\(\int (c+d x) (c^2-d^2 x^2)^{5/2} (A+B x+C x^2+D x^3) \, dx\) [155]

Optimal result

Integrand size = 37, antiderivative size = 363 \[ \int (c+d x) \left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {c^5 \left (10 c^2 C d+10 B c d^2+80 A d^3+3 c^3 D\right ) x \sqrt {c^2-d^2 x^2}}{256 d^3}+\frac {c^3 \left (10 c^2 C d+10 B c d^2+80 A d^3+3 c^3 D\right ) x \left (c^2-d^2 x^2\right )^{3/2}}{384 d^3}+\frac {c \left (10 c^2 C d+10 B c d^2+80 A d^3+3 c^3 D\right ) x \left (c^2-d^2 x^2\right )^{5/2}}{480 d^3}-\frac {\left (c^2 C d+B c d^2+A d^3+c^3 D\right ) \left (c^2-d^2 x^2\right )^{7/2}}{7 d^4}-\frac {\left (10 c C d+10 B d^2+3 c^2 D\right ) x \left (c^2-d^2 x^2\right )^{7/2}}{80 d^3}-\frac {D x^3 \left (c^2-d^2 x^2\right )^{7/2}}{10 d}+\frac {(C d+c D) \left (c^2-d^2 x^2\right )^{9/2}}{9 d^4}+\frac {c^7 \left (10 c^2 C d+10 B c d^2+80 A d^3+3 c^3 D\right ) \arctan \left (\frac {d x}{\sqrt {c^2-d^2 x^2}}\right )}{256 d^4} \] Output:

1/256*c^5*(80*A*d^3+10*B*c*d^2+10*C*c^2*d+3*D*c^3)*x*(-d^2*x^2+c^2)^(1/2)/ 
d^3+1/384*c^3*(80*A*d^3+10*B*c*d^2+10*C*c^2*d+3*D*c^3)*x*(-d^2*x^2+c^2)^(3 
/2)/d^3+1/480*c*(80*A*d^3+10*B*c*d^2+10*C*c^2*d+3*D*c^3)*x*(-d^2*x^2+c^2)^ 
(5/2)/d^3-1/7*(A*d^3+B*c*d^2+C*c^2*d+D*c^3)*(-d^2*x^2+c^2)^(7/2)/d^4-1/80* 
(10*B*d^2+10*C*c*d+3*D*c^2)*x*(-d^2*x^2+c^2)^(7/2)/d^3-1/10*D*x^3*(-d^2*x^ 
2+c^2)^(7/2)/d+1/9*(C*d+D*c)*(-d^2*x^2+c^2)^(9/2)/d^4+1/256*c^7*(80*A*d^3+ 
10*B*c*d^2+10*C*c^2*d+3*D*c^3)*arctan(d*x/(-d^2*x^2+c^2)^(1/2))/d^4
 

Mathematica [A] (verified)

Time = 3.85 (sec) , antiderivative size = 343, normalized size of antiderivative = 0.94 \[ \int (c+d x) \left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right ) \, dx=-\frac {\sqrt {c^2-d^2 x^2} \left (2560 c^9 D+5 c^8 d (512 C+189 D x)+10 c^7 d^2 (1152 B+x (315 C+128 D x))-160 c d^8 x^5 (84 A+x (72 B+7 x (9 C+8 D x)))-32 d^9 x^6 (360 A+7 x (45 B+4 x (10 C+9 D x)))+10 c^6 d^3 (1152 A+x (315 B+x (128 C+63 D x)))+80 c^3 d^6 x^3 (546 A+x (432 B+x (357 C+304 D x)))-60 c^5 d^4 x (924 A+x (576 B+x (413 C+320 D x)))-12 c^4 d^5 x^2 (2880 A+x (2065 B+2 x (800 C+651 D x)))+16 c^2 d^7 x^4 (2160 A+x (1785 B+x (1520 C+1323 D x)))\right )+630 c^7 \left (10 c^2 C d+10 B c d^2+80 A d^3+3 c^3 D\right ) \arctan \left (\frac {d x}{\sqrt {c^2}-\sqrt {c^2-d^2 x^2}}\right )}{80640 d^4} \] Input:

Integrate[(c + d*x)*(c^2 - d^2*x^2)^(5/2)*(A + B*x + C*x^2 + D*x^3),x]
 

Output:

-1/80640*(Sqrt[c^2 - d^2*x^2]*(2560*c^9*D + 5*c^8*d*(512*C + 189*D*x) + 10 
*c^7*d^2*(1152*B + x*(315*C + 128*D*x)) - 160*c*d^8*x^5*(84*A + x*(72*B + 
7*x*(9*C + 8*D*x))) - 32*d^9*x^6*(360*A + 7*x*(45*B + 4*x*(10*C + 9*D*x))) 
 + 10*c^6*d^3*(1152*A + x*(315*B + x*(128*C + 63*D*x))) + 80*c^3*d^6*x^3*( 
546*A + x*(432*B + x*(357*C + 304*D*x))) - 60*c^5*d^4*x*(924*A + x*(576*B 
+ x*(413*C + 320*D*x))) - 12*c^4*d^5*x^2*(2880*A + x*(2065*B + 2*x*(800*C 
+ 651*D*x))) + 16*c^2*d^7*x^4*(2160*A + x*(1785*B + x*(1520*C + 1323*D*x)) 
)) + 630*c^7*(10*c^2*C*d + 10*B*c*d^2 + 80*A*d^3 + 3*c^3*D)*ArcTan[(d*x)/( 
Sqrt[c^2] - Sqrt[c^2 - d^2*x^2])])/d^4
 

Rubi [A] (verified)

Time = 1.24 (sec) , antiderivative size = 310, normalized size of antiderivative = 0.85, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.351, Rules used = {2346, 25, 2346, 25, 2346, 25, 27, 455, 211, 211, 211, 224, 216}

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.

