\(\int \frac {\sqrt {c^2-d^2 x^2} (A+B x+C x^2+D x^3)}{(c+d x)^8} \, dx\) [139]

Optimal result

Integrand size = 39, antiderivative size = 359 \[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^8} \, dx=-\frac {\left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \left (c^2-d^2 x^2\right )^{3/2}}{13 c d^4 (c+d x)^8}+\frac {\left (21 c^2 C d-8 B c d^2-5 A d^3-34 c^3 D\right ) \left (c^2-d^2 x^2\right )^{3/2}}{143 c^2 d^4 (c+d x)^7}-\frac {\left (59 c^2 C d+32 B c d^2+20 A d^3-293 c^3 D\right ) \left (c^2-d^2 x^2\right )^{3/2}}{1287 c^3 d^4 (c+d x)^6}-\frac {\left (59 c^2 C d+32 B c d^2+20 A d^3+136 c^3 D\right ) \left (c^2-d^2 x^2\right )^{3/2}}{3003 c^4 d^4 (c+d x)^5}-\frac {2 \left (59 c^2 C d+32 B c d^2+20 A d^3+136 c^3 D\right ) \left (c^2-d^2 x^2\right )^{3/2}}{15015 c^5 d^4 (c+d x)^4}-\frac {2 \left (59 c^2 C d+32 B c d^2+20 A d^3+136 c^3 D\right ) \left (c^2-d^2 x^2\right )^{3/2}}{45045 c^6 d^4 (c+d x)^3} \] Output:

-1/13*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(-d^2*x^2+c^2)^(3/2)/c/d^4/(d*x+c)^8+1 
/143*(-5*A*d^3-8*B*c*d^2+21*C*c^2*d-34*D*c^3)*(-d^2*x^2+c^2)^(3/2)/c^2/d^4 
/(d*x+c)^7-1/1287*(20*A*d^3+32*B*c*d^2+59*C*c^2*d-293*D*c^3)*(-d^2*x^2+c^2 
)^(3/2)/c^3/d^4/(d*x+c)^6-1/3003*(20*A*d^3+32*B*c*d^2+59*C*c^2*d+136*D*c^3 
)*(-d^2*x^2+c^2)^(3/2)/c^4/d^4/(d*x+c)^5-2/15015*(20*A*d^3+32*B*c*d^2+59*C 
*c^2*d+136*D*c^3)*(-d^2*x^2+c^2)^(3/2)/c^5/d^4/(d*x+c)^4-2/45045*(20*A*d^3 
+32*B*c*d^2+59*C*c^2*d+136*D*c^3)*(-d^2*x^2+c^2)^(3/2)/c^6/d^4/(d*x+c)^3
 

Mathematica [A] (verified)

Time = 2.98 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.60 \[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^8} \, dx=-\frac {(c-d x) \sqrt {c^2-d^2 x^2} \left (118 c^8 D+40 A d^8 x^5+64 c d^7 x^4 (5 A+B x)+16 c^7 d (17 C+59 D x)+2 c^2 d^6 x^3 (590 A+x (256 B+59 C x))+c^6 d^2 (911 B+x (2176 C+3481 D x))+8 c^5 d^3 (775 A+x (911 B+1003 x (C+D x)))+16 c^3 d^5 x^2 (170 A+x (118 B+x (59 C+17 D x)))+c^4 d^4 x (4555 A+x (4352 B+x (3481 C+2176 D x)))\right )}{45045 c^6 d^4 (c+d x)^7} \] Input:

Integrate[(Sqrt[c^2 - d^2*x^2]*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^8,x]
 

Output:

-1/45045*((c - d*x)*Sqrt[c^2 - d^2*x^2]*(118*c^8*D + 40*A*d^8*x^5 + 64*c*d 
^7*x^4*(5*A + B*x) + 16*c^7*d*(17*C + 59*D*x) + 2*c^2*d^6*x^3*(590*A + x*( 
256*B + 59*C*x)) + c^6*d^2*(911*B + x*(2176*C + 3481*D*x)) + 8*c^5*d^3*(77 
5*A + x*(911*B + 1003*x*(C + D*x))) + 16*c^3*d^5*x^2*(170*A + x*(118*B + x 
*(59*C + 17*D*x))) + c^4*d^4*x*(4555*A + x*(4352*B + x*(3481*C + 2176*D*x) 
))))/(c^6*d^4*(c + d*x)^7)
 

Rubi [A] (verified)

Time = 1.15 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.256, Rules used = {2170, 27, 2170, 27, 671, 461, 461, 461, 461, 460}

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.

