\(\int (c+d x)^2 (c^2-d^2 x^2)^{3/2} (A+B x+C x^2+D x^3) \, dx\) [141]

Optimal result

Integrand size = 39, antiderivative size = 362 \[ \int (c+d x)^2 \left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {c^4 \left (11 c^2 C d+16 B c d^2+56 A d^3+6 c^3 D\right ) x \sqrt {c^2-d^2 x^2}}{128 d^3}+\frac {c^2 \left (11 c^2 C d+16 B c d^2+56 A d^3+6 c^3 D\right ) x \left (c^2-d^2 x^2\right )^{3/2}}{192 d^3}-\frac {2 c \left (c^2 C d+B c d^2+A d^3+c^3 D\right ) \left (c^2-d^2 x^2\right )^{5/2}}{5 d^4}-\frac {\left (11 c^2 C d+16 B c d^2+8 A d^3+6 c^3 D\right ) x \left (c^2-d^2 x^2\right )^{5/2}}{48 d^3}-\frac {(C d+2 c D) x^3 \left (c^2-d^2 x^2\right )^{5/2}}{8 d}+\frac {\left (2 c C d+B d^2+3 c^2 D\right ) \left (c^2-d^2 x^2\right )^{7/2}}{7 d^4}-\frac {D \left (c^2-d^2 x^2\right )^{9/2}}{9 d^4}+\frac {c^6 \left (11 c^2 C d+16 B c d^2+56 A d^3+6 c^3 D\right ) \arctan \left (\frac {d x}{\sqrt {c^2-d^2 x^2}}\right )}{128 d^4} \] Output:

1/128*c^4*(56*A*d^3+16*B*c*d^2+11*C*c^2*d+6*D*c^3)*x*(-d^2*x^2+c^2)^(1/2)/ 
d^3+1/192*c^2*(56*A*d^3+16*B*c*d^2+11*C*c^2*d+6*D*c^3)*x*(-d^2*x^2+c^2)^(3 
/2)/d^3-2/5*c*(A*d^3+B*c*d^2+C*c^2*d+D*c^3)*(-d^2*x^2+c^2)^(5/2)/d^4-1/48* 
(8*A*d^3+16*B*c*d^2+11*C*c^2*d+6*D*c^3)*x*(-d^2*x^2+c^2)^(5/2)/d^3-1/8*(C* 
d+2*D*c)*x^3*(-d^2*x^2+c^2)^(5/2)/d+1/7*(B*d^2+2*C*c*d+3*D*c^2)*(-d^2*x^2+ 
c^2)^(7/2)/d^4-1/9*D*(-d^2*x^2+c^2)^(9/2)/d^4+1/128*c^6*(56*A*d^3+16*B*c*d 
^2+11*C*c^2*d+6*D*c^3)*arctan(d*x/(-d^2*x^2+c^2)^(1/2))/d^4
 

Mathematica [A] (verified)

Time = 2.75 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.86 \[ \int (c+d x)^2 \left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=-\frac {\sqrt {c^2-d^2 x^2} \left (3328 c^8 D+18 c^7 d (256 C+105 D x)+c^6 d^2 (10368 B+x (3465 C+1664 D x))-48 c^3 d^5 x^2 \left (672 A+x \left (490 B+384 C x+315 D x^2\right )\right )+96 c d^7 x^4 (168 A+5 x (28 B+3 x (8 C+7 D x)))+80 d^8 x^5 (84 A+x (72 B+7 x (9 C+8 D x)))-8 c^2 d^6 x^3 (210 A+x (144 B+5 x (21 C+16 D x)))+36 c^5 d^3 (448 A+x (140 B+x (64 C+35 D x)))-6 c^4 d^4 x (3780 A+x (2496 B+x (1855 C+1472 D x)))\right )+630 c^6 \left (11 c^2 C d+16 B c d^2+56 A d^3+6 c^3 D\right ) \arctan \left (\frac {d x}{\sqrt {c^2}-\sqrt {c^2-d^2 x^2}}\right )}{40320 d^4} \] Input:

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

Output:

-1/40320*(Sqrt[c^2 - d^2*x^2]*(3328*c^8*D + 18*c^7*d*(256*C + 105*D*x) + c 
^6*d^2*(10368*B + x*(3465*C + 1664*D*x)) - 48*c^3*d^5*x^2*(672*A + x*(490* 
B + 384*C*x + 315*D*x^2)) + 96*c*d^7*x^4*(168*A + 5*x*(28*B + 3*x*(8*C + 7 
*D*x))) + 80*d^8*x^5*(84*A + x*(72*B + 7*x*(9*C + 8*D*x))) - 8*c^2*d^6*x^3 
*(210*A + x*(144*B + 5*x*(21*C + 16*D*x))) + 36*c^5*d^3*(448*A + x*(140*B 
+ x*(64*C + 35*D*x))) - 6*c^4*d^4*x*(3780*A + x*(2496*B + x*(1855*C + 1472 
*D*x)))) + 630*c^6*(11*c^2*C*d + 16*B*c*d^2 + 56*A*d^3 + 6*c^3*D)*ArcTan[( 
d*x)/(Sqrt[c^2] - Sqrt[c^2 - d^2*x^2])])/d^4
 

