\(\int \frac {\sqrt {c^2-d^2 x^2} (A+B x+C x^2+D x^3)}{(c+d x)^6} \, dx\) [137]

Optimal result

Integrand size = 39, antiderivative size = 239 \[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^6} \, dx=-\frac {\left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \left (c^2-d^2 x^2\right )^{3/2}}{9 c d^4 (c+d x)^6}+\frac {\left (5 c^2 C d-2 B c d^2-A d^3-8 c^3 D\right ) \left (c^2-d^2 x^2\right )^{3/2}}{21 c^2 d^4 (c+d x)^5}-\frac {\left (11 c^2 C d+4 B c d^2+2 A d^3-47 c^3 D\right ) \left (c^2-d^2 x^2\right )^{3/2}}{105 c^3 d^4 (c+d x)^4}-\frac {\left (11 c^2 C d+4 B c d^2+2 A d^3+58 c^3 D\right ) \left (c^2-d^2 x^2\right )^{3/2}}{315 c^4 d^4 (c+d x)^3} \] Output:

-1/9*(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)^6+1/ 
21*(-A*d^3-2*B*c*d^2+5*C*c^2*d-8*D*c^3)*(-d^2*x^2+c^2)^(3/2)/c^2/d^4/(d*x+ 
c)^5-1/105*(2*A*d^3+4*B*c*d^2+11*C*c^2*d-47*D*c^3)*(-d^2*x^2+c^2)^(3/2)/c^ 
3/d^4/(d*x+c)^4-1/315*(2*A*d^3+4*B*c*d^2+11*C*c^2*d+58*D*c^3)*(-d^2*x^2+c^ 
2)^(3/2)/c^4/d^4/(d*x+c)^3
 

Mathematica [A] (verified)

Time = 1.58 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.65 \[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^6} \, dx=-\frac {(c-d x) \sqrt {c^2-d^2 x^2} \left (2 c^6 D+2 A d^6 x^3+4 c d^5 x^2 (3 A+B x)+4 c^5 d (C+3 D x)+c^2 d^4 x (33 A+x (24 B+11 C x))+c^4 d^2 (11 B+3 x (8 C+11 D x))+2 c^3 d^3 \left (29 A+x \left (33 B+33 C x+29 D x^2\right )\right )\right )}{315 c^4 d^4 (c+d x)^5} \] Input:

Integrate[(Sqrt[c^2 - d^2*x^2]*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^6,x]
 

Output:

-1/315*((c - d*x)*Sqrt[c^2 - d^2*x^2]*(2*c^6*D + 2*A*d^6*x^3 + 4*c*d^5*x^2 
*(3*A + B*x) + 4*c^5*d*(C + 3*D*x) + c^2*d^4*x*(33*A + x*(24*B + 11*C*x)) 
+ c^4*d^2*(11*B + 3*x*(8*C + 11*D*x)) + 2*c^3*d^3*(29*A + x*(33*B + 33*C*x 
 + 29*D*x^2))))/(c^4*d^4*(c + d*x)^5)
 

Rubi [A] (verified)

Time = 1.05 (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.

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

\(\Big \downarrow \) 2170

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

\(\Big \downarrow \) 2170

\(\displaystyle \frac {\frac {\int \frac {d^6 \left (18 D c^3+5 C d c^2+2 A d^3+d \left (20 D c^2+3 C d c+2 B d^2\right ) x\right ) \sqrt {c^2-d^2 x^2}}{(c+d x)^6}dx}{2 d^4}+\frac {d \left (c^2-d^2 x^2\right )^{3/2} (2 c D+C d)}{2 (c+d x)^5}}{d^5}+\frac {D \left (c^2-d^2 x^2\right )^{3/2}}{d^4 (c+d x)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{2} d^2 \int \frac {\left (18 D c^3+5 C d c^2+2 A d^3+d \left (20 D c^2+3 C d c+2 B d^2\right ) x\right ) \sqrt {c^2-d^2 x^2}}{(c+d x)^6}dx+\frac {d \left (c^2-d^2 x^2\right )^{3/2} (2 c D+C d)}{2 (c+d x)^5}}{d^5}+\frac {D \left (c^2-d^2 x^2\right )^{3/2}}{d^4 (c+d x)^4}\)

