a c^2 d \left (24 C c^2+55 B d c+198 A d^2\right ) b^4-6 a^2 c d^2 \left (8 C c^2+33 B d c-165 A d^2\right ) b^3+a^3 d^3 \left (48 C c^2-275 B d c+198 A d^2\right ) b^2+8 a^4 d^4 (15 c C-11 B d) b+48 a^5 C d^5\right ) x^2}{\sqrt {-b d x^4+(b c-a d) x^2+a c}}dx}{15 b d}-\frac {x \sqrt {x^2 (b c-a d)+a c-b d x^4} \left (24 a^4 C d^4-11 a^3 b d^3 (4 B d+3 c C)-3 a^2 b^2 d^2 \left (-33 A d^2-11 B c d+24 c^2 C\right )-33 a b^3 c d \left (36 A d^2+B c d+c^2 C\right )-3 b d x^2 \left (9 b d (b c-a d) (a c C+11 A b d)+4 (b c-a d)^2 (-6 a C d+11 b B d+6 b c C)+14 a b c d (-6 a C d+11 b B d+6 b c C)\right )+b^4 c^2 \left (99 A d^2+44 B c d+24 c^2 C\right )\right )}{15 b d}}{21 b d}+\frac {x \left (x^2 (b c-a d)+a c-b d x^4\right )^{3/2} \left (3 (3 b d (a c C+11 A b d)-(b c-a d) (-6 a C d+11 b B d+6 b c C))+7 b d x^2 (-6 a C d+11 b B d+6 b c C)\right )}{63 b d}}{11 b d}-\frac {C x \left (x^2 (b c-a d)+a c-b d x^4\right )^{5/2}}{11 b d}\)

