\(\int x^2 \sqrt {a+b x^2} \sqrt {c+d x^2} (e+f x^2)^2 \, dx\) [54]

Optimal result

Integrand size = 35, antiderivative size = 841 \[ \int x^2 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2 \, dx=-\frac {2 \left (8 a^4 d^4 f^2-4 a^3 b d^3 f (6 d e+c f)+3 a^2 b^2 d^2 \left (7 d^2 e^2+5 c d e f-c^2 f^2\right )-a b^3 c d \left (21 d^2 e^2-15 c d e f+4 c^2 f^2\right )+b^4 c^2 \left (21 d^2 e^2-24 c d e f+8 c^2 f^2\right )\right ) x \sqrt {c+d x^2}}{315 b^3 d^4 \sqrt {a+b x^2}}+\frac {\left (8 a^3 d^3 f^2-3 a^2 b d^2 f (8 d e+c f)+3 a b^2 d \left (7 d^2 e^2+4 c d e f-c^2 f^2\right )+b^3 c \left (21 d^2 e^2-24 c d e f+8 c^2 f^2\right )\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{315 b^3 d^3}-\frac {\left (\frac {6 a^2 d f^2}{b}-2 a f (9 d e+c f)-b \left (63 d e^2+18 c e f-\frac {6 c^2 f^2}{d}\right )\right ) x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{315 b d}+\frac {f (18 b d e+b c f+a d f) x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{63 b d}+\frac {1}{9} f^2 x^7 \sqrt {a+b x^2} \sqrt {c+d x^2}+\frac {2 \sqrt {a} \left (8 a^4 d^4 f^2-4 a^3 b d^3 f (6 d e+c f)+3 a^2 b^2 d^2 \left (7 d^2 e^2+5 c d e f-c^2 f^2\right )-a b^3 c d \left (21 d^2 e^2-15 c d e f+4 c^2 f^2\right )+b^4 c^2 \left (21 d^2 e^2-24 c d e f+8 c^2 f^2\right )\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{315 b^{7/2} d^4 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} \left (8 a^3 d^3 f^2-3 a^2 b d^2 f (8 d e+c f)+3 a b^2 d \left (7 d^2 e^2+4 c d e f-c^2 f^2\right )+b^3 c \left (21 d^2 e^2-24 c d e f+8 c^2 f^2\right )\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{315 b^{7/2} d^3 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

-2/315*(8*a^4*d^4*f^2-4*a^3*b*d^3*f*(c*f+6*d*e)+3*a^2*b^2*d^2*(-c^2*f^2+5* 
c*d*e*f+7*d^2*e^2)-a*b^3*c*d*(4*c^2*f^2-15*c*d*e*f+21*d^2*e^2)+b^4*c^2*(8* 
c^2*f^2-24*c*d*e*f+21*d^2*e^2))*x*(d*x^2+c)^(1/2)/b^3/d^4/(b*x^2+a)^(1/2)+ 
1/315*(8*a^3*d^3*f^2-3*a^2*b*d^2*f*(c*f+8*d*e)+3*a*b^2*d*(-c^2*f^2+4*c*d*e 
*f+7*d^2*e^2)+b^3*c*(8*c^2*f^2-24*c*d*e*f+21*d^2*e^2))*x*(b*x^2+a)^(1/2)*( 
d*x^2+c)^(1/2)/b^3/d^3-1/315*(6*a^2*d*f^2/b-2*a*f*(c*f+9*d*e)-b*(63*d*e^2+ 
18*c*e*f-6*c^2*f^2/d))*x^3*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b/d+1/63*f*(a*d 
*f+b*c*f+18*b*d*e)*x^5*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b/d+1/9*f^2*x^7*(b* 
x^2+a)^(1/2)*(d*x^2+c)^(1/2)+2/315*a^(1/2)*(8*a^4*d^4*f^2-4*a^3*b*d^3*f*(c 
*f+6*d*e)+3*a^2*b^2*d^2*(-c^2*f^2+5*c*d*e*f+7*d^2*e^2)-a*b^3*c*d*(4*c^2*f^ 
2-15*c*d*e*f+21*d^2*e^2)+b^4*c^2*(8*c^2*f^2-24*c*d*e*f+21*d^2*e^2))*(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^(7/2)/d^4/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)-1/315*a^(3/2) 
*(8*a^3*d^3*f^2-3*a^2*b*d^2*f*(c*f+8*d*e)+3*a*b^2*d*(-c^2*f^2+4*c*d*e*f+7* 
d^2*e^2)+b^3*c*(8*c^2*f^2-24*c*d*e*f+21*d^2*e^2))*(d*x^2+c)^(1/2)*InverseJ 
acobiAM(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/b^(7/2)/d^3/(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 = 4.23 (sec) , antiderivative size = 570, normalized size of antiderivative = 0.68 \[ \int x^2 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2 \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (8 a^3 d^3 f^2-3 a^2 b d^2 f \left (8 d e+c f+2 d f x^2\right )+a b^2 d \left (-3 c^2 f^2+2 c d f \left (6 e+f x^2\right )+d^2 \left (21 e^2+18 e f x^2+5 f^2 x^4\right )\right )+b^3 \left (8 c^3 f^2-6 c^2 d f \left (4 e+f x^2\right )+c d^2 \left (21 e^2+18 e f x^2+5 f^2 x^4\right )+d^3 x^2 \left (63 e^2+90 e f x^2+35 f^2 x^4\right )\right )\right )+2 i c \left (8 a^4 d^4 f^2-4 a^3 b d^3 f (6 d e+c f)+a b^3 c d \left (-21 d^2 e^2+15 c d e f-4 c^2 f^2\right )+3 a^2 b^2 d^2 \left (7 d^2 e^2+5 c d e f-c^2 f^2\right )+b^4 c^2 \left (21 d^2 e^2-24 c d e f+8 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c (-b c+a d) \left (8 a^3 d^3 f^2+3 a b^2 d^2 e (7 d e-2 c f)+3 a^2 b d^2 f (-8 d e+c f)-2 b^3 c \left (21 d^2 e^2-24 c d e f+8 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{315 b^3 \sqrt {\frac {b}{a}} d^4 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

