\(\int \sqrt {a+b x^2} (c+d x^2)^{3/2} (e+f x^2)^2 \, dx\) [55]

Optimal result

Integrand size = 32, antiderivative size = 849 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2 \, dx=-\frac {\left (16 a^4 d^4 f^2-16 a^3 b d^3 f (3 d e+2 c f)-a b^3 c d \left (147 d^2 e^2+54 c d e f-7 c^2 f^2\right )+3 a^2 b^2 d^2 \left (14 d^2 e^2+38 c d e f+3 c^2 f^2\right )-b^4 c^2 \left (63 d^2 e^2-36 c d e f+8 c^2 f^2\right )\right ) x \sqrt {c+d x^2}}{315 b^3 d^3 \sqrt {a+b x^2}}+\frac {\left (8 a^3 d^3 f^2-3 a^2 b d^2 f (8 d e+5 c f)+2 b^3 c \left (63 d^2 e^2+9 c d e f-2 c^2 f^2\right )+3 a b^2 d \left (7 d^2 e^2+18 c d e f+c^2 f^2\right )\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{315 b^3 d^2}-\frac {\left (6 a^2 d^2 f^2-a b d f (18 d e+11 c f)-3 b^2 \left (21 d^2 e^2+48 c d e f+c^2 f^2\right )\right ) x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{315 b^2 d}+\frac {f (18 b d e+10 b c f+a d f) x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{63 b}+\frac {1}{9} d f^2 x^7 \sqrt {a+b x^2} \sqrt {c+d x^2}+\frac {\sqrt {a} \left (16 a^4 d^4 f^2-16 a^3 b d^3 f (3 d e+2 c f)-a b^3 c d \left (147 d^2 e^2+54 c d e f-7 c^2 f^2\right )+3 a^2 b^2 d^2 \left (14 d^2 e^2+38 c d e f+3 c^2 f^2\right )-b^4 c^2 \left (63 d^2 e^2-36 c d e f+8 c^2 f^2\right )\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{315 b^{7/2} d^3 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} \left (8 a^3 d^3 f^2-3 a^2 b d^2 f (8 d e+5 c f)+3 a b^2 d \left (7 d^2 e^2+18 c d e f+c^2 f^2\right )-b^3 c \left (189 d^2 e^2-18 c d e f+4 c^2 f^2\right )\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{315 b^{7/2} d^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

