\(\int (a+b x^2)^{3/2} (c+d x^2)^{3/2} (e+f x^2) \, dx\) [8]

Optimal result

Integrand size = 30, antiderivative size = 680 \[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right ) \, dx=\frac {\left (8 a^4 d^4 f+5 a b^3 c^2 d (18 d e-5 c f)-2 b^4 c^3 (9 d e-4 c f)+18 a^2 b^2 c d^2 (5 d e+c f)-a^3 b d^3 (18 d e+25 c f)\right ) x \sqrt {c+d x^2}}{315 b^2 d^3 \sqrt {a+b x^2}}-\frac {\left (4 a^3 d^3 f-3 a b^2 c d (27 d e-7 c f)+2 b^3 c^2 (9 d e-4 c f)-9 a^2 b d^2 (d e+c f)\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{315 b^2 d^2}-\frac {\left (18 b c e-72 a d e+17 a c f-\frac {8 b c^2 f}{d}-\frac {3 a^2 d f}{b}\right ) x \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{315 d}+\frac {(9 b d e-4 b c f+3 a d f) x \sqrt {a+b x^2} \left (c+d x^2\right )^{5/2}}{63 d^2}+\frac {f x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2}}{9 d}-\frac {\sqrt {a} \left (8 a^4 d^4 f+5 a b^3 c^2 d (18 d e-5 c f)-2 b^4 c^3 (9 d e-4 c f)+18 a^2 b^2 c d^2 (5 d e+c f)-a^3 b d^3 (18 d e+25 c f)\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{315 b^{5/2} d^3 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {a^{3/2} \left (4 a^3 d^3 f-b^3 c^2 (9 d e-4 c f)+6 a b^2 c d (27 d e-2 c f)-3 a^2 b d^2 (3 d e+4 c f)\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{315 b^{5/2} d^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

1/315*(8*a^4*d^4*f+5*a*b^3*c^2*d*(-5*c*f+18*d*e)-2*b^4*c^3*(-4*c*f+9*d*e)+ 
18*a^2*b^2*c*d^2*(c*f+5*d*e)-a^3*b*d^3*(25*c*f+18*d*e))*x*(d*x^2+c)^(1/2)/ 
b^2/d^3/(b*x^2+a)^(1/2)-1/315*(4*a^3*d^3*f-3*a*b^2*c*d*(-7*c*f+27*d*e)+2*b 
^3*c^2*(-4*c*f+9*d*e)-9*a^2*b*d^2*(c*f+d*e))*x*(b*x^2+a)^(1/2)*(d*x^2+c)^( 
1/2)/b^2/d^2-1/315*(18*b*c*e-72*a*d*e+17*a*c*f-8*b*c^2*f/d-3*a^2*d*f/b)*x* 
(b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/d+1/63*(3*a*d*f-4*b*c*f+9*b*d*e)*x*(b*x^2+ 
a)^(1/2)*(d*x^2+c)^(5/2)/d^2+1/9*f*x*(b*x^2+a)^(3/2)*(d*x^2+c)^(5/2)/d-1/3 
15*a^(1/2)*(8*a^4*d^4*f+5*a*b^3*c^2*d*(-5*c*f+18*d*e)-2*b^4*c^3*(-4*c*f+9* 
d*e)+18*a^2*b^2*c*d^2*(c*f+5*d*e)-a^3*b*d^3*(25*c*f+18*d*e))*(d*x^2+c)^(1/ 
2)*EllipticE(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/2),(1-a*d/b/c)^(1/2))/b^(5/2 
)/d^3/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)+1/315*a^(3/2)*(4*a^3 
*d^3*f-b^3*c^2*(-4*c*f+9*d*e)+6*a*b^2*c*d*(-2*c*f+27*d*e)-3*a^2*b*d^2*(4*c 
*f+3*d*e))*(d*x^2+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*x/a^(1/2)),(1-a* 
d/b/c)^(1/2))/b^(5/2)/d^2/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 8.16 (sec) , antiderivative size = 479, normalized size of antiderivative = 0.70 \[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right ) \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (-4 a^3 d^3 f+3 a^2 b d^2 \left (3 d e+4 c f+d f x^2\right )+b^3 \left (-4 c^3 f+3 c^2 d \left (3 e+f x^2\right )+5 d^3 x^4 \left (9 e+7 f x^2\right )+2 c d^2 x^2 \left (36 e+25 f x^2\right )\right )+a b^2 d \left (12 c^2 f+2 d^2 x^2 \left (36 e+25 f x^2\right )+c d \left (153 e+83 f x^2\right )\right )\right )-i c \left (8 a^4 d^4 f+5 a b^3 c^2 d (18 d e-5 c f)+18 a^2 b^2 c d^2 (5 d e+c f)+2 b^4 c^3 (-9 d e+4 c f)-a^3 b d^3 (18 d e+25 c f)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i c (-b c+a d) \left (4 a^3 d^3 f+2 b^3 c^2 (9 d e-4 c f)-9 a^2 b d^2 (d e+c f)+3 a b^2 c d (-27 d e+7 c f)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{315 a^2 \left (\frac {b}{a}\right )^{5/2} d^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