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

Output:

(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(8*a^3*d^3*f^2 - 3*a^2*b*d^2*f*(8*d 
*e + c*f + 2*d*f*x^2) + a*b^2*d*(-3*c^2*f^2 + 2*c*d*f*(6*e + f*x^2) + d^2* 
(21*e^2 + 18*e*f*x^2 + 5*f^2*x^4)) + b^3*(8*c^3*f^2 - 6*c^2*d*f*(4*e + f*x 
^2) + c*d^2*(21*e^2 + 18*e*f*x^2 + 5*f^2*x^4) + d^3*x^2*(63*e^2 + 90*e*f*x 
^2 + 35*f^2*x^4))) + (2*I)*c*(8*a^4*d^4*f^2 - 4*a^3*b*d^3*f*(6*d*e + c*f) 
+ a*b^3*c*d*(-21*d^2*e^2 + 15*c*d*e*f - 4*c^2*f^2) + 3*a^2*b^2*d^2*(7*d^2* 
e^2 + 5*c*d*e*f - c^2*f^2) + b^4*c^2*(21*d^2*e^2 - 24*c*d*e*f + 8*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)*(8*a^3*d^3*f^2 + 3*a*b^2*d^2*e*(7*d*e 
- 2*c*f) + 3*a^2*b*d^2*f*(-8*d*e + c*f) - 2*b^3*c*(21*d^2*e^2 - 24*c*d*e*f 
 + 8*c^2*f^2))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh 
[Sqrt[b/a]*x], (a*d)/(b*c)])/(315*b^3*Sqrt[b/a]*d^4*Sqrt[a + b*x^2]*Sqrt[c 
 + d*x^2])
 

Rubi [A] (warning: unable to verify)

Time = 1.96 (sec) , antiderivative size = 1103, normalized size of antiderivative = 1.31, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {448, 443, 443, 444, 25, 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.

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

\(\Big \downarrow \) 448

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

\(\Big \downarrow \) 443

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

\(\Big \downarrow \) 443

\(\displaystyle \frac {f \left (\frac {\frac {\int \frac {x^4 \left (a c (18 b d e-5 b c f-5 a d f)-\left (-3 c (3 d e-2 c f) b^2-a d (9 d e+2 c f) b+6 a^2 d^2 f\right ) x^2\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{7 d}+\frac {x^5 \sqrt {a+b x^2} \sqrt {c+d x^2} (-6 a d f+b c f+9 b d e)}{7 d}}{9 b}+\frac {f x^5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{9 b}\right )}{e^2}+e \left (\frac {\frac {\int \frac {x^2 \left (a c (14 b d e-3 b c f-3 a d f)-\left (-c (7 d e-4 c f) b^2-a d (7 d e+2 c f) b+4 a^2 d^2 f\right ) x^2\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 d}+\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (-4 a d f+b c f+7 b d e)}{5 d}}{7 b}+\frac {f x^3 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{7 b}\right )\)

