\(\int \frac {(c^2-d^2 x^2)^{5/2} (A+B x+C x^2+D x^3)}{(c+d x)^{11}} \, dx\) [167]

Optimal result

Integrand size = 39, antiderivative size = 299 \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{11}} \, dx=-\frac {\left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \left (c^2-d^2 x^2\right )^{7/2}}{15 c d^4 (c+d x)^{11}}+\frac {\left (26 c^2 C d-11 B c d^2-4 A d^3-41 c^3 D\right ) \left (c^2-d^2 x^2\right )^{7/2}}{195 c^2 d^4 (c+d x)^{10}}-\frac {\left (39 c^2 C d+11 B c d^2+4 A d^3-154 c^3 D\right ) \left (c^2-d^2 x^2\right )^{7/2}}{715 c^3 d^4 (c+d x)^9}-\frac {\left (78 c^2 C d+22 B c d^2+8 A d^3+407 c^3 D\right ) \left (c^2-d^2 x^2\right )^{7/2}}{6435 c^4 d^4 (c+d x)^8}-\frac {\left (78 c^2 C d+22 B c d^2+8 A d^3+407 c^3 D\right ) \left (c^2-d^2 x^2\right )^{7/2}}{45045 c^5 d^4 (c+d x)^7} \] Output:

-1/15*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(-d^2*x^2+c^2)^(7/2)/c/d^4/(d*x+c)^11+ 
1/195*(-4*A*d^3-11*B*c*d^2+26*C*c^2*d-41*D*c^3)*(-d^2*x^2+c^2)^(7/2)/c^2/d 
^4/(d*x+c)^10-1/715*(4*A*d^3+11*B*c*d^2+39*C*c^2*d-154*D*c^3)*(-d^2*x^2+c^ 
2)^(7/2)/c^3/d^4/(d*x+c)^9-1/6435*(8*A*d^3+22*B*c*d^2+78*C*c^2*d+407*D*c^3 
)*(-d^2*x^2+c^2)^(7/2)/c^4/d^4/(d*x+c)^8-1/45045*(8*A*d^3+22*B*c*d^2+78*C* 
c^2*d+407*D*c^3)*(-d^2*x^2+c^2)^(7/2)/c^5/d^4/(d*x+c)^7
 

Mathematica [A] (verified)

Time = 3.95 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.64 \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{11}} \, dx=-\frac {(c-d x)^3 \sqrt {c^2-d^2 x^2} \left (22 c^7 D+8 A d^7 x^4+22 c d^6 x^3 (4 A+B x)+c^6 (78 C d+242 d D x)+2 c^2 d^5 x^2 (234 A+x (121 B+39 C x))+11 c^5 d^2 (37 B+39 x (2 C+3 D x))+11 c^3 d^4 x \left (148 A+x \left (117 B+78 C x+37 D x^2\right )\right )+c^4 d^3 \left (4243 A+x \left (4477 B+4563 C x+4477 D x^2\right )\right )\right )}{45045 c^5 d^4 (c+d x)^8} \] Input:

Integrate[((c^2 - d^2*x^2)^(5/2)*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^11,x 
]
 

Output:

-1/45045*((c - d*x)^3*Sqrt[c^2 - d^2*x^2]*(22*c^7*D + 8*A*d^7*x^4 + 22*c*d 
^6*x^3*(4*A + B*x) + c^6*(78*C*d + 242*d*D*x) + 2*c^2*d^5*x^2*(234*A + x*( 
121*B + 39*C*x)) + 11*c^5*d^2*(37*B + 39*x*(2*C + 3*D*x)) + 11*c^3*d^4*x*( 
148*A + x*(117*B + 78*C*x + 37*D*x^2)) + c^4*d^3*(4243*A + x*(4477*B + 456 
3*C*x + 4477*D*x^2))))/(c^5*d^4*(c + d*x)^8)
 

Rubi [A] (verified)

Time = 1.20 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.205, Rules used = {2170, 2170, 27, 671, 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.