-1/315*(16*a^4*d^4*f^2-16*a^3*b*d^3*f*(2*c*f+3*d*e)-a*b^3*c*d*(-7*c^2*f^2+ 
54*c*d*e*f+147*d^2*e^2)+3*a^2*b^2*d^2*(3*c^2*f^2+38*c*d*e*f+14*d^2*e^2)-b^ 
4*c^2*(8*c^2*f^2-36*c*d*e*f+63*d^2*e^2))*x*(d*x^2+c)^(1/2)/b^3/d^3/(b*x^2+ 
a)^(1/2)+1/315*(8*a^3*d^3*f^2-3*a^2*b*d^2*f*(5*c*f+8*d*e)+2*b^3*c*(-2*c^2* 
f^2+9*c*d*e*f+63*d^2*e^2)+3*a*b^2*d*(c^2*f^2+18*c*d*e*f+7*d^2*e^2))*x*(b*x 
^2+a)^(1/2)*(d*x^2+c)^(1/2)/b^3/d^2-1/315*(6*a^2*d^2*f^2-a*b*d*f*(11*c*f+1 
8*d*e)-3*b^2*(c^2*f^2+48*c*d*e*f+21*d^2*e^2))*x^3*(b*x^2+a)^(1/2)*(d*x^2+c 
)^(1/2)/b^2/d+1/63*f*(a*d*f+10*b*c*f+18*b*d*e)*x^5*(b*x^2+a)^(1/2)*(d*x^2+ 
c)^(1/2)/b+1/9*d*f^2*x^7*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)+1/315*a^(1/2)*(16 
*a^4*d^4*f^2-16*a^3*b*d^3*f*(2*c*f+3*d*e)-a*b^3*c*d*(-7*c^2*f^2+54*c*d*e*f 
+147*d^2*e^2)+3*a^2*b^2*d^2*(3*c^2*f^2+38*c*d*e*f+14*d^2*e^2)-b^4*c^2*(8*c 
^2*f^2-36*c*d*e*f+63*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^3/(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*(5*c*f+ 
8*d*e)+3*a*b^2*d*(c^2*f^2+18*c*d*e*f+7*d^2*e^2)-b^3*c*(4*c^2*f^2-18*c*d*e* 
f+189*d^2*e^2))*(d*x^2+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*x/a^(1/2)), 
(1-a*d/b/c)^(1/2))/b^(7/2)/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 = 7.78 (sec) , antiderivative size = 584, normalized size of antiderivative = 0.69 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2 \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (8 a^3 d^3 f^2-3 a^2 b d^2 f \left (8 d e+5 c f+2 d f x^2\right )+a b^2 d \left (3 c^2 f^2+c d f \left (54 e+11 f x^2\right )+d^2 \left (21 e^2+18 e f x^2+5 f^2 x^4\right )\right )+b^3 \left (-4 c^3 f^2+3 c^2 d f \left (6 e+f x^2\right )+2 c d^2 \left (63 e^2+72 e f x^2+25 f^2 x^4\right )+d^3 x^2 \left (63 e^2+90 e f x^2+35 f^2 x^4\right )\right )\right )+i c \left (16 a^4 d^4 f^2-16 a^3 b d^3 f (3 d e+2 c f)+b^4 c^2 \left (-63 d^2 e^2+36 c d e f-8 c^2 f^2\right )+3 a^2 b^2 d^2 \left (14 d^2 e^2+38 c d e f+3 c^2 f^2\right )+a b^3 c d \left (-147 d^2 e^2-54 c d e f+7 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i c (b c-a d) \left (8 a^3 d^3 f^2-3 a^2 b d^2 f (8 d e+3 c f)+3 a b^2 d \left (7 d^2 e^2+12 c d e f-c^2 f^2\right )+b^3 c \left (63 d^2 e^2-36 c d e f+8 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{315 b^3 \sqrt {\frac {b}{a}} d^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

Integrate[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)*(8*a^3*d^3*f^2 - 3*a^2*b*d^2*f*(8*d 
*e + 5*c*f + 2*d*f*x^2) + a*b^2*d*(3*c^2*f^2 + c*d*f*(54*e + 11*f*x^2) + d 
^2*(21*e^2 + 18*e*f*x^2 + 5*f^2*x^4)) + b^3*(-4*c^3*f^2 + 3*c^2*d*f*(6*e + 
 f*x^2) + 2*c*d^2*(63*e^2 + 72*e*f*x^2 + 25*f^2*x^4) + d^3*x^2*(63*e^2 + 9 
0*e*f*x^2 + 35*f^2*x^4))) + I*c*(16*a^4*d^4*f^2 - 16*a^3*b*d^3*f*(3*d*e + 
2*c*f) + b^4*c^2*(-63*d^2*e^2 + 36*c*d*e*f - 8*c^2*f^2) + 3*a^2*b^2*d^2*(1 
4*d^2*e^2 + 38*c*d*e*f + 3*c^2*f^2) + a*b^3*c*d*(-147*d^2*e^2 - 54*c*d*e*f 
 + 7*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^2*b*d^2 
*f*(8*d*e + 3*c*f) + 3*a*b^2*d*(7*d^2*e^2 + 12*c*d*e*f - c^2*f^2) + b^3*c* 
(63*d^2*e^2 - 36*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^ 
3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
 

Rubi [A] (verified)

Time = 2.03 (sec) , antiderivative size = 1352, normalized size of antiderivative = 1.59, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {433, 2009}

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.

