\(\int x^4 \sqrt {a+b x^2} (c+d x^2)^{3/2} (e+f x^2)^2 \, dx\) [84]

Optimal result

Integrand size = 35, antiderivative size = 1435 \[ \int x^4 \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2 \, dx =\text {Too large to display} \] Output:

-1/45045*(1280*a^6*d^6*f^2-128*a^5*b*d^5*f*(17*c*f+26*d*e)-a^3*b^3*c*d^3*( 
-203*c^2*f^2+1326*c*d*e*f+4576*d^2*e^2)-8*b^6*c^4*(48*c^2*f^2-156*c*d*e*f+ 
143*d^2*e^2)+a^2*b^4*c^2*d^2*(167*c^2*f^2-754*c*d*e*f+1287*d^2*e^2)+2*a^4* 
b^2*d^4*(207*c^2*f^2+3016*c*d*e*f+1144*d^2*e^2)+a*b^5*c^3*d*(216*c^2*f^2-8 
32*c*d*e*f+1001*d^2*e^2))*x*(d*x^2+c)^(1/2)/b^5/d^5/(b*x^2+a)^(1/2)+1/4504 
5*(640*a^5*d^5*f^2-16*a^4*b*d^4*f*(63*c*f+104*d*e)-2*a*b^4*c^2*d*(-42*c^2* 
f^2+169*c*d*e*f+1518*d^2*e^2)-5*a^2*b^3*c*d^2*(-17*c^2*f^2+78*c*d*e*f+429* 
d^2*e^2)-4*b^5*c^3*(48*c^2*f^2-156*c*d*e*f+143*d^2*e^2)+a^3*b^2*d^3*(111*c 
^2*f^2+2808*c*d*e*f+1144*d^2*e^2))*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b^5/d 
^4-1/45045*(480*a^4*d^4*f^2-32*a^3*b*d^3*f*(23*c*f+39*d*e)+12*b^4*c^2*(-12 
*c^2*f^2+39*c*d*e*f+253*d^2*e^2)+a*b^3*c*d*(57*c^2*f^2+6696*c*d*e*f+5357*d 
^2*e^2)+a^2*b^2*d^2*(63*c^2*f^2+2054*c*d*e*f+858*d^2*e^2))*x^3*(b*x^2+a)^( 
1/2)*(d*x^2+c)^(1/2)/b^4/d^3+1/9009*(80*a^3*d^3*f^2-a^2*b*d^2*f*(121*c*f+2 
08*d*e)+4*b^3*c*(-6*c^2*f^2-327*c*d*e*f+11*d^2*e^2)-2*a*b^2*d*(342*c^2*f^2 
+1217*c*d*e*f+275*d^2*e^2))*x^5*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b^3/d^2-1/ 
1287*(10*a^2*d^2*f^2+a*b*d*f*(183*c*f+172*d*e)-b^2*(-96*c^2*f^2-84*c*d*e*f 
+44*d^2*e^2))*x^7*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b^2/d+2/143*f*(-5*a*d*f- 
4*b*c*f+2*b*d*e)*x^9*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b+1/13*x*(b*x^2+a)^(3 
/2)*(d*x^2+c)^(5/2)*(f*x^2+e)^2/b/d+1/45045*a^(1/2)*(1280*a^6*d^6*f^2-128* 
a^5*b*d^5*f*(17*c*f+26*d*e)-a^3*b^3*c*d^3*(-203*c^2*f^2+1326*c*d*e*f+45...
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 7.82 (sec) , antiderivative size = 1012, normalized size of antiderivative = 0.71 \[ \int x^4 \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2 \, dx =\text {Too large to display} \] Input:

Integrate[x^4*Sqrt[a + b*x^2]*(c + d*x^2)^(3/2)*(e + f*x^2)^2,x]
 

Output:

(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(640*a^5*d^5*f^2 - 16*a^4*b*d^4*f*( 
104*d*e + 63*c*f + 30*d*f*x^2) + a^3*b^2*d^3*(111*c^2*f^2 + 8*c*d*f*(351*e 
 + 92*f*x^2) + 8*d^2*(143*e^2 + 156*e*f*x^2 + 50*f^2*x^4)) + b^5*(-192*c^5 
*f^2 + 48*c^4*d*f*(13*e + 3*f*x^2) - 4*c^3*d^2*(143*e^2 + 117*e*f*x^2 + 30 
*f^2*x^4) + 3*c^2*d^3*x^2*(143*e^2 + 130*e*f*x^2 + 35*f^2*x^4) + 35*d^5*x^ 
6*(143*e^2 + 234*e*f*x^2 + 99*f^2*x^4) + 10*c*d^4*x^4*(715*e^2 + 1092*e*f* 
x^2 + 441*f^2*x^4)) + a*b^4*d*(84*c^4*f^2 - c^3*d*f*(338*e + 57*f*x^2) + 3 
*c^2*d^2*(143*e^2 + 78*e*f*x^2 + 15*f^2*x^4) + 5*d^4*x^4*(143*e^2 + 182*e* 
f*x^2 + 63*f^2*x^4) + c*d^3*x^2*(1573*e^2 + 1690*e*f*x^2 + 525*f^2*x^4)) - 
 a^2*b^3*d^2*(-85*c^3*f^2 + 3*c^2*d*f*(130*e + 21*f*x^2) + 2*d^3*x^2*(429* 
e^2 + 520*e*f*x^2 + 175*f^2*x^4) + c*d^2*(2145*e^2 + 2054*e*f*x^2 + 605*f^ 
2*x^4))) + I*c*(1280*a^6*d^6*f^2 - 128*a^5*b*d^5*f*(26*d*e + 17*c*f) - 8*b 
^6*c^4*(143*d^2*e^2 - 156*c*d*e*f + 48*c^2*f^2) + a^2*b^4*c^2*d^2*(1287*d^ 
2*e^2 - 754*c*d*e*f + 167*c^2*f^2) + a^3*b^3*c*d^3*(-4576*d^2*e^2 - 1326*c 
*d*e*f + 203*c^2*f^2) + 2*a^4*b^2*d^4*(1144*d^2*e^2 + 3016*c*d*e*f + 207*c 
^2*f^2) + a*b^5*c^3*d*(1001*d^2*e^2 - 832*c*d*e*f + 216*c^2*f^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)*(640*a^5*d^5*f^2 - 16*a^4*b*d^4*f*(104*d*e + 33*c*f 
) + a^3*b^2*d^3*(1144*d^2*e^2 + 1560*c*d*e*f - 225*c^2*f^2) + a^2*b^3*c*d^ 
2*(-1287*d^2*e^2 + 624*c*d*e*f - 107*c^2*f^2) + a*b^4*c^2*d*(-429*d^2*e...
 

Rubi [A] (warning: unable to verify)

Time = 3.24 (sec) , antiderivative size = 1816, normalized size of antiderivative = 1.27, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.486, Rules used = {448, 443, 443, 443, 444, 27, 444, 27, 406, 320, 388, 313, 444, 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.

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

\(\Big \downarrow \) 448

\(\displaystyle \frac {f \int x^6 \sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )dx}{e^2}+e \int x^4 \sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )dx\)

\(\Big \downarrow \) 443

\(\displaystyle \frac {f \left (\frac {\int x^6 \sqrt {b x^2+a} \sqrt {d x^2+c} \left ((13 b d e+3 b c f-10 a d f) x^2+c (13 b e-7 a f)\right )dx}{13 b}+\frac {f x^7 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{13 b}\right )}{e^2}+e \left (\frac {\int x^4 \sqrt {b x^2+a} \sqrt {d x^2+c} \left ((11 b d e+3 b c f-8 a d f) x^2+c (11 b e-5 a f)\right )dx}{11 b}+\frac {f x^5 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{11 b}\right )\)

\(\Big \downarrow \) 443

\(\displaystyle \frac {f \left (\frac {\frac {\int \frac {x^6 \sqrt {b x^2+a} \left (\left (3 c (52 d e+c f) b^2-a d (104 d e+111 c f) b+80 a^2 d^2 f\right ) x^2+c \left (70 d f a^2-91 b d e a-98 b c f a+143 b^2 c e\right )\right )}{\sqrt {d x^2+c}}dx}{11 b}+\frac {x^7 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (-10 a d f+3 b c f+13 b d e)}{11 b}}{13 b}+\frac {f x^7 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{13 b}\right )}{e^2}+e \left (\frac {\frac {\int \frac {x^4 \sqrt {b x^2+a} \left (\left (c (110 d e+3 c f) b^2-a d (66 d e+71 c f) b+48 a^2 d^2 f\right ) x^2+c \left (40 d f a^2-55 b d e a-60 b c f a+99 b^2 c e\right )\right )}{\sqrt {d x^2+c}}dx}{9 b}+\frac {x^5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (-8 a d f+3 b c f+11 b d e)}{9 b}}{11 b}+\frac {f x^5 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{11 b}\right )\)