\(\displaystyle \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^8} \, dx\)

\(\Big \downarrow \) 2170

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2170

\(\displaystyle \frac {\frac {\int \frac {d^6 \left (8 D c^3+7 C d c^2+4 A d^3+d \left (12 D c^2+3 C d c+4 B d^2\right ) x\right ) \sqrt {c^2-d^2 x^2}}{(c+d x)^8}dx}{4 d^4}+\frac {C d^2 \left (c^2-d^2 x^2\right )^{3/2}}{4 (c+d x)^7}}{d^5}+\frac {D \left (c^2-d^2 x^2\right )^{3/2}}{3 d^4 (c+d x)^6}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{4} d^2 \int \frac {\left (8 D c^3+7 C d c^2+4 A d^3+d \left (12 D c^2+3 C d c+4 B d^2\right ) x\right ) \sqrt {c^2-d^2 x^2}}{(c+d x)^8}dx+\frac {C d^2 \left (c^2-d^2 x^2\right )^{3/2}}{4 (c+d x)^7}}{d^5}+\frac {D \left (c^2-d^2 x^2\right )^{3/2}}{3 d^4 (c+d x)^6}\)

\(\Big \downarrow \) 671

\(\displaystyle \frac {\frac {1}{4} d^2 \left (\frac {\left (20 A d^3+32 B c d^2+136 c^3 D+59 c^2 C d\right ) \int \frac {\sqrt {c^2-d^2 x^2}}{(c+d x)^7}dx}{13 c}-\frac {4 \left (c^2-d^2 x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{13 c d (c+d x)^8}\right )+\frac {C d^2 \left (c^2-d^2 x^2\right )^{3/2}}{4 (c+d x)^7}}{d^5}+\frac {D \left (c^2-d^2 x^2\right )^{3/2}}{3 d^4 (c+d x)^6}\)

\(\Big \downarrow \) 461

\(\displaystyle \frac {\frac {1}{4} d^2 \left (\frac {\left (20 A d^3+32 B c d^2+136 c^3 D+59 c^2 C d\right ) \left (\frac {4 \int \frac {\sqrt {c^2-d^2 x^2}}{(c+d x)^6}dx}{11 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{11 c d (c+d x)^7}\right )}{13 c}-\frac {4 \left (c^2-d^2 x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{13 c d (c+d x)^8}\right )+\frac {C d^2 \left (c^2-d^2 x^2\right )^{3/2}}{4 (c+d x)^7}}{d^5}+\frac {D \left (c^2-d^2 x^2\right )^{3/2}}{3 d^4 (c+d x)^6}\)

\(\Big \downarrow \) 461

\(\displaystyle \frac {\frac {1}{4} d^2 \left (\frac {\left (20 A d^3+32 B c d^2+136 c^3 D+59 c^2 C d\right ) \left (\frac {4 \left (\frac {\int \frac {\sqrt {c^2-d^2 x^2}}{(c+d x)^5}dx}{3 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{9 c d (c+d x)^6}\right )}{11 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{11 c d (c+d x)^7}\right )}{13 c}-\frac {4 \left (c^2-d^2 x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{13 c d (c+d x)^8}\right )+\frac {C d^2 \left (c^2-d^2 x^2\right )^{3/2}}{4 (c+d x)^7}}{d^5}+\frac {D \left (c^2-d^2 x^2\right )^{3/2}}{3 d^4 (c+d x)^6}\)

\(\Big \downarrow \) 461

\(\displaystyle \frac {\frac {1}{4} d^2 \left (\frac {\left (20 A d^3+32 B c d^2+136 c^3 D+59 c^2 C d\right ) \left (\frac {4 \left (\frac {\frac {2 \int \frac {\sqrt {c^2-d^2 x^2}}{(c+d x)^4}dx}{7 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{7 c d (c+d x)^5}}{3 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{9 c d (c+d x)^6}\right )}{11 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{11 c d (c+d x)^7}\right )}{13 c}-\frac {4 \left (c^2-d^2 x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{13 c d (c+d x)^8}\right )+\frac {C d^2 \left (c^2-d^2 x^2\right )^{3/2}}{4 (c+d x)^7}}{d^5}+\frac {D \left (c^2-d^2 x^2\right )^{3/2}}{3 d^4 (c+d x)^6}\)

\(\Big \downarrow \) 461

\(\displaystyle \frac {\frac {1}{4} d^2 \left (\frac {\left (20 A d^3+32 B c d^2+136 c^3 D+59 c^2 C d\right ) \left (\frac {4 \left (\frac {\frac {2 \left (\frac {\int \frac {\sqrt {c^2-d^2 x^2}}{(c+d x)^3}dx}{5 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{5 c d (c+d x)^4}\right )}{7 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{7 c d (c+d x)^5}}{3 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{9 c d (c+d x)^6}\right )}{11 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{11 c d (c+d x)^7}\right )}{13 c}-\frac {4 \left (c^2-d^2 x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{13 c d (c+d x)^8}\right )+\frac {C d^2 \left (c^2-d^2 x^2\right )^{3/2}}{4 (c+d x)^7}}{d^5}+\frac {D \left (c^2-d^2 x^2\right )^{3/2}}{3 d^4 (c+d x)^6}\)