\(\displaystyle \int (c+d x) \left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right ) \, dx\)

\(\Big \downarrow \) 2346

\(\displaystyle -\frac {\int -\left (c^2-d^2 x^2\right )^{5/2} \left (10 d^2 (C d+c D) x^3+d \left (3 D c^2+10 C d c+10 B d^2\right ) x^2+10 d^2 (B c+A d) x+10 A c d^2\right )dx}{10 d^2}-\frac {D x^3 \left (c^2-d^2 x^2\right )^{7/2}}{10 d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \left (c^2-d^2 x^2\right )^{5/2} \left (10 d^2 (C d+c D) x^3+d \left (3 D c^2+10 C d c+10 B d^2\right ) x^2+10 d^2 (B c+A d) x+10 A c d^2\right )dx}{10 d^2}-\frac {D x^3 \left (c^2-d^2 x^2\right )^{7/2}}{10 d}\)

\(\Big \downarrow \) 2346

\(\displaystyle \frac {-\frac {\int -\left (c^2-d^2 x^2\right )^{5/2} \left (90 A c d^4+9 \left (3 D c^2+10 C d c+10 B d^2\right ) x^2 d^3+10 \left (2 D c^3+2 C d c^2+9 B d^2 c+9 A d^3\right ) x d^2\right )dx}{9 d^2}-\frac {10}{9} x^2 \left (c^2-d^2 x^2\right )^{7/2} (c D+C d)}{10 d^2}-\frac {D x^3 \left (c^2-d^2 x^2\right )^{7/2}}{10 d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \left (c^2-d^2 x^2\right )^{5/2} \left (90 A c d^4+9 \left (3 D c^2+10 C d c+10 B d^2\right ) x^2 d^3+10 \left (2 D c^3+2 C d c^2+9 B d^2 c+9 A d^3\right ) x d^2\right )dx}{9 d^2}-\frac {10}{9} x^2 \left (c^2-d^2 x^2\right )^{7/2} (c D+C d)}{10 d^2}-\frac {D x^3 \left (c^2-d^2 x^2\right )^{7/2}}{10 d}\)

\(\Big \downarrow \) 2346

\(\displaystyle \frac {\frac {-\frac {\int -d^3 \left (9 c \left (3 D c^3+10 C d c^2+10 B d^2 c+80 A d^3\right )+80 d \left (2 D c^3+2 C d c^2+9 B d^2 c+9 A d^3\right ) x\right ) \left (c^2-d^2 x^2\right )^{5/2}dx}{8 d^2}-\frac {9}{8} d x \left (c^2-d^2 x^2\right )^{7/2} \left (10 B d^2+3 c^2 D+10 c C d\right )}{9 d^2}-\frac {10}{9} x^2 \left (c^2-d^2 x^2\right )^{7/2} (c D+C d)}{10 d^2}-\frac {D x^3 \left (c^2-d^2 x^2\right )^{7/2}}{10 d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {\int d^3 \left (9 c \left (3 D c^3+10 C d c^2+10 B d^2 c+80 A d^3\right )+80 d \left (2 D c^3+2 C d c^2+9 B d^2 c+9 A d^3\right ) x\right ) \left (c^2-d^2 x^2\right )^{5/2}dx}{8 d^2}-\frac {9}{8} d x \left (c^2-d^2 x^2\right )^{7/2} \left (10 B d^2+3 c^2 D+10 c C d\right )}{9 d^2}-\frac {10}{9} x^2 \left (c^2-d^2 x^2\right )^{7/2} (c D+C d)}{10 d^2}-\frac {D x^3 \left (c^2-d^2 x^2\right )^{7/2}}{10 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {1}{8} d \int \left (9 c \left (3 D c^3+10 C d c^2+10 B d^2 c+80 A d^3\right )+80 d \left (2 D c^3+2 C d c^2+9 B d^2 c+9 A d^3\right ) x\right ) \left (c^2-d^2 x^2\right )^{5/2}dx-\frac {9}{8} d x \left (c^2-d^2 x^2\right )^{7/2} \left (10 B d^2+3 c^2 D+10 c C d\right )}{9 d^2}-\frac {10}{9} x^2 \left (c^2-d^2 x^2\right )^{7/2} (c D+C d)}{10 d^2}-\frac {D x^3 \left (c^2-d^2 x^2\right )^{7/2}}{10 d}\)