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

\(\Big \downarrow \) 2170

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

\(\Big \downarrow \) 2170

\(\displaystyle \frac {\frac {\int \frac {d^6 \left (77 D c^3+20 C d c^2+6 A d^3+d \left (83 D c^2+14 C d c+6 B d^2\right ) x\right ) \left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^{11}}dx}{3 d^4}+\frac {d \left (c^2-d^2 x^2\right )^{7/2} (5 c D+2 C d)}{3 (c+d x)^{10}}}{2 d^5}+\frac {D \left (c^2-d^2 x^2\right )^{7/2}}{2 d^4 (c+d x)^9}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{3} d^2 \int \frac {\left (77 D c^3+20 C d c^2+6 A d^3+d \left (83 D c^2+14 C d c+6 B d^2\right ) x\right ) \left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^{11}}dx+\frac {d \left (c^2-d^2 x^2\right )^{7/2} (5 c D+2 C d)}{3 (c+d x)^{10}}}{2 d^5}+\frac {D \left (c^2-d^2 x^2\right )^{7/2}}{2 d^4 (c+d x)^9}\)

\(\Big \downarrow \) 671

\(\displaystyle \frac {\frac {1}{3} d^2 \left (\frac {\left (8 A d^3+22 B c d^2+407 c^3 D+78 c^2 C d\right ) \int \frac {\left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^{10}}dx}{5 c}-\frac {2 \left (c^2-d^2 x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 c d (c+d x)^{11}}\right )+\frac {d \left (c^2-d^2 x^2\right )^{7/2} (5 c D+2 C d)}{3 (c+d x)^{10}}}{2 d^5}+\frac {D \left (c^2-d^2 x^2\right )^{7/2}}{2 d^4 (c+d x)^9}\)

\(\Big \downarrow \) 461

\(\displaystyle \frac {\frac {1}{3} d^2 \left (\frac {\left (8 A d^3+22 B c d^2+407 c^3 D+78 c^2 C d\right ) \left (\frac {3 \int \frac {\left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^9}dx}{13 c}-\frac {\left (c^2-d^2 x^2\right )^{7/2}}{13 c d (c+d x)^{10}}\right )}{5 c}-\frac {2 \left (c^2-d^2 x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 c d (c+d x)^{11}}\right )+\frac {d \left (c^2-d^2 x^2\right )^{7/2} (5 c D+2 C d)}{3 (c+d x)^{10}}}{2 d^5}+\frac {D \left (c^2-d^2 x^2\right )^{7/2}}{2 d^4 (c+d x)^9}\)

\(\Big \downarrow \) 461

\(\displaystyle \frac {\frac {1}{3} d^2 \left (\frac {\left (8 A d^3+22 B c d^2+407 c^3 D+78 c^2 C d\right ) \left (\frac {3 \left (\frac {2 \int \frac {\left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^8}dx}{11 c}-\frac {\left (c^2-d^2 x^2\right )^{7/2}}{11 c d (c+d x)^9}\right )}{13 c}-\frac {\left (c^2-d^2 x^2\right )^{7/2}}{13 c d (c+d x)^{10}}\right )}{5 c}-\frac {2 \left (c^2-d^2 x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 c d (c+d x)^{11}}\right )+\frac {d \left (c^2-d^2 x^2\right )^{7/2} (5 c D+2 C d)}{3 (c+d x)^{10}}}{2 d^5}+\frac {D \left (c^2-d^2 x^2\right )^{7/2}}{2 d^4 (c+d x)^9}\)

