\(\int \frac {\sqrt {c^2-d^2 x^2} (A+B x+C x^2+D x^3)}{(c+d x)^7} \, dx\) [138]

Optimal result

Integrand size = 39, antiderivative size = 299 \[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^7} \, dx=-\frac {\left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \left (c^2-d^2 x^2\right )^{3/2}}{11 c d^4 (c+d x)^7}+\frac {\left (18 c^2 C d-7 B c d^2-4 A d^3-29 c^3 D\right ) \left (c^2-d^2 x^2\right )^{3/2}}{99 c^2 d^4 (c+d x)^6}-\frac {\left (15 c^2 C d+7 B c d^2+4 A d^3-70 c^3 D\right ) \left (c^2-d^2 x^2\right )^{3/2}}{231 c^3 d^4 (c+d x)^5}-\frac {\left (30 c^2 C d+14 B c d^2+8 A d^3+91 c^3 D\right ) \left (c^2-d^2 x^2\right )^{3/2}}{1155 c^4 d^4 (c+d x)^4}-\frac {\left (30 c^2 C d+14 B c d^2+8 A d^3+91 c^3 D\right ) \left (c^2-d^2 x^2\right )^{3/2}}{3465 c^5 d^4 (c+d x)^3} \] Output:

-1/11*(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)^7+1 
/99*(-4*A*d^3-7*B*c*d^2+18*C*c^2*d-29*D*c^3)*(-d^2*x^2+c^2)^(3/2)/c^2/d^4/ 
(d*x+c)^6-1/231*(4*A*d^3+7*B*c*d^2+15*C*c^2*d-70*D*c^3)*(-d^2*x^2+c^2)^(3/ 
2)/c^3/d^4/(d*x+c)^5-1/1155*(8*A*d^3+14*B*c*d^2+30*C*c^2*d+91*D*c^3)*(-d^2 
*x^2+c^2)^(3/2)/c^4/d^4/(d*x+c)^4-1/3465*(8*A*d^3+14*B*c*d^2+30*C*c^2*d+91 
*D*c^3)*(-d^2*x^2+c^2)^(3/2)/c^5/d^4/(d*x+c)^3
 

Mathematica [A] (verified)

Time = 2.21 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.63 \[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^7} \, dx=-\frac {(c-d x) \sqrt {c^2-d^2 x^2} \left (14 c^7 D+8 A d^7 x^4+14 c d^6 x^3 (4 A+B x)+c^6 (30 C d+98 d D x)+2 c^2 d^5 x^2 (90 A+x (49 B+15 C x))+7 c^5 d^2 (13 B+15 x (2 C+3 D x))+7 c^3 d^4 x \left (52 A+x \left (45 B+30 C x+13 D x^2\right )\right )+c^4 d^3 \left (547 A+x \left (637 B+675 C x+637 D x^2\right )\right )\right )}{3465 c^5 d^4 (c+d x)^6} \] Input:

Integrate[(Sqrt[c^2 - d^2*x^2]*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^7,x]
 

Output:

-1/3465*((c - d*x)*Sqrt[c^2 - d^2*x^2]*(14*c^7*D + 8*A*d^7*x^4 + 14*c*d^6* 
x^3*(4*A + B*x) + c^6*(30*C*d + 98*d*D*x) + 2*c^2*d^5*x^2*(90*A + x*(49*B 
+ 15*C*x)) + 7*c^5*d^2*(13*B + 15*x*(2*C + 3*D*x)) + 7*c^3*d^4*x*(52*A + x 
*(45*B + 30*C*x + 13*D*x^2)) + c^4*d^3*(547*A + x*(637*B + 675*C*x + 637*D 
*x^2))))/(c^5*d^4*(c + d*x)^6)
 

Rubi [A] (verified)

Time = 1.12 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.08, 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.

