\(\int x^5 (c+d \sqrt {a+b x^2})^{3/2} \, dx\) [255]

Optimal result

Integrand size = 23, antiderivative size = 236 \[ \int x^5 \left (c+d \sqrt {a+b x^2}\right )^{3/2} \, dx=-\frac {2 c \left (c^2-a d^2\right )^2 \left (c+d \sqrt {a+b x^2}\right )^{5/2}}{5 b^3 d^6}+\frac {2 \left (5 c^4-6 a c^2 d^2+a^2 d^4\right ) \left (c+d \sqrt {a+b x^2}\right )^{7/2}}{7 b^3 d^6}-\frac {4 c \left (5 c^2-3 a d^2\right ) \left (c+d \sqrt {a+b x^2}\right )^{9/2}}{9 b^3 d^6}+\frac {4 \left (5 c^2-a d^2\right ) \left (c+d \sqrt {a+b x^2}\right )^{11/2}}{11 b^3 d^6}-\frac {10 c \left (c+d \sqrt {a+b x^2}\right )^{13/2}}{13 b^3 d^6}+\frac {2 \left (c+d \sqrt {a+b x^2}\right )^{15/2}}{15 b^3 d^6} \] Output:

-2/5*c*(-a*d^2+c^2)^2*(c+d*(b*x^2+a)^(1/2))^(5/2)/b^3/d^6+2/7*(a^2*d^4-6*a 
*c^2*d^2+5*c^4)*(c+d*(b*x^2+a)^(1/2))^(7/2)/b^3/d^6-4/9*c*(-3*a*d^2+5*c^2) 
*(c+d*(b*x^2+a)^(1/2))^(9/2)/b^3/d^6+4/11*(-a*d^2+5*c^2)*(c+d*(b*x^2+a)^(1 
/2))^(11/2)/b^3/d^6-10/13*c*(c+d*(b*x^2+a)^(1/2))^(13/2)/b^3/d^6+2/15*(c+d 
*(b*x^2+a)^(1/2))^(15/2)/b^3/d^6
 

Mathematica [A] (verified)

Time = 0.72 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.08 \[ \int x^5 \left (c+d \sqrt {a+b x^2}\right )^{3/2} \, dx=\frac {2 \sqrt {c+d \sqrt {a+b x^2}} \left (-256 c^7+96 c^5 d^2 \left (12 a-b x^2\right )+128 c^6 d \sqrt {a+b x^2}+16 c^4 d^3 \sqrt {a+b x^2} \left (-34 a+5 b x^2\right )+3 c^2 d^5 \sqrt {a+b x^2} \left (320 a^2-88 a b x^2+21 b^2 x^4\right )-2 c^3 d^4 \left (1088 a^2-164 a b x^2+35 b^2 x^4\right )+39 d^7 \sqrt {a+b x^2} \left (32 a^3-24 a^2 b x^2+21 a b^2 x^4+77 b^3 x^6\right )+24 c d^6 \left (128 a^3-19 a^2 b x^2+7 a b^2 x^4+154 b^3 x^6\right )\right )}{45045 b^3 d^6} \] Input:

Integrate[x^5*(c + d*Sqrt[a + b*x^2])^(3/2),x]
 

Output:

(2*Sqrt[c + d*Sqrt[a + b*x^2]]*(-256*c^7 + 96*c^5*d^2*(12*a - b*x^2) + 128 
*c^6*d*Sqrt[a + b*x^2] + 16*c^4*d^3*Sqrt[a + b*x^2]*(-34*a + 5*b*x^2) + 3* 
c^2*d^5*Sqrt[a + b*x^2]*(320*a^2 - 88*a*b*x^2 + 21*b^2*x^4) - 2*c^3*d^4*(1 
088*a^2 - 164*a*b*x^2 + 35*b^2*x^4) + 39*d^7*Sqrt[a + b*x^2]*(32*a^3 - 24* 
a^2*b*x^2 + 21*a*b^2*x^4 + 77*b^3*x^6) + 24*c*d^6*(128*a^3 - 19*a^2*b*x^2 
+ 7*a*b^2*x^4 + 154*b^3*x^6)))/(45045*b^3*d^6)
 

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.94, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {7283, 896, 1732, 522, 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.