\(\Big \downarrow \) 455

\(\displaystyle \frac {\frac {\frac {1}{8} d \left (9 c \left (80 A d^3+10 B c d^2+3 c^3 D+10 c^2 C d\right ) \int \left (c^2-d^2 x^2\right )^{5/2}dx-\frac {80 \left (c^2-d^2 x^2\right )^{7/2} \left (9 A d^3+9 B c d^2+2 c^3 D+2 c^2 C d\right )}{7 d}\right )-\frac {9}{8} d x \left (c^2-d^2 x^2\right )^{7/2} \left (10 B d^2+3 c^2 D+10 c C d\right )}{9 d^2}-\frac {10}{9} x^2 \left (c^2-d^2 x^2\right )^{7/2} (c D+C d)}{10 d^2}-\frac {D x^3 \left (c^2-d^2 x^2\right )^{7/2}}{10 d}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {\frac {1}{8} d \left (9 c \left (80 A d^3+10 B c d^2+3 c^3 D+10 c^2 C d\right ) \left (\frac {5}{6} c^2 \int \left (c^2-d^2 x^2\right )^{3/2}dx+\frac {1}{6} x \left (c^2-d^2 x^2\right )^{5/2}\right )-\frac {80 \left (c^2-d^2 x^2\right )^{7/2} \left (9 A d^3+9 B c d^2+2 c^3 D+2 c^2 C d\right )}{7 d}\right )-\frac {9}{8} d x \left (c^2-d^2 x^2\right )^{7/2} \left (10 B d^2+3 c^2 D+10 c C d\right )}{9 d^2}-\frac {10}{9} x^2 \left (c^2-d^2 x^2\right )^{7/2} (c D+C d)}{10 d^2}-\frac {D x^3 \left (c^2-d^2 x^2\right )^{7/2}}{10 d}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {\frac {1}{8} d \left (9 c \left (80 A d^3+10 B c d^2+3 c^3 D+10 c^2 C d\right ) \left (\frac {5}{6} c^2 \left (\frac {3}{4} c^2 \int \sqrt {c^2-d^2 x^2}dx+\frac {1}{4} x \left (c^2-d^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (c^2-d^2 x^2\right )^{5/2}\right )-\frac {80 \left (c^2-d^2 x^2\right )^{7/2} \left (9 A d^3+9 B c d^2+2 c^3 D+2 c^2 C d\right )}{7 d}\right )-\frac {9}{8} d x \left (c^2-d^2 x^2\right )^{7/2} \left (10 B d^2+3 c^2 D+10 c C d\right )}{9 d^2}-\frac {10}{9} x^2 \left (c^2-d^2 x^2\right )^{7/2} (c D+C d)}{10 d^2}-\frac {D x^3 \left (c^2-d^2 x^2\right )^{7/2}}{10 d}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {\frac {1}{8} d \left (9 c \left (80 A d^3+10 B c d^2+3 c^3 D+10 c^2 C d\right ) \left (\frac {5}{6} c^2 \left (\frac {3}{4} c^2 \left (\frac {1}{2} c^2 \int \frac {1}{\sqrt {c^2-d^2 x^2}}dx+\frac {1}{2} x \sqrt {c^2-d^2 x^2}\right )+\frac {1}{4} x \left (c^2-d^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (c^2-d^2 x^2\right )^{5/2}\right )-\frac {80 \left (c^2-d^2 x^2\right )^{7/2} \left (9 A d^3+9 B c d^2+2 c^3 D+2 c^2 C d\right )}{7 d}\right )-\frac {9}{8} d x \left (c^2-d^2 x^2\right )^{7/2} \left (10 B d^2+3 c^2 D+10 c C d\right )}{9 d^2}-\frac {10}{9} x^2 \left (c^2-d^2 x^2\right )^{7/2} (c D+C d)}{10 d^2}-\frac {D x^3 \left (c^2-d^2 x^2\right )^{7/2}}{10 d}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {\frac {1}{8} d \left (9 c \left (80 A d^3+10 B c d^2+3 c^3 D+10 c^2 C d\right ) \left (\frac {5}{6} c^2 \left (\frac {3}{4} c^2 \left (\frac {1}{2} c^2 \int \frac {1}{\frac {d^2 x^2}{c^2-d^2 x^2}+1}d\frac {x}{\sqrt {c^2-d^2 x^2}}+\frac {1}{2} x \sqrt {c^2-d^2 x^2}\right )+\frac {1}{4} x \left (c^2-d^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (c^2-d^2 x^2\right )^{5/2}\right )-\frac {80 \left (c^2-d^2 x^2\right )^{7/2} \left (9 A d^3+9 B c d^2+2 c^3 D+2 c^2 C d\right )}{7 d}\right )-\frac {9}{8} d x \left (c^2-d^2 x^2\right )^{7/2} \left (10 B d^2+3 c^2 D+10 c C d\right )}{9 d^2}-\frac {10}{9} x^2 \left (c^2-d^2 x^2\right )^{7/2} (c D+C d)}{10 d^2}-\frac {D x^3 \left (c^2-d^2 x^2\right )^{7/2}}{10 d}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {\frac {\frac {1}{8} d \left (9 c \left (\frac {5}{6} c^2 \left (\frac {3}{4} c^2 \left (\frac {c^2 \arctan \left (\frac {d x}{\sqrt {c^2-d^2 x^2}}\right )}{2 d}+\frac {1}{2} x \sqrt {c^2-d^2 x^2}\right )+\frac {1}{4} x \left (c^2-d^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (c^2-d^2 x^2\right )^{5/2}\right ) \left (80 A d^3+10 B c d^2+3 c^3 D+10 c^2 C d\right )-\frac {80 \left (c^2-d^2 x^2\right )^{7/2} \left (9 A d^3+9 B c d^2+2 c^3 D+2 c^2 C d\right )}{7 d}\right )-\frac {9}{8} d x \left (c^2-d^2 x^2\right )^{7/2} \left (10 B d^2+3 c^2 D+10 c C d\right )}{9 d^2}-\frac {10}{9} x^2 \left (c^2-d^2 x^2\right )^{7/2} (c D+C d)}{10 d^2}-\frac {D x^3 \left (c^2-d^2 x^2\right )^{7/2}}{10 d}\)

