\(\int \frac {(c^2-d^2 x^2)^{3/2} (A+B x+C x^2+D x^3)}{(c+d x)^8} \, dx\) [151]

Optimal result

Integrand size = 39, antiderivative size = 239 \[ \int \frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^8} \, dx=-\frac {\left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \left (c^2-d^2 x^2\right )^{5/2}}{11 c d^4 (c+d x)^8}+\frac {\left (19 c^2 C d-8 B c d^2-3 A d^3-30 c^3 D\right ) \left (c^2-d^2 x^2\right )^{5/2}}{99 c^2 d^4 (c+d x)^7}-\frac {\left (61 c^2 C d+16 B c d^2+6 A d^3-237 c^3 D\right ) \left (c^2-d^2 x^2\right )^{5/2}}{693 c^3 d^4 (c+d x)^6}-\frac {\left (61 c^2 C d+16 B c d^2+6 A d^3+456 c^3 D\right ) \left (c^2-d^2 x^2\right )^{5/2}}{3465 c^4 d^4 (c+d x)^5} \] Output:

-1/11*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(-d^2*x^2+c^2)^(5/2)/c/d^4/(d*x+c)^8+1 
/99*(-3*A*d^3-8*B*c*d^2+19*C*c^2*d-30*D*c^3)*(-d^2*x^2+c^2)^(5/2)/c^2/d^4/ 
(d*x+c)^7-1/693*(6*A*d^3+16*B*c*d^2+61*C*c^2*d-237*D*c^3)*(-d^2*x^2+c^2)^( 
5/2)/c^3/d^4/(d*x+c)^6-1/3465*(6*A*d^3+16*B*c*d^2+61*C*c^2*d+456*D*c^3)*(- 
d^2*x^2+c^2)^(5/2)/c^4/d^4/(d*x+c)^5
 

Mathematica [A] (verified)

Time = 2.08 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.65 \[ \int \frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^8} \, dx=-\frac {(c-d x)^2 \sqrt {c^2-d^2 x^2} \left (6 c^6 D+6 A d^6 x^3+16 c d^5 x^2 (3 A+B x)+16 c^5 d (C+3 D x)+c^2 d^4 x (183 A+x (128 B+61 C x))+c^4 d^2 (61 B+x (128 C+183 D x))+8 c^3 d^3 \left (57 A+x \left (61 B+61 C x+57 D x^2\right )\right )\right )}{3465 c^4 d^4 (c+d x)^6} \] Input:

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

Output:

-1/3465*((c - d*x)^2*Sqrt[c^2 - d^2*x^2]*(6*c^6*D + 6*A*d^6*x^3 + 16*c*d^5 
*x^2*(3*A + B*x) + 16*c^5*d*(C + 3*D*x) + c^2*d^4*x*(183*A + x*(128*B + 61 
*C*x)) + c^4*d^2*(61*B + x*(128*C + 183*D*x)) + 8*c^3*d^3*(57*A + x*(61*B 
+ 61*C*x + 57*D*x^2))))/(c^4*d^4*(c + d*x)^6)
 

Rubi [A] (verified)

Time = 1.09 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.16, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {2170, 2170, 27, 671, 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.

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

\(\Big \downarrow \) 2170

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

\(\Big \downarrow \) 2170

\(\displaystyle \frac {\frac {\int \frac {d^6 \left (40 D c^3+7 C d c^2+2 A d^3+d \left (42 D c^2+5 C d c+2 B d^2\right ) x\right ) \left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^8}dx}{2 d^4}+\frac {d \left (c^2-d^2 x^2\right )^{5/2} (4 c D+C d)}{2 (c+d x)^7}}{d^5}+\frac {D \left (c^2-d^2 x^2\right )^{5/2}}{d^4 (c+d x)^6}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{2} d^2 \int \frac {\left (40 D c^3+7 C d c^2+2 A d^3+d \left (42 D c^2+5 C d c+2 B d^2\right ) x\right ) \left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^8}dx+\frac {d \left (c^2-d^2 x^2\right )^{5/2} (4 c D+C d)}{2 (c+d x)^7}}{d^5}+\frac {D \left (c^2-d^2 x^2\right )^{5/2}}{d^4 (c+d x)^6}\)