\(\displaystyle \int x^5 \left (d \sqrt {a+b x^2}+c\right )^{3/2} \, dx\)

\(\Big \downarrow \) 7283

\(\displaystyle \frac {1}{2} \int x^4 \left (c+d \sqrt {b x^2+a}\right )^{3/2}dx^2\)

\(\Big \downarrow \) 896

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

\(\Big \downarrow \) 1732

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

\(\Big \downarrow \) 522

\(\displaystyle \frac {\int \left (\frac {\left (c+d \sqrt {b x^2+a}\right )^{13/2}}{d^5}-\frac {5 c \left (c+d \sqrt {b x^2+a}\right )^{11/2}}{d^5}-\frac {2 \left (a d^2-5 c^2\right ) \left (c+d \sqrt {b x^2+a}\right )^{9/2}}{d^5}-\frac {2 \left (5 c^3-3 a c d^2\right ) \left (c+d \sqrt {b x^2+a}\right )^{7/2}}{d^5}+\frac {\left (5 c^4-6 a d^2 c^2+a^2 d^4\right ) \left (c+d \sqrt {b x^2+a}\right )^{5/2}}{d^5}-\frac {c \left (c^2-a d^2\right )^2 \left (c+d \sqrt {b x^2+a}\right )^{3/2}}{d^5}\right )d\sqrt {b x^2+a}}{b^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {2 \left (a^2 d^4-6 a c^2 d^2+5 c^4\right ) \left (d \sqrt {a+b x^2}+c\right )^{7/2}}{7 d^6}+\frac {4 \left (5 c^2-a d^2\right ) \left (d \sqrt {a+b x^2}+c\right )^{11/2}}{11 d^6}-\frac {4 c \left (5 c^2-3 a d^2\right ) \left (d \sqrt {a+b x^2}+c\right )^{9/2}}{9 d^6}-\frac {2 c \left (c^2-a d^2\right )^2 \left (d \sqrt {a+b x^2}+c\right )^{5/2}}{5 d^6}+\frac {2 \left (d \sqrt {a+b x^2}+c\right )^{15/2}}{15 d^6}-\frac {10 c \left (d \sqrt {a+b x^2}+c\right )^{13/2}}{13 d^6}}{b^3}\)

Input:

Int[x^5*(c + d*Sqrt[a + b*x^2])^(3/2),x]
 

Output:

((-2*c*(c^2 - a*d^2)^2*(c + d*Sqrt[a + b*x^2])^(5/2))/(5*d^6) + (2*(5*c^4 
- 6*a*c^2*d^2 + a^2*d^4)*(c + d*Sqrt[a + b*x^2])^(7/2))/(7*d^6) - (4*c*(5* 
c^2 - 3*a*d^2)*(c + d*Sqrt[a + b*x^2])^(9/2))/(9*d^6) + (4*(5*c^2 - a*d^2) 
*(c + d*Sqrt[a + b*x^2])^(11/2))/(11*d^6) - (10*c*(c + d*Sqrt[a + b*x^2])^ 
(13/2))/(13*d^6) + (2*(c + d*Sqrt[a + b*x^2])^(15/2))/(15*d^6))/b^3
 

Defintions of rubi rules used

rule 522

Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. 
), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], 
x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
 

rule 896

Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coeff 
icient[v, x, 0], d = Coefficient[v, x, 1]}, Simp[1/d^(m + 1)   Subst[Int[Si 
mplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]] /; 
 FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]
 

rule 1732

Int[((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symb 
ol] :> With[{g = Denominator[n]}, Simp[g   Subst[Int[x^(g - 1)*(d + e*x^(g* 
n))^q*(a + c*x^(2*g*n))^p, x], x, x^(1/g)], x]] /; FreeQ[{a, c, d, e, p, q} 
, x] && EqQ[n2, 2*n] && FractionQ[n]
 

rule 2009

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

rule 7283

Int[(u_)*(x_)^(m_.), x_Symbol] :> With[{lst = PowerVariableExpn[u, m + 1, x 
]}, Simp[1/lst[[2]]   Subst[Int[NormalizeIntegrand[Simplify[lst[[1]]/x], x] 
, x], x, (lst[[3]]*x)^lst[[2]]], x] /;  !FalseQ[lst] && NeQ[lst[[2]], m + 1 
]] /; IntegerQ[m] && NeQ[m, -1] && NonsumQ[u] && (GtQ[m, 0] ||  !AlgebraicF 
unctionQ[u, x])
 