\(\Big \downarrow \) 461

\(\displaystyle \frac {\frac {1}{3} d^2 \left (\frac {\left (8 A d^3+22 B c d^2+407 c^3 D+78 c^2 C d\right ) \left (\frac {3 \left (\frac {2 \left (\frac {\int \frac {\left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^7}dx}{9 c}-\frac {\left (c^2-d^2 x^2\right )^{7/2}}{9 c d (c+d x)^8}\right )}{11 c}-\frac {\left (c^2-d^2 x^2\right )^{7/2}}{11 c d (c+d x)^9}\right )}{13 c}-\frac {\left (c^2-d^2 x^2\right )^{7/2}}{13 c d (c+d x)^{10}}\right )}{5 c}-\frac {2 \left (c^2-d^2 x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 c d (c+d x)^{11}}\right )+\frac {d \left (c^2-d^2 x^2\right )^{7/2} (5 c D+2 C d)}{3 (c+d x)^{10}}}{2 d^5}+\frac {D \left (c^2-d^2 x^2\right )^{7/2}}{2 d^4 (c+d x)^9}\)

\(\Big \downarrow \) 460

\(\displaystyle \frac {\frac {1}{3} d^2 \left (\frac {\left (\frac {3 \left (\frac {2 \left (-\frac {\left (c^2-d^2 x^2\right )^{7/2}}{63 c^2 d (c+d x)^7}-\frac {\left (c^2-d^2 x^2\right )^{7/2}}{9 c d (c+d x)^8}\right )}{11 c}-\frac {\left (c^2-d^2 x^2\right )^{7/2}}{11 c d (c+d x)^9}\right )}{13 c}-\frac {\left (c^2-d^2 x^2\right )^{7/2}}{13 c d (c+d x)^{10}}\right ) \left (8 A d^3+22 B c d^2+407 c^3 D+78 c^2 C d\right )}{5 c}-\frac {2 \left (c^2-d^2 x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 c d (c+d x)^{11}}\right )+\frac {d \left (c^2-d^2 x^2\right )^{7/2} (5 c D+2 C d)}{3 (c+d x)^{10}}}{2 d^5}+\frac {D \left (c^2-d^2 x^2\right )^{7/2}}{2 d^4 (c+d x)^9}\)

Input:

Int[((c^2 - d^2*x^2)^(5/2)*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^11,x]
 

Output:

(D*(c^2 - d^2*x^2)^(7/2))/(2*d^4*(c + d*x)^9) + ((d*(2*C*d + 5*c*D)*(c^2 - 
 d^2*x^2)^(7/2))/(3*(c + d*x)^10) + (d^2*((-2*(c^2*C*d - B*c*d^2 + A*d^3 - 
 c^3*D)*(c^2 - d^2*x^2)^(7/2))/(5*c*d*(c + d*x)^11) + ((78*c^2*C*d + 22*B* 
c*d^2 + 8*A*d^3 + 407*c^3*D)*(-1/13*(c^2 - d^2*x^2)^(7/2)/(c*d*(c + d*x)^1 
0) + (3*(-1/11*(c^2 - d^2*x^2)^(7/2)/(c*d*(c + d*x)^9) + (2*(-1/9*(c^2 - d 
^2*x^2)^(7/2)/(c*d*(c + d*x)^8) - (c^2 - d^2*x^2)^(7/2)/(63*c^2*d*(c + d*x 
)^7)))/(11*c)))/(13*c)))/(5*c)))/3)/(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 = 2.84 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.81