\(\Big \downarrow \) 671

\(\displaystyle \frac {\frac {1}{2} d^2 \left (\frac {\left (2 A d^3+4 B c d^2+58 c^3 D+11 c^2 C d\right ) \int \frac {\sqrt {c^2-d^2 x^2}}{(c+d x)^5}dx}{3 c}-\frac {2 \left (c^2-d^2 x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{9 c d (c+d x)^6}\right )+\frac {d \left (c^2-d^2 x^2\right )^{3/2} (2 c D+C d)}{2 (c+d x)^5}}{d^5}+\frac {D \left (c^2-d^2 x^2\right )^{3/2}}{d^4 (c+d x)^4}\)

\(\Big \downarrow \) 461

\(\displaystyle \frac {\frac {1}{2} d^2 \left (\frac {\left (2 A d^3+4 B c d^2+58 c^3 D+11 c^2 C d\right ) \left (\frac {2 \int \frac {\sqrt {c^2-d^2 x^2}}{(c+d x)^4}dx}{7 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{7 c d (c+d x)^5}\right )}{3 c}-\frac {2 \left (c^2-d^2 x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{9 c d (c+d x)^6}\right )+\frac {d \left (c^2-d^2 x^2\right )^{3/2} (2 c D+C d)}{2 (c+d x)^5}}{d^5}+\frac {D \left (c^2-d^2 x^2\right )^{3/2}}{d^4 (c+d x)^4}\)

\(\Big \downarrow \) 461

\(\displaystyle \frac {\frac {1}{2} d^2 \left (\frac {\left (2 A d^3+4 B c d^2+58 c^3 D+11 c^2 C d\right ) \left (\frac {2 \left (\frac {\int \frac {\sqrt {c^2-d^2 x^2}}{(c+d x)^3}dx}{5 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{5 c d (c+d x)^4}\right )}{7 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{7 c d (c+d x)^5}\right )}{3 c}-\frac {2 \left (c^2-d^2 x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{9 c d (c+d x)^6}\right )+\frac {d \left (c^2-d^2 x^2\right )^{3/2} (2 c D+C d)}{2 (c+d x)^5}}{d^5}+\frac {D \left (c^2-d^2 x^2\right )^{3/2}}{d^4 (c+d x)^4}\)

\(\Big \downarrow \) 460

\(\displaystyle \frac {\frac {1}{2} d^2 \left (\frac {\left (\frac {2 \left (-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{15 c^2 d (c+d x)^3}-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{5 c d (c+d x)^4}\right )}{7 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{7 c d (c+d x)^5}\right ) \left (2 A d^3+4 B c d^2+58 c^3 D+11 c^2 C d\right )}{3 c}-\frac {2 \left (c^2-d^2 x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{9 c d (c+d x)^6}\right )+\frac {d \left (c^2-d^2 x^2\right )^{3/2} (2 c D+C d)}{2 (c+d x)^5}}{d^5}+\frac {D \left (c^2-d^2 x^2\right )^{3/2}}{d^4 (c+d x)^4}\)

Input:

Int[(Sqrt[c^2 - d^2*x^2]*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^6,x]
 

Output:

(D*(c^2 - d^2*x^2)^(3/2))/(d^4*(c + d*x)^4) + ((d*(C*d + 2*c*D)*(c^2 - d^2 
*x^2)^(3/2))/(2*(c + d*x)^5) + (d^2*((-2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3* 
D)*(c^2 - d^2*x^2)^(3/2))/(9*c*d*(c + d*x)^6) + ((11*c^2*C*d + 4*B*c*d^2 + 
 2*A*d^3 + 58*c^3*D)*(-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)))/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 = 0.60 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.82