\(\Big \downarrow \) 444

\(\displaystyle \frac {f \left (\frac {\frac {\frac {1}{5} x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (-\frac {6 a^2 d f}{b}+2 a c f+9 a d e-\frac {6 b c^2 f}{d}+9 b c e\right )-\frac {\int -\frac {3 x^2 \left (\left (-4 c^2 (3 d e-2 c f) b^3+3 a c d (2 d e-c f) b^2-3 a^2 d^2 (4 d e+c f) b+8 a^3 d^3 f\right ) x^2+a c \left (-3 c (3 d e-2 c f) b^2-a d (9 d e+2 c f) b+6 a^2 d^2 f\right )\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 b d}}{7 d}+\frac {x^5 \sqrt {a+b x^2} \sqrt {c+d x^2} (-6 a d f+b c f+9 b d e)}{7 d}}{9 b}+\frac {f x^5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{9 b}\right )}{e^2}+e \left (\frac {\frac {\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (-\frac {4 a^2 d f}{b}+2 a c f+7 a d e-\frac {4 b c^2 f}{d}+7 b c e\right )-\frac {\int -\frac {\left (-2 c^2 (7 d e-4 c f) b^3+a c d (14 d e-5 c f) b^2-a^2 d^2 (14 d e+5 c f) b+8 a^3 d^3 f\right ) x^2+a c \left (-c (7 d e-4 c f) b^2-a d (7 d e+2 c f) b+4 a^2 d^2 f\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b d}}{5 d}+\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (-4 a d f+b c f+7 b d e)}{5 d}}{7 b}+\frac {f x^3 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{7 b}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {f \left (\frac {\frac {\frac {1}{5} x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (-\frac {6 a^2 d f}{b}+2 a c f+9 a d e-\frac {6 b c^2 f}{d}+9 b c e\right )-\frac {\int -\frac {3 x^2 \left (\left (-4 c^2 (3 d e-2 c f) b^3+3 a c d (2 d e-c f) b^2-3 a^2 d^2 (4 d e+c f) b+8 a^3 d^3 f\right ) x^2+a c \left (-3 c (3 d e-2 c f) b^2-a d (9 d e+2 c f) b+6 a^2 d^2 f\right )\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 b d}}{7 d}+\frac {x^5 \sqrt {a+b x^2} \sqrt {c+d x^2} (-6 a d f+b c f+9 b d e)}{7 d}}{9 b}+\frac {f x^5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{9 b}\right )}{e^2}+e \left (\frac {\frac {\frac {\int \frac {\left (-2 c^2 (7 d e-4 c f) b^3+a c d (14 d e-5 c f) b^2-a^2 d^2 (14 d e+5 c f) b+8 a^3 d^3 f\right ) x^2+a c \left (-c (7 d e-4 c f) b^2-a d (7 d e+2 c f) b+4 a^2 d^2 f\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b d}+\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (-\frac {4 a^2 d f}{b}+2 a c f+7 a d e-\frac {4 b c^2 f}{d}+7 b c e\right )}{5 d}+\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (-4 a d f+b c f+7 b d e)}{5 d}}{7 b}+\frac {f x^3 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{7 b}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {f \left (\frac {\frac {\frac {3 \int \frac {x^2 \left (\left (-4 c^2 (3 d e-2 c f) b^3+3 a c d (2 d e-c f) b^2-3 a^2 d^2 (4 d e+c f) b+8 a^3 d^3 f\right ) x^2+a c \left (-3 c (3 d e-2 c f) b^2-a d (9 d e+2 c f) b+6 a^2 d^2 f\right )\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 b d}+\frac {1}{5} x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (-\frac {6 a^2 d f}{b}+2 a c f+9 a d e-\frac {6 b c^2 f}{d}+9 b c e\right )}{7 d}+\frac {x^5 \sqrt {a+b x^2} \sqrt {c+d x^2} (-6 a d f+b c f+9 b d e)}{7 d}}{9 b}+\frac {f x^5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{9 b}\right )}{e^2}+e \left (\frac {\frac {\frac {\int \frac {\left (-2 c^2 (7 d e-4 c f) b^3+a c d (14 d e-5 c f) b^2-a^2 d^2 (14 d e+5 c f) b+8 a^3 d^3 f\right ) x^2+a c \left (-c (7 d e-4 c f) b^2-a d (7 d e+2 c f) b+4 a^2 d^2 f\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b d}+\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (-\frac {4 a^2 d f}{b}+2 a c f+7 a d e-\frac {4 b c^2 f}{d}+7 b c e\right )}{5 d}+\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (-4 a d f+b c f+7 b d e)}{5 d}}{7 b}+\frac {f x^3 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{7 b}\right )\)

\(\Big \downarrow \) 406

\(\displaystyle \frac {f \left (\frac {\frac {\frac {3 \int \frac {x^2 \left (\left (-4 c^2 (3 d e-2 c f) b^3+3 a c d (2 d e-c f) b^2-3 a^2 d^2 (4 d e+c f) b+8 a^3 d^3 f\right ) x^2+a c \left (-3 c (3 d e-2 c f) b^2-a d (9 d e+2 c f) b+6 a^2 d^2 f\right )\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 b d}+\frac {1}{5} x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (-\frac {6 a^2 d f}{b}+2 a c f+9 a d e-\frac {6 b c^2 f}{d}+9 b c e\right )}{7 d}+\frac {x^5 \sqrt {a+b x^2} \sqrt {c+d x^2} (-6 a d f+b c f+9 b d e)}{7 d}}{9 b}+\frac {f x^5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{9 b}\right )}{e^2}+e \left (\frac {\frac {\frac {a c \left (4 a^2 d^2 f-a b d (2 c f+7 d e)+b^2 (-c) (7 d e-4 c f)\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\left (8 a^3 d^3 f-a^2 b d^2 (5 c f+14 d e)+a b^2 c d (14 d e-5 c f)-2 b^3 c^2 (7 d e-4 c f)\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b d}+\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (-\frac {4 a^2 d f}{b}+2 a c f+7 a d e-\frac {4 b c^2 f}{d}+7 b c e\right )}{5 d}+\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (-4 a d f+b c f+7 b d e)}{5 d}}{7 b}+\frac {f x^3 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{7 b}\right )\)