method	result	size
gosper	\(-\frac {\left (-d x +c \right ) \left (8 A \,d^{7} x^{4}+22 B c \,d^{6} x^{4}+78 C \,c^{2} d^{5} x^{4}+407 D c^{3} d^{4} x^{4}+88 A c \,d^{6} x^{3}+242 B \,c^{2} d^{5} x^{3}+858 C \,c^{3} d^{4} x^{3}+4477 D c^{4} d^{3} x^{3}+468 A \,c^{2} d^{5} x^{2}+1287 B \,c^{3} d^{4} x^{2}+4563 C \,c^{4} d^{3} x^{2}+1287 D c^{5} d^{2} x^{2}+1628 A \,c^{3} d^{4} x +4477 B \,c^{4} d^{3} x +858 C \,c^{5} d^{2} x +242 D c^{6} d x +4243 A \,c^{4} d^{3}+407 B \,c^{5} d^{2}+78 C \,c^{6} d +22 D c^{7}\right ) \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{45045 \left (d x +c \right )^{10} c^{5} d^{4}}\)	\(243\)
orering	\(-\frac {\left (-d x +c \right ) \left (8 A \,d^{7} x^{4}+22 B c \,d^{6} x^{4}+78 C \,c^{2} d^{5} x^{4}+407 D c^{3} d^{4} x^{4}+88 A c \,d^{6} x^{3}+242 B \,c^{2} d^{5} x^{3}+858 C \,c^{3} d^{4} x^{3}+4477 D c^{4} d^{3} x^{3}+468 A \,c^{2} d^{5} x^{2}+1287 B \,c^{3} d^{4} x^{2}+4563 C \,c^{4} d^{3} x^{2}+1287 D c^{5} d^{2} x^{2}+1628 A \,c^{3} d^{4} x +4477 B \,c^{4} d^{3} x +858 C \,c^{5} d^{2} x +242 D c^{6} d x +4243 A \,c^{4} d^{3}+407 B \,c^{5} d^{2}+78 C \,c^{6} d +22 D c^{7}\right ) \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{45045 \left (d x +c \right )^{10} c^{5} d^{4}}\)	\(243\)
trager	\(-\frac {\left (-8 A \,d^{10} x^{7}-22 B c \,d^{9} x^{7}-78 C \,c^{2} d^{8} x^{7}-407 D c^{3} d^{7} x^{7}-64 A c \,d^{9} x^{6}-176 B \,c^{2} d^{8} x^{6}-624 C \,c^{3} d^{7} x^{6}-3256 D c^{4} d^{6} x^{6}-228 A \,c^{2} d^{8} x^{5}-627 B \,c^{3} d^{7} x^{5}-2223 C \,c^{4} d^{6} x^{5}+10923 D c^{5} d^{5} x^{5}-480 A \,c^{3} d^{7} x^{4}-1320 B \,c^{4} d^{6} x^{4}+10335 C \,c^{5} d^{5} x^{4}-9405 D c^{6} d^{4} x^{4}-675 A \,c^{4} d^{6} x^{3}+9405 B \,c^{5} d^{5} x^{3}-10335 C \,c^{6} d^{4} x^{3}+1320 D c^{7} d^{3} x^{3}+8313 A \,c^{5} d^{5} x^{2}-10923 B \,c^{6} d^{4} x^{2}+2223 C \,c^{7} d^{3} x^{2}+627 D c^{8} d^{2} x^{2}-11101 A \,c^{6} d^{4} x +3256 B \,c^{7} d^{3} x +624 C \,c^{8} d^{2} x +176 D c^{9} d x +4243 A \,c^{7} d^{3}+407 B \,c^{8} d^{2}+78 C \,c^{9} d +22 D c^{10}\right ) \sqrt {-d^{2} x^{2}+c^{2}}}{45045 c^{5} \left (d x +c \right )^{8} d^{4}}\)	\(381\)
default	\(\frac {D \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {7}{2}}}{9 c d \left (x +\frac {c}{d}\right )^{8}}-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {7}{2}}}{63 c^{2} \left (x +\frac {c}{d}\right )^{7}}\right )}{d^{11}}+\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 {7}{2}}}{11 c d \left (x +\frac {c}{d}\right )^{9}}+\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 {7}{2}}}{9 c d \left (x +\frac {c}{d}\right )^{8}}-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {7}{2}}}{63 c^{2} \left (x +\frac {c}{d}\right )^{7}}\right )}{11 c}\right )}{d^{12}}+\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 {7}{2}}}{13 c d \left (x +\frac {c}{d}\right )^{10}}+\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 {7}{2}}}{11 c d \left (x +\frac {c}{d}\right )^{9}}+\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 {7}{2}}}{9 c d \left (x +\frac {c}{d}\right )^{8}}-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {7}{2}}}{63 c^{2} \left (x +\frac {c}{d}\right )^{7}}\right )}{11 c}\right )}{13 c}\right )}{d^{13}}+\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 {7}{2}}}{15 c d \left (x +\frac {c}{d}\right )^{11}}+\frac {4 d \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {7}{2}}}{13 c d \left (x +\frac {c}{d}\right )^{10}}+\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 {7}{2}}}{11 c d \left (x +\frac {c}{d}\right )^{9}}+\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 {7}{2}}}{9 c d \left (x +\frac {c}{d}\right )^{8}}-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {7}{2}}}{63 c^{2} \left (x +\frac {c}{d}\right )^{7}}\right )}{11 c}\right )}{13 c}\right )}{15 c}\right )}{d^{14}}\)	\(733\)

