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

Optimal result

Integrand size = 38, antiderivative size = 993 \[ \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 {\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 )-5 a b^4 c^2 d \left (24 c^2 C-55 B c d+198 A d^2\right )+a^3 b^2 d^3 \left (48 c^2 C+275 B c d+198 A d^2\right )\right ) x \left (c+d x^2\right )}{3465 b^3 d^4 \sqrt {a c+(b c+a d) x^2+b d x^4}}+\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 {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 )-5 a b^4 c^2 d \left (24 c^2 C-55 B c d+198 A d^2\right )+a^3 b^2 d^3 \left (48 c^2 C+275 B c d+198 A d^2\right )\right ) \left (a+b x^2\right ) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3465 \sqrt {a} b^{7/2} d^4 \sqrt {a c+(b c+a d) x^2+b d x^4}}-\frac {\sqrt {a} c \left (24 a^4 C d^4-a^3 b d^3 (57 c C+44 B d)+3 a^2 b^2 d^2 \left (6 c^2 C+44 B c d+33 A d^2\right )+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 (19 c^2 C-44 B c d+594 A d^2\right )\right ) \left (a+b x^2\right ) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{3465 b^{7/2} d^3 \sqrt {a c+(b c+a d) x^2+b d x^4}} \] Output:

-1/3465*(48*a^5*C*d^5-8*a^4*b*d^4*(11*B*d+15*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^2-44*B*c*d+24*C*c^2)-5*a*b^4*c^2*d 
*(198*A*d^2-55*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)) 
*x*(d*x^2+c)/b^3/d^4/(a*c+(a*d+b*c)*x^2+b*d*x^4)^(1/2)+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*(-11*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*c*(48*a^5*C*d^5-8*a^4*b*d^4*(11*B*d+15*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^2-44*B*c*d+24*C*c^2)-5*a*b^4*c^ 
2*d*(198*A*d^2-55*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))*(b*x^2+a)*(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)*EllipticE(b^(1/2)*x/a^(1/2)/ 
(1+b*x^2/a)^(1/2),(1-a*d/b/c)^(1/2))/a^(1/2)/b^(7/2)/d^4/(a*c+(a*d+b*c)*x^ 
2+b*d*x^4)^(1/2)-1/3465*a^(1/2)*c*(24*a^4*C*d^4-a^3*b*d^3*(44*B*d+57*C*c)+ 
3*a^2*b^2*d^2*(33*A*d^2+44*B*c*d+6*C*c^2)+b^4*c^2*(99*A*d^2-44*B*c*d+24*C* 
c^2)-3*a*b^3*c*d*(594*A*d^2-44*B*c*d+19*C*c^2))*(b*x^2+a)*(a*(d*x^2+c)/c/( 
b*x^2+a))^(1/2)*InverseJacobiAM(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(...
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 12.36 (sec) , antiderivative size = 701, normalized size of antiderivative = 0.71 \[ \int \left (A+B x^2+C x^4\right ) \left (a c+(b c+a d) x^2+b d x^4\right )^{3/2} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (24 a^4 C d^4-a^3 b d^3 \left (57 c C+44 B d+18 C d x^2\right )+3 a^2 b^2 d^2 \left (6 c^2 C+2 c d \left (22 B+7 C x^2\right )+d^2 \left (33 A+11 B x^2+5 C x^4\right )\right )+b^4 \left (24 c^4 C-2 c^3 d \left (22 B+9 C x^2\right )+3 c^2 d^2 \left (33 A+11 B x^2+5 C x^4\right )+5 d^4 x^4 \left (99 A+77 B x^2+63 C x^4\right )+2 c d^3 x^2 \left (396 A+275 B x^2+210 C x^4\right )\right )+a b^3 d \left (-57 c^3 C+6 c^2 d \left (22 B+7 C x^2\right )+2 d^3 x^2 \left (396 A+275 B x^2+210 C x^4\right )+c d^2 \left (1683 A+913 B x^2+615 C x^4\right )\right )\right )+i c \left (48 a^5 C d^5-8 a^4 b d^4 (15 c C+11 B d)+6 a^2 b^3 c d^2 \left (8 c^2 C-33 B c d-165 A d^2\right )+2 b^5 c^3 \left (24 c^2 C-44 B c d+99 A d^2\right )-5 a b^4 c^2 d \left (24 c^2 C-55 B c d+198 A d^2\right )+a^3 b^2 d^3 \left (48 c^2 C+275 B c d+198 A d^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c (-b c+a d) \left (24 a^4 C d^4-a^3 b d^3 (39 c C+44 B d)+9 a^2 b^2 d^2 \left (-c^2 C+11 B c d+11 A d^2\right )-2 b^4 c^2 \left (24 c^2 C-44 B c d+99 A d^2\right )+3 a b^3 c d \left (32 c^2 C-77 B c d+297 A d^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{3465 b^3 \sqrt {\frac {b}{a}} d^4 \sqrt {\left (a+b x^2\right ) \left (c+d x^2\right )}} \] 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 + 
 44*B*d + 18*C*d*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 + 6 
3*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) - 5*a*b^4*c^2*d*(24*c^2*C - 55* 
B*c*d + 198*A*d^2) + a^3*b^2*d^3*(48*c^2*C + 275*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 = 1.73 (sec) , antiderivative size = 1247, normalized size of antiderivative = 1.26, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {2207, 25, 1490, 1490, 1511, 27, 1416, 1509}