\(\Big \downarrow \) 443

\(\displaystyle \frac {f \left (\frac {\frac {\frac {\int \frac {x^6 \left (\left (3 c^2 (13 d e-8 c f) b^3+a c d (169 d e+9 c f) b^2-a^2 d^2 (104 d e+121 c f) b+80 a^3 d^3 f\right ) x^2+a c \left (3 c (65 d e-7 c f) b^2-7 a d (13 d e+15 c f) b+70 a^2 d^2 f\right )\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{9 d}+\frac {x^7 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (80 a^2 d^2 f-a b d (111 c f+104 d e)+3 b^2 c (c f+52 d e)\right )}{9 d}}{11 b}+\frac {x^7 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (-10 a d f+3 b c f+13 b d e)}{11 b}}{13 b}+\frac {f x^7 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{13 b}\right )}{e^2}+e \left (\frac {\frac {\frac {\int \frac {x^4 \left (\left (3 c^2 (11 d e-6 c f) b^3+a c d (121 d e+9 c f) b^2-a^2 d^2 (66 d e+79 c f) b+48 a^3 d^3 f\right ) x^2+a c \left (c (143 d e-15 c f) b^2-5 a d (11 d e+13 c f) b+40 a^2 d^2 f\right )\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{7 d}+\frac {x^5 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (48 a^2 d^2 f-a b d (71 c f+66 d e)+b^2 c (3 c f+110 d e)\right )}{7 d}}{9 b}+\frac {x^5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (-8 a d f+3 b c f+11 b d e)}{9 b}}{11 b}+\frac {f x^5 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{11 b}\right )\)

\(\Big \downarrow \) 444

\(\displaystyle e \left (\frac {f \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} x^5}{11 b}+\frac {\frac {(11 b d e+3 b c f-8 a d f) \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^5}{9 b}+\frac {\frac {\left (c (110 d e+3 c f) b^2-a d (66 d e+71 c f) b+48 a^2 d^2 f\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x^5}{7 d}+\frac {\frac {\left (3 c^2 (11 d e-6 c f) b^3+a c d (121 d e+9 c f) b^2-a^2 d^2 (66 d e+79 c f) b+48 a^3 d^3 f\right ) x^3 \sqrt {b x^2+a} \sqrt {d x^2+c}}{5 b d}-\frac {\int \frac {3 x^2 \left (\left (4 c^3 (11 d e-6 c f) b^4-a c^2 d (33 d e-13 c f) b^3+15 a^2 c d^2 (11 d e+c f) b^2-4 a^3 d^3 (22 d e+27 c f) b+64 a^4 d^4 f\right ) x^2+a c \left (3 c^2 (11 d e-6 c f) b^3+a c d (121 d e+9 c f) b^2-a^2 d^2 (66 d e+79 c f) b+48 a^3 d^3 f\right )\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 b d}}{7 d}}{9 b}}{11 b}\right )+\frac {f \left (\frac {f \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} x^7}{13 b}+\frac {\frac {(13 b d e+3 b c f-10 a d f) \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^7}{11 b}+\frac {\frac {\left (3 c (52 d e+c f) b^2-a d (104 d e+111 c f) b+80 a^2 d^2 f\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x^7}{9 d}+\frac {\frac {\left (3 c^2 (13 d e-8 c f) b^3+a c d (169 d e+9 c f) b^2-a^2 d^2 (104 d e+121 c f) b+80 a^3 d^3 f\right ) x^5 \sqrt {b x^2+a} \sqrt {d x^2+c}}{7 b d}-\frac {\int \frac {x^4 \left (\left (18 c^3 (13 d e-8 c f) b^4-3 a c^2 d (39 d e-19 c f) b^3+a^2 c d^2 (1027 d e+63 c f) b^2-16 a^3 d^3 (39 d e+46 c f) b+480 a^4 d^4 f\right ) x^2+5 a c \left (3 c^2 (13 d e-8 c f) b^3+a c d (169 d e+9 c f) b^2-a^2 d^2 (104 d e+121 c f) b+80 a^3 d^3 f\right )\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{7 b d}}{9 d}}{11 b}}{13 b}\right )}{e^2}\)

\(\Big \downarrow \) 27

\(\displaystyle e \left (\frac {f \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} x^5}{11 b}+\frac {\frac {(11 b d e+3 b c f-8 a d f) \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^5}{9 b}+\frac {\frac {\left (c (110 d e+3 c f) b^2-a d (66 d e+71 c f) b+48 a^2 d^2 f\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x^5}{7 d}+\frac {\frac {\left (3 c^2 (11 d e-6 c f) b^3+a c d (121 d e+9 c f) b^2-a^2 d^2 (66 d e+79 c f) b+48 a^3 d^3 f\right ) x^3 \sqrt {b x^2+a} \sqrt {d x^2+c}}{5 b d}-\frac {3 \int \frac {x^2 \left (\left (4 c^3 (11 d e-6 c f) b^4-a c^2 d (33 d e-13 c f) b^3+15 a^2 c d^2 (11 d e+c f) b^2-4 a^3 d^3 (22 d e+27 c f) b+64 a^4 d^4 f\right ) x^2+a c \left (3 c^2 (11 d e-6 c f) b^3+a c d (121 d e+9 c f) b^2-a^2 d^2 (66 d e+79 c f) b+48 a^3 d^3 f\right )\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 b d}}{7 d}}{9 b}}{11 b}\right )+\frac {f \left (\frac {f \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} x^7}{13 b}+\frac {\frac {(13 b d e+3 b c f-10 a d f) \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^7}{11 b}+\frac {\frac {\left (3 c (52 d e+c f) b^2-a d (104 d e+111 c f) b+80 a^2 d^2 f\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x^7}{9 d}+\frac {\frac {\left (3 c^2 (13 d e-8 c f) b^3+a c d (169 d e+9 c f) b^2-a^2 d^2 (104 d e+121 c f) b+80 a^3 d^3 f\right ) x^5 \sqrt {b x^2+a} \sqrt {d x^2+c}}{7 b d}-\frac {\int \frac {x^4 \left (\left (18 c^3 (13 d e-8 c f) b^4-3 a c^2 d (39 d e-19 c f) b^3+a^2 c d^2 (1027 d e+63 c f) b^2-16 a^3 d^3 (39 d e+46 c f) b+480 a^4 d^4 f\right ) x^2+5 a c \left (3 c^2 (13 d e-8 c f) b^3+a c d (169 d e+9 c f) b^2-a^2 d^2 (104 d e+121 c f) b+80 a^3 d^3 f\right )\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{7 b d}}{9 d}}{11 b}}{13 b}\right )}{e^2}\)

\(\Big \downarrow \) 444

\(\displaystyle e \left (\frac {f \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} x^5}{11 b}+\frac {\frac {(11 b d e+3 b c f-8 a d f) \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^5}{9 b}+\frac {\frac {\left (c (110 d e+3 c f) b^2-a d (66 d e+71 c f) b+48 a^2 d^2 f\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x^5}{7 d}+\frac {\frac {\left (3 c^2 (11 d e-6 c f) b^3+a c d (121 d e+9 c f) b^2-a^2 d^2 (66 d e+79 c f) b+48 a^3 d^3 f\right ) x^3 \sqrt {b x^2+a} \sqrt {d x^2+c}}{5 b d}-\frac {3 \left (\frac {\left (4 c^3 (11 d e-6 c f) b^4-a c^2 d (33 d e-13 c f) b^3+15 a^2 c d^2 (11 d e+c f) b^2-4 a^3 d^3 (22 d e+27 c f) b+64 a^4 d^4 f\right ) x \sqrt {b x^2+a} \sqrt {d x^2+c}}{3 b d}-\frac {\int \frac {\left (8 c^4 (11 d e-6 c f) b^5-a c^3 d (77 d e-32 c f) b^4-a^2 c^2 d^2 (99 d e-29 c f) b^3+a^3 c d^3 (352 d e+51 c f) b^2-8 a^4 d^4 (22 d e+29 c f) b+128 a^5 d^5 f\right ) x^2+a c \left (4 c^3 (11 d e-6 c f) b^4-a c^2 d (33 d e-13 c f) b^3+15 a^2 c d^2 (11 d e+c f) b^2-4 a^3 d^3 (22 d e+27 c f) b+64 a^4 d^4 f\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b d}\right )}{5 b d}}{7 d}}{9 b}}{11 b}\right )+\frac {f \left (\frac {f \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} x^7}{13 b}+\frac {\frac {(13 b d e+3 b c f-10 a d f) \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^7}{11 b}+\frac {\frac {\left (3 c (52 d e+c f) b^2-a d (104 d e+111 c f) b+80 a^2 d^2 f\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x^7}{9 d}+\frac {\frac {\left (3 c^2 (13 d e-8 c f) b^3+a c d (169 d e+9 c f) b^2-a^2 d^2 (104 d e+121 c f) b+80 a^3 d^3 f\right ) x^5 \sqrt {b x^2+a} \sqrt {d x^2+c}}{7 b d}-\frac {\frac {\left (18 c^3 (13 d e-8 c f) b^4-3 a c^2 d (39 d e-19 c f) b^3+a^2 c d^2 (1027 d e+63 c f) b^2-16 a^3 d^3 (39 d e+46 c f) b+480 a^4 d^4 f\right ) x^3 \sqrt {b x^2+a} \sqrt {d x^2+c}}{5 b d}-\frac {\int \frac {3 x^2 \left (\left (24 c^4 (13 d e-8 c f) b^5-a c^3 d (169 d e-84 c f) b^4-5 a^2 c^2 d^2 (39 d e-17 c f) b^3+3 a^3 c d^3 (468 d e+37 c f) b^2-16 a^4 d^4 (52 d e+63 c f) b+640 a^5 d^5 f\right ) x^2+a c \left (18 c^3 (13 d e-8 c f) b^4-3 a c^2 d (39 d e-19 c f) b^3+a^2 c d^2 (1027 d e+63 c f) b^2-16 a^3 d^3 (39 d e+46 c f) b+480 a^4 d^4 f\right )\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 b d}}{7 b d}}{9 d}}{11 b}}{13 b}\right )}{e^2}\)