\(\Big \downarrow \) 320

\(\displaystyle e \left (\frac {\frac {\frac {\left (8 a^3 d^3 f-a^2 b d^2 (5 c f+14 d e)+a b^2 c d (14 d e-5 c f)-2 b^3 c^2 (7 d e-4 c f)\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} \left (4 a^2 d^2 f-a b d (2 c f+7 d e)+b^2 (-c) (7 d e-4 c f)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 b d}+\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (-\frac {4 a^2 d f}{b}+2 a c f+7 a d e-\frac {4 b c^2 f}{d}+7 b c e\right )}{5 d}+\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (-4 a d f+b c f+7 b d e)}{5 d}}{7 b}+\frac {f x^3 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{7 b}\right )+\frac {f \left (\frac {\frac {\frac {3 \int \frac {x^2 \left (\left (-4 c^2 (3 d e-2 c f) b^3+3 a c d (2 d e-c f) b^2-3 a^2 d^2 (4 d e+c f) b+8 a^3 d^3 f\right ) x^2+a c \left (-3 c (3 d e-2 c f) b^2-a d (9 d e+2 c f) b+6 a^2 d^2 f\right )\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 b d}+\frac {1}{5} x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (-\frac {6 a^2 d f}{b}+2 a c f+9 a d e-\frac {6 b c^2 f}{d}+9 b c e\right )}{7 d}+\frac {x^5 \sqrt {a+b x^2} \sqrt {c+d x^2} (-6 a d f+b c f+9 b d e)}{7 d}}{9 b}+\frac {f x^5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{9 b}\right )}{e^2}\)

\(\Big \downarrow \) 388

\(\displaystyle e \left (\frac {\frac {\frac {\left (8 a^3 d^3 f-a^2 b d^2 (5 c f+14 d e)+a b^2 c d (14 d e-5 c f)-2 b^3 c^2 (7 d e-4 c f)\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} \left (4 a^2 d^2 f-a b d (2 c f+7 d e)+b^2 (-c) (7 d e-4 c f)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 b d}+\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (-\frac {4 a^2 d f}{b}+2 a c f+7 a d e-\frac {4 b c^2 f}{d}+7 b c e\right )}{5 d}+\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (-4 a d f+b c f+7 b d e)}{5 d}}{7 b}+\frac {f x^3 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{7 b}\right )+\frac {f \left (\frac {\frac {\frac {3 \int \frac {x^2 \left (\left (-4 c^2 (3 d e-2 c f) b^3+3 a c d (2 d e-c f) b^2-3 a^2 d^2 (4 d e+c f) b+8 a^3 d^3 f\right ) x^2+a c \left (-3 c (3 d e-2 c f) b^2-a d (9 d e+2 c f) b+6 a^2 d^2 f\right )\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 b d}+\frac {1}{5} x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (-\frac {6 a^2 d f}{b}+2 a c f+9 a d e-\frac {6 b c^2 f}{d}+9 b c e\right )}{7 d}+\frac {x^5 \sqrt {a+b x^2} \sqrt {c+d x^2} (-6 a d f+b c f+9 b d e)}{7 d}}{9 b}+\frac {f x^5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{9 b}\right )}{e^2}\)

\(\Big \downarrow \) 313

\(\displaystyle \frac {f \left (\frac {\frac {\frac {3 \int \frac {x^2 \left (\left (-4 c^2 (3 d e-2 c f) b^3+3 a c d (2 d e-c f) b^2-3 a^2 d^2 (4 d e+c f) b+8 a^3 d^3 f\right ) x^2+a c \left (-3 c (3 d e-2 c f) b^2-a d (9 d e+2 c f) b+6 a^2 d^2 f\right )\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 b d}+\frac {1}{5} x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (-\frac {6 a^2 d f}{b}+2 a c f+9 a d e-\frac {6 b c^2 f}{d}+9 b c e\right )}{7 d}+\frac {x^5 \sqrt {a+b x^2} \sqrt {c+d x^2} (-6 a d f+b c f+9 b d e)}{7 d}}{9 b}+\frac {f x^5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{9 b}\right )}{e^2}+e \left (\frac {\frac {\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (-\frac {4 a^2 d f}{b}+2 a c f+7 a d e-\frac {4 b c^2 f}{d}+7 b c e\right )+\frac {\frac {c^{3/2} \sqrt {a+b x^2} \left (4 a^2 d^2 f-a b d (2 c f+7 d e)+b^2 (-c) (7 d e-4 c f)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\left (8 a^3 d^3 f-a^2 b d^2 (5 c f+14 d e)+a b^2 c d (14 d e-5 c f)-2 b^3 c^2 (7 d e-4 c f)\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{3 b d}}{5 d}+\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (-4 a d f+b c f+7 b d e)}{5 d}}{7 b}+\frac {f x^3 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{7 b}\right )\)