\(\Big \downarrow \) 671

\(\displaystyle \frac {\frac {1}{2} d^2 \left (\frac {\left (6 A d^3+16 B c d^2+456 c^3 D+61 c^2 C d\right ) \int \frac {\left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^7}dx}{11 c}-\frac {2 \left (c^2-d^2 x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{11 c d (c+d x)^8}\right )+\frac {d \left (c^2-d^2 x^2\right )^{5/2} (4 c D+C d)}{2 (c+d x)^7}}{d^5}+\frac {D \left (c^2-d^2 x^2\right )^{5/2}}{d^4 (c+d x)^6}\)

\(\Big \downarrow \) 461

\(\displaystyle \frac {\frac {1}{2} d^2 \left (\frac {\left (6 A d^3+16 B c d^2+456 c^3 D+61 c^2 C d\right ) \left (\frac {2 \int \frac {\left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^6}dx}{9 c}-\frac {\left (c^2-d^2 x^2\right )^{5/2}}{9 c d (c+d x)^7}\right )}{11 c}-\frac {2 \left (c^2-d^2 x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{11 c d (c+d x)^8}\right )+\frac {d \left (c^2-d^2 x^2\right )^{5/2} (4 c D+C d)}{2 (c+d x)^7}}{d^5}+\frac {D \left (c^2-d^2 x^2\right )^{5/2}}{d^4 (c+d x)^6}\)

\(\Big \downarrow \) 461

\(\displaystyle \frac {\frac {1}{2} d^2 \left (\frac {\left (6 A d^3+16 B c d^2+456 c^3 D+61 c^2 C d\right ) \left (\frac {2 \left (\frac {\int \frac {\left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^5}dx}{7 c}-\frac {\left (c^2-d^2 x^2\right )^{5/2}}{7 c d (c+d x)^6}\right )}{9 c}-\frac {\left (c^2-d^2 x^2\right )^{5/2}}{9 c d (c+d x)^7}\right )}{11 c}-\frac {2 \left (c^2-d^2 x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{11 c d (c+d x)^8}\right )+\frac {d \left (c^2-d^2 x^2\right )^{5/2} (4 c D+C d)}{2 (c+d x)^7}}{d^5}+\frac {D \left (c^2-d^2 x^2\right )^{5/2}}{d^4 (c+d x)^6}\)

\(\Big \downarrow \) 460

\(\displaystyle \frac {\frac {1}{2} d^2 \left (\frac {\left (\frac {2 \left (-\frac {\left (c^2-d^2 x^2\right )^{5/2}}{35 c^2 d (c+d x)^5}-\frac {\left (c^2-d^2 x^2\right )^{5/2}}{7 c d (c+d x)^6}\right )}{9 c}-\frac {\left (c^2-d^2 x^2\right )^{5/2}}{9 c d (c+d x)^7}\right ) \left (6 A d^3+16 B c d^2+456 c^3 D+61 c^2 C d\right )}{11 c}-\frac {2 \left (c^2-d^2 x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{11 c d (c+d x)^8}\right )+\frac {d \left (c^2-d^2 x^2\right )^{5/2} (4 c D+C d)}{2 (c+d x)^7}}{d^5}+\frac {D \left (c^2-d^2 x^2\right )^{5/2}}{d^4 (c+d x)^6}\)

Input:

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

Output:

(D*(c^2 - d^2*x^2)^(5/2))/(d^4*(c + d*x)^6) + ((d*(C*d + 4*c*D)*(c^2 - d^2 
*x^2)^(5/2))/(2*(c + d*x)^7) + (d^2*((-2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3* 
D)*(c^2 - d^2*x^2)^(5/2))/(11*c*d*(c + d*x)^8) + ((61*c^2*C*d + 16*B*c*d^2 
 + 6*A*d^3 + 456*c^3*D)*(-1/9*(c^2 - d^2*x^2)^(5/2)/(c*d*(c + d*x)^7) + (2 
*(-1/7*(c^2 - d^2*x^2)^(5/2)/(c*d*(c + d*x)^6) - (c^2 - d^2*x^2)^(5/2)/(35 
*c^2*d*(c + d*x)^5)))/(9*c)))/(11*c)))/2)/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 = 0.76 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.82