\(\Big \downarrow \) 1514

\(\displaystyle \frac {\frac {\frac {\sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} \int \frac {a c \left (c^2 \left (24 C c^2+44 B d c+99 A d^2\right ) b^4+3 a c d \left (19 C c^2+44 B d c+594 A d^2\right ) b^3+3 a^2 d^2 \left (6 C c^2-44 B d c+33 A d^2\right ) b^2+a^3 d^3 (57 c C-44 B d) b+24 a^4 C d^4\right )-\left (-2 c^3 \left (24 C c^2+44 B d c+99 A d^2\right ) b^5-5 a c^2 d \left (24 C c^2+55 B d c+198 A d^2\right ) b^4-6 a^2 c d^2 \left (8 C c^2+33 B d c-165 A d^2\right ) b^3+a^3 d^3 \left (48 C c^2-275 B d c+198 A d^2\right ) b^2+8 a^4 d^4 (15 c C-11 B d) b+48 a^5 C d^5\right ) x^2}{\sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}}}dx}{15 b d \sqrt {x^2 (b c-a d)+a c-b d x^4}}-\frac {x \sqrt {x^2 (b c-a d)+a c-b d x^4} \left (24 a^4 C d^4-11 a^3 b d^3 (4 B d+3 c C)-3 a^2 b^2 d^2 \left (-33 A d^2-11 B c d+24 c^2 C\right )-33 a b^3 c d \left (36 A d^2+B c d+c^2 C\right )-3 b d x^2 \left (9 b d (b c-a d) (a c C+11 A b d)+4 (b c-a d)^2 (-6 a C d+11 b B d+6 b c C)+14 a b c d (-6 a C d+11 b B d+6 b c C)\right )+b^4 c^2 \left (99 A d^2+44 B c d+24 c^2 C\right )\right )}{15 b d}}{21 b d}+\frac {x \left (x^2 (b c-a d)+a c-b d x^4\right )^{3/2} \left (3 (3 b d (a c C+11 A b d)-(b c-a d) (-6 a C d+11 b B d+6 b c C))+7 b d x^2 (-6 a C d+11 b B d+6 b c C)\right )}{63 b d}}{11 b d}-\frac {C x \left (x^2 (b c-a d)+a c-b d x^4\right )^{5/2}}{11 b d}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {\frac {\frac {\sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} \left (\frac {a (a d+b c) \left (48 a^4 C d^4+8 a^3 b d^3 (12 c C-11 B d)+3 a^2 b^2 d^2 \left (66 A d^2-77 B c d+3 c^2 C\right )-3 a b^3 c d \left (-297 A d^2+33 B c d+13 c^2 C\right )+b^4 \left (-c^2\right ) \left (99 A d^2+44 B c d+24 c^2 C\right )\right ) \int \frac {1}{\sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}}}dx}{b}-\frac {a \left (48 a^5 C d^5+8 a^4 b d^4 (15 c C-11 B d)+a^3 b^2 d^3 \left (198 A d^2-275 B c d+48 c^2 C\right )-6 a^2 b^3 c d^2 \left (-165 A d^2+33 B c d+8 c^2 C\right )-5 a b^4 c^2 d \left (198 A d^2+55 B c d+24 c^2 C\right )-2 b^5 c^3 \left (99 A d^2+44 B c d+24 c^2 C\right )\right ) \int \frac {\sqrt {\frac {b x^2}{a}+1}}{\sqrt {1-\frac {d x^2}{c}}}dx}{b}\right )}{15 b d \sqrt {x^2 (b c-a d)+a c-b d x^4}}-\frac {x \sqrt {x^2 (b c-a d)+a c-b d x^4} \left (24 a^4 C d^4-11 a^3 b d^3 (4 B d+3 c C)-3 a^2 b^2 d^2 \left (-33 A d^2-11 B c d+24 c^2 C\right )-33 a b^3 c d \left (36 A d^2+B c d+c^2 C\right )-3 b d x^2 \left (9 b d (b c-a d) (a c C+11 A b d)+4 (b c-a d)^2 (-6 a C d+11 b B d+6 b c C)+14 a b c d (-6 a C d+11 b B d+6 b c C)\right )+b^4 c^2 \left (99 A d^2+44 B c d+24 c^2 C\right )\right )}{15 b d}}{21 b d}+\frac {x \left (x^2 (b c-a d)+a c-b d x^4\right )^{3/2} \left (3 (3 b d (a c C+11 A b d)-(b c-a d) (-6 a C d+11 b B d+6 b c C))+7 b d x^2 (-6 a C d+11 b B d+6 b c C)\right )}{63 b d}}{11 b d}-\frac {C x \left (x^2 (b c-a d)+a c-b d x^4\right )^{5/2}}{11 b d}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {\frac {\frac {\sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} \left (\frac {a \sqrt {c} (a d+b c) \left (48 a^4 C d^4+8 a^3 b d^3 (12 c C-11 B d)+3 a^2 b^2 d^2 \left (66 A d^2-77 B c d+3 c^2 C\right )-3 a b^3 c d \left (-297 A d^2+33 B c d+13 c^2 C\right )+b^4 \left (-c^2\right ) \left (99 A d^2+44 B c d+24 c^2 C\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{b \sqrt {d}}-\frac {a \left (48 a^5 C d^5+8 a^4 b d^4 (15 c C-11 B d)+a^3 b^2 d^3 \left (198 A d^2-275 B c d+48 c^2 C\right )-6 a^2 b^3 c d^2 \left (-165 A d^2+33 B c d+8 c^2 C\right )-5 a b^4 c^2 d \left (198 A d^2+55 B c d+24 c^2 C\right )-2 b^5 c^3 \left (99 A d^2+44 B c d+24 c^2 C\right )\right ) \int \frac {\sqrt {\frac {b x^2}{a}+1}}{\sqrt {1-\frac {d x^2}{c}}}dx}{b}\right )}{15 b d \sqrt {x^2 (b c-a d)+a c-b d x^4}}-\frac {x \sqrt {x^2 (b c-a d)+a c-b d x^4} \left (24 a^4 C d^4-11 a^3 b d^3 (4 B d+3 c C)-3 a^2 b^2 d^2 \left (-33 A d^2-11 B c d+24 c^2 C\right )-33 a b^3 c d \left (36 A d^2+B c d+c^2 C\right )-3 b d x^2 \left (9 b d (b c-a d) (a c C+11 A b d)+4 (b c-a d)^2 (-6 a C d+11 b B d+6 b c C)+14 a b c d (-6 a C d+11 b B d+6 b c C)\right )+b^4 c^2 \left (99 A d^2+44 B c d+24 c^2 C\right )\right )}{15 b d}}{21 b d}+\frac {x \left (x^2 (b c-a d)+a c-b d x^4\right )^{3/2} \left (3 (3 b d (a c C+11 A b d)-(b c-a d) (-6 a C d+11 b B d+6 b c C))+7 b d x^2 (-6 a C d+11 b B d+6 b c C)\right )}{63 b d}}{11 b d}-\frac {C x \left (x^2 (b c-a d)+a c-b d x^4\right )^{5/2}}{11 b d}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\frac {\frac {\sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} \left (\frac {a \sqrt {c} (a d+b c) \left (48 a^4 C d^4+8 a^3 b d^3 (12 c C-11 B d)+3 a^2 b^2 d^2 \left (66 A d^2-77 B c d+3 c^2 C\right )-3 a b^3 c d \left (-297 A d^2+33 B c d+13 c^2 C\right )+b^4 \left (-c^2\right ) \left (99 A d^2+44 B c d+24 c^2 C\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{b \sqrt {d}}-\frac {a \sqrt {c} \left (48 a^5 C d^5+8 a^4 b d^4 (15 c C-11 B d)+a^3 b^2 d^3 \left (198 A d^2-275 B c d+48 c^2 C\right )-6 a^2 b^3 c d^2 \left (-165 A d^2+33 B c d+8 c^2 C\right )-5 a b^4 c^2 d \left (198 A d^2+55 B c d+24 c^2 C\right )-2 b^5 c^3 \left (99 A d^2+44 B c d+24 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{b \sqrt {d}}\right )}{15 b d \sqrt {x^2 (b c-a d)+a c-b d x^4}}-\frac {x \sqrt {x^2 (b c-a d)+a c-b d x^4} \left (24 a^4 C d^4-11 a^3 b d^3 (4 B d+3 c C)-3 a^2 b^2 d^2 \left (-33 A d^2-11 B c d+24 c^2 C\right )-33 a b^3 c d \left (36 A d^2+B c d+c^2 C\right )-3 b d x^2 \left (9 b d (b c-a d) (a c C+11 A b d)+4 (b c-a d)^2 (-6 a C d+11 b B d+6 b c C)+14 a b c d (-6 a C d+11 b B d+6 b c C)\right )+b^4 c^2 \left (99 A d^2+44 B c d+24 c^2 C\right )\right )}{15 b d}}{21 b d}+\frac {x \left (x^2 (b c-a d)+a c-b d x^4\right )^{3/2} \left (3 (3 b d (a c C+11 A b d)-(b c-a d) (-6 a C d+11 b B d+6 b c C))+7 b d x^2 (-6 a C d+11 b B d+6 b c C)\right )}{63 b d}}{11 b d}-\frac {C x \left (x^2 (b c-a d)+a c-b d x^4\right )^{5/2}}{11 b d}\)