\(\Big \downarrow \) 460

\(\displaystyle \frac {\frac {1}{4} d^2 \left (\frac {\left (\frac {4 \left (\frac {\frac {2 \left (-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{15 c^2 d (c+d x)^3}-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{5 c d (c+d x)^4}\right )}{7 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{7 c d (c+d x)^5}}{3 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{9 c d (c+d x)^6}\right )}{11 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{11 c d (c+d x)^7}\right ) \left (20 A d^3+32 B c d^2+136 c^3 D+59 c^2 C d\right )}{13 c}-\frac {4 \left (c^2-d^2 x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{13 c d (c+d x)^8}\right )+\frac {C d^2 \left (c^2-d^2 x^2\right )^{3/2}}{4 (c+d x)^7}}{d^5}+\frac {D \left (c^2-d^2 x^2\right )^{3/2}}{3 d^4 (c+d x)^6}\)

Input:

Int[(Sqrt[c^2 - d^2*x^2]*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^8,x]
 

Output:

(D*(c^2 - d^2*x^2)^(3/2))/(3*d^4*(c + d*x)^6) + ((C*d^2*(c^2 - d^2*x^2)^(3 
/2))/(4*(c + d*x)^7) + (d^2*((-4*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(c^2 
- d^2*x^2)^(3/2))/(13*c*d*(c + d*x)^8) + ((59*c^2*C*d + 32*B*c*d^2 + 20*A* 
d^3 + 136*c^3*D)*(-1/11*(c^2 - d^2*x^2)^(3/2)/(c*d*(c + d*x)^7) + (4*(-1/9 
*(c^2 - d^2*x^2)^(3/2)/(c*d*(c + d*x)^6) + (-1/7*(c^2 - d^2*x^2)^(3/2)/(c* 
d*(c + d*x)^5) + (2*(-1/5*(c^2 - d^2*x^2)^(3/2)/(c*d*(c + d*x)^4) - (c^2 - 
 d^2*x^2)^(3/2)/(15*c^2*d*(c + d*x)^3)))/(7*c))/(3*c)))/(11*c)))/(13*c)))/ 
4)/d^5
 

Defintions of rubi rules used

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 460

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(-d)*(c + d*x)^n*((a + b*x^2)^(p + 1)/(b*c*n)), x] /; FreeQ[{a, b, c, d, n, 
 p}, x] && EqQ[b*c^2 + a*d^2, 0] && EqQ[n + 2*p + 2, 0]
 

rule 461

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(-d)*(c + d*x)^n*((a + b*x^2)^(p + 1)/(2*b*c*(n + p + 1))), x] + Simp[Simpl 
ify[n + 2*p + 2]/(2*c*(n + p + 1))   Int[(c + d*x)^(n + 1)*(a + b*x^2)^p, x 
], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && ILtQ[Simp 
lify[n + 2*p + 2], 0] && (LtQ[n, -1] || GtQ[n + p, 0])
 

rule 671

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_ 
), x_Symbol] :> Simp[(d*g - e*f)*(d + e*x)^m*((a + c*x^2)^(p + 1)/(2*c*d*(m 
 + p + 1))), x] + Simp[(m*(g*c*d + c*e*f) + 2*e*c*f*(p + 1))/(e*(2*c*d)*(m 
+ p + 1))   Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, 
e, f, g, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && ((LtQ[m, -1] &&  !IGtQ[m + p 
 + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p 
 + 1, 0]
 

rule 2170

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) 
^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si 
mp[1/(b*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ 
b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - 2*e*f*(m + 
 p + q)*(d + e*x)^(q - 2)*(a*e - b*d*x), x], x], x] /; NeQ[m + q + 2*p + 1, 
 0]] /; FreeQ[{a, b, d, e, m, p}, x] && PolyQ[Pq, x] && EqQ[b*d^2 + a*e^2, 
0] &&  !IGtQ[m, 0]
 

Maple [A] (verified)