Rubi [A] (verified)

Time = 1.73 (sec) , antiderivative size = 346, normalized size of antiderivative = 0.96, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2346, 25, 2346, 25, 2346, 25, 2346, 27, 455, 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.

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

\(\Big \downarrow \) 2346

\(\displaystyle -\frac {\int -\left (c^2-d^2 x^2\right )^{3/2} \left (9 d^3 (C d+2 c D) x^4+d^2 \left (13 D c^2+18 C d c+9 B d^2\right ) x^3+9 d^2 \left (C c^2+2 B d c+A d^2\right ) x^2+9 c d^2 (B c+2 A d) x+9 A c^2 d^2\right )dx}{9 d^2}-\frac {1}{9} D x^4 \left (c^2-d^2 x^2\right )^{5/2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \left (c^2-d^2 x^2\right )^{3/2} \left (9 d^3 (C d+2 c D) x^4+d^2 \left (13 D c^2+18 C d c+9 B d^2\right ) x^3+9 d^2 \left (C c^2+2 B d c+A d^2\right ) x^2+9 c d^2 (B c+2 A d) x+9 A c^2 d^2\right )dx}{9 d^2}-\frac {1}{9} D x^4 \left (c^2-d^2 x^2\right )^{5/2}\)

\(\Big \downarrow \) 2346

\(\displaystyle \frac {-\frac {\int -\left (c^2-d^2 x^2\right )^{3/2} \left (8 \left (13 D c^2+18 C d c+9 B d^2\right ) x^3 d^4+72 A c^2 d^4+72 c (B c+2 A d) x d^4+9 \left (6 D c^3+11 C d c^2+16 B d^2 c+8 A d^3\right ) x^2 d^3\right )dx}{8 d^2}-\frac {9}{8} d x^3 \left (c^2-d^2 x^2\right )^{5/2} (2 c D+C d)}{9 d^2}-\frac {1}{9} D x^4 \left (c^2-d^2 x^2\right )^{5/2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \left (c^2-d^2 x^2\right )^{3/2} \left (8 \left (13 D c^2+18 C d c+9 B d^2\right ) x^3 d^4+72 A c^2 d^4+72 c (B c+2 A d) x d^4+9 \left (6 D c^3+11 C d c^2+16 B d^2 c+8 A d^3\right ) x^2 d^3\right )dx}{8 d^2}-\frac {9}{8} d x^3 \left (c^2-d^2 x^2\right )^{5/2} (2 c D+C d)}{9 d^2}-\frac {1}{9} D x^4 \left (c^2-d^2 x^2\right )^{5/2}\)

\(\Big \downarrow \) 2346

\(\displaystyle \frac {\frac {-\frac {\int -\left (c^2-d^2 x^2\right )^{3/2} \left (504 A c^2 d^6+63 \left (6 D c^3+11 C d c^2+16 B d^2 c+8 A d^3\right ) x^2 d^5+8 c \left (26 D c^3+36 C d c^2+81 B d^2 c+126 A d^3\right ) x d^4\right )dx}{7 d^2}-\frac {8}{7} d^2 x^2 \left (c^2-d^2 x^2\right )^{5/2} \left (9 B d^2+13 c^2 D+18 c C d\right )}{8 d^2}-\frac {9}{8} d x^3 \left (c^2-d^2 x^2\right )^{5/2} (2 c D+C d)}{9 d^2}-\frac {1}{9} D x^4 \left (c^2-d^2 x^2\right )^{5/2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {\int \left (c^2-d^2 x^2\right )^{3/2} \left (504 A c^2 d^6+63 \left (6 D c^3+11 C d c^2+16 B d^2 c+8 A d^3\right ) x^2 d^5+8 c \left (26 D c^3+36 C d c^2+81 B d^2 c+126 A d^3\right ) x d^4\right )dx}{7 d^2}-\frac {8}{7} d^2 x^2 \left (c^2-d^2 x^2\right )^{5/2} \left (9 B d^2+13 c^2 D+18 c C d\right )}{8 d^2}-\frac {9}{8} d x^3 \left (c^2-d^2 x^2\right )^{5/2} (2 c D+C d)}{9 d^2}-\frac {1}{9} D x^4 \left (c^2-d^2 x^2\right )^{5/2}\)