Maple [F]

\[\int x^{5} \left (c +d \sqrt {b \,x^{2}+a}\right )^{\frac {3}{2}}d x\]

Input:

int(x^5*(c+d*(b*x^2+a)^(1/2))^(3/2),x)
 

Output:

int(x^5*(c+d*(b*x^2+a)^(1/2))^(3/2),x)
 

Fricas [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.05 \[ \int x^5 \left (c+d \sqrt {a+b x^2}\right )^{3/2} \, dx=\frac {2 \, {\left (3696 \, b^{3} c d^{6} x^{6} + 3072 \, a^{3} c d^{6} - 2176 \, a^{2} c^{3} d^{4} + 1152 \, a c^{5} d^{2} - 256 \, c^{7} + 14 \, {\left (12 \, a b^{2} c d^{6} - 5 \, b^{2} c^{3} d^{4}\right )} x^{4} - 8 \, {\left (57 \, a^{2} b c d^{6} - 41 \, a b c^{3} d^{4} + 12 \, b c^{5} d^{2}\right )} x^{2} + {\left (3003 \, b^{3} d^{7} x^{6} + 1248 \, a^{3} d^{7} + 960 \, a^{2} c^{2} d^{5} - 544 \, a c^{4} d^{3} + 128 \, c^{6} d + 63 \, {\left (13 \, a b^{2} d^{7} + b^{2} c^{2} d^{5}\right )} x^{4} - 8 \, {\left (117 \, a^{2} b d^{7} + 33 \, a b c^{2} d^{5} - 10 \, b c^{4} d^{3}\right )} x^{2}\right )} \sqrt {b x^{2} + a}\right )} \sqrt {\sqrt {b x^{2} + a} d + c}}{45045 \, b^{3} d^{6}} \] Input:

integrate(x^5*(c+d*(b*x^2+a)^(1/2))^(3/2),x, algorithm="fricas")
 

Output:

2/45045*(3696*b^3*c*d^6*x^6 + 3072*a^3*c*d^6 - 2176*a^2*c^3*d^4 + 1152*a*c 
^5*d^2 - 256*c^7 + 14*(12*a*b^2*c*d^6 - 5*b^2*c^3*d^4)*x^4 - 8*(57*a^2*b*c 
*d^6 - 41*a*b*c^3*d^4 + 12*b*c^5*d^2)*x^2 + (3003*b^3*d^7*x^6 + 1248*a^3*d 
^7 + 960*a^2*c^2*d^5 - 544*a*c^4*d^3 + 128*c^6*d + 63*(13*a*b^2*d^7 + b^2* 
c^2*d^5)*x^4 - 8*(117*a^2*b*d^7 + 33*a*b*c^2*d^5 - 10*b*c^4*d^3)*x^2)*sqrt 
(b*x^2 + a))*sqrt(sqrt(b*x^2 + a)*d + c)/(b^3*d^6)
 

Sympy [F]

\[ \int x^5 \left (c+d \sqrt {a+b x^2}\right )^{3/2} \, dx=\int x^{5} \left (c + d \sqrt {a + b x^{2}}\right )^{\frac {3}{2}}\, dx \] Input:

integrate(x**5*(c+d*(b*x**2+a)**(1/2))**(3/2),x)
 

Output:

Integral(x**5*(c + d*sqrt(a + b*x**2))**(3/2), x)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.76 \[ \int x^5 \left (c+d \sqrt {a+b x^2}\right )^{3/2} \, dx=\frac {2 \, {\left (3003 \, {\left (\sqrt {b x^{2} + a} d + c\right )}^{\frac {15}{2}} - 17325 \, {\left (\sqrt {b x^{2} + a} d + c\right )}^{\frac {13}{2}} c - 8190 \, {\left (a d^{2} - 5 \, c^{2}\right )} {\left (\sqrt {b x^{2} + a} d + c\right )}^{\frac {11}{2}} + 10010 \, {\left (3 \, a c d^{2} - 5 \, c^{3}\right )} {\left (\sqrt {b x^{2} + a} d + c\right )}^{\frac {9}{2}} + 6435 \, {\left (a^{2} d^{4} - 6 \, a c^{2} d^{2} + 5 \, c^{4}\right )} {\left (\sqrt {b x^{2} + a} d + c\right )}^{\frac {7}{2}} - 9009 \, {\left (a^{2} c d^{4} - 2 \, a c^{3} d^{2} + c^{5}\right )} {\left (\sqrt {b x^{2} + a} d + c\right )}^{\frac {5}{2}}\right )}}{45045 \, b^{3} d^{6}} \] Input:

integrate(x^5*(c+d*(b*x^2+a)^(1/2))^(3/2),x, algorithm="maxima")
 