Integrate[(a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)*(e + f*x^2),x]
 

Output:

(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(-4*a^3*d^3*f + 3*a^2*b*d^2*(3*d*e 
+ 4*c*f + d*f*x^2) + b^3*(-4*c^3*f + 3*c^2*d*(3*e + f*x^2) + 5*d^3*x^4*(9* 
e + 7*f*x^2) + 2*c*d^2*x^2*(36*e + 25*f*x^2)) + a*b^2*d*(12*c^2*f + 2*d^2* 
x^2*(36*e + 25*f*x^2) + c*d*(153*e + 83*f*x^2))) - I*c*(8*a^4*d^4*f + 5*a* 
b^3*c^2*d*(18*d*e - 5*c*f) + 18*a^2*b^2*c*d^2*(5*d*e + c*f) + 2*b^4*c^3*(- 
9*d*e + 4*c*f) - a^3*b*d^3*(18*d*e + 25*c*f))*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)*(4*a^3*d^3*f + 2*b^3*c^2*(9*d*e - 4*c*f) - 9*a^2*b*d^2*(d*e + c*f) + 
 3*a*b^2*c*d*(-27*d*e + 7*c*f))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*El 
lipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(315*a^2*(b/a)^(5/2)*d^3*Sqr 
t[a + b*x^2]*Sqrt[c + d*x^2])
 

Rubi [A] (verified)

Time = 0.94 (sec) , antiderivative size = 615, normalized size of antiderivative = 0.90, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {403, 403, 403, 27, 403, 406, 320, 388, 313}

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.