\(\Big \downarrow \) 27

\(\displaystyle e \left (\frac {f \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} x^5}{11 b}+\frac {\frac {(11 b d e+3 b c f-8 a d f) \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^5}{9 b}+\frac {\frac {\left (c (110 d e+3 c f) b^2-a d (66 d e+71 c f) b+48 a^2 d^2 f\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x^5}{7 d}+\frac {\frac {\left (3 c^2 (11 d e-6 c f) b^3+a c d (121 d e+9 c f) b^2-a^2 d^2 (66 d e+79 c f) b+48 a^3 d^3 f\right ) x^3 \sqrt {b x^2+a} \sqrt {d x^2+c}}{5 b d}-\frac {3 \left (\frac {\left (4 c^3 (11 d e-6 c f) b^4-a c^2 d (33 d e-13 c f) b^3+15 a^2 c d^2 (11 d e+c f) b^2-4 a^3 d^3 (22 d e+27 c f) b+64 a^4 d^4 f\right ) x \sqrt {b x^2+a} \sqrt {d x^2+c}}{3 b d}-\frac {\int \frac {\left (8 c^4 (11 d e-6 c f) b^5-a c^3 d (77 d e-32 c f) b^4-a^2 c^2 d^2 (99 d e-29 c f) b^3+a^3 c d^3 (352 d e+51 c f) b^2-8 a^4 d^4 (22 d e+29 c f) b+128 a^5 d^5 f\right ) x^2+a c \left (4 c^3 (11 d e-6 c f) b^4-a c^2 d (33 d e-13 c f) b^3+15 a^2 c d^2 (11 d e+c f) b^2-4 a^3 d^3 (22 d e+27 c f) b+64 a^4 d^4 f\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b d}\right )}{5 b d}}{7 d}}{9 b}}{11 b}\right )+\frac {f \left (\frac {f \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} x^7}{13 b}+\frac {\frac {(13 b d e+3 b c f-10 a d f) \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^7}{11 b}+\frac {\frac {\left (3 c (52 d e+c f) b^2-a d (104 d e+111 c f) b+80 a^2 d^2 f\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x^7}{9 d}+\frac {\frac {\left (3 c^2 (13 d e-8 c f) b^3+a c d (169 d e+9 c f) b^2-a^2 d^2 (104 d e+121 c f) b+80 a^3 d^3 f\right ) x^5 \sqrt {b x^2+a} \sqrt {d x^2+c}}{7 b d}-\frac {\frac {\left (18 c^3 (13 d e-8 c f) b^4-3 a c^2 d (39 d e-19 c f) b^3+a^2 c d^2 (1027 d e+63 c f) b^2-16 a^3 d^3 (39 d e+46 c f) b+480 a^4 d^4 f\right ) x^3 \sqrt {b x^2+a} \sqrt {d x^2+c}}{5 b d}-\frac {3 \int \frac {x^2 \left (\left (24 c^4 (13 d e-8 c f) b^5-a c^3 d (169 d e-84 c f) b^4-5 a^2 c^2 d^2 (39 d e-17 c f) b^3+3 a^3 c d^3 (468 d e+37 c f) b^2-16 a^4 d^4 (52 d e+63 c f) b+640 a^5 d^5 f\right ) x^2+a c \left (18 c^3 (13 d e-8 c f) b^4-3 a c^2 d (39 d e-19 c f) b^3+a^2 c d^2 (1027 d e+63 c f) b^2-16 a^3 d^3 (39 d e+46 c f) b+480 a^4 d^4 f\right )\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 b d}}{7 b d}}{9 d}}{11 b}}{13 b}\right )}{e^2}\)

\(\Big \downarrow \) 406

\(\displaystyle e \left (\frac {f \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} x^5}{11 b}+\frac {\frac {(11 b d e+3 b c f-8 a d f) \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^5}{9 b}+\frac {\frac {\left (c (110 d e+3 c f) b^2-a d (66 d e+71 c f) b+48 a^2 d^2 f\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x^5}{7 d}+\frac {\frac {\left (3 c^2 (11 d e-6 c f) b^3+a c d (121 d e+9 c f) b^2-a^2 d^2 (66 d e+79 c f) b+48 a^3 d^3 f\right ) x^3 \sqrt {b x^2+a} \sqrt {d x^2+c}}{5 b d}-\frac {3 \left (\frac {\left (4 c^3 (11 d e-6 c f) b^4-a c^2 d (33 d e-13 c f) b^3+15 a^2 c d^2 (11 d e+c f) b^2-4 a^3 d^3 (22 d e+27 c f) b+64 a^4 d^4 f\right ) x \sqrt {b x^2+a} \sqrt {d x^2+c}}{3 b d}-\frac {a c \left (4 c^3 (11 d e-6 c f) b^4-a c^2 d (33 d e-13 c f) b^3+15 a^2 c d^2 (11 d e+c f) b^2-4 a^3 d^3 (22 d e+27 c f) b+64 a^4 d^4 f\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\left (8 c^4 (11 d e-6 c f) b^5-a c^3 d (77 d e-32 c f) b^4-a^2 c^2 d^2 (99 d e-29 c f) b^3+a^3 c d^3 (352 d e+51 c f) b^2-8 a^4 d^4 (22 d e+29 c f) b+128 a^5 d^5 f\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b d}\right )}{5 b d}}{7 d}}{9 b}}{11 b}\right )+\frac {f \left (\frac {f \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} x^7}{13 b}+\frac {\frac {(13 b d e+3 b c f-10 a d f) \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^7}{11 b}+\frac {\frac {\left (3 c (52 d e+c f) b^2-a d (104 d e+111 c f) b+80 a^2 d^2 f\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x^7}{9 d}+\frac {\frac {\left (3 c^2 (13 d e-8 c f) b^3+a c d (169 d e+9 c f) b^2-a^2 d^2 (104 d e+121 c f) b+80 a^3 d^3 f\right ) x^5 \sqrt {b x^2+a} \sqrt {d x^2+c}}{7 b d}-\frac {\frac {\left (18 c^3 (13 d e-8 c f) b^4-3 a c^2 d (39 d e-19 c f) b^3+a^2 c d^2 (1027 d e+63 c f) b^2-16 a^3 d^3 (39 d e+46 c f) b+480 a^4 d^4 f\right ) x^3 \sqrt {b x^2+a} \sqrt {d x^2+c}}{5 b d}-\frac {3 \int \frac {x^2 \left (\left (24 c^4 (13 d e-8 c f) b^5-a c^3 d (169 d e-84 c f) b^4-5 a^2 c^2 d^2 (39 d e-17 c f) b^3+3 a^3 c d^3 (468 d e+37 c f) b^2-16 a^4 d^4 (52 d e+63 c f) b+640 a^5 d^5 f\right ) x^2+a c \left (18 c^3 (13 d e-8 c f) b^4-3 a c^2 d (39 d e-19 c f) b^3+a^2 c d^2 (1027 d e+63 c f) b^2-16 a^3 d^3 (39 d e+46 c f) b+480 a^4 d^4 f\right )\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 b d}}{7 b d}}{9 d}}{11 b}}{13 b}\right )}{e^2}\)