Time = 0.77 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.81

method	result	size
gosper	\(-\frac {\left (-d x +c \right ) \left (40 A \,d^{8} x^{5}+64 B c \,d^{7} x^{5}+118 C \,c^{2} d^{6} x^{5}+272 D c^{3} d^{5} x^{5}+320 A c \,d^{7} x^{4}+512 B \,c^{2} d^{6} x^{4}+944 C \,c^{3} d^{5} x^{4}+2176 D c^{4} d^{4} x^{4}+1180 A \,c^{2} d^{6} x^{3}+1888 B \,c^{3} d^{5} x^{3}+3481 C \,c^{4} d^{4} x^{3}+8024 D c^{5} d^{3} x^{3}+2720 A \,c^{3} d^{5} x^{2}+4352 B \,c^{4} d^{4} x^{2}+8024 C \,c^{5} d^{3} x^{2}+3481 D c^{6} d^{2} x^{2}+4555 A \,c^{4} d^{4} x +7288 B \,c^{5} d^{3} x +2176 C \,c^{6} d^{2} x +944 D c^{7} d x +6200 A \,c^{5} d^{3}+911 B \,c^{6} d^{2}+272 C \,c^{7} d +118 D c^{8}\right ) \sqrt {-d^{2} x^{2}+c^{2}}}{45045 \left (d x +c \right )^{7} c^{6} d^{4}}\)	\(291\)
orering	\(-\frac {\left (-d x +c \right ) \left (40 A \,d^{8} x^{5}+64 B c \,d^{7} x^{5}+118 C \,c^{2} d^{6} x^{5}+272 D c^{3} d^{5} x^{5}+320 A c \,d^{7} x^{4}+512 B \,c^{2} d^{6} x^{4}+944 C \,c^{3} d^{5} x^{4}+2176 D c^{4} d^{4} x^{4}+1180 A \,c^{2} d^{6} x^{3}+1888 B \,c^{3} d^{5} x^{3}+3481 C \,c^{4} d^{4} x^{3}+8024 D c^{5} d^{3} x^{3}+2720 A \,c^{3} d^{5} x^{2}+4352 B \,c^{4} d^{4} x^{2}+8024 C \,c^{5} d^{3} x^{2}+3481 D c^{6} d^{2} x^{2}+4555 A \,c^{4} d^{4} x +7288 B \,c^{5} d^{3} x +2176 C \,c^{6} d^{2} x +944 D c^{7} d x +6200 A \,c^{5} d^{3}+911 B \,c^{6} d^{2}+272 C \,c^{7} d +118 D c^{8}\right ) \sqrt {-d^{2} x^{2}+c^{2}}}{45045 \left (d x +c \right )^{7} c^{6} d^{4}}\)	\(291\)
trager	\(-\frac {\left (-40 A \,d^{9} x^{6}-64 B c \,d^{8} x^{6}-118 C \,c^{2} d^{7} x^{6}-272 D c^{3} d^{6} x^{6}-280 A c \,d^{8} x^{5}-448 B \,c^{2} d^{7} x^{5}-826 C \,c^{3} d^{6} x^{5}-1904 D c^{4} d^{5} x^{5}-860 A \,c^{2} d^{7} x^{4}-1376 B \,c^{3} d^{6} x^{4}-2537 C \,c^{4} d^{5} x^{4}-5848 D c^{5} d^{4} x^{4}-1540 A \,c^{3} d^{6} x^{3}-2464 B \,c^{4} d^{5} x^{3}-4543 C \,c^{5} d^{4} x^{3}+4543 D c^{6} d^{3} x^{3}-1835 A \,c^{4} d^{5} x^{2}-2936 B \,c^{5} d^{4} x^{2}+5848 C \,c^{6} d^{3} x^{2}+2537 D c^{7} d^{2} x^{2}-1645 A \,c^{5} d^{4} x +6377 B \,c^{6} d^{3} x +1904 C \,c^{7} d^{2} x +826 D c^{8} d x +6200 A \,c^{6} d^{3}+911 B \,c^{7} d^{2}+272 C \,c^{8} d +118 D c^{9}\right ) \sqrt {-d^{2} x^{2}+c^{2}}}{45045 c^{6} \left (d x +c \right )^{7} d^{4}}\)	\(333\)
default	\(\text {Expression too large to display}\)	\(941\)

Input:

int((-d^2*x^2+c^2)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^8,x,method=_RETURNVER 
BOSE)
 

Output:

-1/45045*(-d*x+c)*(40*A*d^8*x^5+64*B*c*d^7*x^5+118*C*c^2*d^6*x^5+272*D*c^3 
*d^5*x^5+320*A*c*d^7*x^4+512*B*c^2*d^6*x^4+944*C*c^3*d^5*x^4+2176*D*c^4*d^ 
4*x^4+1180*A*c^2*d^6*x^3+1888*B*c^3*d^5*x^3+3481*C*c^4*d^4*x^3+8024*D*c^5* 
d^3*x^3+2720*A*c^3*d^5*x^2+4352*B*c^4*d^4*x^2+8024*C*c^5*d^3*x^2+3481*D*c^ 
6*d^2*x^2+4555*A*c^4*d^4*x+7288*B*c^5*d^3*x+2176*C*c^6*d^2*x+944*D*c^7*d*x 
+6200*A*c^5*d^3+911*B*c^6*d^2+272*C*c^7*d+118*D*c^8)*(-d^2*x^2+c^2)^(1/2)/ 
(d*x+c)^7/c^6/d^4
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 687 vs. \(2 (335) = 670\).