Input:

Int[(c + d*x)*(c^2 - d^2*x^2)^(5/2)*(A + B*x + C*x^2 + D*x^3),x]
 

Output:

-1/10*(D*x^3*(c^2 - d^2*x^2)^(7/2))/d + ((-10*(C*d + c*D)*x^2*(c^2 - d^2*x 
^2)^(7/2))/9 + ((-9*d*(10*c*C*d + 10*B*d^2 + 3*c^2*D)*x*(c^2 - d^2*x^2)^(7 
/2))/8 + (d*((-80*(2*c^2*C*d + 9*B*c*d^2 + 9*A*d^3 + 2*c^3*D)*(c^2 - d^2*x 
^2)^(7/2))/(7*d) + 9*c*(10*c^2*C*d + 10*B*c*d^2 + 80*A*d^3 + 3*c^3*D)*((x* 
(c^2 - d^2*x^2)^(5/2))/6 + (5*c^2*((x*(c^2 - d^2*x^2)^(3/2))/4 + (3*c^2*(( 
x*Sqrt[c^2 - d^2*x^2])/2 + (c^2*ArcTan[(d*x)/Sqrt[c^2 - d^2*x^2]])/(2*d))) 
/4))/6)))/8)/(9*d^2))/(10*d^2)
 

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 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 455

Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( 
a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c   Int[(a + b*x^2)^p, x], x] 
/; FreeQ[{a, b, c, d, p}, x] &&  !LeQ[p, -1]
 

rule 2346

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], 
e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x^2)^(p + 1)/(b*( 
q + 2*p + 1))), x] + Simp[1/(b*(q + 2*p + 1))   Int[(a + b*x^2)^p*ExpandToS 
um[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], 
x], x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] &&  !LeQ[p, -1]
 

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 472, normalized size of antiderivative = 1.30

method	result	size
default	\(A c \left (\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 c^{2} \left (\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 c^{2} \left (\frac {x \sqrt {-d^{2} x^{2}+c^{2}}}{2}+\frac {c^{2} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right )}{2 \sqrt {d^{2}}}\right )}{4}\right )}{6}\right )-\frac {\left (A d +B c \right ) \left (-d^{2} x^{2}+c^{2}\right )^{\frac {7}{2}}}{7 d^{2}}+\left (B d +C c \right ) \left (-\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {7}{2}}}{8 d^{2}}+\frac {c^{2} \left (\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 c^{2} \left (\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 c^{2} \left (\frac {x \sqrt {-d^{2} x^{2}+c^{2}}}{2}+\frac {c^{2} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right )}{2 \sqrt {d^{2}}}\right )}{4}\right )}{6}\right )}{8 d^{2}}\right )+\left (C d +D c \right ) \left (-\frac {x^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {7}{2}}}{9 d^{2}}-\frac {2 c^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {7}{2}}}{63 d^{4}}\right )+D d \left (-\frac {x^{3} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {7}{2}}}{10 d^{2}}+\frac {3 c^{2} \left (-\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {7}{2}}}{8 d^{2}}+\frac {c^{2} \left (\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 c^{2} \left (\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 c^{2} \left (\frac {x \sqrt {-d^{2} x^{2}+c^{2}}}{2}+\frac {c^{2} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right )}{2 \sqrt {d^{2}}}\right )}{4}\right )}{6}\right )}{8 d^{2}}\right )}{10 d^{2}}\right )\)	\(472\)

Input:

int((d*x+c)*(-d^2*x^2+c^2)^(5/2)*(D*x^3+C*x^2+B*x+A),x,method=_RETURNVERBO 
SE)
 

Output:

A*c*(1/6*x*(-d^2*x^2+c^2)^(5/2)+5/6*c^2*(1/4*x*(-d^2*x^2+c^2)^(3/2)+3/4*c^ 
2*(1/2*x*(-d^2*x^2+c^2)^(1/2)+1/2*c^2/(d^2)^(1/2)*arctan((d^2)^(1/2)*x/(-d 
^2*x^2+c^2)^(1/2)))))-1/7*(A*d+B*c)/d^2*(-d^2*x^2+c^2)^(7/2)+(B*d+C*c)*(-1 
/8*x*(-d^2*x^2+c^2)^(7/2)/d^2+1/8*c^2/d^2*(1/6*x*(-d^2*x^2+c^2)^(5/2)+5/6* 
c^2*(1/4*x*(-d^2*x^2+c^2)^(3/2)+3/4*c^2*(1/2*x*(-d^2*x^2+c^2)^(1/2)+1/2*c^ 
2/(d^2)^(1/2)*arctan((d^2)^(1/2)*x/(-d^2*x^2+c^2)^(1/2))))))+(C*d+D*c)*(-1 
/9*x^2*(-d^2*x^2+c^2)^(7/2)/d^2-2/63*c^2*(-d^2*x^2+c^2)^(7/2)/d^4)+D*d*(-1 
/10*x^3*(-d^2*x^2+c^2)^(7/2)/d^2+3/10*c^2/d^2*(-1/8*x*(-d^2*x^2+c^2)^(7/2) 
/d^2+1/8*c^2/d^2*(1/6*x*(-d^2*x^2+c^2)^(5/2)+5/6*c^2*(1/4*x*(-d^2*x^2+c^2) 
^(3/2)+3/4*c^2*(1/2*x*(-d^2*x^2+c^2)^(1/2)+1/2*c^2/(d^2)^(1/2)*arctan((d^2 
)^(1/2)*x/(-d^2*x^2+c^2)^(1/2)))))))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.12 \[ \int (c+d x) \left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right ) \, dx=-\frac {630 \, {\left (3 \, D c^{10} + 10 \, C c^{9} d + 10 \, B c^{8} d^{2} + 80 \, A c^{7} d^{3}\right )} \arctan \left (-\frac {c - \sqrt {-d^{2} x^{2} + c^{2}}}{d x}\right ) - {\left (8064 \, D d^{9} x^{9} - 2560 \, D c^{9} - 2560 \, C c^{8} d - 11520 \, B c^{7} d^{2} - 11520 \, A c^{6} d^{3} + 8960 \, {\left (D c d^{8} + C d^{9}\right )} x^{8} - 1008 \, {\left (21 \, D c^{2} d^{7} - 10 \, C c d^{8} - 10 \, B d^{9}\right )} x^{7} - 1280 \, {\left (19 \, D c^{3} d^{6} + 19 \, C c^{2} d^{7} - 9 \, B c d^{8} - 9 \, A d^{9}\right )} x^{6} + 168 \, {\left (93 \, D c^{4} d^{5} - 170 \, C c^{3} d^{6} - 170 \, B c^{2} d^{7} + 80 \, A c d^{8}\right )} x^{5} + 3840 \, {\left (5 \, D c^{5} d^{4} + 5 \, C c^{4} d^{5} - 9 \, B c^{3} d^{6} - 9 \, A c^{2} d^{7}\right )} x^{4} - 210 \, {\left (3 \, D c^{6} d^{3} - 118 \, C c^{5} d^{4} - 118 \, B c^{4} d^{5} + 208 \, A c^{3} d^{6}\right )} x^{3} - 1280 \, {\left (D c^{7} d^{2} + C c^{6} d^{3} - 27 \, B c^{5} d^{4} - 27 \, A c^{4} d^{5}\right )} x^{2} - 315 \, {\left (3 \, D c^{8} d + 10 \, C c^{7} d^{2} + 10 \, B c^{6} d^{3} - 176 \, A c^{5} d^{4}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{80640 \, d^{4}} \] Input:

integrate((d*x+c)*(-d^2*x^2+c^2)^(5/2)*(D*x^3+C*x^2+B*x+A),x, algorithm="f 
ricas")
 

Output:

-1/80640*(630*(3*D*c^10 + 10*C*c^9*d + 10*B*c^8*d^2 + 80*A*c^7*d^3)*arctan 
(-(c - sqrt(-d^2*x^2 + c^2))/(d*x)) - (8064*D*d^9*x^9 - 2560*D*c^9 - 2560* 
C*c^8*d - 11520*B*c^7*d^2 - 11520*A*c^6*d^3 + 8960*(D*c*d^8 + C*d^9)*x^8 - 
 1008*(21*D*c^2*d^7 - 10*C*c*d^8 - 10*B*d^9)*x^7 - 1280*(19*D*c^3*d^6 + 19 
*C*c^2*d^7 - 9*B*c*d^8 - 9*A*d^9)*x^6 + 168*(93*D*c^4*d^5 - 170*C*c^3*d^6 
- 170*B*c^2*d^7 + 80*A*c*d^8)*x^5 + 3840*(5*D*c^5*d^4 + 5*C*c^4*d^5 - 9*B* 
c^3*d^6 - 9*A*c^2*d^7)*x^4 - 210*(3*D*c^6*d^3 - 118*C*c^5*d^4 - 118*B*c^4* 
d^5 + 208*A*c^3*d^6)*x^3 - 1280*(D*c^7*d^2 + C*c^6*d^3 - 27*B*c^5*d^4 - 27 
*A*c^4*d^5)*x^2 - 315*(3*D*c^8*d + 10*C*c^7*d^2 + 10*B*c^6*d^3 - 176*A*c^5 
*d^4)*x)*sqrt(-d^2*x^2 + c^2))/d^4
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1238 vs. \(2 (355) = 710\).