\(\Big \downarrow \) 320

\(\displaystyle e \left (\frac {f \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} x^5}{11 b}+\frac {\frac {(11 b d e+3 b c f-8 a d f) \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^5}{9 b}+\frac {\frac {\left (c (110 d e+3 c f) b^2-a d (66 d e+71 c f) b+48 a^2 d^2 f\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x^5}{7 d}+\frac {\frac {\left (3 c^2 (11 d e-6 c f) b^3+a c d (121 d e+9 c f) b^2-a^2 d^2 (66 d e+79 c f) b+48 a^3 d^3 f\right ) x^3 \sqrt {b x^2+a} \sqrt {d x^2+c}}{5 b d}-\frac {3 \left (\frac {\left (4 c^3 (11 d e-6 c f) b^4-a c^2 d (33 d e-13 c f) b^3+15 a^2 c d^2 (11 d e+c f) b^2-4 a^3 d^3 (22 d e+27 c f) b+64 a^4 d^4 f\right ) x \sqrt {b x^2+a} \sqrt {d x^2+c}}{3 b d}-\frac {\frac {\left (4 c^3 (11 d e-6 c f) b^4-a c^2 d (33 d e-13 c f) b^3+15 a^2 c d^2 (11 d e+c f) b^2-4 a^3 d^3 (22 d e+27 c f) b+64 a^4 d^4 f\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 ) c^{3/2}}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\left (8 c^4 (11 d e-6 c f) b^5-a c^3 d (77 d e-32 c f) b^4-a^2 c^2 d^2 (99 d e-29 c f) b^3+a^3 c d^3 (352 d e+51 c f) b^2-8 a^4 d^4 (22 d e+29 c f) b+128 a^5 d^5 f\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b d}\right )}{5 b d}}{7 d}}{9 b}}{11 b}\right )+\frac {f \left (\frac {f \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} x^7}{13 b}+\frac {\frac {(13 b d e+3 b c f-10 a d f) \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^7}{11 b}+\frac {\frac {\left (3 c (52 d e+c f) b^2-a d (104 d e+111 c f) b+80 a^2 d^2 f\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x^7}{9 d}+\frac {\frac {\left (3 c^2 (13 d e-8 c f) b^3+a c d (169 d e+9 c f) b^2-a^2 d^2 (104 d e+121 c f) b+80 a^3 d^3 f\right ) x^5 \sqrt {b x^2+a} \sqrt {d x^2+c}}{7 b d}-\frac {\frac {\left (18 c^3 (13 d e-8 c f) b^4-3 a c^2 d (39 d e-19 c f) b^3+a^2 c d^2 (1027 d e+63 c f) b^2-16 a^3 d^3 (39 d e+46 c f) b+480 a^4 d^4 f\right ) x^3 \sqrt {b x^2+a} \sqrt {d x^2+c}}{5 b d}-\frac {3 \int \frac {x^2 \left (\left (24 c^4 (13 d e-8 c f) b^5-a c^3 d (169 d e-84 c f) b^4-5 a^2 c^2 d^2 (39 d e-17 c f) b^3+3 a^3 c d^3 (468 d e+37 c f) b^2-16 a^4 d^4 (52 d e+63 c f) b+640 a^5 d^5 f\right ) x^2+a c \left (18 c^3 (13 d e-8 c f) b^4-3 a c^2 d (39 d e-19 c f) b^3+a^2 c d^2 (1027 d e+63 c f) b^2-16 a^3 d^3 (39 d e+46 c f) b+480 a^4 d^4 f\right )\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 b d}}{7 b d}}{9 d}}{11 b}}{13 b}\right )}{e^2}\)

\(\Big \downarrow \) 388

\(\displaystyle e \left (\frac {f \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} x^5}{11 b}+\frac {\frac {(11 b d e+3 b c f-8 a d f) \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^5}{9 b}+\frac {\frac {\left (c (110 d e+3 c f) b^2-a d (66 d e+71 c f) b+48 a^2 d^2 f\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x^5}{7 d}+\frac {\frac {\left (3 c^2 (11 d e-6 c f) b^3+a c d (121 d e+9 c f) b^2-a^2 d^2 (66 d e+79 c f) b+48 a^3 d^3 f\right ) x^3 \sqrt {b x^2+a} \sqrt {d x^2+c}}{5 b d}-\frac {3 \left (\frac {\left (4 c^3 (11 d e-6 c f) b^4-a c^2 d (33 d e-13 c f) b^3+15 a^2 c d^2 (11 d e+c f) b^2-4 a^3 d^3 (22 d e+27 c f) b+64 a^4 d^4 f\right ) x \sqrt {b x^2+a} \sqrt {d x^2+c}}{3 b d}-\frac {\frac {\left (4 c^3 (11 d e-6 c f) b^4-a c^2 d (33 d e-13 c f) b^3+15 a^2 c d^2 (11 d e+c f) b^2-4 a^3 d^3 (22 d e+27 c f) b+64 a^4 d^4 f\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 ) c^{3/2}}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\left (8 c^4 (11 d e-6 c f) b^5-a c^3 d (77 d e-32 c f) b^4-a^2 c^2 d^2 (99 d e-29 c f) b^3+a^3 c d^3 (352 d e+51 c f) b^2-8 a^4 d^4 (22 d e+29 c f) b+128 a^5 d^5 f\right ) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )}{3 b d}\right )}{5 b d}}{7 d}}{9 b}}{11 b}\right )+\frac {f \left (\frac {f \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} x^7}{13 b}+\frac {\frac {(13 b d e+3 b c f-10 a d f) \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^7}{11 b}+\frac {\frac {\left (3 c (52 d e+c f) b^2-a d (104 d e+111 c f) b+80 a^2 d^2 f\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x^7}{9 d}+\frac {\frac {\left (3 c^2 (13 d e-8 c f) b^3+a c d (169 d e+9 c f) b^2-a^2 d^2 (104 d e+121 c f) b+80 a^3 d^3 f\right ) x^5 \sqrt {b x^2+a} \sqrt {d x^2+c}}{7 b d}-\frac {\frac {\left (18 c^3 (13 d e-8 c f) b^4-3 a c^2 d (39 d e-19 c f) b^3+a^2 c d^2 (1027 d e+63 c f) b^2-16 a^3 d^3 (39 d e+46 c f) b+480 a^4 d^4 f\right ) x^3 \sqrt {b x^2+a} \sqrt {d x^2+c}}{5 b d}-\frac {3 \int \frac {x^2 \left (\left (24 c^4 (13 d e-8 c f) b^5-a c^3 d (169 d e-84 c f) b^4-5 a^2 c^2 d^2 (39 d e-17 c f) b^3+3 a^3 c d^3 (468 d e+37 c f) b^2-16 a^4 d^4 (52 d e+63 c f) b+640 a^5 d^5 f\right ) x^2+a c \left (18 c^3 (13 d e-8 c f) b^4-3 a c^2 d (39 d e-19 c f) b^3+a^2 c d^2 (1027 d e+63 c f) b^2-16 a^3 d^3 (39 d e+46 c f) b+480 a^4 d^4 f\right )\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 b d}}{7 b d}}{9 d}}{11 b}}{13 b}\right )}{e^2}\)

\(\Big \downarrow \) 313

\(\displaystyle e \left (\frac {f \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} x^5}{11 b}+\frac {\frac {(11 b d e+3 b c f-8 a d f) \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^5}{9 b}+\frac {\frac {\left (c (110 d e+3 c f) b^2-a d (66 d e+71 c f) b+48 a^2 d^2 f\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x^5}{7 d}+\frac {\frac {\left (3 c^2 (11 d e-6 c f) b^3+a c d (121 d e+9 c f) b^2-a^2 d^2 (66 d e+79 c f) b+48 a^3 d^3 f\right ) x^3 \sqrt {b x^2+a} \sqrt {d x^2+c}}{5 b d}-\frac {3 \left (\frac {\left (4 c^3 (11 d e-6 c f) b^4-a c^2 d (33 d e-13 c f) b^3+15 a^2 c d^2 (11 d e+c f) b^2-4 a^3 d^3 (22 d e+27 c f) b+64 a^4 d^4 f\right ) x \sqrt {b x^2+a} \sqrt {d x^2+c}}{3 b d}-\frac {\frac {\left (4 c^3 (11 d e-6 c f) b^4-a c^2 d (33 d e-13 c f) b^3+15 a^2 c d^2 (11 d e+c f) b^2-4 a^3 d^3 (22 d e+27 c f) b+64 a^4 d^4 f\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 ) c^{3/2}}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\left (8 c^4 (11 d e-6 c f) b^5-a c^3 d (77 d e-32 c f) b^4-a^2 c^2 d^2 (99 d e-29 c f) b^3+a^3 c d^3 (352 d e+51 c f) b^2-8 a^4 d^4 (22 d e+29 c f) b+128 a^5 d^5 f\right ) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{3 b d}\right )}{5 b d}}{7 d}}{9 b}}{11 b}\right )+\frac {f \left (\frac {f \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} x^7}{13 b}+\frac {\frac {(13 b d e+3 b c f-10 a d f) \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^7}{11 b}+\frac {\frac {\left (3 c (52 d e+c f) b^2-a d (104 d e+111 c f) b+80 a^2 d^2 f\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x^7}{9 d}+\frac {\frac {\left (3 c^2 (13 d e-8 c f) b^3+a c d (169 d e+9 c f) b^2-a^2 d^2 (104 d e+121 c f) b+80 a^3 d^3 f\right ) x^5 \sqrt {b x^2+a} \sqrt {d x^2+c}}{7 b d}-\frac {\frac {\left (18 c^3 (13 d e-8 c f) b^4-3 a c^2 d (39 d e-19 c f) b^3+a^2 c d^2 (1027 d e+63 c f) b^2-16 a^3 d^3 (39 d e+46 c f) b+480 a^4 d^4 f\right ) x^3 \sqrt {b x^2+a} \sqrt {d x^2+c}}{5 b d}-\frac {3 \int \frac {x^2 \left (\left (24 c^4 (13 d e-8 c f) b^5-a c^3 d (169 d e-84 c f) b^4-5 a^2 c^2 d^2 (39 d e-17 c f) b^3+3 a^3 c d^3 (468 d e+37 c f) b^2-16 a^4 d^4 (52 d e+63 c f) b+640 a^5 d^5 f\right ) x^2+a c \left (18 c^3 (13 d e-8 c f) b^4-3 a c^2 d (39 d e-19 c f) b^3+a^2 c d^2 (1027 d e+63 c f) b^2-16 a^3 d^3 (39 d e+46 c f) b+480 a^4 d^4 f\right )\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 b d}}{7 b d}}{9 d}}{11 b}}{13 b}\right )}{e^2}\)