Time = 0.46 (sec) , antiderivative size = 687, normalized size of antiderivative = 1.91 \[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^8} \, dx=-\frac {118 \, D c^{10} + 272 \, C c^{9} d + 911 \, B c^{8} d^{2} + 6200 \, A c^{7} d^{3} + {\left (118 \, D c^{3} d^{7} + 272 \, C c^{2} d^{8} + 911 \, B c d^{9} + 6200 \, A d^{10}\right )} x^{7} + 7 \, {\left (118 \, D c^{4} d^{6} + 272 \, C c^{3} d^{7} + 911 \, B c^{2} d^{8} + 6200 \, A c d^{9}\right )} x^{6} + 21 \, {\left (118 \, D c^{5} d^{5} + 272 \, C c^{4} d^{6} + 911 \, B c^{3} d^{7} + 6200 \, A c^{2} d^{8}\right )} x^{5} + 35 \, {\left (118 \, D c^{6} d^{4} + 272 \, C c^{5} d^{5} + 911 \, B c^{4} d^{6} + 6200 \, A c^{3} d^{7}\right )} x^{4} + 35 \, {\left (118 \, D c^{7} d^{3} + 272 \, C c^{6} d^{4} + 911 \, B c^{5} d^{5} + 6200 \, A c^{4} d^{6}\right )} x^{3} + 21 \, {\left (118 \, D c^{8} d^{2} + 272 \, C c^{7} d^{3} + 911 \, B c^{6} d^{4} + 6200 \, A c^{5} d^{5}\right )} x^{2} + 7 \, {\left (118 \, D c^{9} d + 272 \, C c^{8} d^{2} + 911 \, B c^{7} d^{3} + 6200 \, A c^{6} d^{4}\right )} x + {\left (118 \, D c^{9} + 272 \, C c^{8} d + 911 \, B c^{7} d^{2} + 6200 \, A c^{6} d^{3} - 2 \, {\left (136 \, D c^{3} d^{6} + 59 \, C c^{2} d^{7} + 32 \, B c d^{8} + 20 \, A d^{9}\right )} x^{6} - 14 \, {\left (136 \, D c^{4} d^{5} + 59 \, C c^{3} d^{6} + 32 \, B c^{2} d^{7} + 20 \, A c d^{8}\right )} x^{5} - 43 \, {\left (136 \, D c^{5} d^{4} + 59 \, C c^{4} d^{5} + 32 \, B c^{3} d^{6} + 20 \, A c^{2} d^{7}\right )} x^{4} + 77 \, {\left (59 \, D c^{6} d^{3} - 59 \, C c^{5} d^{4} - 32 \, B c^{4} d^{5} - 20 \, A c^{3} d^{6}\right )} x^{3} + {\left (2537 \, D c^{7} d^{2} + 5848 \, C c^{6} d^{3} - 2936 \, B c^{5} d^{4} - 1835 \, A c^{4} d^{5}\right )} x^{2} + 7 \, {\left (118 \, D c^{8} d + 272 \, C c^{7} d^{2} + 911 \, B c^{6} d^{3} - 235 \, A c^{5} d^{4}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{45045 \, {\left (c^{6} d^{11} x^{7} + 7 \, c^{7} d^{10} x^{6} + 21 \, c^{8} d^{9} x^{5} + 35 \, c^{9} d^{8} x^{4} + 35 \, c^{10} d^{7} x^{3} + 21 \, c^{11} d^{6} x^{2} + 7 \, c^{12} d^{5} x + c^{13} d^{4}\right )}} \] Input:

integrate((-d^2*x^2+c^2)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^8,x, algorithm= 
"fricas")
 

Output:

-1/45045*(118*D*c^10 + 272*C*c^9*d + 911*B*c^8*d^2 + 6200*A*c^7*d^3 + (118 
*D*c^3*d^7 + 272*C*c^2*d^8 + 911*B*c*d^9 + 6200*A*d^10)*x^7 + 7*(118*D*c^4 
*d^6 + 272*C*c^3*d^7 + 911*B*c^2*d^8 + 6200*A*c*d^9)*x^6 + 21*(118*D*c^5*d 
^5 + 272*C*c^4*d^6 + 911*B*c^3*d^7 + 6200*A*c^2*d^8)*x^5 + 35*(118*D*c^6*d 
^4 + 272*C*c^5*d^5 + 911*B*c^4*d^6 + 6200*A*c^3*d^7)*x^4 + 35*(118*D*c^7*d 
^3 + 272*C*c^6*d^4 + 911*B*c^5*d^5 + 6200*A*c^4*d^6)*x^3 + 21*(118*D*c^8*d 
^2 + 272*C*c^7*d^3 + 911*B*c^6*d^4 + 6200*A*c^5*d^5)*x^2 + 7*(118*D*c^9*d 
+ 272*C*c^8*d^2 + 911*B*c^7*d^3 + 6200*A*c^6*d^4)*x + (118*D*c^9 + 272*C*c 
^8*d + 911*B*c^7*d^2 + 6200*A*c^6*d^3 - 2*(136*D*c^3*d^6 + 59*C*c^2*d^7 + 
32*B*c*d^8 + 20*A*d^9)*x^6 - 14*(136*D*c^4*d^5 + 59*C*c^3*d^6 + 32*B*c^2*d 
^7 + 20*A*c*d^8)*x^5 - 43*(136*D*c^5*d^4 + 59*C*c^4*d^5 + 32*B*c^3*d^6 + 2 
0*A*c^2*d^7)*x^4 + 77*(59*D*c^6*d^3 - 59*C*c^5*d^4 - 32*B*c^4*d^5 - 20*A*c 
^3*d^6)*x^3 + (2537*D*c^7*d^2 + 5848*C*c^6*d^3 - 2936*B*c^5*d^4 - 1835*A*c 
^4*d^5)*x^2 + 7*(118*D*c^8*d + 272*C*c^7*d^2 + 911*B*c^6*d^3 - 235*A*c^5*d 
^4)*x)*sqrt(-d^2*x^2 + c^2))/(c^6*d^11*x^7 + 7*c^7*d^10*x^6 + 21*c^8*d^9*x 
^5 + 35*c^9*d^8*x^4 + 35*c^10*d^7*x^3 + 21*c^11*d^6*x^2 + 7*c^12*d^5*x + c 
^13*d^4)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^8} \, dx=\text {Timed out} \] Input:

integrate((-d**2*x**2+c**2)**(1/2)*(D*x**3+C*x**2+B*x+A)/(d*x+c)**8,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3800 vs. \(2 (335) = 670\).

Time = 0.10 (sec) , antiderivative size = 3800, normalized size of antiderivative = 10.58 \[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^8} \, dx=\text {Too large to display} \] Input:

integrate((-d^2*x^2+c^2)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^8,x, algorithm= 
"maxima")
 

Output:

2/13*sqrt(-d^2*x^2 + c^2)*D*c^3/(d^11*x^7 + 7*c*d^10*x^6 + 21*c^2*d^9*x^5 
+ 35*c^3*d^8*x^4 + 35*c^4*d^7*x^3 + 21*c^5*d^6*x^2 + 7*c^6*d^5*x + c^7*d^4 
) - 1/143*sqrt(-d^2*x^2 + c^2)*D*c^3/(c*d^10*x^6 + 6*c^2*d^9*x^5 + 15*c^3* 
d^8*x^4 + 20*c^4*d^7*x^3 + 15*c^5*d^6*x^2 + 6*c^6*d^5*x + c^7*d^4) - 5/128 
7*sqrt(-d^2*x^2 + c^2)*D*c^3/(c^2*d^9*x^5 + 5*c^3*d^8*x^4 + 10*c^4*d^7*x^3 
 + 10*c^5*d^6*x^2 + 5*c^6*d^5*x + c^7*d^4) - 20/9009*sqrt(-d^2*x^2 + c^2)* 
D*c^3/(c^3*d^8*x^4 + 4*c^4*d^7*x^3 + 6*c^5*d^6*x^2 + 4*c^6*d^5*x + c^7*d^4 
) - 4/3003*sqrt(-d^2*x^2 + c^2)*D*c^3/(c^4*d^7*x^3 + 3*c^5*d^6*x^2 + 3*c^6 
*d^5*x + c^7*d^4) - 8/9009*sqrt(-d^2*x^2 + c^2)*D*c^3/(c^5*d^6*x^2 + 2*c^6 
*d^5*x + c^7*d^4) - 8/9009*sqrt(-d^2*x^2 + c^2)*D*c^3/(c^6*d^5*x + c^7*d^4 
) - 2/13*sqrt(-d^2*x^2 + c^2)*C*c^2/(d^10*x^7 + 7*c*d^9*x^6 + 21*c^2*d^8*x 
^5 + 35*c^3*d^7*x^4 + 35*c^4*d^6*x^3 + 21*c^5*d^5*x^2 + 7*c^6*d^4*x + c^7* 
d^3) + 1/143*sqrt(-d^2*x^2 + c^2)*C*c^2/(c*d^9*x^6 + 6*c^2*d^8*x^5 + 15*c^ 
3*d^7*x^4 + 20*c^4*d^6*x^3 + 15*c^5*d^5*x^2 + 6*c^6*d^4*x + c^7*d^3) + 5/1 
287*sqrt(-d^2*x^2 + c^2)*C*c^2/(c^2*d^8*x^5 + 5*c^3*d^7*x^4 + 10*c^4*d^6*x 
^3 + 10*c^5*d^5*x^2 + 5*c^6*d^4*x + c^7*d^3) + 20/9009*sqrt(-d^2*x^2 + c^2 
)*C*c^2/(c^3*d^7*x^4 + 4*c^4*d^6*x^3 + 6*c^5*d^5*x^2 + 4*c^6*d^4*x + c^7*d 
^3) + 4/3003*sqrt(-d^2*x^2 + c^2)*C*c^2/(c^4*d^6*x^3 + 3*c^5*d^5*x^2 + 3*c 
^6*d^4*x + c^7*d^3) + 8/9009*sqrt(-d^2*x^2 + c^2)*C*c^2/(c^5*d^5*x^2 + 2*c 
^6*d^4*x + c^7*d^3) + 8/9009*sqrt(-d^2*x^2 + c^2)*C*c^2/(c^6*d^4*x + c^...
                                                                                    