\(\Big \downarrow \) 2346

\(\displaystyle \frac {\frac {\frac {-\frac {\int -3 c d^5 \left (21 c \left (6 D c^3+11 C d c^2+16 B d^2 c+56 A d^3\right )+16 d \left (26 D c^3+36 C d c^2+81 B d^2 c+126 A d^3\right ) x\right ) \left (c^2-d^2 x^2\right )^{3/2}dx}{6 d^2}-\frac {21}{2} d^3 x \left (c^2-d^2 x^2\right )^{5/2} \left (8 A d^3+16 B c d^2+6 c^3 D+11 c^2 C d\right )}{7 d^2}-\frac {8}{7} d^2 x^2 \left (c^2-d^2 x^2\right )^{5/2} \left (9 B d^2+13 c^2 D+18 c C d\right )}{8 d^2}-\frac {9}{8} d x^3 \left (c^2-d^2 x^2\right )^{5/2} (2 c D+C d)}{9 d^2}-\frac {1}{9} D x^4 \left (c^2-d^2 x^2\right )^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\frac {1}{2} c d^3 \int \left (21 c \left (6 D c^3+11 C d c^2+16 B d^2 c+56 A d^3\right )+16 d \left (26 D c^3+36 C d c^2+81 B d^2 c+126 A d^3\right ) x\right ) \left (c^2-d^2 x^2\right )^{3/2}dx-\frac {21}{2} d^3 x \left (c^2-d^2 x^2\right )^{5/2} \left (8 A d^3+16 B c d^2+6 c^3 D+11 c^2 C d\right )}{7 d^2}-\frac {8}{7} d^2 x^2 \left (c^2-d^2 x^2\right )^{5/2} \left (9 B d^2+13 c^2 D+18 c C d\right )}{8 d^2}-\frac {9}{8} d x^3 \left (c^2-d^2 x^2\right )^{5/2} (2 c D+C d)}{9 d^2}-\frac {1}{9} D x^4 \left (c^2-d^2 x^2\right )^{5/2}\)