method	result	size
gosper	\(-\frac {\left (-d x +c \right ) \left (2 A \,d^{6} x^{3}+4 B c \,d^{5} x^{3}+11 C \,c^{2} d^{4} x^{3}+58 D c^{3} d^{3} x^{3}+12 A c \,d^{5} x^{2}+24 B \,c^{2} d^{4} x^{2}+66 C \,c^{3} d^{3} x^{2}+33 D c^{4} d^{2} x^{2}+33 A \,c^{2} d^{4} x +66 B \,c^{3} d^{3} x +24 C \,c^{4} d^{2} x +12 D c^{5} d x +58 A \,c^{3} d^{3}+11 B \,c^{4} d^{2}+4 C \,c^{5} d +2 D c^{6}\right ) \sqrt {-d^{2} x^{2}+c^{2}}}{315 \left (d x +c \right )^{5} c^{4} d^{4}}\)	\(195\)
orering	\(-\frac {\left (-d x +c \right ) \left (2 A \,d^{6} x^{3}+4 B c \,d^{5} x^{3}+11 C \,c^{2} d^{4} x^{3}+58 D c^{3} d^{3} x^{3}+12 A c \,d^{5} x^{2}+24 B \,c^{2} d^{4} x^{2}+66 C \,c^{3} d^{3} x^{2}+33 D c^{4} d^{2} x^{2}+33 A \,c^{2} d^{4} x +66 B \,c^{3} d^{3} x +24 C \,c^{4} d^{2} x +12 D c^{5} d x +58 A \,c^{3} d^{3}+11 B \,c^{4} d^{2}+4 C \,c^{5} d +2 D c^{6}\right ) \sqrt {-d^{2} x^{2}+c^{2}}}{315 \left (d x +c \right )^{5} c^{4} d^{4}}\)	\(195\)
trager	\(-\frac {\left (-2 A \,d^{7} x^{4}-4 B c \,d^{6} x^{4}-11 C \,c^{2} d^{5} x^{4}-58 D c^{3} d^{4} x^{4}-10 A c \,d^{6} x^{3}-20 B \,c^{2} d^{5} x^{3}-55 C \,c^{3} d^{4} x^{3}+25 D c^{4} d^{3} x^{3}-21 A \,c^{2} d^{5} x^{2}-42 B \,c^{3} d^{4} x^{2}+42 C \,c^{4} d^{3} x^{2}+21 D c^{5} d^{2} x^{2}-25 A \,c^{3} d^{4} x +55 B \,c^{4} d^{3} x +20 C \,c^{5} d^{2} x +10 D c^{6} d x +58 A \,c^{4} d^{3}+11 B \,c^{5} d^{2}+4 C \,c^{6} d +2 D c^{7}\right ) \sqrt {-d^{2} x^{2}+c^{2}}}{315 c^{4} \left (d x +c \right )^{5} d^{4}}\)	\(237\)
default	\(-\frac {D \left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {3}{2}}}{3 d^{7} c \left (x +\frac {c}{d}\right )^{3}}+\frac {\left (C d -3 D c \right ) \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {3}{2}}}{5 c d \left (x +\frac {c}{d}\right )^{4}}-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {3}{2}}}{15 c^{2} \left (x +\frac {c}{d}\right )^{3}}\right )}{d^{7}}+\frac {\left (B \,d^{2}-2 C c d +3 D c^{2}\right ) \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {3}{2}}}{7 c d \left (x +\frac {c}{d}\right )^{5}}+\frac {2 d \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {3}{2}}}{5 c d \left (x +\frac {c}{d}\right )^{4}}-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {3}{2}}}{15 c^{2} \left (x +\frac {c}{d}\right )^{3}}\right )}{7 c}\right )}{d^{8}}+\frac {\left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right ) \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {3}{2}}}{9 c d \left (x +\frac {c}{d}\right )^{6}}+\frac {d \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {3}{2}}}{7 c d \left (x +\frac {c}{d}\right )^{5}}+\frac {2 d \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {3}{2}}}{5 c d \left (x +\frac {c}{d}\right )^{4}}-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {3}{2}}}{15 c^{2} \left (x +\frac {c}{d}\right )^{3}}\right )}{7 c}\right )}{3 c}\right )}{d^{9}}\)	\(530\)

Input:

int((-d^2*x^2+c^2)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^6,x,method=_RETURNVER 
BOSE)
 

Output:

-1/315*(-d*x+c)*(2*A*d^6*x^3+4*B*c*d^5*x^3+11*C*c^2*d^4*x^3+58*D*c^3*d^3*x 
^3+12*A*c*d^5*x^2+24*B*c^2*d^4*x^2+66*C*c^3*d^3*x^2+33*D*c^4*d^2*x^2+33*A* 
c^2*d^4*x+66*B*c^3*d^3*x+24*C*c^4*d^2*x+12*D*c^5*d*x+58*A*c^3*d^3+11*B*c^4 
*d^2+4*C*c^5*d+2*D*c^6)*(-d^2*x^2+c^2)^(1/2)/(d*x+c)^5/c^4/d^4
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 497 vs. \(2 (222) = 444\).