                                                                                    
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1470 vs. \(2 (335) = 670\).

Time = 0.14 (sec) , antiderivative size = 1470, normalized size of antiderivative = 4.09 \[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^8} \, dx=\text {Too large to display} \] Input:

integrate((-d^2*x^2+c^2)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^8,x, algorithm= 
"giac")
 

Output:

2/45045*(118*D*c^3 + 272*C*c^2*d + 911*B*c*d^2 + 6200*A*d^3 + 11843*(c*d + 
 sqrt(-d^2*x^2 + c^2)*abs(d))*B*c/x + 1534*(c*d + sqrt(-d^2*x^2 + c^2)*abs 
(d))*D*c^3/(d^2*x) + 3536*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))*C*c^2/(d*x) 
+ 35555*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))*A*d/x + 9204*(c*d + sqrt(-d^2* 
x^2 + c^2)*abs(d))^2*D*c^3/(d^4*x^2) + 21216*(c*d + sqrt(-d^2*x^2 + c^2)*a 
bs(d))^2*C*c^2/(d^3*x^2) + 26013*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^2*B*c 
/(d^2*x^2) + 258375*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^2*A/(d*x^2) + 3374 
8*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^3*D*c^3/(d^6*x^3) + 17732*(c*d + sqr 
t(-d^2*x^2 + c^2)*abs(d))^3*C*c^2/(d^5*x^3) + 155441*(c*d + sqrt(-d^2*x^2 
+ c^2)*abs(d))^3*B*c/(d^4*x^3) + 857285*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d) 
)^3*A/(d^3*x^3) - 5720*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^4*D*c^3/(d^8*x^ 
4) + 134420*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^4*C*c^2/(d^7*x^4) + 275990 
*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^4*B*c/(d^6*x^4) + 2255825*(c*d + sqrt 
(-d^2*x^2 + c^2)*abs(d))^4*A/(d^5*x^4) + 133848*(c*d + sqrt(-d^2*x^2 + c^2 
)*abs(d))^5*D*c^3/(d^10*x^5) + 97812*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^5 
*C*c^2/(d^9*x^5) + 640926*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^5*B*c/(d^8*x 
^5) + 3925350*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^5*A/(d^7*x^5) - 1716*(c* 
d + sqrt(-d^2*x^2 + c^2)*abs(d))^6*D*c^3/(d^12*x^6) + 310596*(c*d + sqrt(- 
d^2*x^2 + c^2)*abs(d))^6*C*c^2/(d^11*x^6) + 704418*(c*d + sqrt(-d^2*x^2 + 
c^2)*abs(d))^6*B*c/(d^10*x^6) + 5383950*(c*d + sqrt(-d^2*x^2 + c^2)*abs...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^8} \, dx=\int \frac {\sqrt {c^2-d^2\,x^2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^8} \,d x \] Input:

int(((c^2 - d^2*x^2)^(1/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^8,x)
 

Output:

int(((c^2 - d^2*x^2)^(1/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^8, x)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 901, normalized size of antiderivative = 2.51 \[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^8} \, dx=\frac {-6930 a \,c^{7} d +390 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{8} x -770 a \,d^{8} x^{7}+2535 c^{8} d \,x^{2}+22035 c^{7} d^{2} x^{3}+22035 c^{6} d^{3} x^{4}+2535 c^{5} d^{4} x^{5}+390 c^{4} d^{5} x^{6}-975 \sqrt {-d^{2} x^{2}+c^{2}}\, b c \,d^{6} x^{6}-15041 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{3} d^{4} x^{4}-5914 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{2} d^{5} x^{5}-20684 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{4} d^{3} x^{3}+4100 \sqrt {-d^{2} x^{2}+c^{2}}\, a c \,d^{6} x^{5}+911 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{6} d x -16601 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{5} d^{2} x^{2}+2735 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{5} d^{2} x +9115 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{4} d^{3} x^{2}+13060 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{3} d^{4} x^{3}+10090 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{2} d^{5} x^{4}+28444 b \,c^{6} d^{2} x^{2}+31413 b \,c^{5} d^{3} x^{3}+30797 b \,c^{4} d^{4} x^{4}+18203 b \,c^{3} d^{5} x^{5}+5993 b \,c^{2} d^{6} x^{6}+847 b c \,d^{7} x^{7}+6930 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{6} d +690 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,d^{7} x^{6}+2535 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{7} d \,x^{2}-7800 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{6} d^{2} x^{3}-14235 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{5} d^{3} x^{4}-5070 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{4} d^{4} x^{5}-780 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{3} d^{5} x^{6}+2735 a \,c^{6} d^{2} x -15140 a \,c^{5} d^{3} x^{2}-25845 a \,c^{4} d^{4} x^{3}-26230 a \,c^{3} d^{5} x^{4}-15910 a \,c^{2} d^{6} x^{5}-5350 a c \,d^{7} x^{6}+911 b \,c^{7} d x +390 c^{9} x}{45045 c^{6} d^{2} \left (\sqrt {-d^{2} x^{2}+c^{2}}\, c^{6}+6 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{5} d x +15 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{4} d^{2} x^{2}+20 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{3} d^{3} x^{3}+15 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{2} d^{4} x^{4}+6 \sqrt {-d^{2} x^{2}+c^{2}}\, c \,d^{5} x^{5}+\sqrt {-d^{2} x^{2}+c^{2}}\, d^{6} x^{6}-c^{7}-7 c^{6} d x -21 c^{5} d^{2} x^{2}-35 c^{4} d^{3} x^{3}-35 c^{3} d^{4} x^{4}-21 c^{2} d^{5} x^{5}-7 c \,d^{6} x^{6}-d^{7} x^{7}\right )} \] Input:

int((-d^2*x^2+c^2)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^8,x)
 

Output:

(6930*sqrt(c**2 - d**2*x**2)*a*c**6*d + 2735*sqrt(c**2 - d**2*x**2)*a*c**5 
*d**2*x + 9115*sqrt(c**2 - d**2*x**2)*a*c**4*d**3*x**2 + 13060*sqrt(c**2 - 
 d**2*x**2)*a*c**3*d**4*x**3 + 10090*sqrt(c**2 - d**2*x**2)*a*c**2*d**5*x* 
*4 + 4100*sqrt(c**2 - d**2*x**2)*a*c*d**6*x**5 + 690*sqrt(c**2 - d**2*x**2 
)*a*d**7*x**6 + 911*sqrt(c**2 - d**2*x**2)*b*c**6*d*x - 16601*sqrt(c**2 - 
d**2*x**2)*b*c**5*d**2*x**2 - 20684*sqrt(c**2 - d**2*x**2)*b*c**4*d**3*x** 
3 - 15041*sqrt(c**2 - d**2*x**2)*b*c**3*d**4*x**4 - 5914*sqrt(c**2 - d**2* 
x**2)*b*c**2*d**5*x**5 - 975*sqrt(c**2 - d**2*x**2)*b*c*d**6*x**6 + 390*sq 
rt(c**2 - d**2*x**2)*c**8*x + 2535*sqrt(c**2 - d**2*x**2)*c**7*d*x**2 - 78 
00*sqrt(c**2 - d**2*x**2)*c**6*d**2*x**3 - 14235*sqrt(c**2 - d**2*x**2)*c* 
*5*d**3*x**4 - 5070*sqrt(c**2 - d**2*x**2)*c**4*d**4*x**5 - 780*sqrt(c**2 
- d**2*x**2)*c**3*d**5*x**6 - 6930*a*c**7*d + 2735*a*c**6*d**2*x - 15140*a 
*c**5*d**3*x**2 - 25845*a*c**4*d**4*x**3 - 26230*a*c**3*d**5*x**4 - 15910* 
a*c**2*d**6*x**5 - 5350*a*c*d**7*x**6 - 770*a*d**8*x**7 + 911*b*c**7*d*x + 
 28444*b*c**6*d**2*x**2 + 31413*b*c**5*d**3*x**3 + 30797*b*c**4*d**4*x**4 
+ 18203*b*c**3*d**5*x**5 + 5993*b*c**2*d**6*x**6 + 847*b*c*d**7*x**7 + 390 
*c**9*x + 2535*c**8*d*x**2 + 22035*c**7*d**2*x**3 + 22035*c**6*d**3*x**4 + 
 2535*c**5*d**4*x**5 + 390*c**4*d**5*x**6)/(45045*c**6*d**2*(sqrt(c**2 - d 
**2*x**2)*c**6 + 6*sqrt(c**2 - d**2*x**2)*c**5*d*x + 15*sqrt(c**2 - d**2*x 
**2)*c**4*d**2*x**2 + 20*sqrt(c**2 - d**2*x**2)*c**3*d**3*x**3 + 15*sqr...