Input:

Int[(A + B*x^2 + C*x^4)*(a*c + (b*c - a*d)*x^2 - b*d*x^4)^(3/2),x]
 

Output:

-1/11*(C*x*(a*c + (b*c - a*d)*x^2 - b*d*x^4)^(5/2))/(b*d) + ((x*(3*(3*b*d* 
(a*c*C + 11*A*b*d) - (b*c - a*d)*(6*b*c*C + 11*b*B*d - 6*a*C*d)) + 7*b*d*( 
6*b*c*C + 11*b*B*d - 6*a*C*d)*x^2)*(a*c + (b*c - a*d)*x^2 - b*d*x^4)^(3/2) 
)/(63*b*d) + (-1/15*(x*(24*a^4*C*d^4 - 11*a^3*b*d^3*(3*c*C + 4*B*d) - 3*a^ 
2*b^2*d^2*(24*c^2*C - 11*B*c*d - 33*A*d^2) - 33*a*b^3*c*d*(c^2*C + B*c*d + 
 36*A*d^2) + b^4*c^2*(24*c^2*C + 44*B*c*d + 99*A*d^2) - 3*b*d*(9*b*d*(b*c 
- a*d)*(a*c*C + 11*A*b*d) + 14*a*b*c*d*(6*b*c*C + 11*b*B*d - 6*a*C*d) + 4* 
(b*c - a*d)^2*(6*b*c*C + 11*b*B*d - 6*a*C*d))*x^2)*Sqrt[a*c + (b*c - a*d)* 
x^2 - b*d*x^4])/(b*d) + (Sqrt[1 + (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*(-((a*Sqr 
t[c]*(48*a^5*C*d^5 + 8*a^4*b*d^4*(15*c*C - 11*B*d) - 6*a^2*b^3*c*d^2*(8*c^ 
2*C + 33*B*c*d - 165*A*d^2) - 2*b^5*c^3*(24*c^2*C + 44*B*c*d + 99*A*d^2) + 
 a^3*b^2*d^3*(48*c^2*C - 275*B*c*d + 198*A*d^2) - 5*a*b^4*c^2*d*(24*c^2*C 
+ 55*B*c*d + 198*A*d^2))*EllipticE[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((b*c)/(a 
*d))])/(b*Sqrt[d])) + (a*Sqrt[c]*(b*c + a*d)*(48*a^4*C*d^4 + 8*a^3*b*d^3*( 
12*c*C - 11*B*d) - 3*a*b^3*c*d*(13*c^2*C + 33*B*c*d - 297*A*d^2) + 3*a^2*b 
^2*d^2*(3*c^2*C - 77*B*c*d + 66*A*d^2) - b^4*c^2*(24*c^2*C + 44*B*c*d + 99 
*A*d^2))*EllipticF[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((b*c)/(a*d))])/(b*Sqrt[d 
])))/(15*b*d*Sqrt[a*c + (b*c - a*d)*x^2 - b*d*x^4]))/(21*b*d))/(11*b*d)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 399

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) 
^2]), x_Symbol] :> Simp[f/b   Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + 
 Simp[(b*e - a*f)/b   Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr 
eeQ[{a, b, c, d, e, f}, x] &&  !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && 
(PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
 

rule 1490

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb 
ol] :> Simp[x*(2*b*e*p + c*d*(4*p + 3) + c*e*(4*p + 1)*x^2)*((a + b*x^2 + c 
*x^4)^p/(c*(4*p + 1)*(4*p + 3))), x] + Simp[2*(p/(c*(4*p + 1)*(4*p + 3))) 
 Int[Simp[2*a*c*d*(4*p + 3) - a*b*e + (2*a*c*e*(4*p + 1) + b*c*d*(4*p + 3) 
- b^2*e*(2*p + 1))*x^2, x]*(a + b*x^2 + c*x^4)^(p - 1), x], x] /; FreeQ[{a, 
 b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && 
 GtQ[p, 0] && FractionQ[p] && IntegerQ[2*p]
 

rule 1514

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[Sqrt[1 + 2*c*(x^2/(b - q))]*(Sqrt 
[1 + 2*c*(x^2/(b + q))]/Sqrt[a + b*x^2 + c*x^4])   Int[(d + e*x^2)/(Sqrt[1 
+ 2*c*(x^2/(b - q))]*Sqrt[1 + 2*c*(x^2/(b + q))]), x], x]] /; FreeQ[{a, b, 
c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[c/a]
 

rule 2207

Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{n = 
 Expon[Px, x^2], e = Coeff[Px, x^2, Expon[Px, x^2]]}, Simp[e*x^(2*n - 3)*(( 
a + b*x^2 + c*x^4)^(p + 1)/(c*(2*n + 4*p + 1))), x] + Simp[1/(c*(2*n + 4*p 
+ 1))   Int[(a + b*x^2 + c*x^4)^p*ExpandToSum[c*(2*n + 4*p + 1)*Px - a*e*(2 
*n - 3)*x^(2*n - 4) - b*e*(2*n + 2*p - 1)*x^(2*n - 2) - c*e*(2*n + 4*p + 1) 
*x^(2*n), x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Px, x^2] && Expon[ 
Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] &&  !LtQ[p, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1969\) vs. \(2(776)=1552\).