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

\(\Big \downarrow \) 2170

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

\(\Big \downarrow \) 2170

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 671

\(\displaystyle \frac {d^2 \left (\frac {\left (8 A d^3+14 B c d^2+91 c^3 D+30 c^2 C d\right ) \int \frac {\sqrt {c^2-d^2 x^2}}{(c+d x)^6}dx}{11 c}-\frac {2 \left (c^2-d^2 x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{11 c d (c+d x)^7}\right )+\frac {d \left (c^2-d^2 x^2\right )^{3/2} (c D+2 C d)}{3 (c+d x)^6}}{2 d^5}+\frac {D \left (c^2-d^2 x^2\right )^{3/2}}{2 d^4 (c+d x)^5}\)

\(\Big \downarrow \) 461

\(\displaystyle \frac {d^2 \left (\frac {\left (8 A d^3+14 B c d^2+91 c^3 D+30 c^2 C d\right ) \left (\frac {\int \frac {\sqrt {c^2-d^2 x^2}}{(c+d x)^5}dx}{3 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{9 c d (c+d x)^6}\right )}{11 c}-\frac {2 \left (c^2-d^2 x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{11 c d (c+d x)^7}\right )+\frac {d \left (c^2-d^2 x^2\right )^{3/2} (c D+2 C d)}{3 (c+d x)^6}}{2 d^5}+\frac {D \left (c^2-d^2 x^2\right )^{3/2}}{2 d^4 (c+d x)^5}\)

\(\Big \downarrow \) 461

\(\displaystyle \frac {d^2 \left (\frac {\left (8 A d^3+14 B c d^2+91 c^3 D+30 c^2 C d\right ) \left (\frac {\frac {2 \int \frac {\sqrt {c^2-d^2 x^2}}{(c+d x)^4}dx}{7 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{7 c d (c+d x)^5}}{3 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{9 c d (c+d x)^6}\right )}{11 c}-\frac {2 \left (c^2-d^2 x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{11 c d (c+d x)^7}\right )+\frac {d \left (c^2-d^2 x^2\right )^{3/2} (c D+2 C d)}{3 (c+d x)^6}}{2 d^5}+\frac {D \left (c^2-d^2 x^2\right )^{3/2}}{2 d^4 (c+d x)^5}\)

\(\Big \downarrow \) 461

\(\displaystyle \frac {d^2 \left (\frac {\left (8 A d^3+14 B c d^2+91 c^3 D+30 c^2 C d\right ) \left (\frac {\frac {2 \left (\frac {\int \frac {\sqrt {c^2-d^2 x^2}}{(c+d x)^3}dx}{5 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{5 c d (c+d x)^4}\right )}{7 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{7 c d (c+d x)^5}}{3 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{9 c d (c+d x)^6}\right )}{11 c}-\frac {2 \left (c^2-d^2 x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{11 c d (c+d x)^7}\right )+\frac {d \left (c^2-d^2 x^2\right )^{3/2} (c D+2 C d)}{3 (c+d x)^6}}{2 d^5}+\frac {D \left (c^2-d^2 x^2\right )^{3/2}}{2 d^4 (c+d x)^5}\)

\(\Big \downarrow \) 460

\(\displaystyle \frac {d^2 \left (\frac {\left (\frac {\frac {2 \left (-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{15 c^2 d (c+d x)^3}-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{5 c d (c+d x)^4}\right )}{7 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{7 c d (c+d x)^5}}{3 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{9 c d (c+d x)^6}\right ) \left (8 A d^3+14 B c d^2+91 c^3 D+30 c^2 C d\right )}{11 c}-\frac {2 \left (c^2-d^2 x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{11 c d (c+d x)^7}\right )+\frac {d \left (c^2-d^2 x^2\right )^{3/2} (c D+2 C d)}{3 (c+d x)^6}}{2 d^5}+\frac {D \left (c^2-d^2 x^2\right )^{3/2}}{2 d^4 (c+d x)^5}\)

Input:

Int[(Sqrt[c^2 - d^2*x^2]*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^7,x]
 

Output:

(D*(c^2 - d^2*x^2)^(3/2))/(2*d^4*(c + d*x)^5) + ((d*(2*C*d + c*D)*(c^2 - d 
^2*x^2)^(3/2))/(3*(c + d*x)^6) + 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))/(11*c*d*(c + d*x)^7) + ((30*c^2*C*d + 14*B*c*d^ 
2 + 8*A*d^3 + 91*c^3*D)*(-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)))/(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 = 0.67 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.81

method	result	size
gosper	\(-\frac {\left (-d x +c \right ) \left (8 A \,d^{7} x^{4}+14 B c \,d^{6} x^{4}+30 C \,c^{2} d^{5} x^{4}+91 D c^{3} d^{4} x^{4}+56 A c \,d^{6} x^{3}+98 B \,c^{2} d^{5} x^{3}+210 C \,c^{3} d^{4} x^{3}+637 D c^{4} d^{3} x^{3}+180 A \,c^{2} d^{5} x^{2}+315 B \,c^{3} d^{4} x^{2}+675 C \,c^{4} d^{3} x^{2}+315 D c^{5} d^{2} x^{2}+364 A \,c^{3} d^{4} x +637 B \,c^{4} d^{3} x +210 C \,c^{5} d^{2} x +98 D c^{6} d x +547 A \,c^{4} d^{3}+91 B \,c^{5} d^{2}+30 C \,c^{6} d +14 D c^{7}\right ) \sqrt {-d^{2} x^{2}+c^{2}}}{3465 \left (d x +c \right )^{6} c^{5} d^{4}}\)	\(243\)
orering	\(-\frac {\left (-d x +c \right ) \left (8 A \,d^{7} x^{4}+14 B c \,d^{6} x^{4}+30 C \,c^{2} d^{5} x^{4}+91 D c^{3} d^{4} x^{4}+56 A c \,d^{6} x^{3}+98 B \,c^{2} d^{5} x^{3}+210 C \,c^{3} d^{4} x^{3}+637 D c^{4} d^{3} x^{3}+180 A \,c^{2} d^{5} x^{2}+315 B \,c^{3} d^{4} x^{2}+675 C \,c^{4} d^{3} x^{2}+315 D c^{5} d^{2} x^{2}+364 A \,c^{3} d^{4} x +637 B \,c^{4} d^{3} x +210 C \,c^{5} d^{2} x +98 D c^{6} d x +547 A \,c^{4} d^{3}+91 B \,c^{5} d^{2}+30 C \,c^{6} d +14 D c^{7}\right ) \sqrt {-d^{2} x^{2}+c^{2}}}{3465 \left (d x +c \right )^{6} c^{5} d^{4}}\)	\(243\)
trager	\(-\frac {\left (-8 A \,d^{8} x^{5}-14 B c \,d^{7} x^{5}-30 C \,c^{2} d^{6} x^{5}-91 D c^{3} d^{5} x^{5}-48 A c \,d^{7} x^{4}-84 B \,c^{2} d^{6} x^{4}-180 C \,c^{3} d^{5} x^{4}-546 D c^{4} d^{4} x^{4}-124 A \,c^{2} d^{6} x^{3}-217 B \,c^{3} d^{5} x^{3}-465 C \,c^{4} d^{4} x^{3}+322 D c^{5} d^{3} x^{3}-184 A \,c^{3} d^{5} x^{2}-322 B \,c^{4} d^{4} x^{2}+465 C \,c^{5} d^{3} x^{2}+217 D c^{6} d^{2} x^{2}-183 A \,c^{4} d^{4} x +546 B \,c^{5} d^{3} x +180 C \,c^{6} d^{2} x +84 D c^{7} d x +547 A \,c^{5} d^{3}+91 B \,c^{6} d^{2}+30 C \,c^{7} d +14 D c^{8}\right ) \sqrt {-d^{2} x^{2}+c^{2}}}{3465 c^{5} \left (d x +c \right )^{6} d^{4}}\)	\(285\)
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 {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 (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}}}{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 (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}}}{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}}+\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}}}{11 c d \left (x +\frac {c}{d}\right )^{7}}+\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 {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 )}{11 c}\right )}{d^{10}}\)	\(733\)