\(\Big \downarrow \) 444

\(\displaystyle e \left (\frac {f \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^3}{7 b}+\frac {\frac {(7 b d e+b c f-4 a d f) \sqrt {b x^2+a} \sqrt {d x^2+c} x^3}{5 d}+\frac {\frac {1}{3} \left (-\frac {4 d f a^2}{b}+7 d e a+2 c f a+7 b c e-\frac {4 b c^2 f}{d}\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x+\frac {\frac {\left (-c (7 d e-4 c f) b^2-a d (7 d e+2 c f) b+4 a^2 d^2 f\right ) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) c^{3/2}}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\left (-2 c^2 (7 d e-4 c f) b^3+a c d (14 d e-5 c f) b^2-a^2 d^2 (14 d e+5 c f) b+8 a^3 d^3 f\right ) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{3 b d}}{5 d}}{7 b}\right )+\frac {f \left (\frac {f \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^5}{9 b}+\frac {\frac {(9 b d e+b c f-6 a d f) \sqrt {b x^2+a} \sqrt {d x^2+c} x^5}{7 d}+\frac {\frac {1}{5} \left (-\frac {6 d f a^2}{b}+9 d e a+2 c f a+9 b c e-\frac {6 b c^2 f}{d}\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x^3+\frac {3 \left (\frac {\left (-4 c^2 (3 d e-2 c f) b^3+3 a c d (2 d e-c f) b^2-3 a^2 d^2 (4 d e+c f) b+8 a^3 d^3 f\right ) x \sqrt {b x^2+a} \sqrt {d x^2+c}}{3 b d}-\frac {\int \frac {\left (-8 c^3 (3 d e-2 c f) b^4+a c^2 d (15 d e-8 c f) b^3+3 a^2 c d^2 (5 d e-2 c f) b^2-8 a^3 d^3 (3 d e+c f) b+16 a^4 d^4 f\right ) x^2+a c \left (-4 c^2 (3 d e-2 c f) b^3+3 a c d (2 d e-c f) b^2-3 a^2 d^2 (4 d e+c f) b+8 a^3 d^3 f\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b d}\right )}{5 b d}}{7 d}}{9 b}\right )}{e^2}\)

\(\Big \downarrow \) 406

\(\displaystyle e \left (\frac {f \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^3}{7 b}+\frac {\frac {(7 b d e+b c f-4 a d f) \sqrt {b x^2+a} \sqrt {d x^2+c} x^3}{5 d}+\frac {\frac {1}{3} \left (-\frac {4 d f a^2}{b}+7 d e a+2 c f a+7 b c e-\frac {4 b c^2 f}{d}\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x+\frac {\frac {\left (-c (7 d e-4 c f) b^2-a d (7 d e+2 c f) b+4 a^2 d^2 f\right ) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) c^{3/2}}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\left (-2 c^2 (7 d e-4 c f) b^3+a c d (14 d e-5 c f) b^2-a^2 d^2 (14 d e+5 c f) b+8 a^3 d^3 f\right ) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{3 b d}}{5 d}}{7 b}\right )+\frac {f \left (\frac {f \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^5}{9 b}+\frac {\frac {(9 b d e+b c f-6 a d f) \sqrt {b x^2+a} \sqrt {d x^2+c} x^5}{7 d}+\frac {\frac {1}{5} \left (-\frac {6 d f a^2}{b}+9 d e a+2 c f a+9 b c e-\frac {6 b c^2 f}{d}\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x^3+\frac {3 \left (\frac {\left (-4 c^2 (3 d e-2 c f) b^3+3 a c d (2 d e-c f) b^2-3 a^2 d^2 (4 d e+c f) b+8 a^3 d^3 f\right ) x \sqrt {b x^2+a} \sqrt {d x^2+c}}{3 b d}-\frac {a c \left (-4 c^2 (3 d e-2 c f) b^3+3 a c d (2 d e-c f) b^2-3 a^2 d^2 (4 d e+c f) b+8 a^3 d^3 f\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\left (-8 c^3 (3 d e-2 c f) b^4+a c^2 d (15 d e-8 c f) b^3+3 a^2 c d^2 (5 d e-2 c f) b^2-8 a^3 d^3 (3 d e+c f) b+16 a^4 d^4 f\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b d}\right )}{5 b d}}{7 d}}{9 b}\right )}{e^2}\)