\(\Big \downarrow \) 444

\(\displaystyle e \left (\frac {f \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} x^5}{11 b}+\frac {\frac {(11 b d e+3 b c f-8 a d f) \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^5}{9 b}+\frac {\frac {\left (c (110 d e+3 c f) b^2-a d (66 d e+71 c f) b+48 a^2 d^2 f\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x^5}{7 d}+\frac {\frac {\left (3 c^2 (11 d e-6 c f) b^3+a c d (121 d e+9 c f) b^2-a^2 d^2 (66 d e+79 c f) b+48 a^3 d^3 f\right ) x^3 \sqrt {b x^2+a} \sqrt {d x^2+c}}{5 b d}-\frac {3 \left (\frac {\left (4 c^3 (11 d e-6 c f) b^4-a c^2 d (33 d e-13 c f) b^3+15 a^2 c d^2 (11 d e+c f) b^2-4 a^3 d^3 (22 d e+27 c f) b+64 a^4 d^4 f\right ) x \sqrt {b x^2+a} \sqrt {d x^2+c}}{3 b d}-\frac {\frac {\left (4 c^3 (11 d e-6 c f) b^4-a c^2 d (33 d e-13 c f) b^3+15 a^2 c d^2 (11 d e+c f) b^2-4 a^3 d^3 (22 d e+27 c f) b+64 a^4 d^4 f\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 ) c^{3/2}}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\left (8 c^4 (11 d e-6 c f) b^5-a c^3 d (77 d e-32 c f) b^4-a^2 c^2 d^2 (99 d e-29 c f) b^3+a^3 c d^3 (352 d e+51 c f) b^2-8 a^4 d^4 (22 d e+29 c f) b+128 a^5 d^5 f\right ) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{3 b d}\right )}{5 b d}}{7 d}}{9 b}}{11 b}\right )+\frac {f \left (\frac {f \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} x^7}{13 b}+\frac {\frac {(13 b d e+3 b c f-10 a d f) \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^7}{11 b}+\frac {\frac {\left (3 c (52 d e+c f) b^2-a d (104 d e+111 c f) b+80 a^2 d^2 f\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x^7}{9 d}+\frac {\frac {\left (3 c^2 (13 d e-8 c f) b^3+a c d (169 d e+9 c f) b^2-a^2 d^2 (104 d e+121 c f) b+80 a^3 d^3 f\right ) x^5 \sqrt {b x^2+a} \sqrt {d x^2+c}}{7 b d}-\frac {\frac {\left (18 c^3 (13 d e-8 c f) b^4-3 a c^2 d (39 d e-19 c f) b^3+a^2 c d^2 (1027 d e+63 c f) b^2-16 a^3 d^3 (39 d e+46 c f) b+480 a^4 d^4 f\right ) x^3 \sqrt {b x^2+a} \sqrt {d x^2+c}}{5 b d}-\frac {3 \left (\frac {\left (24 c^4 (13 d e-8 c f) b^5-a c^3 d (169 d e-84 c f) b^4-5 a^2 c^2 d^2 (39 d e-17 c f) b^3+3 a^3 c d^3 (468 d e+37 c f) b^2-16 a^4 d^4 (52 d e+63 c f) b+640 a^5 d^5 f\right ) x \sqrt {b x^2+a} \sqrt {d x^2+c}}{3 b d}-\frac {\int \frac {\left (48 c^5 (13 d e-8 c f) b^6-8 a c^4 d (52 d e-27 c f) b^5-a^2 c^3 d^2 (377 d e-167 c f) b^4-a^3 c^2 d^3 (663 d e-203 c f) b^3+2 a^4 c d^4 (1508 d e+207 c f) b^2-128 a^5 d^5 (13 d e+17 c f) b+1280 a^6 d^6 f\right ) x^2+a c \left (24 c^4 (13 d e-8 c f) b^5-a c^3 d (169 d e-84 c f) b^4-5 a^2 c^2 d^2 (39 d e-17 c f) b^3+3 a^3 c d^3 (468 d e+37 c f) b^2-16 a^4 d^4 (52 d e+63 c f) b+640 a^5 d^5 f\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b d}\right )}{5 b d}}{7 b d}}{9 d}}{11 b}}{13 b}\right )}{e^2}\)

\(\Big \downarrow \) 406

\(\displaystyle e \left (\frac {f \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} x^5}{11 b}+\frac {\frac {(11 b d e+3 b c f-8 a d f) \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^5}{9 b}+\frac {\frac {\left (c (110 d e+3 c f) b^2-a d (66 d e+71 c f) b+48 a^2 d^2 f\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x^5}{7 d}+\frac {\frac {\left (3 c^2 (11 d e-6 c f) b^3+a c d (121 d e+9 c f) b^2-a^2 d^2 (66 d e+79 c f) b+48 a^3 d^3 f\right ) x^3 \sqrt {b x^2+a} \sqrt {d x^2+c}}{5 b d}-\frac {3 \left (\frac {\left (4 c^3 (11 d e-6 c f) b^4-a c^2 d (33 d e-13 c f) b^3+15 a^2 c d^2 (11 d e+c f) b^2-4 a^3 d^3 (22 d e+27 c f) b+64 a^4 d^4 f\right ) x \sqrt {b x^2+a} \sqrt {d x^2+c}}{3 b d}-\frac {\frac {\left (4 c^3 (11 d e-6 c f) b^4-a c^2 d (33 d e-13 c f) b^3+15 a^2 c d^2 (11 d e+c f) b^2-4 a^3 d^3 (22 d e+27 c f) b+64 a^4 d^4 f\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 ) c^{3/2}}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\left (8 c^4 (11 d e-6 c f) b^5-a c^3 d (77 d e-32 c f) b^4-a^2 c^2 d^2 (99 d e-29 c f) b^3+a^3 c d^3 (352 d e+51 c f) b^2-8 a^4 d^4 (22 d e+29 c f) b+128 a^5 d^5 f\right ) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{3 b d}\right )}{5 b d}}{7 d}}{9 b}}{11 b}\right )+\frac {f \left (\frac {f \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} x^7}{13 b}+\frac {\frac {(13 b d e+3 b c f-10 a d f) \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^7}{11 b}+\frac {\frac {\left (3 c (52 d e+c f) b^2-a d (104 d e+111 c f) b+80 a^2 d^2 f\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x^7}{9 d}+\frac {\frac {\left (3 c^2 (13 d e-8 c f) b^3+a c d (169 d e+9 c f) b^2-a^2 d^2 (104 d e+121 c f) b+80 a^3 d^3 f\right ) x^5 \sqrt {b x^2+a} \sqrt {d x^2+c}}{7 b d}-\frac {\frac {\left (18 c^3 (13 d e-8 c f) b^4-3 a c^2 d (39 d e-19 c f) b^3+a^2 c d^2 (1027 d e+63 c f) b^2-16 a^3 d^3 (39 d e+46 c f) b+480 a^4 d^4 f\right ) x^3 \sqrt {b x^2+a} \sqrt {d x^2+c}}{5 b d}-\frac {3 \left (\frac {\left (24 c^4 (13 d e-8 c f) b^5-a c^3 d (169 d e-84 c f) b^4-5 a^2 c^2 d^2 (39 d e-17 c f) b^3+3 a^3 c d^3 (468 d e+37 c f) b^2-16 a^4 d^4 (52 d e+63 c f) b+640 a^5 d^5 f\right ) x \sqrt {b x^2+a} \sqrt {d x^2+c}}{3 b d}-\frac {a c \left (24 c^4 (13 d e-8 c f) b^5-a c^3 d (169 d e-84 c f) b^4-5 a^2 c^2 d^2 (39 d e-17 c f) b^3+3 a^3 c d^3 (468 d e+37 c f) b^2-16 a^4 d^4 (52 d e+63 c f) b+640 a^5 d^5 f\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\left (48 c^5 (13 d e-8 c f) b^6-8 a c^4 d (52 d e-27 c f) b^5-a^2 c^3 d^2 (377 d e-167 c f) b^4-a^3 c^2 d^3 (663 d e-203 c f) b^3+2 a^4 c d^4 (1508 d e+207 c f) b^2-128 a^5 d^5 (13 d e+17 c f) b+1280 a^6 d^6 f\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b d}\right )}{5 b d}}{7 b d}}{9 d}}{11 b}}{13 b}\right )}{e^2}\)