Input:

int((-d^2*x^2+c^2)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^7,x,method=_RETURNVER 
BOSE)
 

Output:

-1/3465*(-d*x+c)*(8*A*d^7*x^4+14*B*c*d^6*x^4+30*C*c^2*d^5*x^4+91*D*c^3*d^4 
*x^4+56*A*c*d^6*x^3+98*B*c^2*d^5*x^3+210*C*c^3*d^4*x^3+637*D*c^4*d^3*x^3+1 
80*A*c^2*d^5*x^2+315*B*c^3*d^4*x^2+675*C*c^4*d^3*x^2+315*D*c^5*d^2*x^2+364 
*A*c^3*d^4*x+637*B*c^4*d^3*x+210*C*c^5*d^2*x+98*D*c^6*d*x+547*A*c^4*d^3+91 
*B*c^5*d^2+30*C*c^6*d+14*D*c^7)*(-d^2*x^2+c^2)^(1/2)/(d*x+c)^6/c^5/d^4
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 591 vs. \(2 (279) = 558\).

Time = 0.28 (sec) , antiderivative size = 591, normalized size of antiderivative = 1.98 \[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^7} \, dx=-\frac {14 \, D c^{9} + 30 \, C c^{8} d + 91 \, B c^{7} d^{2} + 547 \, A c^{6} d^{3} + {\left (14 \, D c^{3} d^{6} + 30 \, C c^{2} d^{7} + 91 \, B c d^{8} + 547 \, A d^{9}\right )} x^{6} + 6 \, {\left (14 \, D c^{4} d^{5} + 30 \, C c^{3} d^{6} + 91 \, B c^{2} d^{7} + 547 \, A c d^{8}\right )} x^{5} + 15 \, {\left (14 \, D c^{5} d^{4} + 30 \, C c^{4} d^{5} + 91 \, B c^{3} d^{6} + 547 \, A c^{2} d^{7}\right )} x^{4} + 20 \, {\left (14 \, D c^{6} d^{3} + 30 \, C c^{5} d^{4} + 91 \, B c^{4} d^{5} + 547 \, A c^{3} d^{6}\right )} x^{3} + 15 \, {\left (14 \, D c^{7} d^{2} + 30 \, C c^{6} d^{3} + 91 \, B c^{5} d^{4} + 547 \, A c^{4} d^{5}\right )} x^{2} + 6 \, {\left (14 \, D c^{8} d + 30 \, C c^{7} d^{2} + 91 \, B c^{6} d^{3} + 547 \, A c^{5} d^{4}\right )} x + {\left (14 \, D c^{8} + 30 \, C c^{7} d + 91 \, B c^{6} d^{2} + 547 \, A c^{5} d^{3} - {\left (91 \, D c^{3} d^{5} + 30 \, C c^{2} d^{6} + 14 \, B c d^{7} + 8 \, A d^{8}\right )} x^{5} - 6 \, {\left (91 \, D c^{4} d^{4} + 30 \, C c^{3} d^{5} + 14 \, B c^{2} d^{6} + 8 \, A c d^{7}\right )} x^{4} + {\left (322 \, D c^{5} d^{3} - 465 \, C c^{4} d^{4} - 217 \, B c^{3} d^{5} - 124 \, A c^{2} d^{6}\right )} x^{3} + {\left (217 \, D c^{6} d^{2} + 465 \, C c^{5} d^{3} - 322 \, B c^{4} d^{4} - 184 \, A c^{3} d^{5}\right )} x^{2} + 3 \, {\left (28 \, D c^{7} d + 60 \, C c^{6} d^{2} + 182 \, B c^{5} d^{3} - 61 \, A c^{4} d^{4}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{3465 \, {\left (c^{5} d^{10} x^{6} + 6 \, c^{6} d^{9} x^{5} + 15 \, c^{7} d^{8} x^{4} + 20 \, c^{8} d^{7} x^{3} + 15 \, c^{9} d^{6} x^{2} + 6 \, c^{10} d^{5} x + c^{11} d^{4}\right )}} \] Input:

integrate((-d^2*x^2+c^2)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^7,x, algorithm= 
"fricas")
 