\(\Big \downarrow \) 320

\(\displaystyle e \left (\frac {f \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^3}{7 b}+\frac {\frac {(7 b d e+b c f-4 a d f) \sqrt {b x^2+a} \sqrt {d x^2+c} x^3}{5 d}+\frac {\frac {1}{3} \left (-\frac {4 d f a^2}{b}+7 d e a+2 c f a+7 b c e-\frac {4 b c^2 f}{d}\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x+\frac {\frac {\left (-c (7 d e-4 c f) b^2-a d (7 d e+2 c f) b+4 a^2 d^2 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 (-2 c^2 (7 d e-4 c f) b^3+a c d (14 d e-5 c f) b^2-a^2 d^2 (14 d e+5 c f) b+8 a^3 d^3 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}}{5 d}}{7 b}\right )+\frac {f \left (\frac {f \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^5}{9 b}+\frac {\frac {(9 b d e+b c f-6 a d f) \sqrt {b x^2+a} \sqrt {d x^2+c} x^5}{7 d}+\frac {\frac {1}{5} \left (-\frac {6 d f a^2}{b}+9 d e a+2 c f a+9 b c e-\frac {6 b c^2 f}{d}\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x^3+\frac {3 \left (\frac {\left (-4 c^2 (3 d e-2 c f) b^3+3 a c d (2 d e-c f) b^2-3 a^2 d^2 (4 d e+c f) b+8 a^3 d^3 f\right ) x \sqrt {b x^2+a} \sqrt {d x^2+c}}{3 b d}-\frac {\frac {\left (-4 c^2 (3 d e-2 c f) b^3+3 a c d (2 d e-c f) b^2-3 a^2 d^2 (4 d e+c f) b+8 a^3 d^3 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^3 (3 d e-2 c f) b^4+a c^2 d (15 d e-8 c f) b^3+3 a^2 c d^2 (5 d e-2 c f) b^2-8 a^3 d^3 (3 d e+c f) b+16 a^4 d^4 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}\right )}{e^2}\)

\(\Big \downarrow \) 388

\(\displaystyle e \left (\frac {f \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^3}{7 b}+\frac {\frac {(7 b d e+b c f-4 a d f) \sqrt {b x^2+a} \sqrt {d x^2+c} x^3}{5 d}+\frac {\frac {1}{3} \left (-\frac {4 d f a^2}{b}+7 d e a+2 c f a+7 b c e-\frac {4 b c^2 f}{d}\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x+\frac {\frac {\left (-c (7 d e-4 c f) b^2-a d (7 d e+2 c f) b+4 a^2 d^2 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 (-2 c^2 (7 d e-4 c f) b^3+a c d (14 d e-5 c f) b^2-a^2 d^2 (14 d e+5 c f) b+8 a^3 d^3 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}}{5 d}}{7 b}\right )+\frac {f \left (\frac {f \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^5}{9 b}+\frac {\frac {(9 b d e+b c f-6 a d f) \sqrt {b x^2+a} \sqrt {d x^2+c} x^5}{7 d}+\frac {\frac {1}{5} \left (-\frac {6 d f a^2}{b}+9 d e a+2 c f a+9 b c e-\frac {6 b c^2 f}{d}\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x^3+\frac {3 \left (\frac {\left (-4 c^2 (3 d e-2 c f) b^3+3 a c d (2 d e-c f) b^2-3 a^2 d^2 (4 d e+c f) b+8 a^3 d^3 f\right ) x \sqrt {b x^2+a} \sqrt {d x^2+c}}{3 b d}-\frac {\frac {\left (-4 c^2 (3 d e-2 c f) b^3+3 a c d (2 d e-c f) b^2-3 a^2 d^2 (4 d e+c f) b+8 a^3 d^3 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^3 (3 d e-2 c f) b^4+a c^2 d (15 d e-8 c f) b^3+3 a^2 c d^2 (5 d e-2 c f) b^2-8 a^3 d^3 (3 d e+c f) b+16 a^4 d^4 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}\right )}{e^2}\)

\(\Big \downarrow \) 313

\(\displaystyle e \left (\frac {f \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^3}{7 b}+\frac {\frac {(7 b d e+b c f-4 a d f) \sqrt {b x^2+a} \sqrt {d x^2+c} x^3}{5 d}+\frac {\frac {1}{3} \left (-\frac {4 d f a^2}{b}+7 d e a+2 c f a+7 b c e-\frac {4 b c^2 f}{d}\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x+\frac {\frac {\left (-c (7 d e-4 c f) b^2-a d (7 d e+2 c f) b+4 a^2 d^2 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 (-2 c^2 (7 d e-4 c f) b^3+a c d (14 d e-5 c f) b^2-a^2 d^2 (14 d e+5 c f) b+8 a^3 d^3 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}}{5 d}}{7 b}\right )+\frac {f \left (\frac {f \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^5}{9 b}+\frac {\frac {(9 b d e+b c f-6 a d f) \sqrt {b x^2+a} \sqrt {d x^2+c} x^5}{7 d}+\frac {\frac {1}{5} \left (-\frac {6 d f a^2}{b}+9 d e a+2 c f a+9 b c e-\frac {6 b c^2 f}{d}\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x^3+\frac {3 \left (\frac {\left (-4 c^2 (3 d e-2 c f) b^3+3 a c d (2 d e-c f) b^2-3 a^2 d^2 (4 d e+c f) b+8 a^3 d^3 f\right ) x \sqrt {b x^2+a} \sqrt {d x^2+c}}{3 b d}-\frac {\frac {\left (-4 c^2 (3 d e-2 c f) b^3+3 a c d (2 d e-c f) b^2-3 a^2 d^2 (4 d e+c f) b+8 a^3 d^3 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^3 (3 d e-2 c f) b^4+a c^2 d (15 d e-8 c f) b^3+3 a^2 c d^2 (5 d e-2 c f) b^2-8 a^3 d^3 (3 d e+c f) b+16 a^4 d^4 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}\right )}{e^2}\)