\(\Big \downarrow \) 455

\(\displaystyle \frac {\frac {\frac {\frac {1}{2} c d^3 \left (21 c \left (56 A d^3+16 B c d^2+6 c^3 D+11 c^2 C d\right ) \int \left (c^2-d^2 x^2\right )^{3/2}dx-\frac {16 \left (c^2-d^2 x^2\right )^{5/2} \left (126 A d^3+81 B c d^2+26 c^3 D+36 c^2 C d\right )}{5 d}\right )-\frac {21}{2} d^3 x \left (c^2-d^2 x^2\right )^{5/2} \left (8 A d^3+16 B c d^2+6 c^3 D+11 c^2 C d\right )}{7 d^2}-\frac {8}{7} d^2 x^2 \left (c^2-d^2 x^2\right )^{5/2} \left (9 B d^2+13 c^2 D+18 c C d\right )}{8 d^2}-\frac {9}{8} d x^3 \left (c^2-d^2 x^2\right )^{5/2} (2 c D+C d)}{9 d^2}-\frac {1}{9} D x^4 \left (c^2-d^2 x^2\right )^{5/2}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {\frac {\frac {1}{2} c d^3 \left (21 c \left (56 A d^3+16 B c d^2+6 c^3 D+11 c^2 C d\right ) \left (\frac {3}{4} c^2 \int \sqrt {c^2-d^2 x^2}dx+\frac {1}{4} x \left (c^2-d^2 x^2\right )^{3/2}\right )-\frac {16 \left (c^2-d^2 x^2\right )^{5/2} \left (126 A d^3+81 B c d^2+26 c^3 D+36 c^2 C d\right )}{5 d}\right )-\frac {21}{2} d^3 x \left (c^2-d^2 x^2\right )^{5/2} \left (8 A d^3+16 B c d^2+6 c^3 D+11 c^2 C d\right )}{7 d^2}-\frac {8}{7} d^2 x^2 \left (c^2-d^2 x^2\right )^{5/2} \left (9 B d^2+13 c^2 D+18 c C d\right )}{8 d^2}-\frac {9}{8} d x^3 \left (c^2-d^2 x^2\right )^{5/2} (2 c D+C d)}{9 d^2}-\frac {1}{9} D x^4 \left (c^2-d^2 x^2\right )^{5/2}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {\frac {\frac {1}{2} c d^3 \left (21 c \left (56 A d^3+16 B c d^2+6 c^3 D+11 c^2 C d\right ) \left (\frac {3}{4} c^2 \left (\frac {1}{2} c^2 \int \frac {1}{\sqrt {c^2-d^2 x^2}}dx+\frac {1}{2} x \sqrt {c^2-d^2 x^2}\right )+\frac {1}{4} x \left (c^2-d^2 x^2\right )^{3/2}\right )-\frac {16 \left (c^2-d^2 x^2\right )^{5/2} \left (126 A d^3+81 B c d^2+26 c^3 D+36 c^2 C d\right )}{5 d}\right )-\frac {21}{2} d^3 x \left (c^2-d^2 x^2\right )^{5/2} \left (8 A d^3+16 B c d^2+6 c^3 D+11 c^2 C d\right )}{7 d^2}-\frac {8}{7} d^2 x^2 \left (c^2-d^2 x^2\right )^{5/2} \left (9 B d^2+13 c^2 D+18 c C d\right )}{8 d^2}-\frac {9}{8} d x^3 \left (c^2-d^2 x^2\right )^{5/2} (2 c D+C d)}{9 d^2}-\frac {1}{9} D x^4 \left (c^2-d^2 x^2\right )^{5/2}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {\frac {\frac {1}{2} c d^3 \left (21 c \left (56 A d^3+16 B c d^2+6 c^3 D+11 c^2 C d\right ) \left (\frac {3}{4} c^2 \left (\frac {1}{2} c^2 \int \frac {1}{\frac {d^2 x^2}{c^2-d^2 x^2}+1}d\frac {x}{\sqrt {c^2-d^2 x^2}}+\frac {1}{2} x \sqrt {c^2-d^2 x^2}\right )+\frac {1}{4} x \left (c^2-d^2 x^2\right )^{3/2}\right )-\frac {16 \left (c^2-d^2 x^2\right )^{5/2} \left (126 A d^3+81 B c d^2+26 c^3 D+36 c^2 C d\right )}{5 d}\right )-\frac {21}{2} d^3 x \left (c^2-d^2 x^2\right )^{5/2} \left (8 A d^3+16 B c d^2+6 c^3 D+11 c^2 C d\right )}{7 d^2}-\frac {8}{7} d^2 x^2 \left (c^2-d^2 x^2\right )^{5/2} \left (9 B d^2+13 c^2 D+18 c C d\right )}{8 d^2}-\frac {9}{8} d x^3 \left (c^2-d^2 x^2\right )^{5/2} (2 c D+C d)}{9 d^2}-\frac {1}{9} D x^4 \left (c^2-d^2 x^2\right )^{5/2}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {\frac {\frac {\frac {1}{2} c d^3 \left (21 c \left (\frac {3}{4} c^2 \left (\frac {c^2 \arctan \left (\frac {d x}{\sqrt {c^2-d^2 x^2}}\right )}{2 d}+\frac {1}{2} x \sqrt {c^2-d^2 x^2}\right )+\frac {1}{4} x \left (c^2-d^2 x^2\right )^{3/2}\right ) \left (56 A d^3+16 B c d^2+6 c^3 D+11 c^2 C d\right )-\frac {16 \left (c^2-d^2 x^2\right )^{5/2} \left (126 A d^3+81 B c d^2+26 c^3 D+36 c^2 C d\right )}{5 d}\right )-\frac {21}{2} d^3 x \left (c^2-d^2 x^2\right )^{5/2} \left (8 A d^3+16 B c d^2+6 c^3 D+11 c^2 C d\right )}{7 d^2}-\frac {8}{7} d^2 x^2 \left (c^2-d^2 x^2\right )^{5/2} \left (9 B d^2+13 c^2 D+18 c C d\right )}{8 d^2}-\frac {9}{8} d x^3 \left (c^2-d^2 x^2\right )^{5/2} (2 c D+C d)}{9 d^2}-\frac {1}{9} D x^4 \left (c^2-d^2 x^2\right )^{5/2}\)

Input:

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

Output:

-1/9*(D*x^4*(c^2 - d^2*x^2)^(5/2)) + ((-9*d*(C*d + 2*c*D)*x^3*(c^2 - d^2*x 
^2)^(5/2))/8 + ((-8*d^2*(18*c*C*d + 9*B*d^2 + 13*c^2*D)*x^2*(c^2 - d^2*x^2 
)^(5/2))/7 + ((-21*d^3*(11*c^2*C*d + 16*B*c*d^2 + 8*A*d^3 + 6*c^3*D)*x*(c^ 
2 - d^2*x^2)^(5/2))/2 + (c*d^3*((-16*(36*c^2*C*d + 81*B*c*d^2 + 126*A*d^3 
+ 26*c^3*D)*(c^2 - d^2*x^2)^(5/2))/(5*d) + 21*c*(11*c^2*C*d + 16*B*c*d^2 + 
 56*A*d^3 + 6*c^3*D)*((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)))/2)/(7*d 
^2))/(8*d^2))/(9*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 = 0.50 (sec) , antiderivative size = 513, normalized size of antiderivative = 1.42