Time = 0.17 (sec) , antiderivative size = 497, normalized size of antiderivative = 2.08 \[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^6} \, dx=-\frac {2 \, D c^{8} + 4 \, C c^{7} d + 11 \, B c^{6} d^{2} + 58 \, A c^{5} d^{3} + {\left (2 \, D c^{3} d^{5} + 4 \, C c^{2} d^{6} + 11 \, B c d^{7} + 58 \, A d^{8}\right )} x^{5} + 5 \, {\left (2 \, D c^{4} d^{4} + 4 \, C c^{3} d^{5} + 11 \, B c^{2} d^{6} + 58 \, A c d^{7}\right )} x^{4} + 10 \, {\left (2 \, D c^{5} d^{3} + 4 \, C c^{4} d^{4} + 11 \, B c^{3} d^{5} + 58 \, A c^{2} d^{6}\right )} x^{3} + 10 \, {\left (2 \, D c^{6} d^{2} + 4 \, C c^{5} d^{3} + 11 \, B c^{4} d^{4} + 58 \, A c^{3} d^{5}\right )} x^{2} + 5 \, {\left (2 \, D c^{7} d + 4 \, C c^{6} d^{2} + 11 \, B c^{5} d^{3} + 58 \, A c^{4} d^{4}\right )} x + {\left (2 \, D c^{7} + 4 \, C c^{6} d + 11 \, B c^{5} d^{2} + 58 \, A c^{4} d^{3} - {\left (58 \, D c^{3} d^{4} + 11 \, C c^{2} d^{5} + 4 \, B c d^{6} + 2 \, A d^{7}\right )} x^{4} + 5 \, {\left (5 \, D c^{4} d^{3} - 11 \, C c^{3} d^{4} - 4 \, B c^{2} d^{5} - 2 \, A c d^{6}\right )} x^{3} + 21 \, {\left (D c^{5} d^{2} + 2 \, C c^{4} d^{3} - 2 \, B c^{3} d^{4} - A c^{2} d^{5}\right )} x^{2} + 5 \, {\left (2 \, D c^{6} d + 4 \, C c^{5} d^{2} + 11 \, B c^{4} d^{3} - 5 \, A c^{3} d^{4}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{315 \, {\left (c^{4} d^{9} x^{5} + 5 \, c^{5} d^{8} x^{4} + 10 \, c^{6} d^{7} x^{3} + 10 \, c^{7} d^{6} x^{2} + 5 \, c^{8} d^{5} x + c^{9} d^{4}\right )}} \] Input:

integrate((-d^2*x^2+c^2)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^6,x, algorithm= 
"fricas")
 

Output:

-1/315*(2*D*c^8 + 4*C*c^7*d + 11*B*c^6*d^2 + 58*A*c^5*d^3 + (2*D*c^3*d^5 + 
 4*C*c^2*d^6 + 11*B*c*d^7 + 58*A*d^8)*x^5 + 5*(2*D*c^4*d^4 + 4*C*c^3*d^5 + 
 11*B*c^2*d^6 + 58*A*c*d^7)*x^4 + 10*(2*D*c^5*d^3 + 4*C*c^4*d^4 + 11*B*c^3 
*d^5 + 58*A*c^2*d^6)*x^3 + 10*(2*D*c^6*d^2 + 4*C*c^5*d^3 + 11*B*c^4*d^4 + 
58*A*c^3*d^5)*x^2 + 5*(2*D*c^7*d + 4*C*c^6*d^2 + 11*B*c^5*d^3 + 58*A*c^4*d 
^4)*x + (2*D*c^7 + 4*C*c^6*d + 11*B*c^5*d^2 + 58*A*c^4*d^3 - (58*D*c^3*d^4 
 + 11*C*c^2*d^5 + 4*B*c*d^6 + 2*A*d^7)*x^4 + 5*(5*D*c^4*d^3 - 11*C*c^3*d^4 
 - 4*B*c^2*d^5 - 2*A*c*d^6)*x^3 + 21*(D*c^5*d^2 + 2*C*c^4*d^3 - 2*B*c^3*d^ 
4 - A*c^2*d^5)*x^2 + 5*(2*D*c^6*d + 4*C*c^5*d^2 + 11*B*c^4*d^3 - 5*A*c^3*d 
^4)*x)*sqrt(-d^2*x^2 + c^2))/(c^4*d^9*x^5 + 5*c^5*d^8*x^4 + 10*c^6*d^7*x^3 
 + 10*c^7*d^6*x^2 + 5*c^8*d^5*x + c^9*d^4)
 

Sympy [F]

\[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^6} \, dx=\int \frac {\sqrt {- \left (- c + d x\right ) \left (c + d x\right )} \left (A + B x + C x^{2} + D x^{3}\right )}{\left (c + d x\right )^{6}}\, dx \] Input:

integrate((-d**2*x**2+c**2)**(1/2)*(D*x**3+C*x**2+B*x+A)/(d*x+c)**6,x)
 

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2090 vs. \(2 (222) = 444\).