Time = 12.10 (sec) , antiderivative size = 1970, normalized size of antiderivative = 2.40

method	result	size
risch	\(\text {Expression too large to display}\)	\(1970\)
elliptic	\(\text {Expression too large to display}\)	\(2279\)
default	\(\text {Expression too large to display}\)	\(3125\)

Input:

int((C*x^4+B*x^2+A)*(a*c+(-a*d+b*c)*x^2-b*d*x^4)^(3/2),x,method=_RETURNVER 
BOSE)
 

Output:

-1/3465/b^3/d^3*x*(315*C*b^4*d^4*x^8+385*B*b^4*d^4*x^6+420*C*a*b^3*d^4*x^6 
-420*C*b^4*c*d^3*x^6+495*A*b^4*d^4*x^4+550*B*a*b^3*d^4*x^4-550*B*b^4*c*d^3 
*x^4+15*C*a^2*b^2*d^4*x^4-615*C*a*b^3*c*d^3*x^4+15*C*b^4*c^2*d^2*x^4+792*A 
*a*b^3*d^4*x^2-792*A*b^4*c*d^3*x^2+33*B*a^2*b^2*d^4*x^2-913*B*a*b^3*c*d^3* 
x^2+33*B*b^4*c^2*d^2*x^2-18*C*a^3*b*d^4*x^2-42*C*a^2*b^2*c*d^3*x^2+42*C*a* 
b^3*c^2*d^2*x^2+18*C*b^4*c^3*d*x^2+99*A*a^2*b^2*d^4-1683*A*a*b^3*c*d^3+99* 
A*b^4*c^2*d^2-44*B*a^3*b*d^4-132*B*a^2*b^2*c*d^3+132*B*a*b^3*c^2*d^2+44*B* 
b^4*c^3*d+24*C*a^4*d^4+57*C*a^3*b*c*d^3+18*C*a^2*b^2*c^2*d^2+57*C*a*b^3*c^ 
3*d+24*C*b^4*c^4)*(b*x^2+a)*(-d*x^2+c)/(-(b*x^2+a)*(d*x^2-c))^(1/2)+1/3465 
/b^3/d^3*((198*A*a^3*b^2*d^5+990*A*a^2*b^3*c*d^4-990*A*a*b^4*c^2*d^3-198*A 
*b^5*c^3*d^2-88*B*a^4*b*d^5-275*B*a^3*b^2*c*d^4-198*B*a^2*b^3*c^2*d^3-275* 
B*a*b^4*c^3*d^2-88*B*b^5*c^4*d+48*C*a^5*d^5+120*C*a^4*b*c*d^4+48*C*a^3*b^2 
*c^2*d^3-48*C*a^2*b^3*c^3*d^2-120*C*a*b^4*c^4*d-48*C*b^5*c^5)*a/(d/c)^(1/2 
)*(1-d*x^2/c)^(1/2)*(1+b*x^2/a)^(1/2)/(-b*d*x^4-a*d*x^2+b*c*x^2+a*c)^(1/2) 
/b*(EllipticF(x*(d/c)^(1/2),(-1-(-a*d+b*c)/a/d)^(1/2))-EllipticE(x*(d/c)^( 
1/2),(-1-(-a*d+b*c)/a/d)^(1/2)))+24*C*a*b^4*c^5/(d/c)^(1/2)*(1-d*x^2/c)^(1 
/2)*(1+b*x^2/a)^(1/2)/(-b*d*x^4-a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x*(d/ 
c)^(1/2),(-1-(-a*d+b*c)/a/d)^(1/2))+24*C*a^5*c*d^4/(d/c)^(1/2)*(1-d*x^2/c) 
^(1/2)*(1+b*x^2/a)^(1/2)/(-b*d*x^4-a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x* 
(d/c)^(1/2),(-1-(-a*d+b*c)/a/d)^(1/2))+99*A*a*b^4*c^3*d^2/(d/c)^(1/2)*(...
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 1012, normalized size of antiderivative = 1.23 \[ \int \left (A+B x^2+C x^4\right ) \left (a c+(b c-a d) x^2-b d x^4\right )^{3/2} \, dx =\text {Too large to display} \] Input:

integrate((C*x^4+B*x^2+A)*(a*c+(-a*d+b*c)*x^2-b*d*x^4)^(3/2),x, algorithm= 
"fricas")
 

Output:

-1/3465*((48*C*b^5*c^6 + 8*(15*C*a*b^4 + 11*B*b^5)*c^5*d + (48*C*a^2*b^3 + 
 275*B*a*b^4 + 198*A*b^5)*c^4*d^2 - 6*(8*C*a^3*b^2 - 33*B*a^2*b^3 - 165*A* 
a*b^4)*c^3*d^3 - 5*(24*C*a^4*b - 55*B*a^3*b^2 + 198*A*a^2*b^3)*c^2*d^4 - 2 
*(24*C*a^5 - 44*B*a^4*b + 99*A*a^3*b^2)*c*d^5)*sqrt(-b*d)*x*sqrt(c/d)*elli 
ptic_e(arcsin(sqrt(c/d)/x), -a*d/(b*c)) - (48*C*b^5*c^6 + 8*(15*C*a*b^4 + 
11*B*b^5)*c^5*d + (48*C*a^2*b^3 + (275*B + 24*C)*a*b^4 + 198*A*b^5)*c^4*d^ 
2 - (48*C*a^3*b^2 - 3*(66*B + 19*C)*a^2*b^3 - 22*(45*A + 2*B)*a*b^4)*c^3*d 
^3 - (120*C*a^4*b - (275*B + 18*C)*a^3*b^2 + 66*(15*A - 2*B)*a^2*b^3 - 99* 
A*a*b^4)*c^2*d^4 - (48*C*a^5 - (88*B + 57*C)*a^4*b + 66*(3*A + 2*B)*a^3*b^ 
2 - 1782*A*a^2*b^3)*c*d^5 + (24*C*a^5 - 44*B*a^4*b + 99*A*a^3*b^2)*d^6)*sq 
rt(-b*d)*x*sqrt(c/d)*elliptic_f(arcsin(sqrt(c/d)/x), -a*d/(b*c)) + (315*C* 
b^5*d^6*x^10 + 48*C*b^5*c^5*d - 35*(12*C*b^5*c*d^5 - (12*C*a*b^4 + 11*B*b^ 
5)*d^6)*x^8 + 8*(15*C*a*b^4 + 11*B*b^5)*c^4*d^2 + (48*C*a^2*b^3 + 275*B*a* 
b^4 + 198*A*b^5)*c^3*d^3 - 6*(8*C*a^3*b^2 - 33*B*a^2*b^3 - 165*A*a*b^4)*c^ 
2*d^4 - 5*(24*C*a^4*b - 55*B*a^3*b^2 + 198*A*a^2*b^3)*c*d^5 - 2*(24*C*a^5 
- 44*B*a^4*b + 99*A*a^3*b^2)*d^6 + 5*(3*C*b^5*c^2*d^4 - (123*C*a*b^4 + 110 
*B*b^5)*c*d^5 + (3*C*a^2*b^3 + 110*B*a*b^4 + 99*A*b^5)*d^6)*x^6 + (18*C*b^ 
5*c^3*d^3 + 3*(14*C*a*b^4 + 11*B*b^5)*c^2*d^4 - (42*C*a^2*b^3 + 913*B*a*b^ 
4 + 792*A*b^5)*c*d^5 - 3*(6*C*a^3*b^2 - 11*B*a^2*b^3 - 264*A*a*b^4)*d^6)*x 
^4 + (24*C*b^5*c^4*d^2 + (57*C*a*b^4 + 44*B*b^5)*c^3*d^3 + 3*(6*C*a^2*b...
 