method	result	size
gosper	\(-\frac {\left (-d x +c \right ) \left (6 A \,d^{6} x^{3}+16 B c \,d^{5} x^{3}+61 C \,c^{2} d^{4} x^{3}+456 D c^{3} d^{3} x^{3}+48 A c \,d^{5} x^{2}+128 B \,c^{2} d^{4} x^{2}+488 C \,c^{3} d^{3} x^{2}+183 D c^{4} d^{2} x^{2}+183 A \,c^{2} d^{4} x +488 B \,c^{3} d^{3} x +128 C \,c^{4} d^{2} x +48 D c^{5} d x +456 A \,c^{3} d^{3}+61 B \,c^{4} d^{2}+16 C \,c^{5} d +6 D c^{6}\right ) \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{3465 \left (d x +c \right )^{7} c^{4} d^{4}}\)	\(195\)
orering	\(-\frac {\left (-d x +c \right ) \left (6 A \,d^{6} x^{3}+16 B c \,d^{5} x^{3}+61 C \,c^{2} d^{4} x^{3}+456 D c^{3} d^{3} x^{3}+48 A c \,d^{5} x^{2}+128 B \,c^{2} d^{4} x^{2}+488 C \,c^{3} d^{3} x^{2}+183 D c^{4} d^{2} x^{2}+183 A \,c^{2} d^{4} x +488 B \,c^{3} d^{3} x +128 C \,c^{4} d^{2} x +48 D c^{5} d x +456 A \,c^{3} d^{3}+61 B \,c^{4} d^{2}+16 C \,c^{5} d +6 D c^{6}\right ) \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{3465 \left (d x +c \right )^{7} c^{4} d^{4}}\)	\(195\)
trager	\(-\frac {\left (6 A \,d^{8} x^{5}+16 B c \,d^{7} x^{5}+61 C \,c^{2} d^{6} x^{5}+456 D c^{3} d^{5} x^{5}+36 A c \,d^{7} x^{4}+96 B \,c^{2} d^{6} x^{4}+366 C \,c^{3} d^{5} x^{4}-729 D c^{4} d^{4} x^{4}+93 A \,c^{2} d^{6} x^{3}+248 B \,c^{3} d^{5} x^{3}-787 C \,c^{4} d^{4} x^{3}+138 D c^{5} d^{3} x^{3}+138 A \,c^{3} d^{5} x^{2}-787 B \,c^{4} d^{4} x^{2}+248 C \,c^{5} d^{3} x^{2}+93 D c^{6} d^{2} x^{2}-729 A \,c^{4} d^{4} x +366 B \,c^{5} d^{3} x +96 C \,c^{6} d^{2} x +36 D c^{7} d x +456 A \,c^{5} d^{3}+61 B \,c^{6} d^{2}+16 C \,c^{7} d +6 D c^{8}\right ) \sqrt {-d^{2} x^{2}+c^{2}}}{3465 c^{4} \left (d x +c \right )^{6} d^{4}}\)	\(285\)
default	\(-\frac {D \left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {5}{2}}}{5 d^{9} c \left (x +\frac {c}{d}\right )^{5}}+\frac {\left (C d -3 D c \right ) \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {5}{2}}}{7 c d \left (x +\frac {c}{d}\right )^{6}}-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {5}{2}}}{35 c^{2} \left (x +\frac {c}{d}\right )^{5}}\right )}{d^{9}}+\frac {\left (B \,d^{2}-2 C c d +3 D c^{2}\right ) \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {5}{2}}}{9 c d \left (x +\frac {c}{d}\right )^{7}}+\frac {2 d \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {5}{2}}}{7 c d \left (x +\frac {c}{d}\right )^{6}}-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {5}{2}}}{35 c^{2} \left (x +\frac {c}{d}\right )^{5}}\right )}{9 c}\right )}{d^{10}}+\frac {\left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right ) \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {5}{2}}}{11 c d \left (x +\frac {c}{d}\right )^{8}}+\frac {3 d \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {5}{2}}}{9 c d \left (x +\frac {c}{d}\right )^{7}}+\frac {2 d \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {5}{2}}}{7 c d \left (x +\frac {c}{d}\right )^{6}}-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {5}{2}}}{35 c^{2} \left (x +\frac {c}{d}\right )^{5}}\right )}{9 c}\right )}{11 c}\right )}{d^{11}}\)	\(530\)

Input:

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

Output:

-1/3465*(-d*x+c)*(6*A*d^6*x^3+16*B*c*d^5*x^3+61*C*c^2*d^4*x^3+456*D*c^3*d^ 
3*x^3+48*A*c*d^5*x^2+128*B*c^2*d^4*x^2+488*C*c^3*d^3*x^2+183*D*c^4*d^2*x^2 
+183*A*c^2*d^4*x+488*B*c^3*d^3*x+128*C*c^4*d^2*x+48*D*c^5*d*x+456*A*c^3*d^ 
3+61*B*c^4*d^2+16*C*c^5*d+6*D*c^6)*(-d^2*x^2+c^2)^(3/2)/(d*x+c)^7/c^4/d^4
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 590 vs. \(2 (223) = 446\).

Time = 0.30 (sec) , antiderivative size = 590, normalized size of antiderivative = 2.47 \[ \int \frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^8} \, dx=-\frac {6 \, D c^{9} + 16 \, C c^{8} d + 61 \, B c^{7} d^{2} + 456 \, A c^{6} d^{3} + {\left (6 \, D c^{3} d^{6} + 16 \, C c^{2} d^{7} + 61 \, B c d^{8} + 456 \, A d^{9}\right )} x^{6} + 6 \, {\left (6 \, D c^{4} d^{5} + 16 \, C c^{3} d^{6} + 61 \, B c^{2} d^{7} + 456 \, A c d^{8}\right )} x^{5} + 15 \, {\left (6 \, D c^{5} d^{4} + 16 \, C c^{4} d^{5} + 61 \, B c^{3} d^{6} + 456 \, A c^{2} d^{7}\right )} x^{4} + 20 \, {\left (6 \, D c^{6} d^{3} + 16 \, C c^{5} d^{4} + 61 \, B c^{4} d^{5} + 456 \, A c^{3} d^{6}\right )} x^{3} + 15 \, {\left (6 \, D c^{7} d^{2} + 16 \, C c^{6} d^{3} + 61 \, B c^{5} d^{4} + 456 \, A c^{4} d^{5}\right )} x^{2} + 6 \, {\left (6 \, D c^{8} d + 16 \, C c^{7} d^{2} + 61 \, B c^{6} d^{3} + 456 \, A c^{5} d^{4}\right )} x + {\left (6 \, D c^{8} + 16 \, C c^{7} d + 61 \, B c^{6} d^{2} + 456 \, A c^{5} d^{3} + {\left (456 \, D c^{3} d^{5} + 61 \, C c^{2} d^{6} + 16 \, B c d^{7} + 6 \, A d^{8}\right )} x^{5} - 3 \, {\left (243 \, D c^{4} d^{4} - 122 \, C c^{3} d^{5} - 32 \, B c^{2} d^{6} - 12 \, A c d^{7}\right )} x^{4} + {\left (138 \, D c^{5} d^{3} - 787 \, C c^{4} d^{4} + 248 \, B c^{3} d^{5} + 93 \, A c^{2} d^{6}\right )} x^{3} + {\left (93 \, D c^{6} d^{2} + 248 \, C c^{5} d^{3} - 787 \, B c^{4} d^{4} + 138 \, A c^{3} d^{5}\right )} x^{2} + 3 \, {\left (12 \, D c^{7} d + 32 \, C c^{6} d^{2} + 122 \, B c^{5} d^{3} - 243 \, A c^{4} d^{4}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{3465 \, {\left (c^{4} d^{10} x^{6} + 6 \, c^{5} d^{9} x^{5} + 15 \, c^{6} d^{8} x^{4} + 20 \, c^{7} d^{7} x^{3} + 15 \, c^{8} d^{6} x^{2} + 6 \, c^{9} d^{5} x + c^{10} d^{4}\right )}} \] Input:

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

Output:

-1/3465*(6*D*c^9 + 16*C*c^8*d + 61*B*c^7*d^2 + 456*A*c^6*d^3 + (6*D*c^3*d^ 
6 + 16*C*c^2*d^7 + 61*B*c*d^8 + 456*A*d^9)*x^6 + 6*(6*D*c^4*d^5 + 16*C*c^3 
*d^6 + 61*B*c^2*d^7 + 456*A*c*d^8)*x^5 + 15*(6*D*c^5*d^4 + 16*C*c^4*d^5 + 
61*B*c^3*d^6 + 456*A*c^2*d^7)*x^4 + 20*(6*D*c^6*d^3 + 16*C*c^5*d^4 + 61*B* 
c^4*d^5 + 456*A*c^3*d^6)*x^3 + 15*(6*D*c^7*d^2 + 16*C*c^6*d^3 + 61*B*c^5*d 
^4 + 456*A*c^4*d^5)*x^2 + 6*(6*D*c^8*d + 16*C*c^7*d^2 + 61*B*c^6*d^3 + 456 
*A*c^5*d^4)*x + (6*D*c^8 + 16*C*c^7*d + 61*B*c^6*d^2 + 456*A*c^5*d^3 + (45 
6*D*c^3*d^5 + 61*C*c^2*d^6 + 16*B*c*d^7 + 6*A*d^8)*x^5 - 3*(243*D*c^4*d^4 
- 122*C*c^3*d^5 - 32*B*c^2*d^6 - 12*A*c*d^7)*x^4 + (138*D*c^5*d^3 - 787*C* 
c^4*d^4 + 248*B*c^3*d^5 + 93*A*c^2*d^6)*x^3 + (93*D*c^6*d^2 + 248*C*c^5*d^ 
3 - 787*B*c^4*d^4 + 138*A*c^3*d^5)*x^2 + 3*(12*D*c^7*d + 32*C*c^6*d^2 + 12 
2*B*c^5*d^3 - 243*A*c^4*d^4)*x)*sqrt(-d^2*x^2 + c^2))/(c^4*d^10*x^6 + 6*c^ 
5*d^9*x^5 + 15*c^6*d^8*x^4 + 20*c^7*d^7*x^3 + 15*c^8*d^6*x^2 + 6*c^9*d^5*x 
 + c^10*d^4)
 

Sympy [F(-1)]

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

integrate((-d**2*x**2+c**2)**(3/2)*(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. 3298 vs. \(2 (223) = 446\).

Time = 0.12 (sec) , antiderivative size = 3298, normalized size of antiderivative = 13.80 \[ \int \frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^8} \, dx=\text {Too large to display} \] Input:

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

Output:

1/4*(-d^2*x^2 + c^2)^(3/2)*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) - 3/22*sqrt(-d^2*x^2 + c^2)*D*c^4/(d^10*x^6 + 6*c*d^9*x^5 + 15*c^2*d^8* 
x^4 + 20*c^3*d^7*x^3 + 15*c^4*d^6*x^2 + 6*c^5*d^5*x + c^6*d^4) - 1/4*(-d^2 
*x^2 + c^2)^(3/2)*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) - (-d^2 
*x^2 + c^2)^(3/2)*D*c^2/(d^10*x^6 + 6*c*d^9*x^5 + 15*c^2*d^8*x^4 + 20*c^3* 
d^7*x^3 + 15*c^4*d^6*x^2 + 6*c^5*d^5*x + c^6*d^4) + 3/22*sqrt(-d^2*x^2 + c 
^2)*C*c^3/(d^9*x^6 + 6*c*d^8*x^5 + 15*c^2*d^7*x^4 + 20*c^3*d^6*x^3 + 15*c^ 
4*d^5*x^2 + 6*c^5*d^4*x + c^6*d^3) + 89/132*sqrt(-d^2*x^2 + c^2)*D*c^3/(d^ 
9*x^5 + 5*c*d^8*x^4 + 10*c^2*d^7*x^3 + 10*c^3*d^6*x^2 + 5*c^4*d^5*x + c^5* 
d^4) + 1/231*sqrt(-d^2*x^2 + c^2)*D*c^3/(c*d^8*x^4 + 4*c^2*d^7*x^3 + 6*c^3 
*d^6*x^2 + 4*c^4*d^5*x + c^5*d^4) + 1/385*sqrt(-d^2*x^2 + c^2)*D*c^3/(c^2* 
d^7*x^3 + 3*c^3*d^6*x^2 + 3*c^4*d^5*x + c^5*d^4) + 2/1155*sqrt(-d^2*x^2 + 
c^2)*D*c^3/(c^3*d^6*x^2 + 2*c^4*d^5*x + c^5*d^4) + 2/1155*sqrt(-d^2*x^2 + 
c^2)*D*c^3/(c^4*d^5*x + c^5*d^4) + 1/4*(-d^2*x^2 + c^2)^(3/2)*B*c/(d^9*x^7 
 + 7*c*d^8*x^6 + 21*c^2*d^7*x^5 + 35*c^3*d^6*x^4 + 35*c^4*d^5*x^3 + 21*c^5 
*d^4*x^2 + 7*c^6*d^3*x + c^7*d^2) + 2/3*(-d^2*x^2 + c^2)^(3/2)*C*c/(d^9*x^ 
6 + 6*c*d^8*x^5 + 15*c^2*d^7*x^4 + 20*c^3*d^6*x^3 + 15*c^4*d^5*x^2 + 6*c^5 
*d^4*x + c^6*d^3) + 3/2*(-d^2*x^2 + c^2)^(3/2)*D*c/(d^9*x^5 + 5*c*d^8*x...
                                                                                    