\(\displaystyle \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right ) \, dx\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {\int \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} \left ((9 b d e+3 b c f-4 a d f) x^2+c (9 b e-a f)\right )dx}{9 b}+\frac {f x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}}{9 b}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {\frac {\int \frac {\left (b x^2+a\right )^{3/2} \left (\left (3 c (24 d e+c f) b^2-a d (18 d e+17 c f) b+8 a^2 d^2 f\right ) x^2+c \left (4 d f a^2-9 b d e a-10 b c f a+63 b^2 c e\right )\right )}{\sqrt {d x^2+c}}dx}{7 b}+\frac {x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} (-4 a d f+3 b c f+9 b d e)}{7 b}}{9 b}+\frac {f x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}}{9 b}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {\frac {\frac {\int \frac {3 \sqrt {b x^2+a} \left (\left (c^2 (9 d e-4 c f) b^3+9 a c d (9 d e+c f) b^2-3 a^2 d^2 (6 d e+7 c f) b+8 a^3 d^3 f\right ) x^2+a c \left (c (81 d e-c f) b^2-a d (9 d e+11 c f) b+4 a^2 d^2 f\right )\right )}{\sqrt {d x^2+c}}dx}{5 d}+\frac {x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (8 a^2 d^2 f-a b d (17 c f+18 d e)+3 b^2 c (c f+24 d e)\right )}{5 d}}{7 b}+\frac {x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} (-4 a d f+3 b c f+9 b d e)}{7 b}}{9 b}+\frac {f x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}}{9 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {3 \int \frac {\sqrt {b x^2+a} \left (\left (c^2 (9 d e-4 c f) b^3+9 a c d (9 d e+c f) b^2-3 a^2 d^2 (6 d e+7 c f) b+8 a^3 d^3 f\right ) x^2+a c \left (c (81 d e-c f) b^2-a d (9 d e+11 c f) b+4 a^2 d^2 f\right )\right )}{\sqrt {d x^2+c}}dx}{5 d}+\frac {x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (8 a^2 d^2 f-a b d (17 c f+18 d e)+3 b^2 c (c f+24 d e)\right )}{5 d}}{7 b}+\frac {x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} (-4 a d f+3 b c f+9 b d e)}{7 b}}{9 b}+\frac {f x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}}{9 b}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {\int \frac {\left (-2 c^3 (9 d e-4 c f) b^4+5 a c^2 d (18 d e-5 c f) b^3+18 a^2 c d^2 (5 d e+c f) b^2-a^3 d^3 (18 d e+25 c f) b+8 a^4 d^4 f\right ) x^2+a c \left (-c^2 (9 d e-4 c f) b^3+6 a c d (27 d e-2 c f) b^2-3 a^2 d^2 (3 d e+4 c f) b+4 a^3 d^3 f\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (8 a^3 d^3 f-3 a^2 b d^2 (7 c f+6 d e)+9 a b^2 c d (c f+9 d e)+b^3 c^2 (9 d e-4 c f)\right )}{3 d}\right )}{5 d}+\frac {x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (8 a^2 d^2 f-a b d (17 c f+18 d e)+3 b^2 c (c f+24 d e)\right )}{5 d}}{7 b}+\frac {x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} (-4 a d f+3 b c f+9 b d e)}{7 b}}{9 b}+\frac {f x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}}{9 b}\)

\(\Big \downarrow \) 406

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {a c \left (4 a^3 d^3 f-3 a^2 b d^2 (4 c f+3 d e)+6 a b^2 c d (27 d e-2 c f)+b^3 \left (-c^2\right ) (9 d e-4 c f)\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\left (8 a^4 d^4 f-a^3 b d^3 (25 c f+18 d e)+18 a^2 b^2 c d^2 (c f+5 d e)+5 a b^3 c^2 d (18 d e-5 c f)-2 b^4 c^3 (9 d e-4 c f)\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (8 a^3 d^3 f-3 a^2 b d^2 (7 c f+6 d e)+9 a b^2 c d (c f+9 d e)+b^3 c^2 (9 d e-4 c f)\right )}{3 d}\right )}{5 d}+\frac {x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (8 a^2 d^2 f-a b d (17 c f+18 d e)+3 b^2 c (c f+24 d e)\right )}{5 d}}{7 b}+\frac {x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} (-4 a d f+3 b c f+9 b d e)}{7 b}}{9 b}+\frac {f x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}}{9 b}\)

\(\Big \downarrow \) 320

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {\left (8 a^4 d^4 f-a^3 b d^3 (25 c f+18 d e)+18 a^2 b^2 c d^2 (c f+5 d e)+5 a b^3 c^2 d (18 d e-5 c f)-2 b^4 c^3 (9 d e-4 c f)\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} \left (4 a^3 d^3 f-3 a^2 b d^2 (4 c f+3 d e)+6 a b^2 c d (27 d e-2 c f)+b^3 \left (-c^2\right ) (9 d e-4 c f)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 d}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (8 a^3 d^3 f-3 a^2 b d^2 (7 c f+6 d e)+9 a b^2 c d (c f+9 d e)+b^3 c^2 (9 d e-4 c f)\right )}{3 d}\right )}{5 d}+\frac {x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (8 a^2 d^2 f-a b d (17 c f+18 d e)+3 b^2 c (c f+24 d e)\right )}{5 d}}{7 b}+\frac {x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} (-4 a d f+3 b c f+9 b d e)}{7 b}}{9 b}+\frac {f x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}}{9 b}\)