Input:

int((-d^2*x^2+c^2)^(5/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^11,x,method=_RETURNVE 
RBOSE)
 

Output:

-1/45045*(-d*x+c)*(8*A*d^7*x^4+22*B*c*d^6*x^4+78*C*c^2*d^5*x^4+407*D*c^3*d 
^4*x^4+88*A*c*d^6*x^3+242*B*c^2*d^5*x^3+858*C*c^3*d^4*x^3+4477*D*c^4*d^3*x 
^3+468*A*c^2*d^5*x^2+1287*B*c^3*d^4*x^2+4563*C*c^4*d^3*x^2+1287*D*c^5*d^2* 
x^2+1628*A*c^3*d^4*x+4477*B*c^4*d^3*x+858*C*c^5*d^2*x+242*D*c^6*d*x+4243*A 
*c^4*d^3+407*B*c^5*d^2+78*C*c^6*d+22*D*c^7)*(-d^2*x^2+c^2)^(5/2)/(d*x+c)^1 
0/c^5/d^4
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 782 vs. \(2 (279) = 558\).

Time = 0.91 (sec) , antiderivative size = 782, normalized size of antiderivative = 2.62 \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{11}} \, dx =\text {Too large to display} \] Input:

integrate((-d^2*x^2+c^2)^(5/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^11,x, algorithm 
="fricas")
 

Output:

-1/45045*(22*D*c^11 + 78*C*c^10*d + 407*B*c^9*d^2 + 4243*A*c^8*d^3 + (22*D 
*c^3*d^8 + 78*C*c^2*d^9 + 407*B*c*d^10 + 4243*A*d^11)*x^8 + 8*(22*D*c^4*d^ 
7 + 78*C*c^3*d^8 + 407*B*c^2*d^9 + 4243*A*c*d^10)*x^7 + 28*(22*D*c^5*d^6 + 
 78*C*c^4*d^7 + 407*B*c^3*d^8 + 4243*A*c^2*d^9)*x^6 + 56*(22*D*c^6*d^5 + 7 
8*C*c^5*d^6 + 407*B*c^4*d^7 + 4243*A*c^3*d^8)*x^5 + 70*(22*D*c^7*d^4 + 78* 
C*c^6*d^5 + 407*B*c^5*d^6 + 4243*A*c^4*d^7)*x^4 + 56*(22*D*c^8*d^3 + 78*C* 
c^7*d^4 + 407*B*c^6*d^5 + 4243*A*c^5*d^6)*x^3 + 28*(22*D*c^9*d^2 + 78*C*c^ 
8*d^3 + 407*B*c^7*d^4 + 4243*A*c^6*d^5)*x^2 + 8*(22*D*c^10*d + 78*C*c^9*d^ 
2 + 407*B*c^8*d^3 + 4243*A*c^7*d^4)*x + (22*D*c^10 + 78*C*c^9*d + 407*B*c^ 
8*d^2 + 4243*A*c^7*d^3 - (407*D*c^3*d^7 + 78*C*c^2*d^8 + 22*B*c*d^9 + 8*A* 
d^10)*x^7 - 8*(407*D*c^4*d^6 + 78*C*c^3*d^7 + 22*B*c^2*d^8 + 8*A*c*d^9)*x^ 
6 + 3*(3641*D*c^5*d^5 - 741*C*c^4*d^6 - 209*B*c^3*d^7 - 76*A*c^2*d^8)*x^5 
- 15*(627*D*c^6*d^4 - 689*C*c^5*d^5 + 88*B*c^4*d^6 + 32*A*c^3*d^7)*x^4 + 1 
5*(88*D*c^7*d^3 - 689*C*c^6*d^4 + 627*B*c^5*d^5 - 45*A*c^4*d^6)*x^3 + 3*(2 
09*D*c^8*d^2 + 741*C*c^7*d^3 - 3641*B*c^6*d^4 + 2771*A*c^5*d^5)*x^2 + (176 
*D*c^9*d + 624*C*c^8*d^2 + 3256*B*c^7*d^3 - 11101*A*c^6*d^4)*x)*sqrt(-d^2* 
x^2 + c^2))/(c^5*d^12*x^8 + 8*c^6*d^11*x^7 + 28*c^7*d^10*x^6 + 56*c^8*d^9* 
x^5 + 70*c^9*d^8*x^4 + 56*c^10*d^7*x^3 + 28*c^11*d^6*x^2 + 8*c^12*d^5*x + 
c^13*d^4)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{11}} \, dx=\text {Timed out} \] Input:

integrate((-d**2*x**2+c**2)**(5/2)*(D*x**3+C*x**2+B*x+A)/(d*x+c)**11,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5999 vs. \(2 (279) = 558\).

Time = 0.17 (sec) , antiderivative size = 5999, normalized size of antiderivative = 20.06 \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{11}} \, dx=\text {Too large to display} \] Input:

integrate((-d^2*x^2+c^2)^(5/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^11,x, algorithm 
="maxima")
 

Output:

1/5*(-d^2*x^2 + c^2)^(5/2)*D*c^3/(d^14*x^10 + 10*c*d^13*x^9 + 45*c^2*d^12* 
x^8 + 120*c^3*d^11*x^7 + 210*c^4*d^10*x^6 + 252*c^5*d^9*x^5 + 210*c^6*d^8* 
x^4 + 120*c^7*d^7*x^3 + 45*c^8*d^6*x^2 + 10*c^9*d^5*x + c^10*d^4) - 1/6*(- 
d^2*x^2 + c^2)^(3/2)*D*c^4/(d^13*x^9 + 9*c*d^12*x^8 + 36*c^2*d^11*x^7 + 84 
*c^3*d^10*x^6 + 126*c^4*d^9*x^5 + 126*c^5*d^8*x^4 + 84*c^6*d^7*x^3 + 36*c^ 
7*d^6*x^2 + 9*c^8*d^5*x + c^9*d^4) + 1/15*sqrt(-d^2*x^2 + c^2)*D*c^5/(d^12 
*x^8 + 8*c*d^11*x^7 + 28*c^2*d^10*x^6 + 56*c^3*d^9*x^5 + 70*c^4*d^8*x^4 + 
56*c^5*d^7*x^3 + 28*c^6*d^6*x^2 + 8*c^7*d^5*x + c^8*d^4) - 1/5*(-d^2*x^2 + 
 c^2)^(5/2)*C*c^2/(d^13*x^10 + 10*c*d^12*x^9 + 45*c^2*d^11*x^8 + 120*c^3*d 
^10*x^7 + 210*c^4*d^9*x^6 + 252*c^5*d^8*x^5 + 210*c^6*d^7*x^4 + 120*c^7*d^ 
6*x^3 + 45*c^8*d^5*x^2 + 10*c^9*d^4*x + c^10*d^3) - 3/4*(-d^2*x^2 + c^2)^( 
5/2)*D*c^2/(d^13*x^9 + 9*c*d^12*x^8 + 36*c^2*d^11*x^7 + 84*c^3*d^10*x^6 + 
126*c^4*d^9*x^5 + 126*c^5*d^8*x^4 + 84*c^6*d^7*x^3 + 36*c^7*d^6*x^2 + 9*c^ 
8*d^5*x + c^9*d^4) + 1/6*(-d^2*x^2 + c^2)^(3/2)*C*c^3/(d^12*x^9 + 9*c*d^11 
*x^8 + 36*c^2*d^10*x^7 + 84*c^3*d^9*x^6 + 126*c^4*d^8*x^5 + 126*c^5*d^7*x^ 
4 + 84*c^6*d^6*x^3 + 36*c^7*d^5*x^2 + 9*c^8*d^4*x + c^9*d^3) + 3/4*(-d^2*x 
^2 + c^2)^(3/2)*D*c^3/(d^12*x^8 + 8*c*d^11*x^7 + 28*c^2*d^10*x^6 + 56*c^3* 
d^9*x^5 + 70*c^4*d^8*x^4 + 56*c^5*d^7*x^3 + 28*c^6*d^6*x^2 + 8*c^7*d^5*x + 
 c^8*d^4) - 1/15*sqrt(-d^2*x^2 + c^2)*C*c^4/(d^11*x^8 + 8*c*d^10*x^7 + 28* 
c^2*d^9*x^6 + 56*c^3*d^8*x^5 + 70*c^4*d^7*x^4 + 56*c^5*d^6*x^3 + 28*c^6...
                                                                                    