\(\Big \downarrow \) 320

\(\displaystyle e \left (\frac {f \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} x^5}{11 b}+\frac {\frac {(11 b d e+3 b c f-8 a d f) \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^5}{9 b}+\frac {\frac {\left (c (110 d e+3 c f) b^2-a d (66 d e+71 c f) b+48 a^2 d^2 f\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x^5}{7 d}+\frac {\frac {\left (3 c^2 (11 d e-6 c f) b^3+a c d (121 d e+9 c f) b^2-a^2 d^2 (66 d e+79 c f) b+48 a^3 d^3 f\right ) x^3 \sqrt {b x^2+a} \sqrt {d x^2+c}}{5 b d}-\frac {3 \left (\frac {\left (4 c^3 (11 d e-6 c f) b^4-a c^2 d (33 d e-13 c f) b^3+15 a^2 c d^2 (11 d e+c f) b^2-4 a^3 d^3 (22 d e+27 c f) b+64 a^4 d^4 f\right ) x \sqrt {b x^2+a} \sqrt {d x^2+c}}{3 b d}-\frac {\frac {\left (4 c^3 (11 d e-6 c f) b^4-a c^2 d (33 d e-13 c f) b^3+15 a^2 c d^2 (11 d e+c f) b^2-4 a^3 d^3 (22 d e+27 c f) b+64 a^4 d^4 f\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 ) c^{3/2}}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\left (8 c^4 (11 d e-6 c f) b^5-a c^3 d (77 d e-32 c f) b^4-a^2 c^2 d^2 (99 d e-29 c f) b^3+a^3 c d^3 (352 d e+51 c f) b^2-8 a^4 d^4 (22 d e+29 c f) b+128 a^5 d^5 f\right ) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{3 b d}\right )}{5 b d}}{7 d}}{9 b}}{11 b}\right )+\frac {f \left (\frac {f \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} x^7}{13 b}+\frac {\frac {(13 b d e+3 b c f-10 a d f) \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^7}{11 b}+\frac {\frac {\left (3 c (52 d e+c f) b^2-a d (104 d e+111 c f) b+80 a^2 d^2 f\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x^7}{9 d}+\frac {\frac {\left (3 c^2 (13 d e-8 c f) b^3+a c d (169 d e+9 c f) b^2-a^2 d^2 (104 d e+121 c f) b+80 a^3 d^3 f\right ) x^5 \sqrt {b x^2+a} \sqrt {d x^2+c}}{7 b d}-\frac {\frac {\left (18 c^3 (13 d e-8 c f) b^4-3 a c^2 d (39 d e-19 c f) b^3+a^2 c d^2 (1027 d e+63 c f) b^2-16 a^3 d^3 (39 d e+46 c f) b+480 a^4 d^4 f\right ) x^3 \sqrt {b x^2+a} \sqrt {d x^2+c}}{5 b d}-\frac {3 \left (\frac {\left (24 c^4 (13 d e-8 c f) b^5-a c^3 d (169 d e-84 c f) b^4-5 a^2 c^2 d^2 (39 d e-17 c f) b^3+3 a^3 c d^3 (468 d e+37 c f) b^2-16 a^4 d^4 (52 d e+63 c f) b+640 a^5 d^5 f\right ) x \sqrt {b x^2+a} \sqrt {d x^2+c}}{3 b d}-\frac {\frac {\left (24 c^4 (13 d e-8 c f) b^5-a c^3 d (169 d e-84 c f) b^4-5 a^2 c^2 d^2 (39 d e-17 c f) b^3+3 a^3 c d^3 (468 d e+37 c f) b^2-16 a^4 d^4 (52 d e+63 c f) b+640 a^5 d^5 f\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 ) c^{3/2}}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\left (48 c^5 (13 d e-8 c f) b^6-8 a c^4 d (52 d e-27 c f) b^5-a^2 c^3 d^2 (377 d e-167 c f) b^4-a^3 c^2 d^3 (663 d e-203 c f) b^3+2 a^4 c d^4 (1508 d e+207 c f) b^2-128 a^5 d^5 (13 d e+17 c f) b+1280 a^6 d^6 f\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b d}\right )}{5 b d}}{7 b d}}{9 d}}{11 b}}{13 b}\right )}{e^2}\)

\(\Big \downarrow \) 388

\(\displaystyle e \left (\frac {f \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} x^5}{11 b}+\frac {\frac {(11 b d e+3 b c f-8 a d f) \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^5}{9 b}+\frac {\frac {\left (c (110 d e+3 c f) b^2-a d (66 d e+71 c f) b+48 a^2 d^2 f\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x^5}{7 d}+\frac {\frac {\left (3 c^2 (11 d e-6 c f) b^3+a c d (121 d e+9 c f) b^2-a^2 d^2 (66 d e+79 c f) b+48 a^3 d^3 f\right ) x^3 \sqrt {b x^2+a} \sqrt {d x^2+c}}{5 b d}-\frac {3 \left (\frac {\left (4 c^3 (11 d e-6 c f) b^4-a c^2 d (33 d e-13 c f) b^3+15 a^2 c d^2 (11 d e+c f) b^2-4 a^3 d^3 (22 d e+27 c f) b+64 a^4 d^4 f\right ) x \sqrt {b x^2+a} \sqrt {d x^2+c}}{3 b d}-\frac {\frac {\left (4 c^3 (11 d e-6 c f) b^4-a c^2 d (33 d e-13 c f) b^3+15 a^2 c d^2 (11 d e+c f) b^2-4 a^3 d^3 (22 d e+27 c f) b+64 a^4 d^4 f\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 ) c^{3/2}}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\left (8 c^4 (11 d e-6 c f) b^5-a c^3 d (77 d e-32 c f) b^4-a^2 c^2 d^2 (99 d e-29 c f) b^3+a^3 c d^3 (352 d e+51 c f) b^2-8 a^4 d^4 (22 d e+29 c f) b+128 a^5 d^5 f\right ) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{3 b d}\right )}{5 b d}}{7 d}}{9 b}}{11 b}\right )+\frac {f \left (\frac {f \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} x^7}{13 b}+\frac {\frac {(13 b d e+3 b c f-10 a d f) \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^7}{11 b}+\frac {\frac {\left (3 c (52 d e+c f) b^2-a d (104 d e+111 c f) b+80 a^2 d^2 f\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x^7}{9 d}+\frac {\frac {\left (3 c^2 (13 d e-8 c f) b^3+a c d (169 d e+9 c f) b^2-a^2 d^2 (104 d e+121 c f) b+80 a^3 d^3 f\right ) x^5 \sqrt {b x^2+a} \sqrt {d x^2+c}}{7 b d}-\frac {\frac {\left (18 c^3 (13 d e-8 c f) b^4-3 a c^2 d (39 d e-19 c f) b^3+a^2 c d^2 (1027 d e+63 c f) b^2-16 a^3 d^3 (39 d e+46 c f) b+480 a^4 d^4 f\right ) x^3 \sqrt {b x^2+a} \sqrt {d x^2+c}}{5 b d}-\frac {3 \left (\frac {\left (24 c^4 (13 d e-8 c f) b^5-a c^3 d (169 d e-84 c f) b^4-5 a^2 c^2 d^2 (39 d e-17 c f) b^3+3 a^3 c d^3 (468 d e+37 c f) b^2-16 a^4 d^4 (52 d e+63 c f) b+640 a^5 d^5 f\right ) x \sqrt {b x^2+a} \sqrt {d x^2+c}}{3 b d}-\frac {\frac {\left (24 c^4 (13 d e-8 c f) b^5-a c^3 d (169 d e-84 c f) b^4-5 a^2 c^2 d^2 (39 d e-17 c f) b^3+3 a^3 c d^3 (468 d e+37 c f) b^2-16 a^4 d^4 (52 d e+63 c f) b+640 a^5 d^5 f\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 ) c^{3/2}}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\left (48 c^5 (13 d e-8 c f) b^6-8 a c^4 d (52 d e-27 c f) b^5-a^2 c^3 d^2 (377 d e-167 c f) b^4-a^3 c^2 d^3 (663 d e-203 c f) b^3+2 a^4 c d^4 (1508 d e+207 c f) b^2-128 a^5 d^5 (13 d e+17 c f) b+1280 a^6 d^6 f\right ) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )}{3 b d}\right )}{5 b d}}{7 b d}}{9 d}}{11 b}}{13 b}\right )}{e^2}\)