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

\(\Big \downarrow \) 433

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{9} f^2 \sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} x^5+\frac {(3 b c+a d) f^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x^5}{63 b}+\frac {2}{7} e f \sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} x^3+\frac {\left (-\frac {6 d a^2}{b}+11 c a+\frac {3 b c^2}{d}\right ) f^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x^3}{315 b}+\frac {2 (3 b c+a d) e f \sqrt {b x^2+a} \sqrt {d x^2+c} x^3}{35 b}+\frac {d e^2 \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x}{5 b}+\frac {2 (3 b c-a d) e^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x}{15 b}-\frac {\left (4 b^3 c^3-3 a b^2 d c^2+15 a^2 b d^2 c-8 a^3 d^3\right ) f^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x}{315 b^3 d^2}+\frac {2 \left (-\frac {4 d a^2}{b}+9 c a+\frac {3 b c^2}{d}\right ) e f \sqrt {b x^2+a} \sqrt {d x^2+c} x}{105 b}+\frac {\left (3 b^2 c^2+7 a b d c-2 a^2 d^2\right ) e^2 \sqrt {b x^2+a} x}{15 b^2 \sqrt {d x^2+c}}+\frac {\left (8 b^4 c^4-7 a b^3 d c^3-9 a^2 b^2 d^2 c^2+32 a^3 b d^3 c-16 a^4 d^4\right ) f^2 \sqrt {b x^2+a} x}{315 b^4 d^2 \sqrt {d x^2+c}}-\frac {2 (2 b c-a d) \left (3 b^2 c^2-3 a b d c+8 a^2 d^2\right ) e f \sqrt {b x^2+a} x}{105 b^3 d \sqrt {d x^2+c}}-\frac {\sqrt {c} \left (3 b^2 c^2+7 a b d c-2 a^2 d^2\right ) e^2 \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 b^2 \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\sqrt {c} \left (8 b^4 c^4-7 a b^3 d c^3-9 a^2 b^2 d^2 c^2+32 a^3 b d^3 c-16 a^4 d^4\right ) f^2 \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{315 b^4 d^{5/2} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {2 \sqrt {c} (2 b c-a d) \left (3 b^2 c^2-3 a b d c+8 a^2 d^2\right ) e f \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{105 b^3 d^{3/2} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {c^{3/2} (9 b c-a d) e^2 \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{15 b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {c^{3/2} \left (4 b^3 c^3-3 a b^2 d c^2+15 a^2 b d^2 c-8 a^3 d^3\right ) f^2 \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{315 b^3 d^{5/2} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {2 c^{3/2} \left (3 b^2 c^2+9 a b d c-4 a^2 d^2\right ) e f \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{105 b^2 d^{3/2} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\)

Input:

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

Output:

((3*b^2*c^2 + 7*a*b*c*d - 2*a^2*d^2)*e^2*x*Sqrt[a + b*x^2])/(15*b^2*Sqrt[c 
 + d*x^2]) - (2*(2*b*c - a*d)*(3*b^2*c^2 - 3*a*b*c*d + 8*a^2*d^2)*e*f*x*Sq 
rt[a + b*x^2])/(105*b^3*d*Sqrt[c + d*x^2]) + ((8*b^4*c^4 - 7*a*b^3*c^3*d - 
 9*a^2*b^2*c^2*d^2 + 32*a^3*b*c*d^3 - 16*a^4*d^4)*f^2*x*Sqrt[a + b*x^2])/( 
315*b^4*d^2*Sqrt[c + d*x^2]) + (2*(3*b*c - a*d)*e^2*x*Sqrt[a + b*x^2]*Sqrt 
[c + d*x^2])/(15*b) + (2*(9*a*c + (3*b*c^2)/d - (4*a^2*d)/b)*e*f*x*Sqrt[a 
+ b*x^2]*Sqrt[c + d*x^2])/(105*b) - ((4*b^3*c^3 - 3*a*b^2*c^2*d + 15*a^2*b 
*c*d^2 - 8*a^3*d^3)*f^2*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(315*b^3*d^2) + 
 (2*(3*b*c + a*d)*e*f*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(35*b) + ((11*a 
*c + (3*b*c^2)/d - (6*a^2*d)/b)*f^2*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/( 
315*b) + ((3*b*c + a*d)*f^2*x^5*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(63*b) + 
(d*e^2*x*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(5*b) + (2*e*f*x^3*Sqrt[a + b* 
x^2]*(c + d*x^2)^(3/2))/7 + (f^2*x^5*Sqrt[a + b*x^2]*(c + d*x^2)^(3/2))/9 
- (Sqrt[c]*(3*b^2*c^2 + 7*a*b*c*d - 2*a^2*d^2)*e^2*Sqrt[a + b*x^2]*Ellipti 
cE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*b^2*Sqrt[d]*Sqrt[(c* 
(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (2*Sqrt[c]*(2*b*c - a*d)* 
(3*b^2*c^2 - 3*a*b*c*d + 8*a^2*d^2)*e*f*Sqrt[a + b*x^2]*EllipticE[ArcTan[( 
Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(105*b^3*d^(3/2)*Sqrt[(c*(a + b*x^2 
))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (Sqrt[c]*(8*b^4*c^4 - 7*a*b^3*c^3*d 
 - 9*a^2*b^2*c^2*d^2 + 32*a^3*b*c*d^3 - 16*a^4*d^4)*f^2*Sqrt[a + b*x^2]...
 

Defintions of rubi rules used

rule 433

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

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 11.60 (sec) , antiderivative size = 1300, normalized size of antiderivative = 1.53

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1300\)
risch	\(\text {Expression too large to display}\)	\(1494\)
default	\(\text {Expression too large to display}\)	\(2491\)

Input:

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

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*d*x^7 
*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+1/7*(a*d^2*f^2+2*b*c*d*f^2+2*b*d^2*e* 
f-1/9*f^2*d*(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 
*(11/9*a*c*d*f^2+2*a*d^2*e*f+b*c^2*f^2+4*b*c*d*e*f+b*d^2*e^2-1/7*(a*d^2*f^ 
2+2*b*c*d*f^2+2*b*d^2*e*f-1/9*f^2*d*(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*(a*c^2*f^2+4*a*c*e*f*d+a*d^2*e 
^2+2*b*c^2*e*f+2*b*c*d*e^2-5/7*(a*d^2*f^2+2*b*c*d*f^2+2*b*d^2*e*f-1/9*f^2* 
d*(8*a*d+8*b*c))/b/d*a*c-1/5*(11/9*a*c*d*f^2+2*a*d^2*e*f+b*c^2*f^2+4*b*c*d 
*e*f+b*d^2*e^2-1/7*(a*d^2*f^2+2*b*c*d*f^2+2*b*d^2*e*f-1/9*f^2*d*(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)+(a*c^2*e^2-1/3*(a*c^2*f^2+4*a*c*e*f*d+a*d^2*e^2+2*b*c^2*e*f+2*b 
*c*d*e^2-5/7*(a*d^2*f^2+2*b*c*d*f^2+2*b*d^2*e*f-1/9*f^2*d*(8*a*d+8*b*c))/b 
/d*a*c-1/5*(11/9*a*c*d*f^2+2*a*d^2*e*f+b*c^2*f^2+4*b*c*d*e*f+b*d^2*e^2-1/7 
*(a*d^2*f^2+2*b*c*d*f^2+2*b*d^2*e*f-1/9*f^2*d*(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))-(2*a*c^2*e*f+2*a*c*e^2*d+b*c^2*e^2-3/5*(11/9*a*c*d*f 
^2+2*a*d^2*e*f+b*c^2*f^2+4*b*c*d*e*f+b*d^2*e^2-1/7*(a*d^2*f^2+2*b*c*d*f^2+ 
2*b*d^2*e*f-1/9*f^2*d*(8*a*d+8*b*c))/b/d*(6*a*d+6*b*c))/b/d*a*c-1/3*(a*c^2 
*f^2+4*a*c*e*f*d+a*d^2*e^2+2*b*c^2*e*f+2*b*c*d*e^2-5/7*(a*d^2*f^2+2*b*c...
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 881, normalized size of antiderivative = 1.04 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2 \, dx =\text {Too large to display} \] Input:

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

Output:

-1/315*((21*(3*b^4*c^3*d^2 + 7*a*b^3*c^2*d^3 - 2*a^2*b^2*c*d^4)*e^2 - 6*(6 
*b^4*c^4*d - 9*a*b^3*c^3*d^2 + 19*a^2*b^2*c^2*d^3 - 8*a^3*b*c*d^4)*e*f + ( 
8*b^4*c^5 - 7*a*b^3*c^4*d - 9*a^2*b^2*c^3*d^2 + 32*a^3*b*c^2*d^3 - 16*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*(3*b^4*c^3*d^2 + 7*a*b^3*c^2*d^3 - a^2*b^2*d^5 - (2*a^2*b^2 - 9 
*a*b^3)*c*d^4)*e^2 - 6*(6*b^4*c^4*d - 9*a*b^3*c^3*d^2 - 4*a^3*b*d^5 + (19* 
a^2*b^2 + 3*a*b^3)*c^2*d^3 - (8*a^3*b - 9*a^2*b^2)*c*d^4)*e*f + (8*b^4*c^5 
 - 7*a*b^3*c^4*d - 8*a^4*d^5 - (9*a^2*b^2 - 4*a*b^3)*c^3*d^2 + (32*a^3*b - 
 3*a^2*b^2)*c^2*d^3 - (16*a^4 - 15*a^3*b)*c*d^4)*f^2)*sqrt(b*d)*x*sqrt(-c/ 
d)*elliptic_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 + (10*b^4*c*d^4 + a*b^3*d^5)*f^2)*x^6 + (63*b^4*d^5*e^2 + 1 
8*(8*b^4*c*d^4 + a*b^3*d^5)*e*f + (3*b^4*c^2*d^3 + 11*a*b^3*c*d^4 - 6*a^2* 
b^2*d^5)*f^2)*x^4 + 21*(3*b^4*c^2*d^3 + 7*a*b^3*c*d^4 - 2*a^2*b^2*d^5)*e^2 
 - 6*(6*b^4*c^3*d^2 - 9*a*b^3*c^2*d^3 + 19*a^2*b^2*c*d^4 - 8*a^3*b*d^5)*e* 
f + (8*b^4*c^4*d - 7*a*b^3*c^3*d^2 - 9*a^2*b^2*c^2*d^3 + 32*a^3*b*c*d^4 - 
16*a^4*d^5)*f^2 + (21*(6*b^4*c*d^4 + a*b^3*d^5)*e^2 + 6*(3*b^4*c^2*d^3 + 9 
*a*b^3*c*d^4 - 4*a^2*b^2*d^5)*e*f - (4*b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 15* 
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^4*x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

int((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)*(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 - 15*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 + 54*s 
qrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*c*d**2*e*f*x + 11*sqrt(c + d*x**2) 
*sqrt(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 - 
4*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c**3*f**2*x + 18*sqrt(c + d*x**2) 
*sqrt(a + b*x**2)*b**3*c**2*d*e*f*x + 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)* 
b**3*c**2*d*f**2*x**3 + 126*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c*d**2* 
e**2*x + 144*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c*d**2*e*f*x**3 + 50*s 
qrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c*d**2*f**2*x**5 + 63*sqrt(c + d*x** 
2)*sqrt(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 + 32*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 - 9*int((sqrt(c + d*x**2)*sqrt(a...