\(\Big \downarrow \) 388

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {\left (8 a^4 d^4 f-a^3 b d^3 (25 c f+18 d e)+18 a^2 b^2 c d^2 (c f+5 d e)+5 a b^3 c^2 d (18 d e-5 c f)-2 b^4 c^3 (9 d e-4 c f)\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} \left (4 a^3 d^3 f-3 a^2 b d^2 (4 c f+3 d e)+6 a b^2 c d (27 d e-2 c f)+b^3 \left (-c^2\right ) (9 d e-4 c f)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 d}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (8 a^3 d^3 f-3 a^2 b d^2 (7 c f+6 d e)+9 a b^2 c d (c f+9 d e)+b^3 c^2 (9 d e-4 c f)\right )}{3 d}\right )}{5 d}+\frac {x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (8 a^2 d^2 f-a b d (17 c f+18 d e)+3 b^2 c (c f+24 d e)\right )}{5 d}}{7 b}+\frac {x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} (-4 a d f+3 b c f+9 b d e)}{7 b}}{9 b}+\frac {f x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}}{9 b}\)

\(\Big \downarrow \) 313

\(\displaystyle \frac {\frac {\frac {x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (8 a^2 d^2 f-a b d (17 c f+18 d e)+3 b^2 c (c f+24 d e)\right )}{5 d}+\frac {3 \left (\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (8 a^3 d^3 f-3 a^2 b d^2 (7 c f+6 d e)+9 a b^2 c d (c f+9 d e)+b^3 c^2 (9 d e-4 c f)\right )}{3 d}+\frac {\frac {c^{3/2} \sqrt {a+b x^2} \left (4 a^3 d^3 f-3 a^2 b d^2 (4 c f+3 d e)+6 a b^2 c d (27 d e-2 c f)+b^3 \left (-c^2\right ) (9 d e-4 c f)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\left (8 a^4 d^4 f-a^3 b d^3 (25 c f+18 d e)+18 a^2 b^2 c d^2 (c f+5 d e)+5 a b^3 c^2 d (18 d e-5 c f)-2 b^4 c^3 (9 d e-4 c f)\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{3 d}\right )}{5 d}}{7 b}+\frac {x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} (-4 a d f+3 b c f+9 b d e)}{7 b}}{9 b}+\frac {f x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}}{9 b}\)

Input:

Int[(a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)*(e + f*x^2),x]
 

Output:

(f*x*(a + b*x^2)^(5/2)*(c + d*x^2)^(3/2))/(9*b) + (((9*b*d*e + 3*b*c*f - 4 
*a*d*f)*x*(a + b*x^2)^(5/2)*Sqrt[c + d*x^2])/(7*b) + (((8*a^2*d^2*f + 3*b^ 
2*c*(24*d*e + c*f) - a*b*d*(18*d*e + 17*c*f))*x*(a + b*x^2)^(3/2)*Sqrt[c + 
 d*x^2])/(5*d) + (3*(((8*a^3*d^3*f + b^3*c^2*(9*d*e - 4*c*f) + 9*a*b^2*c*d 
*(9*d*e + c*f) - 3*a^2*b*d^2*(6*d*e + 7*c*f))*x*Sqrt[a + b*x^2]*Sqrt[c + d 
*x^2])/(3*d) + ((8*a^4*d^4*f + 5*a*b^3*c^2*d*(18*d*e - 5*c*f) - 2*b^4*c^3* 
(9*d*e - 4*c*f) + 18*a^2*b^2*c*d^2*(5*d*e + c*f) - a^3*b*d^3*(18*d*e + 25* 
c*f))*((x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[a + b*x^2]* 
EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b*Sqrt[d]*Sqrt[( 
c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])) + (c^(3/2)*(4*a^3*d^3*f 
- b^3*c^2*(9*d*e - 4*c*f) + 6*a*b^2*c*d*(27*d*e - 2*c*f) - 3*a^2*b*d^2*(3* 
d*e + 4*c*f))*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - ( 
b*c)/(a*d)])/(Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2 
]))/(3*d)))/(5*d))/(7*b))/(9*b)
 