Time = 0.89 (sec) , antiderivative size = 1238, normalized size of antiderivative = 3.41 \[ \int (c+d x) \left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)*(-d**2*x**2+c**2)**(5/2)*(D*x**3+C*x**2+B*x+A),x)
 

Output:

Piecewise((sqrt(c**2 - d**2*x**2)*(D*d**5*x**9/10 - x**8*(-C*d**7 - D*c*d* 
*6)/(9*d**2) - x**7*(-B*d**7 - C*c*d**6 + 21*D*c**2*d**5/10)/(8*d**2) - x* 
*6*(-A*d**7 - B*c*d**6 + 3*C*c**2*d**5 + 3*D*c**3*d**4 + 8*c**2*(-C*d**7 - 
 D*c*d**6)/(9*d**2))/(7*d**2) - x**5*(-A*c*d**6 + 3*B*c**2*d**5 + 3*C*c**3 
*d**4 - 3*D*c**4*d**3 + 7*c**2*(-B*d**7 - C*c*d**6 + 21*D*c**2*d**5/10)/(8 
*d**2))/(6*d**2) - x**4*(3*A*c**2*d**5 + 3*B*c**3*d**4 - 3*C*c**4*d**3 - 3 
*D*c**5*d**2 + 6*c**2*(-A*d**7 - B*c*d**6 + 3*C*c**2*d**5 + 3*D*c**3*d**4 
+ 8*c**2*(-C*d**7 - D*c*d**6)/(9*d**2))/(7*d**2))/(5*d**2) - x**3*(3*A*c** 
3*d**4 - 3*B*c**4*d**3 - 3*C*c**5*d**2 + D*c**6*d + 5*c**2*(-A*c*d**6 + 3* 
B*c**2*d**5 + 3*C*c**3*d**4 - 3*D*c**4*d**3 + 7*c**2*(-B*d**7 - C*c*d**6 + 
 21*D*c**2*d**5/10)/(8*d**2))/(6*d**2))/(4*d**2) - x**2*(-3*A*c**4*d**3 - 
3*B*c**5*d**2 + C*c**6*d + D*c**7 + 4*c**2*(3*A*c**2*d**5 + 3*B*c**3*d**4 
- 3*C*c**4*d**3 - 3*D*c**5*d**2 + 6*c**2*(-A*d**7 - B*c*d**6 + 3*C*c**2*d* 
*5 + 3*D*c**3*d**4 + 8*c**2*(-C*d**7 - D*c*d**6)/(9*d**2))/(7*d**2))/(5*d* 
*2))/(3*d**2) - x*(-3*A*c**5*d**2 + B*c**6*d + C*c**7 + 3*c**2*(3*A*c**3*d 
**4 - 3*B*c**4*d**3 - 3*C*c**5*d**2 + D*c**6*d + 5*c**2*(-A*c*d**6 + 3*B*c 
**2*d**5 + 3*C*c**3*d**4 - 3*D*c**4*d**3 + 7*c**2*(-B*d**7 - C*c*d**6 + 21 
*D*c**2*d**5/10)/(8*d**2))/(6*d**2))/(4*d**2))/(2*d**2) - (A*c**6*d + B*c* 
*7 + 2*c**2*(-3*A*c**4*d**3 - 3*B*c**5*d**2 + C*c**6*d + D*c**7 + 4*c**2*( 
3*A*c**2*d**5 + 3*B*c**3*d**4 - 3*C*c**4*d**3 - 3*D*c**5*d**2 + 6*c**2*...
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.24 \[ \int (c+d x) \left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {3 \, D c^{10} \arcsin \left (\frac {d x}{c}\right )}{256 \, d^{4}} + \frac {5 \, A c^{7} \arcsin \left (\frac {d x}{c}\right )}{16 \, d} + \frac {5}{16} \, \sqrt {-d^{2} x^{2} + c^{2}} A c^{5} x + \frac {3 \, \sqrt {-d^{2} x^{2} + c^{2}} D c^{8} x}{256 \, d^{3}} + \frac {5 \, {\left (C c + B d\right )} c^{8} \arcsin \left (\frac {d x}{c}\right )}{128 \, d^{3}} + \frac {5}{24} \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} A c^{3} x + \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} D c^{6} x}{128 \, d^{3}} + \frac {5 \, \sqrt {-d^{2} x^{2} + c^{2}} {\left (C c + B d\right )} c^{6} x}{128 \, d^{2}} - \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {7}{2}} D x^{3}}{10 \, d} + \frac {1}{6} \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} A c x + \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} D c^{4} x}{160 \, d^{3}} + \frac {5 \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} {\left (C c + B d\right )} c^{4} x}{192 \, d^{2}} - \frac {3 \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {7}{2}} D c^{2} x}{80 \, d^{3}} + \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} {\left (C c + B d\right )} c^{2} x}{48 \, d^{2}} - \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {7}{2}} {\left (D c + C d\right )} x^{2}}{9 \, d^{2}} - \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {7}{2}} B c}{7 \, d^{2}} - \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {7}{2}} A}{7 \, d} - \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {7}{2}} {\left (C c + B d\right )} x}{8 \, d^{2}} - \frac {2 \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {7}{2}} {\left (D c + C d\right )} c^{2}}{63 \, d^{4}} \] Input:

integrate((d*x+c)*(-d^2*x^2+c^2)^(5/2)*(D*x^3+C*x^2+B*x+A),x, algorithm="m 
axima")
 

Output:

3/256*D*c^10*arcsin(d*x/c)/d^4 + 5/16*A*c^7*arcsin(d*x/c)/d + 5/16*sqrt(-d 
^2*x^2 + c^2)*A*c^5*x + 3/256*sqrt(-d^2*x^2 + c^2)*D*c^8*x/d^3 + 5/128*(C* 
c + B*d)*c^8*arcsin(d*x/c)/d^3 + 5/24*(-d^2*x^2 + c^2)^(3/2)*A*c^3*x + 1/1 
28*(-d^2*x^2 + c^2)^(3/2)*D*c^6*x/d^3 + 5/128*sqrt(-d^2*x^2 + c^2)*(C*c + 
B*d)*c^6*x/d^2 - 1/10*(-d^2*x^2 + c^2)^(7/2)*D*x^3/d + 1/6*(-d^2*x^2 + c^2 
)^(5/2)*A*c*x + 1/160*(-d^2*x^2 + c^2)^(5/2)*D*c^4*x/d^3 + 5/192*(-d^2*x^2 
 + c^2)^(3/2)*(C*c + B*d)*c^4*x/d^2 - 3/80*(-d^2*x^2 + c^2)^(7/2)*D*c^2*x/ 
d^3 + 1/48*(-d^2*x^2 + c^2)^(5/2)*(C*c + B*d)*c^2*x/d^2 - 1/9*(-d^2*x^2 + 
c^2)^(7/2)*(D*c + C*d)*x^2/d^2 - 1/7*(-d^2*x^2 + c^2)^(7/2)*B*c/d^2 - 1/7* 
(-d^2*x^2 + c^2)^(7/2)*A/d - 1/8*(-d^2*x^2 + c^2)^(7/2)*(C*c + B*d)*x/d^2 
- 2/63*(-d^2*x^2 + c^2)^(7/2)*(D*c + C*d)*c^2/d^4
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.20 \[ \int (c+d x) \left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {1}{80640} \, \sqrt {-d^{2} x^{2} + c^{2}} {\left ({\left (2 \, {\left ({\left (4 \, {\left ({\left (2 \, {\left (7 \, {\left (8 \, {\left (9 \, D d^{5} x + \frac {10 \, {\left (D c d^{20} + C d^{21}\right )}}{d^{16}}\right )} x - \frac {9 \, {\left (21 \, D c^{2} d^{19} - 10 \, C c d^{20} - 10 \, B d^{21}\right )}}{d^{16}}\right )} x - \frac {80 \, {\left (19 \, D c^{3} d^{18} + 19 \, C c^{2} d^{19} - 9 \, B c d^{20} - 9 \, A d^{21}\right )}}{d^{16}}\right )} x + \frac {21 \, {\left (93 \, D c^{4} d^{17} - 170 \, C c^{3} d^{18} - 170 \, B c^{2} d^{19} + 80 \, A c d^{20}\right )}}{d^{16}}\right )} x + \frac {480 \, {\left (5 \, D c^{5} d^{16} + 5 \, C c^{4} d^{17} - 9 \, B c^{3} d^{18} - 9 \, A c^{2} d^{19}\right )}}{d^{16}}\right )} x - \frac {105 \, {\left (3 \, D c^{6} d^{15} - 118 \, C c^{5} d^{16} - 118 \, B c^{4} d^{17} + 208 \, A c^{3} d^{18}\right )}}{d^{16}}\right )} x - \frac {640 \, {\left (D c^{7} d^{14} + C c^{6} d^{15} - 27 \, B c^{5} d^{16} - 27 \, A c^{4} d^{17}\right )}}{d^{16}}\right )} x - \frac {315 \, {\left (3 \, D c^{8} d^{13} + 10 \, C c^{7} d^{14} + 10 \, B c^{6} d^{15} - 176 \, A c^{5} d^{16}\right )}}{d^{16}}\right )} x - \frac {1280 \, {\left (2 \, D c^{9} d^{12} + 2 \, C c^{8} d^{13} + 9 \, B c^{7} d^{14} + 9 \, A c^{6} d^{15}\right )}}{d^{16}}\right )} + \frac {{\left (3 \, D c^{10} + 10 \, C c^{9} d + 10 \, B c^{8} d^{2} + 80 \, A c^{7} d^{3}\right )} \arcsin \left (\frac {d x}{c}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right )}{256 \, d^{3} {\left | d \right |}} \] Input:

integrate((d*x+c)*(-d^2*x^2+c^2)^(5/2)*(D*x^3+C*x^2+B*x+A),x, algorithm="g 
iac")
 

Output:

1/80640*sqrt(-d^2*x^2 + c^2)*((2*((4*((2*(7*(8*(9*D*d^5*x + 10*(D*c*d^20 + 
 C*d^21)/d^16)*x - 9*(21*D*c^2*d^19 - 10*C*c*d^20 - 10*B*d^21)/d^16)*x - 8 
0*(19*D*c^3*d^18 + 19*C*c^2*d^19 - 9*B*c*d^20 - 9*A*d^21)/d^16)*x + 21*(93 
*D*c^4*d^17 - 170*C*c^3*d^18 - 170*B*c^2*d^19 + 80*A*c*d^20)/d^16)*x + 480 
*(5*D*c^5*d^16 + 5*C*c^4*d^17 - 9*B*c^3*d^18 - 9*A*c^2*d^19)/d^16)*x - 105 
*(3*D*c^6*d^15 - 118*C*c^5*d^16 - 118*B*c^4*d^17 + 208*A*c^3*d^18)/d^16)*x 
 - 640*(D*c^7*d^14 + C*c^6*d^15 - 27*B*c^5*d^16 - 27*A*c^4*d^17)/d^16)*x - 
 315*(3*D*c^8*d^13 + 10*C*c^7*d^14 + 10*B*c^6*d^15 - 176*A*c^5*d^16)/d^16) 
*x - 1280*(2*D*c^9*d^12 + 2*C*c^8*d^13 + 9*B*c^7*d^14 + 9*A*c^6*d^15)/d^16 
) + 1/256*(3*D*c^10 + 10*C*c^9*d + 10*B*c^8*d^2 + 80*A*c^7*d^3)*arcsin(d*x 
/c)*sgn(c)*sgn(d)/(d^3*abs(d))
 