method	result	size
default	\(A \,c^{2} \left (\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 c^{2} \left (\frac {x \sqrt {-d^{2} x^{2}+c^{2}}}{2}+\frac {c^{2} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right )}{2 \sqrt {d^{2}}}\right )}{4}\right )-\frac {c \left (2 A d +B c \right ) \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{5 d^{2}}+d \left (C d +2 D c \right ) \left (-\frac {x^{3} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{8 d^{2}}+\frac {3 c^{2} \left (-\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{6 d^{2}}+\frac {c^{2} \left (\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 c^{2} \left (\frac {x \sqrt {-d^{2} x^{2}+c^{2}}}{2}+\frac {c^{2} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right )}{2 \sqrt {d^{2}}}\right )}{4}\right )}{6 d^{2}}\right )}{8 d^{2}}\right )+\left (A \,d^{2}+2 B c d +C \,c^{2}\right ) \left (-\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{6 d^{2}}+\frac {c^{2} \left (\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 c^{2} \left (\frac {x \sqrt {-d^{2} x^{2}+c^{2}}}{2}+\frac {c^{2} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right )}{2 \sqrt {d^{2}}}\right )}{4}\right )}{6 d^{2}}\right )+\left (B \,d^{2}+2 C c d +D c^{2}\right ) \left (-\frac {x^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{7 d^{2}}-\frac {2 c^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{35 d^{4}}\right )+D d^{2} \left (-\frac {x^{4} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{9 d^{2}}+\frac {4 c^{2} \left (-\frac {x^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{7 d^{2}}-\frac {2 c^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{35 d^{4}}\right )}{9 d^{2}}\right )\)	\(513\)

Input:

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

Output:

A*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))))-1/5*c*(2*A*d+ 
B*c)*(-d^2*x^2+c^2)^(5/2)/d^2+d*(C*d+2*D*c)*(-1/8*x^3*(-d^2*x^2+c^2)^(5/2) 
/d^2+3/8*c^2/d^2*(-1/6*x*(-d^2*x^2+c^2)^(5/2)/d^2+1/6*c^2/d^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)*ar 
ctan((d^2)^(1/2)*x/(-d^2*x^2+c^2)^(1/2))))))+(A*d^2+2*B*c*d+C*c^2)*(-1/6*x 
*(-d^2*x^2+c^2)^(5/2)/d^2+1/6*c^2/d^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)))))+(B*d^2+2*C*c*d+D*c^2)*(-1/7*x^2*(-d^2*x^2+c^2)^(5/2)/d 
^2-2/35*c^2*(-d^2*x^2+c^2)^(5/2)/d^4)+D*d^2*(-1/9*x^4*(-d^2*x^2+c^2)^(5/2) 
/d^2+4/9*c^2/d^2*(-1/7*x^2*(-d^2*x^2+c^2)^(5/2)/d^2-2/35*c^2*(-d^2*x^2+c^2 
)^(5/2)/d^4))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.01 \[ \int (c+d x)^2 \left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=-\frac {630 \, {\left (6 \, D c^{9} + 11 \, C c^{8} d + 16 \, B c^{7} d^{2} + 56 \, A c^{6} d^{3}\right )} \arctan \left (-\frac {c - \sqrt {-d^{2} x^{2} + c^{2}}}{d x}\right ) + {\left (4480 \, D d^{8} x^{8} + 3328 \, D c^{8} + 4608 \, C c^{7} d + 10368 \, B c^{6} d^{2} + 16128 \, A c^{5} d^{3} + 5040 \, {\left (2 \, D c d^{7} + C d^{8}\right )} x^{7} - 640 \, {\left (D c^{2} d^{6} - 18 \, C c d^{7} - 9 \, B d^{8}\right )} x^{6} - 840 \, {\left (18 \, D c^{3} d^{5} + C c^{2} d^{6} - 16 \, B c d^{7} - 8 \, A d^{8}\right )} x^{5} - 384 \, {\left (23 \, D c^{4} d^{4} + 48 \, C c^{3} d^{5} + 3 \, B c^{2} d^{6} - 42 \, A c d^{7}\right )} x^{4} + 210 \, {\left (6 \, D c^{5} d^{3} - 53 \, C c^{4} d^{4} - 112 \, B c^{3} d^{5} - 8 \, A c^{2} d^{6}\right )} x^{3} + 128 \, {\left (13 \, D c^{6} d^{2} + 18 \, C c^{5} d^{3} - 117 \, B c^{4} d^{4} - 252 \, A c^{3} d^{5}\right )} x^{2} + 315 \, {\left (6 \, D c^{7} d + 11 \, C c^{6} d^{2} + 16 \, B c^{5} d^{3} - 72 \, A c^{4} d^{4}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{40320 \, d^{4}} \] Input:

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

Output:

-1/40320*(630*(6*D*c^9 + 11*C*c^8*d + 16*B*c^7*d^2 + 56*A*c^6*d^3)*arctan( 
-(c - sqrt(-d^2*x^2 + c^2))/(d*x)) + (4480*D*d^8*x^8 + 3328*D*c^8 + 4608*C 
*c^7*d + 10368*B*c^6*d^2 + 16128*A*c^5*d^3 + 5040*(2*D*c*d^7 + C*d^8)*x^7 
- 640*(D*c^2*d^6 - 18*C*c*d^7 - 9*B*d^8)*x^6 - 840*(18*D*c^3*d^5 + C*c^2*d 
^6 - 16*B*c*d^7 - 8*A*d^8)*x^5 - 384*(23*D*c^4*d^4 + 48*C*c^3*d^5 + 3*B*c^ 
2*d^6 - 42*A*c*d^7)*x^4 + 210*(6*D*c^5*d^3 - 53*C*c^4*d^4 - 112*B*c^3*d^5 
- 8*A*c^2*d^6)*x^3 + 128*(13*D*c^6*d^2 + 18*C*c^5*d^3 - 117*B*c^4*d^4 - 25 
2*A*c^3*d^5)*x^2 + 315*(6*D*c^7*d + 11*C*c^6*d^2 + 16*B*c^5*d^3 - 72*A*c^4 
*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. 1044 vs. \(2 (352) = 704\).

Time = 0.79 (sec) , antiderivative size = 1044, normalized size of antiderivative = 2.88 \[ \int (c+d x)^2 \left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 554, normalized size of antiderivative = 1.53 \[ \int (c+d x)^2 \left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=-\frac {1}{9} \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} D x^{4} + \frac {3 \, A c^{6} \arcsin \left (\frac {d x}{c}\right )}{8 \, d} + \frac {3}{8} \, \sqrt {-d^{2} x^{2} + c^{2}} A c^{4} x + \frac {1}{4} \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} A c^{2} x - \frac {4 \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} D c^{2} x^{2}}{63 \, d^{2}} + \frac {3 \, {\left (2 \, D c d + C d^{2}\right )} c^{8} \arcsin \left (\frac {d x}{c}\right )}{128 \, d^{5}} + \frac {{\left (C c^{2} + 2 \, B c d + A d^{2}\right )} c^{6} \arcsin \left (\frac {d x}{c}\right )}{16 \, d^{3}} + \frac {3 \, \sqrt {-d^{2} x^{2} + c^{2}} {\left (2 \, D c d + C d^{2}\right )} c^{6} x}{128 \, d^{4}} + \frac {\sqrt {-d^{2} x^{2} + c^{2}} {\left (C c^{2} + 2 \, B c d + A d^{2}\right )} c^{4} x}{16 \, d^{2}} - \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} {\left (2 \, D c d + C d^{2}\right )} x^{3}}{8 \, d^{2}} - \frac {8 \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} D c^{4}}{315 \, d^{4}} - \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} B c^{2}}{5 \, d^{2}} - \frac {2 \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} A c}{5 \, d} + \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} {\left (2 \, D c d + C d^{2}\right )} c^{4} x}{64 \, d^{4}} + \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} {\left (C c^{2} + 2 \, B c d + A d^{2}\right )} c^{2} x}{24 \, d^{2}} - \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} {\left (D c^{2} + 2 \, C c d + B d^{2}\right )} x^{2}}{7 \, d^{2}} - \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} {\left (2 \, D c d + C d^{2}\right )} c^{2} x}{16 \, d^{4}} - \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} {\left (C c^{2} + 2 \, B c d + A d^{2}\right )} x}{6 \, d^{2}} - \frac {2 \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} {\left (D c^{2} + 2 \, C c d + B d^{2}\right )} c^{2}}{35 \, d^{4}} \] Input:

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

Output:

-1/9*(-d^2*x^2 + c^2)^(5/2)*D*x^4 + 3/8*A*c^6*arcsin(d*x/c)/d + 3/8*sqrt(- 
d^2*x^2 + c^2)*A*c^4*x + 1/4*(-d^2*x^2 + c^2)^(3/2)*A*c^2*x - 4/63*(-d^2*x 
^2 + c^2)^(5/2)*D*c^2*x^2/d^2 + 3/128*(2*D*c*d + C*d^2)*c^8*arcsin(d*x/c)/ 
d^5 + 1/16*(C*c^2 + 2*B*c*d + A*d^2)*c^6*arcsin(d*x/c)/d^3 + 3/128*sqrt(-d 
^2*x^2 + c^2)*(2*D*c*d + C*d^2)*c^6*x/d^4 + 1/16*sqrt(-d^2*x^2 + c^2)*(C*c 
^2 + 2*B*c*d + A*d^2)*c^4*x/d^2 - 1/8*(-d^2*x^2 + c^2)^(5/2)*(2*D*c*d + C* 
d^2)*x^3/d^2 - 8/315*(-d^2*x^2 + c^2)^(5/2)*D*c^4/d^4 - 1/5*(-d^2*x^2 + c^ 
2)^(5/2)*B*c^2/d^2 - 2/5*(-d^2*x^2 + c^2)^(5/2)*A*c/d + 1/64*(-d^2*x^2 + c 
^2)^(3/2)*(2*D*c*d + C*d^2)*c^4*x/d^4 + 1/24*(-d^2*x^2 + c^2)^(3/2)*(C*c^2 
 + 2*B*c*d + A*d^2)*c^2*x/d^2 - 1/7*(-d^2*x^2 + c^2)^(5/2)*(D*c^2 + 2*C*c* 
d + B*d^2)*x^2/d^2 - 1/16*(-d^2*x^2 + c^2)^(5/2)*(2*D*c*d + C*d^2)*c^2*x/d 
^4 - 1/6*(-d^2*x^2 + c^2)^(5/2)*(C*c^2 + 2*B*c*d + A*d^2)*x/d^2 - 2/35*(-d 
^2*x^2 + c^2)^(5/2)*(D*c^2 + 2*C*c*d + B*d^2)*c^2/d^4
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.08 \[ \int (c+d x)^2 \left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=-\frac {1}{40320} \, \sqrt {-d^{2} x^{2} + c^{2}} {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (2 \, {\left (7 \, {\left (8 \, D d^{4} x + \frac {9 \, {\left (2 \, D c d^{17} + C d^{18}\right )}}{d^{14}}\right )} x - \frac {8 \, {\left (D c^{2} d^{16} - 18 \, C c d^{17} - 9 \, B d^{18}\right )}}{d^{14}}\right )} x - \frac {21 \, {\left (18 \, D c^{3} d^{15} + C c^{2} d^{16} - 16 \, B c d^{17} - 8 \, A d^{18}\right )}}{d^{14}}\right )} x - \frac {48 \, {\left (23 \, D c^{4} d^{14} + 48 \, C c^{3} d^{15} + 3 \, B c^{2} d^{16} - 42 \, A c d^{17}\right )}}{d^{14}}\right )} x + \frac {105 \, {\left (6 \, D c^{5} d^{13} - 53 \, C c^{4} d^{14} - 112 \, B c^{3} d^{15} - 8 \, A c^{2} d^{16}\right )}}{d^{14}}\right )} x + \frac {64 \, {\left (13 \, D c^{6} d^{12} + 18 \, C c^{5} d^{13} - 117 \, B c^{4} d^{14} - 252 \, A c^{3} d^{15}\right )}}{d^{14}}\right )} x + \frac {315 \, {\left (6 \, D c^{7} d^{11} + 11 \, C c^{6} d^{12} + 16 \, B c^{5} d^{13} - 72 \, A c^{4} d^{14}\right )}}{d^{14}}\right )} x + \frac {128 \, {\left (26 \, D c^{8} d^{10} + 36 \, C c^{7} d^{11} + 81 \, B c^{6} d^{12} + 126 \, A c^{5} d^{13}\right )}}{d^{14}}\right )} + \frac {{\left (6 \, D c^{9} + 11 \, C c^{8} d + 16 \, B c^{7} d^{2} + 56 \, A c^{6} d^{3}\right )} \arcsin \left (\frac {d x}{c}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right )}{128 \, d^{3} {\left | d \right |}} \] Input:

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

Output:

-1/40320*sqrt(-d^2*x^2 + c^2)*((2*((4*(5*(2*(7*(8*D*d^4*x + 9*(2*D*c*d^17 
+ C*d^18)/d^14)*x - 8*(D*c^2*d^16 - 18*C*c*d^17 - 9*B*d^18)/d^14)*x - 21*( 
18*D*c^3*d^15 + C*c^2*d^16 - 16*B*c*d^17 - 8*A*d^18)/d^14)*x - 48*(23*D*c^ 
4*d^14 + 48*C*c^3*d^15 + 3*B*c^2*d^16 - 42*A*c*d^17)/d^14)*x + 105*(6*D*c^ 
5*d^13 - 53*C*c^4*d^14 - 112*B*c^3*d^15 - 8*A*c^2*d^16)/d^14)*x + 64*(13*D 
*c^6*d^12 + 18*C*c^5*d^13 - 117*B*c^4*d^14 - 252*A*c^3*d^15)/d^14)*x + 315 
*(6*D*c^7*d^11 + 11*C*c^6*d^12 + 16*B*c^5*d^13 - 72*A*c^4*d^14)/d^14)*x + 
128*(26*D*c^8*d^10 + 36*C*c^7*d^11 + 81*B*c^6*d^12 + 126*A*c^5*d^13)/d^14) 
 + 1/128*(6*D*c^9 + 11*C*c^8*d + 16*B*c^7*d^2 + 56*A*c^6*d^3)*arcsin(d*x/c 
)*sgn(c)*sgn(d)/(d^3*abs(d))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 573, normalized size of antiderivative = 1.58 \[ \int (c+d x)^2 \left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {-4480 \sqrt {-d^{2} x^{2}+c^{2}}\, d^{8} x^{8}+16128 a \,c^{6} d^{2}+10368 b \,c^{7} d +5355 \mathit {asin} \left (\frac {d x}{c}\right ) c^{9}-7936 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{8}+7936 c^{9}+22680 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{4} d^{3} x +32256 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{3} d^{4} x^{2}+1680 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{2} d^{5} x^{3}-16128 \sqrt {-d^{2} x^{2}+c^{2}}\, a c \,d^{6} x^{4}-5040 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{5} d^{2} x +14976 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{4} d^{3} x^{2}+23520 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{3} d^{4} x^{3}+1152 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{2} d^{5} x^{4}-13440 \sqrt {-d^{2} x^{2}+c^{2}}\, b c \,d^{6} x^{5}+17640 \mathit {asin} \left (\frac {d x}{c}\right ) a \,c^{6} d^{2}+5040 \mathit {asin} \left (\frac {d x}{c}\right ) b \,c^{7} d -16128 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{5} d^{2}-6720 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,d^{7} x^{5}-10368 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{6} d -5760 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,d^{7} x^{6}-5355 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{7} d x -3968 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{6} d^{2} x^{2}+9870 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{5} d^{3} x^{3}+27264 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{4} d^{4} x^{4}+15960 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{3} d^{5} x^{5}-10880 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{2} d^{6} x^{6}-15120 \sqrt {-d^{2} x^{2}+c^{2}}\, c \,d^{7} x^{7}}{40320 d^{3}} \] Input:

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

Output:

(17640*asin((d*x)/c)*a*c**6*d**2 + 5040*asin((d*x)/c)*b*c**7*d + 5355*asin 
((d*x)/c)*c**9 - 16128*sqrt(c**2 - d**2*x**2)*a*c**5*d**2 + 22680*sqrt(c** 
2 - d**2*x**2)*a*c**4*d**3*x + 32256*sqrt(c**2 - d**2*x**2)*a*c**3*d**4*x* 
*2 + 1680*sqrt(c**2 - d**2*x**2)*a*c**2*d**5*x**3 - 16128*sqrt(c**2 - d**2 
*x**2)*a*c*d**6*x**4 - 6720*sqrt(c**2 - d**2*x**2)*a*d**7*x**5 - 10368*sqr 
t(c**2 - d**2*x**2)*b*c**6*d - 5040*sqrt(c**2 - d**2*x**2)*b*c**5*d**2*x + 
 14976*sqrt(c**2 - d**2*x**2)*b*c**4*d**3*x**2 + 23520*sqrt(c**2 - d**2*x* 
*2)*b*c**3*d**4*x**3 + 1152*sqrt(c**2 - d**2*x**2)*b*c**2*d**5*x**4 - 1344 
0*sqrt(c**2 - d**2*x**2)*b*c*d**6*x**5 - 5760*sqrt(c**2 - d**2*x**2)*b*d** 
7*x**6 - 7936*sqrt(c**2 - d**2*x**2)*c**8 - 5355*sqrt(c**2 - d**2*x**2)*c* 
*7*d*x - 3968*sqrt(c**2 - d**2*x**2)*c**6*d**2*x**2 + 9870*sqrt(c**2 - d** 
2*x**2)*c**5*d**3*x**3 + 27264*sqrt(c**2 - d**2*x**2)*c**4*d**4*x**4 + 159 
60*sqrt(c**2 - d**2*x**2)*c**3*d**5*x**5 - 10880*sqrt(c**2 - d**2*x**2)*c* 
*2*d**6*x**6 - 15120*sqrt(c**2 - d**2*x**2)*c*d**7*x**7 - 4480*sqrt(c**2 - 
 d**2*x**2)*d**8*x**8 + 16128*a*c**6*d**2 + 10368*b*c**7*d + 7936*c**9)/(4 
0320*d**3)