Defintions of rubi rules used

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 313

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim 
p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c 
+ d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ 
[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
 

rule 320

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

rule 388

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] 
 :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b   Int[Sqrt[ 
a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]
 

rule 403

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2), x_Symbol] :> Simp[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + 
 q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1))   Int[(a + b*x^2)^p*(c 
+ d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + 
f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, 
 d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 0]
 

rule 406

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2), x_Symbol] :> Simp[e   Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim 
p[f   Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, 
f, p, q}, x]
 

Maple [A] (verified)

Time = 10.81 (sec) , antiderivative size = 1164, normalized size of antiderivative = 1.71

method	result	size
risch	\(\text {Expression too large to display}\)	\(1164\)
elliptic	\(\text {Expression too large to display}\)	\(1314\)
default	\(\text {Expression too large to display}\)	\(1846\)

Input:

int((b*x^2+a)^(3/2)*(d*x^2+c)^(3/2)*(f*x^2+e),x,method=_RETURNVERBOSE)
 

Output:

-1/315/b^2/d^2*x*(-35*b^3*d^3*f*x^6-50*a*b^2*d^3*f*x^4-50*b^3*c*d^2*f*x^4- 
45*b^3*d^3*e*x^4-3*a^2*b*d^3*f*x^2-83*a*b^2*c*d^2*f*x^2-72*a*b^2*d^3*e*x^2 
-3*b^3*c^2*d*f*x^2-72*b^3*c*d^2*e*x^2+4*a^3*d^3*f-12*a^2*b*c*d^2*f-9*a^2*b 
*d^3*e-12*a*b^2*c^2*d*f-153*a*b^2*c*d^2*e+4*b^3*c^3*f-9*b^3*c^2*d*e)*(b*x^ 
2+a)^(1/2)*(d*x^2+c)^(1/2)+1/315/b^2/d^2*(-(8*a^4*d^4*f-25*a^3*b*c*d^3*f-1 
8*a^3*b*d^4*e+18*a^2*b^2*c^2*d^2*f+90*a^2*b^2*c*d^3*e-25*a*b^3*c^3*d*f+90* 
a*b^3*c^2*d^2*e+8*b^4*c^4*f-18*b^4*c^3*d*e)*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)))+4*a*b^3*c^4*f/(-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))+4*a^4*c*d^3*f/(-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))-9*a*b^3*c^3*d*e/(-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))+162*a^2*b^2*c^2*d^2*e/(-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))-12*a^2*b^2*c^3*d*f/(-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))-9*a^3*b*c*d^3*e/(-b/a)^(1/2)*(1...
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 679, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right ) \, dx=\frac {\sqrt {b d} {\left (18 \, {\left (b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} - 5 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4}\right )} e - {\left (8 \, b^{4} c^{5} - 25 \, a b^{3} c^{4} d + 18 \, a^{2} b^{2} c^{3} d^{2} - 25 \, a^{3} b c^{2} d^{3} + 8 \, a^{4} c d^{4}\right )} f\right )} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - \sqrt {b d} {\left (9 \, {\left (2 \, b^{4} c^{4} d - 10 \, a b^{3} c^{3} d^{2} + a^{3} b d^{5} - {\left (10 \, a^{2} b^{2} - a b^{3}\right )} c^{2} d^{3} + 2 \, {\left (a^{3} b - 9 \, a^{2} b^{2}\right )} c d^{4}\right )} e - {\left (8 \, b^{4} c^{5} - 25 \, a b^{3} c^{4} d + 4 \, a^{4} d^{5} + 2 \, {\left (9 \, a^{2} b^{2} + 2 \, a b^{3}\right )} c^{3} d^{2} - {\left (25 \, a^{3} b + 12 \, a^{2} b^{2}\right )} c^{2} d^{3} + 4 \, {\left (2 \, a^{4} - 3 \, a^{3} b\right )} c d^{4}\right )} f\right )} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) + {\left (35 \, b^{4} d^{5} f x^{8} + 5 \, {\left (9 \, b^{4} d^{5} e + 10 \, {\left (b^{4} c d^{4} + a b^{3} d^{5}\right )} f\right )} x^{6} + {\left (72 \, {\left (b^{4} c d^{4} + a b^{3} d^{5}\right )} e + {\left (3 \, b^{4} c^{2} d^{3} + 83 \, a b^{3} c d^{4} + 3 \, a^{2} b^{2} d^{5}\right )} f\right )} x^{4} + {\left (9 \, {\left (b^{4} c^{2} d^{3} + 17 \, a b^{3} c d^{4} + a^{2} b^{2} d^{5}\right )} e - 4 \, {\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} - 3 \, a^{2} b^{2} c d^{4} + a^{3} b d^{5}\right )} f\right )} x^{2} - 18 \, {\left (b^{4} c^{3} d^{2} - 5 \, a b^{3} c^{2} d^{3} - 5 \, a^{2} b^{2} c d^{4} + a^{3} b d^{5}\right )} e + {\left (8 \, b^{4} c^{4} d - 25 \, a b^{3} c^{3} d^{2} + 18 \, a^{2} b^{2} c^{2} d^{3} - 25 \, a^{3} b c d^{4} + 8 \, a^{4} d^{5}\right )} f\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{315 \, b^{3} d^{4} x} \] Input:

integrate((b*x^2+a)^(3/2)*(d*x^2+c)^(3/2)*(f*x^2+e),x, algorithm="fricas")
 

Output:

1/315*(sqrt(b*d)*(18*(b^4*c^4*d - 5*a*b^3*c^3*d^2 - 5*a^2*b^2*c^2*d^3 + a^ 
3*b*c*d^4)*e - (8*b^4*c^5 - 25*a*b^3*c^4*d + 18*a^2*b^2*c^3*d^2 - 25*a^3*b 
*c^2*d^3 + 8*a^4*c*d^4)*f)*x*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), a 
*d/(b*c)) - sqrt(b*d)*(9*(2*b^4*c^4*d - 10*a*b^3*c^3*d^2 + a^3*b*d^5 - (10 
*a^2*b^2 - a*b^3)*c^2*d^3 + 2*(a^3*b - 9*a^2*b^2)*c*d^4)*e - (8*b^4*c^5 - 
25*a*b^3*c^4*d + 4*a^4*d^5 + 2*(9*a^2*b^2 + 2*a*b^3)*c^3*d^2 - (25*a^3*b + 
 12*a^2*b^2)*c^2*d^3 + 4*(2*a^4 - 3*a^3*b)*c*d^4)*f)*x*sqrt(-c/d)*elliptic 
_f(arcsin(sqrt(-c/d)/x), a*d/(b*c)) + (35*b^4*d^5*f*x^8 + 5*(9*b^4*d^5*e + 
 10*(b^4*c*d^4 + a*b^3*d^5)*f)*x^6 + (72*(b^4*c*d^4 + a*b^3*d^5)*e + (3*b^ 
4*c^2*d^3 + 83*a*b^3*c*d^4 + 3*a^2*b^2*d^5)*f)*x^4 + (9*(b^4*c^2*d^3 + 17* 
a*b^3*c*d^4 + a^2*b^2*d^5)*e - 4*(b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 - 3*a^2*b^ 
2*c*d^4 + a^3*b*d^5)*f)*x^2 - 18*(b^4*c^3*d^2 - 5*a*b^3*c^2*d^3 - 5*a^2*b^ 
2*c*d^4 + a^3*b*d^5)*e + (8*b^4*c^4*d - 25*a*b^3*c^3*d^2 + 18*a^2*b^2*c^2* 
d^3 - 25*a^3*b*c*d^4 + 8*a^4*d^5)*f)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(b^3 
*d^4*x)
 