Input:

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

Output:

e*((f*x^3*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(7*b) + (((7*b*d*e + b*c*f - 
4*a*d*f)*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(5*d) + (((7*b*c*e + 7*a*d*e 
 + 2*a*c*f - (4*b*c^2*f)/d - (4*a^2*d*f)/b)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x 
^2])/3 + ((8*a^3*d^3*f + a*b^2*c*d*(14*d*e - 5*c*f) - 2*b^3*c^2*(7*d*e - 4 
*c*f) - a^2*b*d^2*(14*d*e + 5*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^2*d^2*f - b^2*c*(7*d*e - 4*c*f) - a*b*d*(7*d*e + 2*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*d))/(7*b)) + (f*((f*x^5*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(9*b) + 
(((9*b*d*e + b*c*f - 6*a*d*f)*x^5*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(7*d) + 
 (((9*b*c*e + 9*a*d*e + 2*a*c*f - (6*b*c^2*f)/d - (6*a^2*d*f)/b)*x^3*Sqrt[ 
a + b*x^2]*Sqrt[c + d*x^2])/5 + (3*(((8*a^3*d^3*f - 4*b^3*c^2*(3*d*e - 2*c 
*f) + 3*a*b^2*c*d*(2*d*e - c*f) - 3*a^2*b*d^2*(4*d*e + c*f))*x*Sqrt[a + b* 
x^2]*Sqrt[c + d*x^2])/(3*b*d) - ((16*a^4*d^4*f + a*b^3*c^2*d*(15*d*e - 8*c 
*f) - 8*b^4*c^3*(3*d*e - 2*c*f) + 3*a^2*b^2*c*d^2*(5*d*e - 2*c*f) - 8*a^3* 
b*d^3*(3*d*e + c*f))*((x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[c]*S 
qrt[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^(...
 

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 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 = 7.88 (sec) , antiderivative size = 1075, normalized size of antiderivative = 1.28

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1075\)
risch	\(\text {Expression too large to display}\)	\(1494\)
default	\(\text {Expression too large to display}\)	\(2491\)

Input:

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

Output:

((b*x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(1/9*f^2*x^7*( 
b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+1/7*(a*d*f^2+b*c*f^2+2*d*b*e*f-1/9*f^2* 
(8*a*d+8*b*c))/b/d*x^5*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+1/5*(2/9*a*c*f^ 
2+2*a*d*e*f+2*b*c*e*f+b*d*e^2-1/7*(a*d*f^2+b*c*f^2+2*d*b*e*f-1/9*f^2*(8*a* 
d+8*b*c))/b/d*(6*a*d+6*b*c))/b/d*x^3*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+1 
/3*(2*a*c*e*f+a*d*e^2+b*c*e^2-5/7*(a*d*f^2+b*c*f^2+2*d*b*e*f-1/9*f^2*(8*a* 
d+8*b*c))/b/d*a*c-1/5*(2/9*a*c*f^2+2*a*d*e*f+2*b*c*e*f+b*d*e^2-1/7*(a*d*f^ 
2+b*c*f^2+2*d*b*e*f-1/9*f^2*(8*a*d+8*b*c))/b/d*(6*a*d+6*b*c))/b/d*(4*a*d+4 
*b*c))/b/d*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)-1/3*(2*a*c*e*f+a*d*e^2+b* 
c*e^2-5/7*(a*d*f^2+b*c*f^2+2*d*b*e*f-1/9*f^2*(8*a*d+8*b*c))/b/d*a*c-1/5*(2 
/9*a*c*f^2+2*a*d*e*f+2*b*c*e*f+b*d*e^2-1/7*(a*d*f^2+b*c*f^2+2*d*b*e*f-1/9* 
f^2*(8*a*d+8*b*c))/b/d*(6*a*d+6*b*c))/b/d*(4*a*d+4*b*c))/b/d*a*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 
)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-(a*c*e^2-3/5*(2/9*a*c 
*f^2+2*a*d*e*f+2*b*c*e*f+b*d*e^2-1/7*(a*d*f^2+b*c*f^2+2*d*b*e*f-1/9*f^2*(8 
*a*d+8*b*c))/b/d*(6*a*d+6*b*c))/b/d*a*c-1/3*(2*a*c*e*f+a*d*e^2+b*c*e^2-5/7 
*(a*d*f^2+b*c*f^2+2*d*b*e*f-1/9*f^2*(8*a*d+8*b*c))/b/d*a*c-1/5*(2/9*a*c*f^ 
2+2*a*d*e*f+2*b*c*e*f+b*d*e^2-1/7*(a*d*f^2+b*c*f^2+2*d*b*e*f-1/9*f^2*(8*a* 
d+8*b*c))/b/d*(6*a*d+6*b*c))/b/d*(4*a*d+4*b*c))/b/d*(2*a*d+2*b*c))*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*...
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 873, normalized size of antiderivative = 1.04 \[ \int x^2 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2 \, dx =\text {Too large to display} \] Input:

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

Output:

1/315*(2*(21*(b^4*c^3*d^2 - a*b^3*c^2*d^3 + a^2*b^2*c*d^4)*e^2 - 3*(8*b^4* 
c^4*d - 5*a*b^3*c^3*d^2 - 5*a^2*b^2*c^2*d^3 + 8*a^3*b*c*d^4)*e*f + (8*b^4* 
c^5 - 4*a*b^3*c^4*d - 3*a^2*b^2*c^3*d^2 - 4*a^3*b*c^2*d^3 + 8*a^4*c*d^4)*f 
^2)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - ( 
21*(2*b^4*c^3*d^2 - 2*a*b^3*c^2*d^3 + a^2*b^2*d^5 + (2*a^2*b^2 + a*b^3)*c* 
d^4)*e^2 - 6*(8*b^4*c^4*d - 5*a*b^3*c^3*d^2 + 4*a^3*b*d^5 - (5*a^2*b^2 - 4 
*a*b^3)*c^2*d^3 + 2*(4*a^3*b - a^2*b^2)*c*d^4)*e*f + (16*b^4*c^5 - 8*a*b^3 
*c^4*d + 8*a^4*d^5 - 2*(3*a^2*b^2 - 4*a*b^3)*c^3*d^2 - (8*a^3*b + 3*a^2*b^ 
2)*c^2*d^3 + (16*a^4 - 3*a^3*b)*c*d^4)*f^2)*sqrt(b*d)*x*sqrt(-c/d)*ellipti 
c_f(arcsin(sqrt(-c/d)/x), a*d/(b*c)) + (35*b^4*d^5*f^2*x^8 + 5*(18*b^4*d^5 
*e*f + (b^4*c*d^4 + a*b^3*d^5)*f^2)*x^6 + (63*b^4*d^5*e^2 + 18*(b^4*c*d^4 
+ a*b^3*d^5)*e*f - 2*(3*b^4*c^2*d^3 - a*b^3*c*d^4 + 3*a^2*b^2*d^5)*f^2)*x^ 
4 - 42*(b^4*c^2*d^3 - a*b^3*c*d^4 + a^2*b^2*d^5)*e^2 + 6*(8*b^4*c^3*d^2 - 
5*a*b^3*c^2*d^3 - 5*a^2*b^2*c*d^4 + 8*a^3*b*d^5)*e*f - 2*(8*b^4*c^4*d - 4* 
a*b^3*c^3*d^2 - 3*a^2*b^2*c^2*d^3 - 4*a^3*b*c*d^4 + 8*a^4*d^5)*f^2 + (21*( 
b^4*c*d^4 + a*b^3*d^5)*e^2 - 12*(2*b^4*c^2*d^3 - a*b^3*c*d^4 + 2*a^2*b^2*d 
^5)*e*f + (8*b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 - 3*a^2*b^2*c*d^4 + 8*a^3*b*d^5 
)*f^2)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(b^4*d^5*x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(8*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**3*d**3*f**2*x - 3*sqrt(c + d*x**2) 
*sqrt(a + b*x**2)*a**2*b*c*d**2*f**2*x - 24*sqrt(c + d*x**2)*sqrt(a + b*x* 
*2)*a**2*b*d**3*e*f*x - 6*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*b*d**3*f* 
*2*x**3 - 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*c**2*d*f**2*x + 12*sq 
rt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*c*d**2*e*f*x + 2*sqrt(c + d*x**2)*s 
qrt(a + b*x**2)*a*b**2*c*d**2*f**2*x**3 + 21*sqrt(c + d*x**2)*sqrt(a + b*x 
**2)*a*b**2*d**3*e**2*x + 18*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*d**3 
*e*f*x**3 + 5*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*d**3*f**2*x**5 + 8* 
sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c**3*f**2*x - 24*sqrt(c + d*x**2)*s 
qrt(a + b*x**2)*b**3*c**2*d*e*f*x - 6*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b* 
*3*c**2*d*f**2*x**3 + 21*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c*d**2*e** 
2*x + 18*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c*d**2*e*f*x**3 + 5*sqrt(c 
 + d*x**2)*sqrt(a + b*x**2)*b**3*c*d**2*f**2*x**5 + 63*sqrt(c + d*x**2)*sq 
rt(a + b*x**2)*b**3*d**3*e**2*x**3 + 90*sqrt(c + d*x**2)*sqrt(a + b*x**2)* 
b**3*d**3*e*f*x**5 + 35*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*d**3*f**2*x 
**7 - 16*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**2 + 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**3*b*c*d**3*f**2 + 
 48*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*f + 6*int((sqrt(c + d*x**2)*sqrt(a + b*x...