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.

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

\(\Big \downarrow \) 2207

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1490

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

\(\Big \downarrow \) 1490

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

\(\Big \downarrow \) 1511

\(\displaystyle \frac {C x \left (b d x^4+(b c+a d) x^2+a c\right )^{5/2}}{11 b d}-\frac {\frac {x \left (7 b d (6 b c C+6 a d C-11 b B d) x^2+3 (3 b d (a c C-11 A b d)+(b c+a d) (6 b c C+6 a d C-11 b B d))\right ) \left (b d x^4+(b c+a d) x^2+a c\right )^{3/2}}{63 b d}+\frac {\frac {-\frac {\sqrt {a} \sqrt {c} \left (-2 c^2 \left (24 C c^2-44 B d c+99 A d^2\right ) b^4+3 \sqrt {a} c^{3/2} \sqrt {d} \left (24 C c^2-44 B d c+99 A d^2\right ) b^{7/2}+3 a c d \left (8 C c^2-33 B d c+198 A d^2\right ) b^3-9 a^{3/2} \sqrt {c} d^{3/2} \left (7 C c^2-22 B d c-33 A d^2\right ) b^{5/2}+9 a^2 d^2 \left (6 C c^2-11 B d c-22 A d^2\right ) b^2-3 a^{5/2} \sqrt {c} d^{5/2} (21 c C+44 B d) b^{3/2}+8 a^3 d^3 (3 c C+11 B d) b+72 a^{7/2} \sqrt {c} C d^{7/2} \sqrt {b}-48 a^4 C d^4\right ) \int \frac {1}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx \left (\sqrt {b} \sqrt {c}+\sqrt {a} \sqrt {d}\right )^2}{\sqrt {b} \sqrt {d}}-\frac {\sqrt {a} \sqrt {c} \left (2 c^3 \left (24 C c^2-44 B d c+99 A d^2\right ) b^5-5 a c^2 d \left (24 C c^2-55 B d c+198 A d^2\right ) b^4+6 a^2 c d^2 \left (8 C c^2-33 B d c-165 A d^2\right ) b^3+a^3 d^3 \left (48 C c^2+275 B d c+198 A d^2\right ) b^2-8 a^4 d^4 (15 c C+11 B d) b+48 a^5 C d^5\right ) \int \frac {\sqrt {a} \sqrt {c}-\sqrt {b} \sqrt {d} x^2}{\sqrt {a} \sqrt {c} \sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}}{15 b d}-\frac {x \left (c^2 \left (24 C c^2-44 B d c+99 A d^2\right ) b^4+33 a c d \left (C c^2-B d c+36 A d^2\right ) b^3-3 a^2 d^2 \left (24 C c^2+11 B d c-33 A d^2\right ) b^2-3 d \left (-4 (6 b c C+6 a d C-11 b B d) (b c+a d)^2+9 b d (a c C-11 A b d) (b c+a d)+14 a b c d (6 b c C+6 a d C-11 b B d)\right ) x^2 b+11 a^3 d^3 (3 c C-4 B d) b+24 a^4 C d^4\right ) \sqrt {b d x^4+(b c+a d) x^2+a c}}{15 b d}}{21 b d}}{11 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {C x \left (b d x^4+(b c+a d) x^2+a c\right )^{5/2}}{11 b d}-\frac {\frac {x \left (7 b d (6 b c C+6 a d C-11 b B d) x^2+3 (3 b d (a c C-11 A b d)+(b c+a d) (6 b c C+6 a d C-11 b B d))\right ) \left (b d x^4+(b c+a d) x^2+a c\right )^{3/2}}{63 b d}+\frac {\frac {-\frac {\sqrt {a} \sqrt {c} \left (-2 c^2 \left (24 C c^2-44 B d c+99 A d^2\right ) b^4+3 \sqrt {a} c^{3/2} \sqrt {d} \left (24 C c^2-44 B d c+99 A d^2\right ) b^{7/2}+3 a c d \left (8 C c^2-33 B d c+198 A d^2\right ) b^3-9 a^{3/2} \sqrt {c} d^{3/2} \left (7 C c^2-22 B d c-33 A d^2\right ) b^{5/2}+9 a^2 d^2 \left (6 C c^2-11 B d c-22 A d^2\right ) b^2-3 a^{5/2} \sqrt {c} d^{5/2} (21 c C+44 B