Output:

2/45045*(3003*(sqrt(b*x^2 + a)*d + c)^(15/2) - 17325*(sqrt(b*x^2 + a)*d + 
c)^(13/2)*c - 8190*(a*d^2 - 5*c^2)*(sqrt(b*x^2 + a)*d + c)^(11/2) + 10010* 
(3*a*c*d^2 - 5*c^3)*(sqrt(b*x^2 + a)*d + c)^(9/2) + 6435*(a^2*d^4 - 6*a*c^ 
2*d^2 + 5*c^4)*(sqrt(b*x^2 + a)*d + c)^(7/2) - 9009*(a^2*c*d^4 - 2*a*c^3*d 
^2 + c^5)*(sqrt(b*x^2 + a)*d + c)^(5/2))/(b^3*d^6)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2794 vs. \(2 (200) = 400\).

Time = 0.26 (sec) , antiderivative size = 2794, normalized size of antiderivative = 11.84 \[ \int x^5 \left (c+d \sqrt {a+b x^2}\right )^{3/2} \, dx=\text {Too large to display} \] Input:

integrate(x^5*(c+d*(b*x^2+a)^(1/2))^(3/2),x, algorithm="giac")
 

Output:

2/45045*(143*(315*sqrt(sqrt(b*x^2 + a)*d + c)*a^2*d^4 - 126*(sqrt(b*x^2 + 
a)*d + c)^(5/2)*a*d^2 + 420*(sqrt(b*x^2 + a)*d + c)^(3/2)*a*c*d^2 - 630*sq 
rt(sqrt(b*x^2 + a)*d + c)*a*c^2*d^2 + 35*(sqrt(b*x^2 + a)*d + c)^(9/2) - 1 
80*(sqrt(b*x^2 + a)*d + c)^(7/2)*c + 378*(sqrt(b*x^2 + a)*d + c)^(5/2)*c^2 
 - 420*(sqrt(b*x^2 + a)*d + c)^(3/2)*c^3 + 315*sqrt(sqrt(b*x^2 + a)*d + c) 
*c^4)*a*c/(b^2*d^3) + 13*(1155*(sqrt(b*x^2 + a)*d + c)^(3/2)*a^2*d^4*sgn(( 
sqrt(b*x^2 + a)*d + c)*d - c*d) - 3465*sqrt(sqrt(b*x^2 + a)*d + c)*a^2*c*d 
^4*sgn((sqrt(b*x^2 + a)*d + c)*d - c*d) - 990*(sqrt(b*x^2 + a)*d + c)^(7/2 
)*a*d^2*sgn((sqrt(b*x^2 + a)*d + c)*d - c*d) + 4158*(sqrt(b*x^2 + a)*d + c 
)^(5/2)*a*c*d^2*sgn((sqrt(b*x^2 + a)*d + c)*d - c*d) - 6930*(sqrt(b*x^2 + 
a)*d + c)^(3/2)*a*c^2*d^2*sgn((sqrt(b*x^2 + a)*d + c)*d - c*d) + 6930*sqrt 
(sqrt(b*x^2 + a)*d + c)*a*c^3*d^2*sgn((sqrt(b*x^2 + a)*d + c)*d - c*d) + 3 
15*(sqrt(b*x^2 + a)*d + c)^(11/2)*sgn((sqrt(b*x^2 + a)*d + c)*d - c*d) - 1 
925*(sqrt(b*x^2 + a)*d + c)^(9/2)*c*sgn((sqrt(b*x^2 + a)*d + c)*d - c*d) + 
 4950*(sqrt(b*x^2 + a)*d + c)^(7/2)*c^2*sgn((sqrt(b*x^2 + a)*d + c)*d - c* 
d) - 6930*(sqrt(b*x^2 + a)*d + c)^(5/2)*c^3*sgn((sqrt(b*x^2 + a)*d + c)*d 
- c*d) + 5775*(sqrt(b*x^2 + a)*d + c)^(3/2)*c^4*sgn((sqrt(b*x^2 + a)*d + c 
)*d - c*d) - 3465*sqrt(sqrt(b*x^2 + a)*d + c)*c^5*sgn((sqrt(b*x^2 + a)*d + 
 c)*d - c*d))*a/(b^2*d^3) + 13*(1155*(sqrt(b*x^2 + a)*d + c)^(3/2)*a^2*d^4 
*sgn((sqrt(b*x^2 + a)*d + c)*d - c*d) - 3465*sqrt(sqrt(b*x^2 + a)*d + c...
 