                                                                                    
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1200 vs. \(2 (223) = 446\).

Time = 0.17 (sec) , antiderivative size = 1200, normalized size of antiderivative = 5.02 \[ \int \frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^8} \, dx=\text {Too large to display} \] Input:

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

Output:

2/3465*(6*D*c^3 + 16*C*c^2*d + 61*B*c*d^2 + 456*A*d^3 + 671*(c*d + sqrt(-d 
^2*x^2 + c^2)*abs(d))*B*c/x + 66*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))*D*c^3 
/(d^2*x) + 176*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))*C*c^2/(d*x) + 1551*(c*d 
 + sqrt(-d^2*x^2 + c^2)*abs(d))*A*d/x + 330*(c*d + sqrt(-d^2*x^2 + c^2)*ab 
s(d))^2*D*c^3/(d^4*x^2) + 880*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^2*C*c^2/ 
(d^3*x^2) - 110*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^2*B*c/(d^2*x^2) + 1468 
5*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^2*A/(d*x^2) + 990*(c*d + sqrt(-d^2*x 
^2 + c^2)*abs(d))^3*D*c^3/(d^6*x^3) - 1980*(c*d + sqrt(-d^2*x^2 + c^2)*abs 
(d))^3*C*c^2/(d^5*x^3) + 8910*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^3*B*c/(d 
^4*x^3) + 33660*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^3*A/(d^3*x^3) - 4950*( 
c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^4*D*c^3/(d^8*x^4) + 9900*(c*d + sqrt(-d 
^2*x^2 + c^2)*abs(d))^4*C*c^2/(d^7*x^4) + 3960*(c*d + sqrt(-d^2*x^2 + c^2) 
*abs(d))^4*B*c/(d^6*x^4) + 81180*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^4*A/( 
d^5*x^4) + 15246*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^5*D*c^3/(d^10*x^5) - 
5544*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^5*C*c^2/(d^9*x^5) + 22176*(c*d + 
sqrt(-d^2*x^2 + c^2)*abs(d))^5*B*c/(d^8*x^5) + 98406*(c*d + sqrt(-d^2*x^2 
+ c^2)*abs(d))^5*A/(d^7*x^5) - 12474*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^6 
*D*c^3/(d^12*x^6) + 12936*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^6*C*c^2/(d^1 
1*x^6) + 6006*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^6*B*c/(d^10*x^6) + 11226 
6*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^6*A/(d^9*x^6) + 6930*(c*d + sqrt(...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 768, normalized size of antiderivative = 3.21 \[ \int \frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^8} \, dx=\frac {-630 a \,c^{6} d +22 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{7} x +121 c^{7} d \,x^{2}+1430 c^{6} d^{2} x^{3}+44 c^{5} d^{3} x^{4}-748 c^{4} d^{4} x^{5}+539 c^{3} d^{5} x^{6}+22 c^{8} x +141 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{4} d^{2} x +1878 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{3} d^{3} x^{2}+1833 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{2} d^{4} x^{3}+906 \sqrt {-d^{2} x^{2}+c^{2}}\, a c \,d^{5} x^{4}+61 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{5} d x -1397 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{4} d^{2} x^{2}-362 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{3} d^{3} x^{3}-209 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{2} d^{4} x^{4}-45 \sqrt {-d^{2} x^{2}+c^{2}}\, b c \,d^{5} x^{5}+121 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{6} d \,x^{2}-869 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{5} d^{2} x^{3}-473 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{4} d^{3} x^{4}+495 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{3} d^{4} x^{5}+141 a \,c^{5} d^{2} x -3477 a \,c^{4} d^{3} x^{2}-3435 a \,c^{3} d^{4} x^{3}-2553 a \,c^{2} d^{5} x^{4}-1014 a c \,d^{6} x^{5}+61 b \,c^{6} d x +2068 b \,c^{5} d^{2} x^{2}+185 b \,c^{4} d^{3} x^{3}+1067 b \,c^{3} d^{4} x^{4}+446 b \,c^{2} d^{5} x^{5}+77 b c \,d^{6} x^{6}+630 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{5} d +180 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,d^{6} x^{5}-168 a \,d^{7} x^{6}}{3465 c^{4} d^{2} \left (\sqrt {-d^{2} x^{2}+c^{2}}\, c^{5}+5 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{4} d x +10 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{3} d^{2} x^{2}+10 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{2} d^{3} x^{3}+5 \sqrt {-d^{2} x^{2}+c^{2}}\, c \,d^{4} x^{4}+\sqrt {-d^{2} x^{2}+c^{2}}\, d^{5} x^{5}-c^{6}-6 c^{5} d x -15 c^{4} d^{2} x^{2}-20 c^{3} d^{3} x^{3}-15 c^{2} d^{4} x^{4}-6 c \,d^{5} x^{5}-d^{6} x^{6}\right )} \] Input:

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

Output:

(630*sqrt(c**2 - d**2*x**2)*a*c**5*d + 141*sqrt(c**2 - d**2*x**2)*a*c**4*d 
**2*x + 1878*sqrt(c**2 - d**2*x**2)*a*c**3*d**3*x**2 + 1833*sqrt(c**2 - d* 
*2*x**2)*a*c**2*d**4*x**3 + 906*sqrt(c**2 - d**2*x**2)*a*c*d**5*x**4 + 180 
*sqrt(c**2 - d**2*x**2)*a*d**6*x**5 + 61*sqrt(c**2 - d**2*x**2)*b*c**5*d*x 
 - 1397*sqrt(c**2 - d**2*x**2)*b*c**4*d**2*x**2 - 362*sqrt(c**2 - d**2*x** 
2)*b*c**3*d**3*x**3 - 209*sqrt(c**2 - d**2*x**2)*b*c**2*d**4*x**4 - 45*sqr 
t(c**2 - d**2*x**2)*b*c*d**5*x**5 + 22*sqrt(c**2 - d**2*x**2)*c**7*x + 121 
*sqrt(c**2 - d**2*x**2)*c**6*d*x**2 - 869*sqrt(c**2 - d**2*x**2)*c**5*d**2 
*x**3 - 473*sqrt(c**2 - d**2*x**2)*c**4*d**3*x**4 + 495*sqrt(c**2 - d**2*x 
**2)*c**3*d**4*x**5 - 630*a*c**6*d + 141*a*c**5*d**2*x - 3477*a*c**4*d**3* 
x**2 - 3435*a*c**3*d**4*x**3 - 2553*a*c**2*d**5*x**4 - 1014*a*c*d**6*x**5 
- 168*a*d**7*x**6 + 61*b*c**6*d*x + 2068*b*c**5*d**2*x**2 + 185*b*c**4*d** 
3*x**3 + 1067*b*c**3*d**4*x**4 + 446*b*c**2*d**5*x**5 + 77*b*c*d**6*x**6 + 
 22*c**8*x + 121*c**7*d*x**2 + 1430*c**6*d**2*x**3 + 44*c**5*d**3*x**4 - 7 
48*c**4*d**4*x**5 + 539*c**3*d**5*x**6)/(3465*c**4*d**2*(sqrt(c**2 - d**2* 
x**2)*c**5 + 5*sqrt(c**2 - d**2*x**2)*c**4*d*x + 10*sqrt(c**2 - d**2*x**2) 
*c**3*d**2*x**2 + 10*sqrt(c**2 - d**2*x**2)*c**2*d**3*x**3 + 5*sqrt(c**2 - 
 d**2*x**2)*c*d**4*x**4 + sqrt(c**2 - d**2*x**2)*d**5*x**5 - c**6 - 6*c**5 
*d*x - 15*c**4*d**2*x**2 - 20*c**3*d**3*x**3 - 15*c**2*d**4*x**4 - 6*c*d** 
5*x**5 - d**6*x**6))