d) b^{3/2}+8 a^3 d^3 (3 c C+11 B d) b+72 a^{7/2} \sqrt {c} C d^{7/2} \sqrt {b}-48 a^4 C d^4\right ) \int \frac {1}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx \left (\sqrt {b} \sqrt {c}+\sqrt {a} \sqrt {d}\right )^2}{\sqrt {b} \sqrt {d}}-\frac {\left (2 c^3 \left (24 C c^2-44 B d c+99 A d^2\right ) b^5-5 a c^2 d \left (24 C c^2-55 B d c+198 A d^2\right ) b^4+6 a^2 c d^2 \left (8 C c^2-33 B d c-165 A d^2\right ) b^3+a^3 d^3 \left (48 C c^2+275 B d c+198 A d^2\right ) b^2-8 a^4 d^4 (15 c C+11 B d) b+48 a^5 C d^5\right ) \int \frac {\sqrt {a} \sqrt {c}-\sqrt {b} \sqrt {d} x^2}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}}{15 b d}-\frac {x \left (c^2 \left (24 C c^2-44 B d c+99 A d^2\right ) b^4+33 a c d \left (C c^2-B d c+36 A d^2\right ) b^3-3 a^2 d^2 \left (24 C c^2+11 B d c-33 A d^2\right ) b^2-3 d \left (-4 (6 b c C+6 a d C-11 b B d) (b c+a d)^2+9 b d (a c C-11 A b d) (b c+a d)+14 a b c d (6 b c C+6 a d C-11 b B d)\right ) x^2 b+11 a^3 d^3 (3 c C-4 B d) b+24 a^4 C d^4\right ) \sqrt {b d x^4+(b c+a d) x^2+a c}}{15 b d}}{21 b d}}{11 b d}\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {C x \left (b d x^4+(b c+a d) x^2+a c\right )^{5/2}}{11 b d}-\frac {\frac {x \left (7 b d (6 b c C+6 a d C-11 b B d) x^2+3 (3 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 ) \left (b d x^4+(b c+a d) x^2+a c\right )^{3/2}}{63 b d}+\frac {\frac {-\frac {\sqrt [4]{a} \sqrt [4]{c} \left (-2 c^2 \left (24 C c^2-44 B d c+99 A d^2\right ) b^4+3 \sqrt {a} c^{3/2} \sqrt {d} \left (24 C c^2-44 B d c+99 A d^2\right ) b^{7/2}+3 a c d \left (8 C c^2-33 B d c+198 A d^2\right ) b^3-9 a^{3/2} \sqrt {c} d^{3/2} \left (7 C c^2-22 B d c-33 A d^2\right ) b^{5/2}+9 a^2 d^2 \left (6 C c^2-11 B d c-22 A d^2\right ) b^2-3 a^{5/2} \sqrt {c} d^{5/2} (21 c C+44 B d) b^{3/2}+8 a^3 d^3 (3 c C+11 B d) b+72 a^{7/2} \sqrt {c} C d^{7/2} \sqrt {b}-48 a^4 C d^4\right ) \left (\sqrt {b} \sqrt {d} x^2+\sqrt {a} \sqrt {c}\right ) \sqrt {\frac {b d x^4+(b c+a d) x^2+a c}{\left (\sqrt {b} \sqrt {d} x^2+\sqrt {a} \sqrt {c}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{d} x}{\sqrt [4]{a} \sqrt [4]{c}}\right ),\frac {1}{4} \left (2-\frac {b c+a d}{\sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}}\right )\right ) \left (\sqrt {b} \sqrt {c}+\sqrt {a} \sqrt {d}\right )^2}{2 b^{3/4} d^{3/4} \sqrt {b d x^4+(b c+a d) x^2+a c}}-\frac {\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 ) \int \frac {\sqrt {a} \sqrt {c}-\sqrt {b} \sqrt {d} x^2}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}}{15 b d}-\frac {x \left (c^2 \left (24 C c^2-44 B d c+99 A d^2\right ) b^4+33 a c d \left (C c^2-B d c+36 A d^2\right ) b^3-3 a^2 d^2 \left (24 C c^2+11 B d c-33 A d^2\right ) b^2-3 d \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 b+11 a^3 d^3 (3 c C-4 B d) b+24 a^4 C d^4\right ) \sqrt {b d x^4+(b c+a d) x^2+a c}}{15 b d}}{21 b d}}{11 b d}\)