                                                                                    
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1740 vs. \(2 (279) = 558\).

Time = 0.17 (sec) , antiderivative size = 1740, normalized size of antiderivative = 5.82 \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{11}} \, dx=\text {Too large to display} \] Input:

integrate((-d^2*x^2+c^2)^(5/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^11,x, algorithm 
="giac")
 

Output:

2/45045*(22*D*c^3 + 78*C*c^2*d + 407*B*c*d^2 + 4243*A*d^3 + 6105*(c*d + sq 
rt(-d^2*x^2 + c^2)*abs(d))*B*c/x + 330*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d)) 
*D*c^3/(d^2*x) + 1170*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))*C*c^2/(d*x) + 18 
600*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))*A*d/x + 2310*(c*d + sqrt(-d^2*x^2 
+ c^2)*abs(d))^2*D*c^3/(d^4*x^2) + 8190*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d) 
)^2*C*c^2/(d^3*x^2) - 2310*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^2*B*c/(d^2* 
x^2) + 265335*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^2*A/(d*x^2) + 10010*(c*d 
 + sqrt(-d^2*x^2 + c^2)*abs(d))^3*D*c^3/(d^6*x^3) - 24570*(c*d + sqrt(-d^2 
*x^2 + c^2)*abs(d))^3*C*c^2/(d^5*x^3) + 170170*(c*d + sqrt(-d^2*x^2 + c^2) 
*abs(d))^3*B*c/(d^4*x^3) + 864500*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^3*A/ 
(d^3*x^3) - 60060*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^4*D*c^3/(d^8*x^4) + 
196560*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^4*C*c^2/(d^7*x^4) + 105105*(c*d 
 + sqrt(-d^2*x^2 + c^2)*abs(d))^4*B*c/(d^6*x^4) + 3088995*(c*d + sqrt(-d^2 
*x^2 + c^2)*abs(d))^4*A/(d^5*x^4) + 300300*(c*d + sqrt(-d^2*x^2 + c^2)*abs 
(d))^5*D*c^3/(d^10*x^5) - 180180*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^5*C*c 
^2/(d^9*x^5) + 915915*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^5*B*c/(d^8*x^5) 
+ 6066060*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^5*A/(d^7*x^5) - 460460*(c*d 
+ sqrt(-d^2*x^2 + c^2)*abs(d))^6*D*c^3/(d^12*x^6) + 660660*(c*d + sqrt(-d^ 
2*x^2 + c^2)*abs(d))^6*C*c^2/(d^11*x^6) + 580580*(c*d + sqrt(-d^2*x^2 + c^ 
2)*abs(d))^6*B*c/(d^10*x^6) + 11026015*(c*d + sqrt(-d^2*x^2 + c^2)*abs(...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{11}} \, dx=\int \frac {{\left (c^2-d^2\,x^2\right )}^{5/2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^{11}} \,d x \] Input:

int(((c^2 - d^2*x^2)^(5/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^11,x)
 

Output:

int(((c^2 - d^2*x^2)^(5/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^11, x)
 

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 1056, normalized size of antiderivative = 3.53 \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{11}} \, dx =\text {Too large to display} \] Input:

int((-d^2*x^2+c^2)^(5/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^11,x)
 

Output:

(6006*sqrt(c**2 - d**2*x**2)*a*c**7*d + 1240*sqrt(c**2 - d**2*x**2)*a*c**6 
*d**2*x + 45336*sqrt(c**2 - d**2*x**2)*a*c**5*d**3*x**2 + 61030*sqrt(c**2 
- d**2*x**2)*a*c**4*d**4*x**3 + 61225*sqrt(c**2 - d**2*x**2)*a*c**3*d**5*x 
**4 + 36795*sqrt(c**2 - d**2*x**2)*a*c**2*d**6*x**5 + 12277*sqrt(c**2 - d* 
*2*x**2)*a*c*d**7*x**6 + 1755*sqrt(c**2 - d**2*x**2)*a*d**8*x**7 + 407*sqr 
t(c**2 - d**2*x**2)*b*c**7*d*x - 19470*sqrt(c**2 - d**2*x**2)*b*c**6*d**2* 
x**2 - 4840*sqrt(c**2 - d**2*x**2)*b*c**5*d**3*x**3 - 15565*sqrt(c**2 - d* 
*2*x**2)*b*c**4*d**4*x**4 - 9174*sqrt(c**2 - d**2*x**2)*b*c**3*d**5*x**5 - 
 3025*sqrt(c**2 - d**2*x**2)*b*c**2*d**6*x**6 - 429*sqrt(c**2 - d**2*x**2) 
*b*c*d**7*x**7 + 100*sqrt(c**2 - d**2*x**2)*c**9*x + 750*sqrt(c**2 - d**2* 
x**2)*c**8*d*x**2 - 12515*sqrt(c**2 - d**2*x**2)*c**7*d**2*x**3 - 2570*sqr 
t(c**2 - d**2*x**2)*c**6*d**3*x**4 + 6600*sqrt(c**2 - d**2*x**2)*c**5*d**4 
*x**5 - 4580*sqrt(c**2 - d**2*x**2)*c**4*d**5*x**6 - 585*sqrt(c**2 - d**2* 
x**2)*c**3*d**6*x**7 - 6006*a*c**8*d + 1240*a*c**7*d**2*x - 68778*a*c**6*d 
**3*x**2 - 89740*a*c**5*d**4*x**3 - 123605*a*c**4*d**5*x**4 - 98980*a*c**3 
*d**6*x**5 - 49528*a*c**2*d**7*x**6 - 14160*a*c*d**8*x**7 - 1771*a*d**9*x* 
*8 + 407*b*c**8*d*x + 25575*b*c**7*d**2*x**2 + 2464*b*c**6*d**3*x**3 + 392 
15*b*c**5*d**4*x**4 + 22099*b*c**4*d**5*x**5 + 10945*b*c**3*d**6*x**6 + 31 
02*b*c**2*d**7*x**7 + 385*b*c*d**8*x**8 + 100*c**10*x + 750*c**9*d*x**2 + 
17465*c**8*d**2*x**3 - 2945*c**7*d**3*x**4 - 2170*c**6*d**4*x**5 + 1538...