Mupad [F(-1)]

Timed out. \[ \int x^5 \left (c+d \sqrt {a+b x^2}\right )^{3/2} \, dx=\int x^5\,{\left (c+d\,\sqrt {b\,x^2+a}\right )}^{3/2} \,d x \] Input:

int(x^5*(c + d*(a + b*x^2)^(1/2))^(3/2),x)
 

Output:

int(x^5*(c + d*(a + b*x^2)^(1/2))^(3/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 770, normalized size of antiderivative = 3.26 \[ \int x^5 \left (c+d \sqrt {a+b x^2}\right )^{3/2} \, dx=\frac {2 \sqrt {\sqrt {b}\, \sqrt {b \,x^{2}+a}\, d x +\sqrt {b \,x^{2}+a}\, c +\sqrt {b}\, c x +a d +b d \,x^{2}}\, \sqrt {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}\, \left (-256 \sqrt {b \,x^{2}+a}\, c^{7}+1248 a^{4} d^{7}+256 \sqrt {b}\, c^{7} x +960 a^{3} c^{2} d^{5}-544 a^{2} c^{4} d^{3}+128 a \,c^{6} d +3003 b^{4} d^{7} x^{8}+264 \sqrt {b}\, \sqrt {b \,x^{2}+a}\, a b \,c^{2} d^{5} x^{3}-960 \sqrt {b}\, \sqrt {b \,x^{2}+a}\, a^{2} c^{2} d^{5} x -819 \sqrt {b}\, \sqrt {b \,x^{2}+a}\, a \,b^{2} d^{7} x^{5}+544 \sqrt {b}\, \sqrt {b \,x^{2}+a}\, a \,c^{4} d^{3} x -63 \sqrt {b}\, \sqrt {b \,x^{2}+a}\, b^{2} c^{2} d^{5} x^{5}-80 \sqrt {b}\, \sqrt {b \,x^{2}+a}\, b \,c^{4} d^{3} x^{3}-456 \sqrt {b \,x^{2}+a}\, a^{2} b c \,d^{6} x^{2}+168 \sqrt {b \,x^{2}+a}\, a \,b^{2} c \,d^{6} x^{4}+328 \sqrt {b \,x^{2}+a}\, a b \,c^{3} d^{4} x^{2}+456 \sqrt {b}\, a^{2} b c \,d^{6} x^{3}-168 \sqrt {b}\, a \,b^{2} c \,d^{6} x^{5}-328 \sqrt {b}\, a b \,c^{3} d^{4} x^{3}-1248 \sqrt {b}\, \sqrt {b \,x^{2}+a}\, a^{3} d^{7} x -3003 \sqrt {b}\, \sqrt {b \,x^{2}+a}\, b^{3} d^{7} x^{7}-128 \sqrt {b}\, \sqrt {b \,x^{2}+a}\, c^{6} d x +3696 \sqrt {b \,x^{2}+a}\, b^{3} c \,d^{6} x^{6}-70 \sqrt {b \,x^{2}+a}\, b^{2} c^{3} d^{4} x^{4}-96 \sqrt {b \,x^{2}+a}\, b \,c^{5} d^{2} x^{2}-3072 \sqrt {b}\, a^{3} c \,d^{6} x +2176 \sqrt {b}\, a^{2} c^{3} d^{4} x -1152 \sqrt {b}\, a \,c^{5} d^{2} x -3696 \sqrt {b}\, b^{3} c \,d^{6} x^{7}+70 \sqrt {b}\, b^{2} c^{3} d^{4} x^{5}+96 \sqrt {b}\, b \,c^{5} d^{2} x^{3}+696 a^{2} b \,c^{2} d^{5} x^{2}-201 a \,b^{2} c^{2} d^{5} x^{4}-464 a b \,c^{4} d^{3} x^{2}+936 \sqrt {b}\, \sqrt {b \,x^{2}+a}\, a^{2} b \,d^{7} x^{3}+3072 \sqrt {b \,x^{2}+a}\, a^{3} c \,d^{6}-2176 \sqrt {b \,x^{2}+a}\, a^{2} c^{3} d^{4}+1152 \sqrt {b \,x^{2}+a}\, a \,c^{5} d^{2}+312 a^{3} b \,d^{7} x^{2}-117 a^{2} b^{2} d^{7} x^{4}+3822 a \,b^{3} d^{7} x^{6}+63 b^{3} c^{2} d^{5} x^{6}+80 b^{2} c^{4} d^{3} x^{4}+128 b \,c^{6} d \,x^{2}\right )}{45045 a \,b^{3} d^{6}} \] Input:

int(x^5*(c+d*(b*x^2+a)^(1/2))^(3/2),x)
 

Output:

(2*sqrt(sqrt(b)*sqrt(a + b*x**2)*d*x + sqrt(a + b*x**2)*c + sqrt(b)*c*x + 
a*d + b*d*x**2)*sqrt(sqrt(a + b*x**2) + sqrt(b)*x)*( - 1248*sqrt(b)*sqrt(a 
 + b*x**2)*a**3*d**7*x + 936*sqrt(b)*sqrt(a + b*x**2)*a**2*b*d**7*x**3 - 9 
60*sqrt(b)*sqrt(a + b*x**2)*a**2*c**2*d**5*x - 819*sqrt(b)*sqrt(a + b*x**2 
)*a*b**2*d**7*x**5 + 264*sqrt(b)*sqrt(a + b*x**2)*a*b*c**2*d**5*x**3 + 544 
*sqrt(b)*sqrt(a + b*x**2)*a*c**4*d**3*x - 3003*sqrt(b)*sqrt(a + b*x**2)*b* 
*3*d**7*x**7 - 63*sqrt(b)*sqrt(a + b*x**2)*b**2*c**2*d**5*x**5 - 80*sqrt(b 
)*sqrt(a + b*x**2)*b*c**4*d**3*x**3 - 128*sqrt(b)*sqrt(a + b*x**2)*c**6*d* 
x + 3072*sqrt(a + b*x**2)*a**3*c*d**6 - 456*sqrt(a + b*x**2)*a**2*b*c*d**6 
*x**2 - 2176*sqrt(a + b*x**2)*a**2*c**3*d**4 + 168*sqrt(a + b*x**2)*a*b**2 
*c*d**6*x**4 + 328*sqrt(a + b*x**2)*a*b*c**3*d**4*x**2 + 1152*sqrt(a + b*x 
**2)*a*c**5*d**2 + 3696*sqrt(a + b*x**2)*b**3*c*d**6*x**6 - 70*sqrt(a + b* 
x**2)*b**2*c**3*d**4*x**4 - 96*sqrt(a + b*x**2)*b*c**5*d**2*x**2 - 256*sqr 
t(a + b*x**2)*c**7 - 3072*sqrt(b)*a**3*c*d**6*x + 456*sqrt(b)*a**2*b*c*d** 
6*x**3 + 2176*sqrt(b)*a**2*c**3*d**4*x - 168*sqrt(b)*a*b**2*c*d**6*x**5 - 
328*sqrt(b)*a*b*c**3*d**4*x**3 - 1152*sqrt(b)*a*c**5*d**2*x - 3696*sqrt(b) 
*b**3*c*d**6*x**7 + 70*sqrt(b)*b**2*c**3*d**4*x**5 + 96*sqrt(b)*b*c**5*d** 
2*x**3 + 256*sqrt(b)*c**7*x + 1248*a**4*d**7 + 312*a**3*b*d**7*x**2 + 960* 
a**3*c**2*d**5 - 117*a**2*b**2*d**7*x**4 + 696*a**2*b*c**2*d**5*x**2 - 544 
*a**2*c**4*d**3 + 3822*a*b**3*d**7*x**6 - 201*a*b**2*c**2*d**5*x**4 - 4...