\(\Big \downarrow \) 1509

\(\displaystyle \frac {C x \left (b d x^4+(b c+a d) x^2+a c\right )^{5/2}}{11 b d}-\frac {\frac {x \left (7 b d (6 b c C+6 a d C-11 b B d) x^2+3 (3 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 ) \left (b d x^4+(b c+a d) x^2+a c\right )^{3/2}}{63 b d}+\frac {\frac {-\frac {\sqrt [4]{a} \sqrt [4]{c} \left (-2 c^2 \left (24 C c^2-44 B d c+99 A d^2\right ) b^4+3 \sqrt {a} c^{3/2} \sqrt {d} \left (24 C c^2-44 B d c+99 A d^2\right ) b^{7/2}+3 a c d \left (8 C c^2-33 B d c+198 A d^2\right ) b^3-9 a^{3/2} \sqrt {c} d^{3/2} \left (7 C c^2-22 B d c-33 A d^2\right ) b^{5/2}+9 a^2 d^2 \left (6 C c^2-11 B d c-22 A d^2\right ) b^2-3 a^{5/2} \sqrt {c} d^{5/2} (21 c C+44 B d) b^{3/2}+8 a^3 d^3 (3 c C+11 B d) b+72 a^{7/2} \sqrt {c} C d^{7/2} \sqrt {b}-48 a^4 C d^4\right ) \left (\sqrt {b} \sqrt {d} x^2+\sqrt {a} \sqrt {c}\right ) \sqrt {\frac {b d x^4+(b c+a d) x^2+a c}{\left (\sqrt {b} \sqrt {d} x^2+\sqrt {a} \sqrt {c}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{d} x}{\sqrt [4]{a} \sqrt [4]{c}}\right ),\frac {1}{4} \left (2-\frac {b c+a d}{\sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}}\right )\right ) \left (\sqrt {b} \sqrt {c}+\sqrt {a} \sqrt {d}\right )^2}{2 b^{3/4} d^{3/4} \sqrt {b d x^4+(b c+a d) x^2+a c}}-\frac {\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 ) \left (\frac {\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {b} \sqrt {d} x^2+\sqrt {a} \sqrt {c}\right ) \sqrt {\frac {b d x^4+(b c+a d) x^2+a c}{\left (\sqrt {b} \sqrt {d} x^2+\sqrt {a} \sqrt {c}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{d} x}{\sqrt [4]{a} \sqrt [4]{c}}\right )|\frac {1}{4} \left (2-\frac {b c+a d}{\sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}}\right )\right )}{\sqrt [4]{b} \sqrt [4]{d} \sqrt {b d x^4+(b c+a d) x^2+a c}}-\frac {x \sqrt {b d x^4+(b c+a d) x^2+a c}}{\sqrt {b} \sqrt {d} x^2+\sqrt {a} \sqrt {c}}\right )}{\sqrt {b} \sqrt {d}}}{15 b d}-\frac {x \left (c^2 \left (24 C c^2-44 B d c+99 A d^2\right ) b^4+33 a c d \left (C c^2-B d c+36 A d^2\right ) b^3-3 a^2 d^2 \left (24 C c^2+11 B d c-33 A d^2\right ) b^2-3 d \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 b+11 a^3 d^3 (3 c C-4 B d) b+24 a^4 C d^4\right ) \sqrt {b d x^4+(b c+a d) x^2+a c}}{15 b d}}{21 b d}}{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:

(C*x*(a*c + (b*c + a*d)*x^2 + b*d*x^4)^(5/2))/(11*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) + (-(((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) - 5*a*b^4*c^2*d*(24*c^2*C - 55*B*c*d + 198*A*d^2) + a^3 
*b^2*d^3*(48*c^2*C + 275*B*c*d + 198*A*d^2))*(-((x*Sqrt[a*c + (b*c + a*d)* 
x^2 + b*d*x^4])/(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt[d]*x^2)) + (a^(1/4)*c^(1/4 
)*(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt[d]*x^2)*Sqrt[(a*c + (b*c + a*d)*x^2 + b* 
d*x^4)/(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt[d]*x^2)^2]*EllipticE[2*ArcTan[(b^(1 
/4)*d^(1/4)*x)/(a^(1/4)*c^(1/4))], (2 - (b*c + a*d)/(Sqrt[a]*Sqrt[b]*Sqrt[ 
c]*Sqrt[d]))/4])/(b^(1/4)*d^(1/4)*Sqrt[a*c + (b*c + a*d)*x^2 + b*d*x^4]))) 
/(Sqrt[b]*Sqrt[d])) - (a^(1/4)*c^(1/4)*(Sqrt[b]*Sqrt[c] + Sqrt[a]*Sqrt[d]) 
^2*(72*a^(7/2)*Sqrt[b]*Sqrt[c]*C*d^(7/2) - 48*a^4*C*d^4 + 8*a^3*b*d^3*(3*c 
*C + 11*B*d) - 3*a^(5/2)*b^(3/2)*Sqrt[c]*d^(5/2)*(21*c*C + 44*B*d) - 9*...
 

Defintions of rubi rules used

rule 25

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

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 1416

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c 
/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ 
(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) 
], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

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 1509

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

rule 1511

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q   Int[1/Sqrt[a + b*x^2 + c*x^ 
4], x], x] - Simp[e/q   Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; 
NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos 
Q[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. \(1926\) vs. \(2(962)=1924\).

Time = 10.88 (sec) , antiderivative size = 1927, normalized size of antiderivative = 1.94

method	result	size
risch	\(\text {Expression too large to display}\)	\(1927\)
elliptic	\(\text {Expression too large to display}\)	\(2262\)
default	\(\text {Expression too large to display}\)	\(3080\)

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=_RETURNVERB 
OSE)
 

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)*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)))+24*C*a*b^4*c^5/(-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))+24*C*a^5*c*d^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))+99*A*a*b^4*c^3*d^2/(-b/a)^(1/2)*(1...
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 1009, normalized size of antiderivative = 1.02 \[ \int \left (A+B x^2+C x^4\right ) \left (a c+(b c+a d) x^2+b d x^4\right )^{3/2} \, dx =\text {Too large to display} \] 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)*ellip 
tic_e(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - (48*C*b^5*c^6 - 8*(15*C*a*b^4 + 1 
1*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)*sqr 
t(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. \[ \int \left (A+B x^2+C x^4\right ) \left (a c+(b c+a d) x^2+b d x^4\right )^{3/2} \, dx=\int {\left (b\,d\,x^4+\left (a\,d+b\,c\right )\,x^2+a\,c\right )}^{3/2}\,\left (C\,x^4+B\,x^2+A\right ) \,d x \] 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*sqrt(c 
 + d*x**2)*sqrt(a + b*x**2)*a**2*b**3*d**4*x**3 + 18*sqrt(c + d*x**2)*sqrt 
(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 + 1 
705*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 + 4 
20*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*sqrt(c 
 + d*x**2)*sqrt(a + b*x**2)*b**4*c*d**4*x**9 - 48*int((sqrt(c + d*x**2)...