Mupad [F(-1)]

Timed out. \[ \int (c+d x) \left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\int {\left (c^2-d^2\,x^2\right )}^{5/2}\,\left (c+d\,x\right )\,\left (A+B\,x+C\,x^2+x^3\,D\right ) \,d x \] Input:

int((c^2 - d^2*x^2)^(5/2)*(c + d*x)*(A + B*x + C*x^2 + x^3*D),x)
 

Output:

int((c^2 - d^2*x^2)^(5/2)*(c + d*x)*(A + B*x + C*x^2 + x^3*D), x)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 647, normalized size of antiderivative = 1.78 \[ \int (c+d x) \left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {25200 \mathit {asin} \left (\frac {d x}{c}\right ) a \,c^{7} d^{2}+3150 \mathit {asin} \left (\frac {d x}{c}\right ) b \,c^{8} d -11520 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{6} d^{2}+11520 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,d^{8} x^{6}-11520 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{7} d +10080 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,d^{8} x^{7}-4095 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{8} d x -2560 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{7} d^{2} x^{2}+24150 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{6} d^{3} x^{3}+38400 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{5} d^{4} x^{4}-12936 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{4} d^{5} x^{5}-48640 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{3} d^{6} x^{6}-11088 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{2} d^{7} x^{7}+17920 \sqrt {-d^{2} x^{2}+c^{2}}\, c \,d^{8} x^{8}+8064 \sqrt {-d^{2} x^{2}+c^{2}}\, d^{9} x^{9}+11520 a \,c^{7} d^{2}+11520 b \,c^{8} d +55440 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{5} d^{3} x +34560 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{4} d^{4} x^{2}-43680 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{3} d^{5} x^{3}-34560 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{2} d^{6} x^{4}+13440 \sqrt {-d^{2} x^{2}+c^{2}}\, a c \,d^{7} x^{5}-3150 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{6} d^{2} x +34560 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{5} d^{3} x^{2}+24780 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{4} d^{4} x^{3}-34560 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{3} d^{5} x^{4}-28560 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{2} d^{6} x^{5}+11520 \sqrt {-d^{2} x^{2}+c^{2}}\, b c \,d^{7} x^{6}+4095 \mathit {asin} \left (\frac {d x}{c}\right ) c^{10}-5120 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{9}+5120 c^{10}}{80640 d^{3}} \] Input:

int((d*x+c)*(-d^2*x^2+c^2)^(5/2)*(D*x^3+C*x^2+B*x+A),x)
 

Output:

(25200*asin((d*x)/c)*a*c**7*d**2 + 3150*asin((d*x)/c)*b*c**8*d + 4095*asin 
((d*x)/c)*c**10 - 11520*sqrt(c**2 - d**2*x**2)*a*c**6*d**2 + 55440*sqrt(c* 
*2 - d**2*x**2)*a*c**5*d**3*x + 34560*sqrt(c**2 - d**2*x**2)*a*c**4*d**4*x 
**2 - 43680*sqrt(c**2 - d**2*x**2)*a*c**3*d**5*x**3 - 34560*sqrt(c**2 - d* 
*2*x**2)*a*c**2*d**6*x**4 + 13440*sqrt(c**2 - d**2*x**2)*a*c*d**7*x**5 + 1 
1520*sqrt(c**2 - d**2*x**2)*a*d**8*x**6 - 11520*sqrt(c**2 - d**2*x**2)*b*c 
**7*d - 3150*sqrt(c**2 - d**2*x**2)*b*c**6*d**2*x + 34560*sqrt(c**2 - d**2 
*x**2)*b*c**5*d**3*x**2 + 24780*sqrt(c**2 - d**2*x**2)*b*c**4*d**4*x**3 - 
34560*sqrt(c**2 - d**2*x**2)*b*c**3*d**5*x**4 - 28560*sqrt(c**2 - d**2*x** 
2)*b*c**2*d**6*x**5 + 11520*sqrt(c**2 - d**2*x**2)*b*c*d**7*x**6 + 10080*s 
qrt(c**2 - d**2*x**2)*b*d**8*x**7 - 5120*sqrt(c**2 - d**2*x**2)*c**9 - 409 
5*sqrt(c**2 - d**2*x**2)*c**8*d*x - 2560*sqrt(c**2 - d**2*x**2)*c**7*d**2* 
x**2 + 24150*sqrt(c**2 - d**2*x**2)*c**6*d**3*x**3 + 38400*sqrt(c**2 - d** 
2*x**2)*c**5*d**4*x**4 - 12936*sqrt(c**2 - d**2*x**2)*c**4*d**5*x**5 - 486 
40*sqrt(c**2 - d**2*x**2)*c**3*d**6*x**6 - 11088*sqrt(c**2 - d**2*x**2)*c* 
*2*d**7*x**7 + 17920*sqrt(c**2 - d**2*x**2)*c*d**8*x**8 + 8064*sqrt(c**2 - 
 d**2*x**2)*d**9*x**9 + 11520*a*c**7*d**2 + 11520*b*c**8*d + 5120*c**10)/( 
80640*d**3)