Sympy [F]

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

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

Output:

Integral((-(a + b*x**2)*(-c + d*x**2))**(3/2)*(A + B*x**2 + C*x**4), x)
 

Maxima [F]

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

integrate((C*x^4+B*x^2+A)*(a*c+(-a*d+b*c)*x^2-b*d*x^4)^(3/2),x, algorithm= 
"maxima")
 

Output:

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

Giac [F]

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

integrate((C*x^4+B*x^2+A)*(a*c+(-a*d+b*c)*x^2-b*d*x^4)^(3/2),x, algorithm= 
"giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 24*sqrt(c - d*x**2)*sqrt(a + b*x**2)*a**4*c*d**4*x - 55*sqrt(c - d*x** 
2)*sqrt(a + b*x**2)*a**3*b**2*d**4*x - 57*sqrt(c - d*x**2)*sqrt(a + b*x**2 
)*a**3*b*c**2*d**3*x + 18*sqrt(c - d*x**2)*sqrt(a + b*x**2)*a**3*b*c*d**4* 
x**3 + 1815*sqrt(c - d*x**2)*sqrt(a + b*x**2)*a**2*b**3*c*d**3*x - 825*sqr 
t(c - d*x**2)*sqrt(a + b*x**2)*a**2*b**3*d**4*x**3 - 18*sqrt(c - d*x**2)*s 
qrt(a + b*x**2)*a**2*b**2*c**3*d**2*x + 42*sqrt(c - d*x**2)*sqrt(a + b*x** 
2)*a**2*b**2*c**2*d**3*x**3 - 15*sqrt(c - d*x**2)*sqrt(a + b*x**2)*a**2*b* 
*2*c*d**4*x**5 - 231*sqrt(c - d*x**2)*sqrt(a + b*x**2)*a*b**4*c**2*d**2*x 
+ 1705*sqrt(c - d*x**2)*sqrt(a + b*x**2)*a*b**4*c*d**3*x**3 - 1045*sqrt(c 
- d*x**2)*sqrt(a + b*x**2)*a*b**4*d**4*x**5 - 57*sqrt(c - d*x**2)*sqrt(a + 
 b*x**2)*a*b**3*c**4*d*x - 42*sqrt(c - d*x**2)*sqrt(a + b*x**2)*a*b**3*c** 
3*d**2*x**3 + 615*sqrt(c - d*x**2)*sqrt(a + b*x**2)*a*b**3*c**2*d**3*x**5 
- 420*sqrt(c - d*x**2)*sqrt(a + b*x**2)*a*b**3*c*d**4*x**7 - 44*sqrt(c - d 
*x**2)*sqrt(a + b*x**2)*b**5*c**3*d*x - 33*sqrt(c - d*x**2)*sqrt(a + b*x** 
2)*b**5*c**2*d**2*x**3 + 550*sqrt(c - d*x**2)*sqrt(a + b*x**2)*b**5*c*d**3 
*x**5 - 385*sqrt(c - d*x**2)*sqrt(a + b*x**2)*b**5*d**4*x**7 - 24*sqrt(c - 
 d*x**2)*sqrt(a + b*x**2)*b**4*c**5*x - 18*sqrt(c - d*x**2)*sqrt(a + b*x** 
2)*b**4*c**4*d*x**3 - 15*sqrt(c - d*x**2)*sqrt(a + b*x**2)*b**4*c**3*d**2* 
x**5 + 420*sqrt(c - d*x**2)*sqrt(a + b*x**2)*b**4*c**2*d**3*x**7 - 315*sqr 
t(c - d*x**2)*sqrt(a + b*x**2)*b**4*c*d**4*x**9 - 48*int((sqrt(c - d*x*...