Sympy [F]

\[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right ) \, dx=\int \left (a + b x^{2}\right )^{\frac {3}{2}} \left (c + d x^{2}\right )^{\frac {3}{2}} \left (e + f x^{2}\right )\, dx \] Input:

integrate((b*x**2+a)**(3/2)*(d*x**2+c)**(3/2)*(f*x**2+e),x)
 

Output:

Integral((a + b*x**2)**(3/2)*(c + d*x**2)**(3/2)*(e + f*x**2), x)
 

Maxima [F]

\[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right ) \, dx=\int { {\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )} \,d x } \] Input:

integrate((b*x^2+a)^(3/2)*(d*x^2+c)^(3/2)*(f*x^2+e),x, algorithm="maxima")
 

Output:

integrate((b*x^2 + a)^(3/2)*(d*x^2 + c)^(3/2)*(f*x^2 + e), x)
 

Giac [F]

\[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right ) \, dx=\int { {\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )} \,d x } \] Input:

integrate((b*x^2+a)^(3/2)*(d*x^2+c)^(3/2)*(f*x^2+e),x, algorithm="giac")
 

Output:

integrate((b*x^2 + a)^(3/2)*(d*x^2 + c)^(3/2)*(f*x^2 + e), x)
 

Mupad [F(-1)]

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

int((a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)*(e + f*x^2),x)
 

Output:

int((a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)*(e + f*x^2), x)
 

Reduce [F]

\[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right ) \, dx =\text {Too large to display} \] Input:

int((b*x^2+a)^(3/2)*(d*x^2+c)^(3/2)*(f*x^2+e),x)
 

Output:

( - 4*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**3*d**3*f*x + 12*sqrt(c + d*x**2 
)*sqrt(a + b*x**2)*a**2*b*c*d**2*f*x + 9*sqrt(c + d*x**2)*sqrt(a + b*x**2) 
*a**2*b*d**3*e*x + 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*b*d**3*f*x**3 
+ 12*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*c**2*d*f*x + 153*sqrt(c + d* 
x**2)*sqrt(a + b*x**2)*a*b**2*c*d**2*e*x + 83*sqrt(c + d*x**2)*sqrt(a + b* 
x**2)*a*b**2*c*d**2*f*x**3 + 72*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*d 
**3*e*x**3 + 50*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*d**3*f*x**5 - 4*s 
qrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c**3*f*x + 9*sqrt(c + d*x**2)*sqrt(a 
 + b*x**2)*b**3*c**2*d*e*x + 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c**2 
*d*f*x**3 + 72*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c*d**2*e*x**3 + 50*s 
qrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c*d**2*f*x**5 + 45*sqrt(c + d*x**2)* 
sqrt(a + b*x**2)*b**3*d**3*e*x**5 + 35*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b 
**3*d**3*f*x**7 + 8*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a* 
d*x**2 + b*c*x**2 + b*d*x**4),x)*a**4*d**4*f - 25*int((sqrt(c + d*x**2)*sq 
rt(a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a**3*b*c*d* 
*3*f - 18*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x**2 + b 
*c*x**2 + b*d*x**4),x)*a**3*b*d**4*e + 18*int((sqrt(c + d*x**2)*sqrt(a + b 
*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a**2*b**2*c**2*d**2 
*f + 90*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c 
*x**2 + b*d*x**4),x)*a**2*b**2*c*d**3*e - 25*int((sqrt(c + d*x**2)*sqrt...