Time = 0.07 (sec) , antiderivative size = 2090, normalized size of antiderivative = 8.74 \[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^6} \, dx=\text {Too large to display} \] Input:

integrate((-d^2*x^2+c^2)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^6,x, algorithm= 
"maxima")
 

Output:

2/9*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 + 1 
0*c^3*d^6*x^2 + 5*c^4*d^5*x + c^5*d^4) - 1/63*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) - 1/105 
*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) - 2/315*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) - 2/315*sqrt(-d^2*x^2 + c^2)*D*c^3/(c^4*d^5*x + c^5*d^4) - 2/9*sqrt 
(-d^2*x^2 + c^2)*C*c^2/(d^8*x^5 + 5*c*d^7*x^4 + 10*c^2*d^6*x^3 + 10*c^3*d^ 
5*x^2 + 5*c^4*d^4*x + c^5*d^3) + 1/63*sqrt(-d^2*x^2 + c^2)*C*c^2/(c*d^7*x^ 
4 + 4*c^2*d^6*x^3 + 6*c^3*d^5*x^2 + 4*c^4*d^4*x + c^5*d^3) + 1/105*sqrt(-d 
^2*x^2 + c^2)*C*c^2/(c^2*d^6*x^3 + 3*c^3*d^5*x^2 + 3*c^4*d^4*x + c^5*d^3) 
+ 2/315*sqrt(-d^2*x^2 + c^2)*C*c^2/(c^3*d^5*x^2 + 2*c^4*d^4*x + c^5*d^3) + 
 2/315*sqrt(-d^2*x^2 + c^2)*C*c^2/(c^4*d^4*x + c^5*d^3) - 6/7*sqrt(-d^2*x^ 
2 + c^2)*D*c^2/(d^8*x^4 + 4*c*d^7*x^3 + 6*c^2*d^6*x^2 + 4*c^3*d^5*x + c^4* 
d^4) + 3/35*sqrt(-d^2*x^2 + c^2)*D*c^2/(c*d^7*x^3 + 3*c^2*d^6*x^2 + 3*c^3* 
d^5*x + c^4*d^4) + 2/35*sqrt(-d^2*x^2 + c^2)*D*c^2/(c^2*d^6*x^2 + 2*c^3*d^ 
5*x + c^4*d^4) + 2/35*sqrt(-d^2*x^2 + c^2)*D*c^2/(c^3*d^5*x + c^4*d^4) + 2 
/9*sqrt(-d^2*x^2 + c^2)*B*c/(d^7*x^5 + 5*c*d^6*x^4 + 10*c^2*d^5*x^3 + 10*c 
^3*d^4*x^2 + 5*c^4*d^3*x + c^5*d^2) - 1/63*sqrt(-d^2*x^2 + c^2)*B*c/(c*d^6 
*x^4 + 4*c^2*d^5*x^3 + 6*c^3*d^4*x^2 + 4*c^4*d^3*x + c^5*d^2) - 1/105*sqrt 
(-d^2*x^2 + c^2)*B*c/(c^2*d^5*x^3 + 3*c^3*d^4*x^2 + 3*c^4*d^3*x + c^5*d...
                                                                                    
                                                                                    
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 895 vs. \(2 (222) = 444\).

Time = 0.15 (sec) , antiderivative size = 895, normalized size of antiderivative = 3.74 \[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^6} \, dx =\text {Too large to display} \] Input:

integrate((-d^2*x^2+c^2)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^6,x, algorithm= 
"giac")
 

Output:

2/315*(2*D*c^3 + 4*C*c^2*d + 11*B*c*d^2 + 58*A*d^3 + 99*(c*d + sqrt(-d^2*x 
^2 + c^2)*abs(d))*B*c/x + 18*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))*D*c^3/(d^ 
2*x) + 36*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))*C*c^2/(d*x) + 207*(c*d + sqr 
t(-d^2*x^2 + c^2)*abs(d))*A*d/x + 72*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^2 
*D*c^3/(d^4*x^2) + 144*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^2*C*c^2/(d^3*x^ 
2) + 81*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^2*B*c/(d^2*x^2) + 1143*(c*d + 
sqrt(-d^2*x^2 + c^2)*abs(d))^2*A/(d*x^2) + 168*(c*d + sqrt(-d^2*x^2 + c^2) 
*abs(d))^3*D*c^3/(d^6*x^3) - 84*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^3*C*c^ 
2/(d^5*x^3) + 609*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^3*B*c/(d^4*x^3) + 22 
47*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^3*A/(d^3*x^3) - 378*(c*d + sqrt(-d^ 
2*x^2 + c^2)*abs(d))^4*D*c^3/(d^8*x^4) + 504*(c*d + sqrt(-d^2*x^2 + c^2)*a 
bs(d))^4*C*c^2/(d^7*x^4) + 441*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^4*B*c/( 
d^6*x^4) + 3843*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^4*A/(d^5*x^4) + 630*(c 
*d + sqrt(-d^2*x^2 + c^2)*abs(d))^5*D*c^3/(d^10*x^5) + 945*(c*d + sqrt(-d^ 
2*x^2 + c^2)*abs(d))^5*B*c/(d^8*x^5) + 3465*(c*d + sqrt(-d^2*x^2 + c^2)*ab 
s(d))^5*A/(d^7*x^5) + 420*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^6*C*c^2/(d^1 
1*x^6) + 315*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^6*B*c/(d^10*x^6) + 2625*( 
c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^6*A/(d^9*x^6) + 315*(c*d + sqrt(-d^2*x^ 
2 + c^2)*abs(d))^7*B*c/(d^12*x^7) + 945*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d) 
)^7*A/(d^11*x^7) + 315*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^8*A/(d^13*x^...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^6} \, dx=\int \frac {\sqrt {c^2-d^2\,x^2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^6} \,d x \] Input:

int(((c^2 - d^2*x^2)^(1/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^6,x)
 

Output:

int(((c^2 - d^2*x^2)^(1/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^6, x)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 510, normalized size of antiderivative = 2.13 \[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^6} \, dx =\text {Too large to display} \] Input:

int((-d^2*x^2+c^2)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^6,x)
 

Output:

(2*(35*tan(asin((d*x)/c)/2)**9*a*d**2 + 315*tan(asin((d*x)/c)/2)**7*a*d**2 
 - 315*tan(asin((d*x)/c)/2)**7*b*c*d + 315*tan(asin((d*x)/c)/2)**6*a*d**2 
- 315*tan(asin((d*x)/c)/2)**6*b*c*d - 420*tan(asin((d*x)/c)/2)**6*c**3 + 9 
45*tan(asin((d*x)/c)/2)**5*a*d**2 - 945*tan(asin((d*x)/c)/2)**5*b*c*d - 63 
0*tan(asin((d*x)/c)/2)**5*c**3 + 567*tan(asin((d*x)/c)/2)**4*a*d**2 - 441* 
tan(asin((d*x)/c)/2)**4*b*c*d - 126*tan(asin((d*x)/c)/2)**4*c**3 + 693*tan 
(asin((d*x)/c)/2)**3*a*d**2 - 609*tan(asin((d*x)/c)/2)**3*b*c*d - 84*tan(a 
sin((d*x)/c)/2)**3*c**3 + 117*tan(asin((d*x)/c)/2)**2*a*d**2 - 81*tan(asin 
((d*x)/c)/2)**2*b*c*d - 216*tan(asin((d*x)/c)/2)**2*c**3 + 108*tan(asin((d 
*x)/c)/2)*a*d**2 - 99*tan(asin((d*x)/c)/2)*b*c*d - 54*tan(asin((d*x)/c)/2) 
*c**3 - 23*a*d**2 - 11*b*c*d - 6*c**3))/(315*c**4*d**3*(tan(asin((d*x)/c)/ 
2)**9 + 9*tan(asin((d*x)/c)/2)**8 + 36*tan(asin((d*x)/c)/2)**7 + 84*tan(as 
in((d*x)/c)/2)**6 + 126*tan(asin((d*x)/c)/2)**5 + 126*tan(asin((d*x)/c)/2) 
**4 + 84*tan(asin((d*x)/c)/2)**3 + 36*tan(asin((d*x)/c)/2)**2 + 9*tan(asin 
((d*x)/c)/2) + 1))