\(\Big \downarrow \) 313

\(\displaystyle e \left (\frac {f \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} x^5}{11 b}+\frac {\frac {(11 b d e+3 b c f-8 a d f) \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^5}{9 b}+\frac {\frac {\left (c (110 d e+3 c f) b^2-a d (66 d e+71 c f) b+48 a^2 d^2 f\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x^5}{7 d}+\frac {\frac {\left (3 c^2 (11 d e-6 c f) b^3+a c d (121 d e+9 c f) b^2-a^2 d^2 (66 d e+79 c f) b+48 a^3 d^3 f\right ) x^3 \sqrt {b x^2+a} \sqrt {d x^2+c}}{5 b d}-\frac {3 \left (\frac {\left (4 c^3 (11 d e-6 c f) b^4-a c^2 d (33 d e-13 c f) b^3+15 a^2 c d^2 (11 d e+c f) b^2-4 a^3 d^3 (22 d e+27 c f) b+64 a^4 d^4 f\right ) x \sqrt {b x^2+a} \sqrt {d x^2+c}}{3 b d}-\frac {\frac {\left (4 c^3 (11 d e-6 c f) b^4-a c^2 d (33 d e-13 c f) b^3+15 a^2 c d^2 (11 d e+c f) b^2-4 a^3 d^3 (22 d e+27 c f) b+64 a^4 d^4 f\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 ) c^{3/2}}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\left (8 c^4 (11 d e-6 c f) b^5-a c^3 d (77 d e-32 c f) b^4-a^2 c^2 d^2 (99 d e-29 c f) b^3+a^3 c d^3 (352 d e+51 c f) b^2-8 a^4 d^4 (22 d e+29 c f) b+128 a^5 d^5 f\right ) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{3 b d}\right )}{5 b d}}{7 d}}{9 b}}{11 b}\right )+\frac {f \left (\frac {f \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} x^7}{13 b}+\frac {\frac {(13 b d e+3 b c f-10 a d f) \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^7}{11 b}+\frac {\frac {\left (3 c (52 d e+c f) b^2-a d (104 d e+111 c f) b+80 a^2 d^2 f\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x^7}{9 d}+\frac {\frac {\left (3 c^2 (13 d e-8 c f) b^3+a c d (169 d e+9 c f) b^2-a^2 d^2 (104 d e+121 c f) b+80 a^3 d^3 f\right ) x^5 \sqrt {b x^2+a} \sqrt {d x^2+c}}{7 b d}-\frac {\frac {\left (18 c^3 (13 d e-8 c f) b^4-3 a c^2 d (39 d e-19 c f) b^3+a^2 c d^2 (1027 d e+63 c f) b^2-16 a^3 d^3 (39 d e+46 c f) b+480 a^4 d^4 f\right ) x^3 \sqrt {b x^2+a} \sqrt {d x^2+c}}{5 b d}-\frac {3 \left (\frac {\left (24 c^4 (13 d e-8 c f) b^5-a c^3 d (169 d e-84 c f) b^4-5 a^2 c^2 d^2 (39 d e-17 c f) b^3+3 a^3 c d^3 (468 d e+37 c f) b^2-16 a^4 d^4 (52 d e+63 c f) b+640 a^5 d^5 f\right ) x \sqrt {b x^2+a} \sqrt {d x^2+c}}{3 b d}-\frac {\frac {\left (24 c^4 (13 d e-8 c f) b^5-a c^3 d (169 d e-84 c f) b^4-5 a^2 c^2 d^2 (39 d e-17 c f) b^3+3 a^3 c d^3 (468 d e+37 c f) b^2-16 a^4 d^4 (52 d e+63 c f) b+640 a^5 d^5 f\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 ) c^{3/2}}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\left (48 c^5 (13 d e-8 c f) b^6-8 a c^4 d (52 d e-27 c f) b^5-a^2 c^3 d^2 (377 d e-167 c f) b^4-a^3 c^2 d^3 (663 d e-203 c f) b^3+2 a^4 c d^4 (1508 d e+207 c f) b^2-128 a^5 d^5 (13 d e+17 c f) b+1280 a^6 d^6 f\right ) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{3 b d}\right )}{5 b d}}{7 b d}}{9 d}}{11 b}}{13 b}\right )}{e^2}\)

Input:

Int[x^4*Sqrt[a + b*x^2]*(c + d*x^2)^(3/2)*(e + f*x^2)^2,x]
 

Output:

e*((f*x^5*(a + b*x^2)^(3/2)*(c + d*x^2)^(3/2))/(11*b) + (((11*b*d*e + 3*b* 
c*f - 8*a*d*f)*x^5*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(9*b) + (((48*a^2*d^ 
2*f + b^2*c*(110*d*e + 3*c*f) - a*b*d*(66*d*e + 71*c*f))*x^5*Sqrt[a + b*x^ 
2]*Sqrt[c + d*x^2])/(7*d) + (((48*a^3*d^3*f + 3*b^3*c^2*(11*d*e - 6*c*f) + 
 a*b^2*c*d*(121*d*e + 9*c*f) - a^2*b*d^2*(66*d*e + 79*c*f))*x^3*Sqrt[a + b 
*x^2]*Sqrt[c + d*x^2])/(5*b*d) - (3*(((64*a^4*d^4*f - a*b^3*c^2*d*(33*d*e 
- 13*c*f) + 4*b^4*c^3*(11*d*e - 6*c*f) + 15*a^2*b^2*c*d^2*(11*d*e + c*f) - 
 4*a^3*b*d^3*(22*d*e + 27*c*f))*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(3*b*d) 
 - ((128*a^5*d^5*f - a*b^4*c^3*d*(77*d*e - 32*c*f) - a^2*b^3*c^2*d^2*(99*d 
*e - 29*c*f) + 8*b^5*c^4*(11*d*e - 6*c*f) - 8*a^4*b*d^4*(22*d*e + 29*c*f) 
+ a^3*b^2*c*d^3*(352*d*e + 51*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)*(64*a^4*d^4*f - a*b^3*c^2*d*(33*d*e - 13*c*f) + 4*b^4*c^3 
*(11*d*e - 6*c*f) + 15*a^2*b^2*c*d^2*(11*d*e + c*f) - 4*a^3*b*d^3*(22*d*e 
+ 27*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*b*d)))/(5*b*d))/(7*d))/(9*b))/(11*b)) + (f*((f*x^7*(a + b*x^2)^(3/2)*( 
c + d*x^2)^(3/2))/(13*b) + (((13*b*d*e + 3*b*c*f - 10*a*d*f)*x^7*(a + b*x^ 
2)^(3/2)*Sqrt[c + d*x^2])/(11*b) + (((80*a^2*d^2*f + 3*b^2*c*(52*d*e + ...
 

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 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]
 

rule 443

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

rule 444

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

rule 448

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q 
_.)*((e_) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Simp[e   Int[(g*x)^m*(a + b*x 
^2)^p*(c + d*x^2)^q*(e + f*x^2)^(r - 1), x], x] + Simp[f/e^2   Int[(g*x)^(m 
 + 2)*(a + b*x^2)^p*(c + d*x^2)^q*(e + f*x^2)^(r - 1), x], x] /; FreeQ[{a, 
b, c, d, e, f, g, m, p, q}, x] && IGtQ[r, 0]
 

Maple [A] (verified)

Time = 20.77 (sec) , antiderivative size = 2657, normalized size of antiderivative = 1.85

method	result	size
risch	\(\text {Expression too large to display}\)	\(2657\)
elliptic	\(\text {Expression too large to display}\)	\(3316\)
default	\(\text {Expression too large to display}\)	\(4274\)

Input:

int(x^4*(b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)*(f*x^2+e)^2,x,method=_RETURNVERBOS 
E)
 

Output:

1/45045*x/d^4*(3465*b^5*d^5*f^2*x^10+315*a*b^4*d^5*f^2*x^8+4410*b^5*c*d^4* 
f^2*x^8+8190*b^5*d^5*e*f*x^8-350*a^2*b^3*d^5*f^2*x^6+525*a*b^4*c*d^4*f^2*x 
^6+910*a*b^4*d^5*e*f*x^6+105*b^5*c^2*d^3*f^2*x^6+10920*b^5*c*d^4*e*f*x^6+5 
005*b^5*d^5*e^2*x^6+400*a^3*b^2*d^5*f^2*x^4-605*a^2*b^3*c*d^4*f^2*x^4-1040 
*a^2*b^3*d^5*e*f*x^4+45*a*b^4*c^2*d^3*f^2*x^4+1690*a*b^4*c*d^4*e*f*x^4+715 
*a*b^4*d^5*e^2*x^4-120*b^5*c^3*d^2*f^2*x^4+390*b^5*c^2*d^3*e*f*x^4+7150*b^ 
5*c*d^4*e^2*x^4-480*a^4*b*d^5*f^2*x^2+736*a^3*b^2*c*d^4*f^2*x^2+1248*a^3*b 
^2*d^5*e*f*x^2-63*a^2*b^3*c^2*d^3*f^2*x^2-2054*a^2*b^3*c*d^4*e*f*x^2-858*a 
^2*b^3*d^5*e^2*x^2-57*a*b^4*c^3*d^2*f^2*x^2+234*a*b^4*c^2*d^3*e*f*x^2+1573 
*a*b^4*c*d^4*e^2*x^2+144*b^5*c^4*d*f^2*x^2-468*b^5*c^3*d^2*e*f*x^2+429*b^5 
*c^2*d^3*e^2*x^2+640*a^5*d^5*f^2-1008*a^4*b*c*d^4*f^2-1664*a^4*b*d^5*e*f+1 
11*a^3*b^2*c^2*d^3*f^2+2808*a^3*b^2*c*d^4*e*f+1144*a^3*b^2*d^5*e^2+85*a^2* 
b^3*c^3*d^2*f^2-390*a^2*b^3*c^2*d^3*e*f-2145*a^2*b^3*c*d^4*e^2+84*a*b^4*c^ 
4*d*f^2-338*a*b^4*c^3*d^2*e*f+429*a*b^4*c^2*d^3*e^2-192*b^5*c^5*f^2+624*b^ 
5*c^4*d*e*f-572*b^5*c^3*d^2*e^2)*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b^5-1/450 
45/d^4/b^5*(-(1280*a^6*d^6*f^2-2176*a^5*b*c*d^5*f^2-3328*a^5*b*d^6*e*f+414 
*a^4*b^2*c^2*d^4*f^2+6032*a^4*b^2*c*d^5*e*f+2288*a^4*b^2*d^6*e^2+203*a^3*b 
^3*c^3*d^3*f^2-1326*a^3*b^3*c^2*d^4*e*f-4576*a^3*b^3*c*d^5*e^2+167*a^2*b^4 
*c^4*d^2*f^2-754*a^2*b^4*c^3*d^3*e*f+1287*a^2*b^4*c^2*d^4*e^2+216*a*b^5*c^ 
5*d*f^2-832*a*b^5*c^4*d^2*e*f+1001*a*b^5*c^3*d^3*e^2-384*b^6*c^6*f^2+12...
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 1561, normalized size of antiderivative = 1.09 \[ \int x^4 \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2 \, dx=\text {Too large to display} \] Input:

integrate(x^4*(b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)*(f*x^2+e)^2,x, algorithm="fr 
icas")
 

Output:

-1/45045*((143*(8*b^6*c^5*d^2 - 7*a*b^5*c^4*d^3 - 9*a^2*b^4*c^3*d^4 + 32*a 
^3*b^3*c^2*d^5 - 16*a^4*b^2*c*d^6)*e^2 - 26*(48*b^6*c^6*d - 32*a*b^5*c^5*d 
^2 - 29*a^2*b^4*c^4*d^3 - 51*a^3*b^3*c^3*d^4 + 232*a^4*b^2*c^2*d^5 - 128*a 
^5*b*c*d^6)*e*f + (384*b^6*c^7 - 216*a*b^5*c^6*d - 167*a^2*b^4*c^5*d^2 - 2 
03*a^3*b^3*c^4*d^3 - 414*a^4*b^2*c^3*d^4 + 2176*a^5*b*c^2*d^5 - 1280*a^6*c 
*d^6)*f^2)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), a*d/(b* 
c)) - (143*(8*b^6*c^5*d^2 - 7*a*b^5*c^4*d^3 - 8*a^4*b^2*d^7 - (9*a^2*b^4 - 
 4*a*b^5)*c^3*d^4 + (32*a^3*b^3 - 3*a^2*b^4)*c^2*d^5 - (16*a^4*b^2 - 15*a^ 
3*b^3)*c*d^6)*e^2 - 26*(48*b^6*c^6*d - 32*a*b^5*c^5*d^2 - 64*a^5*b*d^7 - ( 
29*a^2*b^4 - 24*a*b^5)*c^4*d^3 - (51*a^3*b^3 + 13*a^2*b^4)*c^3*d^4 + (232* 
a^4*b^2 - 15*a^3*b^3)*c^2*d^5 - 4*(32*a^5*b - 27*a^4*b^2)*c*d^6)*e*f + (38 
4*b^6*c^7 - 216*a*b^5*c^6*d - 640*a^6*d^7 - (167*a^2*b^4 - 192*a*b^5)*c^5* 
d^2 - 7*(29*a^3*b^3 + 12*a^2*b^4)*c^4*d^3 - (414*a^4*b^2 + 85*a^3*b^3)*c^3 
*d^4 + (2176*a^5*b - 111*a^4*b^2)*c^2*d^5 - 16*(80*a^6 - 63*a^5*b)*c*d^6)* 
f^2)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_f(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - 
(3465*b^6*d^7*f^2*x^12 + 315*(26*b^6*d^7*e*f + (14*b^6*c*d^6 + a*b^5*d^7)* 
f^2)*x^10 + 35*(143*b^6*d^7*e^2 + 26*(12*b^6*c*d^6 + a*b^5*d^7)*e*f + (3*b 
^6*c^2*d^5 + 15*a*b^5*c*d^6 - 10*a^2*b^4*d^7)*f^2)*x^8 + 5*(143*(10*b^6*c* 
d^6 + a*b^5*d^7)*e^2 + 26*(3*b^6*c^2*d^5 + 13*a*b^5*c*d^6 - 8*a^2*b^4*d^7) 
*e*f - (24*b^6*c^3*d^4 - 9*a*b^5*c^2*d^5 + 121*a^2*b^4*c*d^6 - 80*a^3*b...
 

Sympy [F]

\[ \int x^4 \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2 \, dx=\int x^{4} \sqrt {a + b x^{2}} \left (c + d x^{2}\right )^{\frac {3}{2}} \left (e + f x^{2}\right )^{2}\, dx \] Input:

integrate(x**4*(b*x**2+a)**(1/2)*(d*x**2+c)**(3/2)*(f*x**2+e)**2,x)
 

Output:

Integral(x**4*sqrt(a + b*x**2)*(c + d*x**2)**(3/2)*(e + f*x**2)**2, x)
 

Maxima [F]

\[ \int x^4 \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2 \, dx=\int { \sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{2} x^{4} \,d x } \] Input:

integrate(x^4*(b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)*(f*x^2+e)^2,x, algorithm="ma 
xima")
 

Output:

integrate(sqrt(b*x^2 + a)*(d*x^2 + c)^(3/2)*(f*x^2 + e)^2*x^4, x)
 

Giac [F]

\[ \int x^4 \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2 \, dx=\int { \sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{2} x^{4} \,d x } \] Input:

integrate(x^4*(b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)*(f*x^2+e)^2,x, algorithm="gi 
ac")
 

Output:

integrate(sqrt(b*x^2 + a)*(d*x^2 + c)^(3/2)*(f*x^2 + e)^2*x^4, x)
 

Mupad [F(-1)]

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

int(x^4*(a + b*x^2)^(1/2)*(c + d*x^2)^(3/2)*(e + f*x^2)^2,x)
                                                                                    
                                                                                    
 

Output:

int(x^4*(a + b*x^2)^(1/2)*(c + d*x^2)^(3/2)*(e + f*x^2)^2, x)
 

Reduce [F]

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

int(x^4*(b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)*(f*x^2+e)^2,x)
 

Output:

(640*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**5*d**5*f**2*x - 1008*sqrt(c + d* 
x**2)*sqrt(a + b*x**2)*a**4*b*c*d**4*f**2*x - 1664*sqrt(c + d*x**2)*sqrt(a 
 + b*x**2)*a**4*b*d**5*e*f*x - 480*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**4* 
b*d**5*f**2*x**3 + 111*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**3*b**2*c**2*d* 
*3*f**2*x + 2808*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**3*b**2*c*d**4*e*f*x 
+ 736*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**3*b**2*c*d**4*f**2*x**3 + 1144* 
sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**3*b**2*d**5*e**2*x + 1248*sqrt(c + d* 
x**2)*sqrt(a + b*x**2)*a**3*b**2*d**5*e*f*x**3 + 400*sqrt(c + d*x**2)*sqrt 
(a + b*x**2)*a**3*b**2*d**5*f**2*x**5 + 85*sqrt(c + d*x**2)*sqrt(a + b*x** 
2)*a**2*b**3*c**3*d**2*f**2*x - 390*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2 
*b**3*c**2*d**3*e*f*x - 63*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*b**3*c** 
2*d**3*f**2*x**3 - 2145*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*b**3*c*d**4 
*e**2*x - 2054*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*b**3*c*d**4*e*f*x**3 
 - 605*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*b**3*c*d**4*f**2*x**5 - 858* 
sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*b**3*d**5*e**2*x**3 - 1040*sqrt(c + 
 d*x**2)*sqrt(a + b*x**2)*a**2*b**3*d**5*e*f*x**5 - 350*sqrt(c + d*x**2)*s 
qrt(a + b*x**2)*a**2*b**3*d**5*f**2*x**7 + 84*sqrt(c + d*x**2)*sqrt(a + b* 
x**2)*a*b**4*c**4*d*f**2*x - 338*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**4* 
c**3*d**2*e*f*x - 57*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**4*c**3*d**2*f* 
*2*x**3 + 429*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**4*c**2*d**3*e**2*x...