Output:

-1/3465*(14*D*c^9 + 30*C*c^8*d + 91*B*c^7*d^2 + 547*A*c^6*d^3 + (14*D*c^3* 
d^6 + 30*C*c^2*d^7 + 91*B*c*d^8 + 547*A*d^9)*x^6 + 6*(14*D*c^4*d^5 + 30*C* 
c^3*d^6 + 91*B*c^2*d^7 + 547*A*c*d^8)*x^5 + 15*(14*D*c^5*d^4 + 30*C*c^4*d^ 
5 + 91*B*c^3*d^6 + 547*A*c^2*d^7)*x^4 + 20*(14*D*c^6*d^3 + 30*C*c^5*d^4 + 
91*B*c^4*d^5 + 547*A*c^3*d^6)*x^3 + 15*(14*D*c^7*d^2 + 30*C*c^6*d^3 + 91*B 
*c^5*d^4 + 547*A*c^4*d^5)*x^2 + 6*(14*D*c^8*d + 30*C*c^7*d^2 + 91*B*c^6*d^ 
3 + 547*A*c^5*d^4)*x + (14*D*c^8 + 30*C*c^7*d + 91*B*c^6*d^2 + 547*A*c^5*d 
^3 - (91*D*c^3*d^5 + 30*C*c^2*d^6 + 14*B*c*d^7 + 8*A*d^8)*x^5 - 6*(91*D*c^ 
4*d^4 + 30*C*c^3*d^5 + 14*B*c^2*d^6 + 8*A*c*d^7)*x^4 + (322*D*c^5*d^3 - 46 
5*C*c^4*d^4 - 217*B*c^3*d^5 - 124*A*c^2*d^6)*x^3 + (217*D*c^6*d^2 + 465*C* 
c^5*d^3 - 322*B*c^4*d^4 - 184*A*c^3*d^5)*x^2 + 3*(28*D*c^7*d + 60*C*c^6*d^ 
2 + 182*B*c^5*d^3 - 61*A*c^4*d^4)*x)*sqrt(-d^2*x^2 + c^2))/(c^5*d^10*x^6 + 
 6*c^6*d^9*x^5 + 15*c^7*d^8*x^4 + 20*c^8*d^7*x^3 + 15*c^9*d^6*x^2 + 6*c^10 
*d^5*x + c^11*d^4)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^7} \, dx=\text {Timed out} \] Input:

integrate((-d**2*x**2+c**2)**(1/2)*(D*x**3+C*x**2+B*x+A)/(d*x+c)**7,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2890 vs. \(2 (279) = 558\).

Time = 0.09 (sec) , antiderivative size = 2890, normalized size of antiderivative = 9.67 \[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^7} \, dx=\text {Too large to display} \] Input:

integrate((-d^2*x^2+c^2)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^7,x, algorithm= 
"maxima")
 

Output:

2/11*sqrt(-d^2*x^2 + c^2)*D*c^3/(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) - 1/99*sqrt(-d^2 
*x^2 + c^2)*D*c^3/(c*d^9*x^5 + 5*c^2*d^8*x^4 + 10*c^3*d^7*x^3 + 10*c^4*d^6 
*x^2 + 5*c^5*d^5*x + c^6*d^4) - 4/693*sqrt(-d^2*x^2 + c^2)*D*c^3/(c^2*d^8* 
x^4 + 4*c^3*d^7*x^3 + 6*c^4*d^6*x^2 + 4*c^5*d^5*x + c^6*d^4) - 4/1155*sqrt 
(-d^2*x^2 + c^2)*D*c^3/(c^3*d^7*x^3 + 3*c^4*d^6*x^2 + 3*c^5*d^5*x + c^6*d^ 
4) - 8/3465*sqrt(-d^2*x^2 + c^2)*D*c^3/(c^4*d^6*x^2 + 2*c^5*d^5*x + c^6*d^ 
4) - 8/3465*sqrt(-d^2*x^2 + c^2)*D*c^3/(c^5*d^5*x + c^6*d^4) - 2/11*sqrt(- 
d^2*x^2 + c^2)*C*c^2/(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) + 1/99*sqrt(-d^2*x^2 + c^2)* 
C*c^2/(c*d^8*x^5 + 5*c^2*d^7*x^4 + 10*c^3*d^6*x^3 + 10*c^4*d^5*x^2 + 5*c^5 
*d^4*x + c^6*d^3) + 4/693*sqrt(-d^2*x^2 + c^2)*C*c^2/(c^2*d^7*x^4 + 4*c^3* 
d^6*x^3 + 6*c^4*d^5*x^2 + 4*c^5*d^4*x + c^6*d^3) + 4/1155*sqrt(-d^2*x^2 + 
c^2)*C*c^2/(c^3*d^6*x^3 + 3*c^4*d^5*x^2 + 3*c^5*d^4*x + c^6*d^3) + 8/3465* 
sqrt(-d^2*x^2 + c^2)*C*c^2/(c^4*d^5*x^2 + 2*c^5*d^4*x + c^6*d^3) + 8/3465* 
sqrt(-d^2*x^2 + c^2)*C*c^2/(c^5*d^4*x + c^6*d^3) - 2/3*sqrt(-d^2*x^2 + c^2 
)*D*c^2/(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) + 1/21*sqrt(-d^2*x^2 + c^2)*D*c^2/(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) + 1/35*sqrt(-d^2*x^2 + c^2)*D* 
c^2/(c^2*d^7*x^3 + 3*c^3*d^6*x^2 + 3*c^4*d^5*x + c^5*d^4) + 2/105*sqrt(...
                                                                                    
                                                                                    
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1200 vs. \(2 (279) = 558\).

Time = 0.14 (sec) , antiderivative size = 1200, normalized size of antiderivative = 4.01 \[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^7} \, dx=\text {Too large to display} \] Input:

integrate((-d^2*x^2+c^2)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^7,x, algorithm= 
"giac")
 

Output:

2/3465*(14*D*c^3 + 30*C*c^2*d + 91*B*c*d^2 + 547*A*d^3 + 1001*(c*d + sqrt( 
-d^2*x^2 + c^2)*abs(d))*B*c/x + 154*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))*D* 
c^3/(d^2*x) + 330*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))*C*c^2/(d*x) + 2552*( 
c*d + sqrt(-d^2*x^2 + c^2)*abs(d))*A*d/x + 770*(c*d + sqrt(-d^2*x^2 + c^2) 
*abs(d))^2*D*c^3/(d^4*x^2) + 1650*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^2*C* 
c^2/(d^3*x^2) + 1540*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^2*B*c/(d^2*x^2) + 
 16225*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^2*A/(d*x^2) + 2310*(c*d + sqrt( 
-d^2*x^2 + c^2)*abs(d))^3*D*c^3/(d^6*x^3) + 330*(c*d + sqrt(-d^2*x^2 + c^2 
)*abs(d))^3*C*c^2/(d^5*x^3) + 9240*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^3*B 
*c/(d^4*x^3) + 42900*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^3*A/(d^3*x^3) - 2 
310*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^4*D*c^3/(d^8*x^4) + 7590*(c*d + sq 
rt(-d^2*x^2 + c^2)*abs(d))^4*C*c^2/(d^7*x^4) + 11550*(c*d + sqrt(-d^2*x^2 
+ c^2)*abs(d))^4*B*c/(d^6*x^4) + 92730*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d)) 
^4*A/(d^5*x^4) + 7854*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^5*D*c^3/(d^10*x^ 
5) + 2310*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^5*C*c^2/(d^9*x^5) + 24486*(c 
*d + sqrt(-d^2*x^2 + c^2)*abs(d))^5*B*c/(d^8*x^5) + 122892*(c*d + sqrt(-d^ 
2*x^2 + c^2)*abs(d))^5*A/(d^7*x^5) - 1386*(c*d + sqrt(-d^2*x^2 + c^2)*abs( 
d))^6*D*c^3/(d^12*x^6) + 11550*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^6*C*c^2 
/(d^11*x^6) + 17556*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^6*B*c/(d^10*x^6) + 
 129822*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^6*A/(d^9*x^6) + 6930*(c*d +...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^7} \, dx=\int \frac {\sqrt {c^2-d^2\,x^2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^7} \,d x \] Input:

int(((c^2 - d^2*x^2)^(1/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^7,x)
 

Output:

int(((c^2 - d^2*x^2)^(1/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^7, x)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 768, normalized size of antiderivative = 2.57 \[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^7} \, dx=\frac {-630 a \,c^{6} d +44 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{7} x +242 c^{7} d \,x^{2}+1705 c^{6} d^{2} x^{3}+1243 c^{5} d^{3} x^{4}-341 c^{4} d^{4} x^{5}-77 c^{3} d^{5} x^{6}+44 c^{8} x +232 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{4} d^{2} x +646 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{3} d^{3} x^{2}+706 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{2} d^{4} x^{3}+367 \sqrt {-d^{2} x^{2}+c^{2}}\, a c \,d^{5} x^{4}+91 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{5} d x -1232 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{4} d^{2} x^{2}-1127 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{3} d^{3} x^{3}-539 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{2} d^{4} x^{4}-105 \sqrt {-d^{2} x^{2}+c^{2}}\, b c \,d^{5} x^{5}+242 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{6} d \,x^{2}-583 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{5} d^{2} x^{3}-946 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{4} d^{3} x^{4}-165 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{3} d^{4} x^{5}+232 a \,c^{5} d^{2} x -1244 a \,c^{4} d^{3} x^{2}-1720 a \,c^{3} d^{4} x^{3}-1321 a \,c^{2} d^{5} x^{4}-538 a c \,d^{6} x^{5}+91 b \,c^{6} d x +2233 b \,c^{5} d^{2} x^{2}+1715 b \,c^{4} d^{3} x^{3}+1232 b \,c^{3} d^{4} x^{4}+476 b \,c^{2} d^{5} x^{5}+77 b c \,d^{6} x^{6}+630 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{5} d +75 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,d^{6} x^{5}-91 a \,d^{7} x^{6}}{3465 c^{5} d^{2} \left (\sqrt {-d^{2} x^{2}+c^{2}}\, c^{5}+5 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{4} d x +10 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{3} d^{2} x^{2}+10 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{2} d^{3} x^{3}+5 \sqrt {-d^{2} x^{2}+c^{2}}\, c \,d^{4} x^{4}+\sqrt {-d^{2} x^{2}+c^{2}}\, d^{5} x^{5}-c^{6}-6 c^{5} d x -15 c^{4} d^{2} x^{2}-20 c^{3} d^{3} x^{3}-15 c^{2} d^{4} x^{4}-6 c \,d^{5} x^{5}-d^{6} x^{6}\right )} \] Input:

int((-d^2*x^2+c^2)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^7,x)
 

Output:

(630*sqrt(c**2 - d**2*x**2)*a*c**5*d + 232*sqrt(c**2 - d**2*x**2)*a*c**4*d 
**2*x + 646*sqrt(c**2 - d**2*x**2)*a*c**3*d**3*x**2 + 706*sqrt(c**2 - d**2 
*x**2)*a*c**2*d**4*x**3 + 367*sqrt(c**2 - d**2*x**2)*a*c*d**5*x**4 + 75*sq 
rt(c**2 - d**2*x**2)*a*d**6*x**5 + 91*sqrt(c**2 - d**2*x**2)*b*c**5*d*x - 
1232*sqrt(c**2 - d**2*x**2)*b*c**4*d**2*x**2 - 1127*sqrt(c**2 - d**2*x**2) 
*b*c**3*d**3*x**3 - 539*sqrt(c**2 - d**2*x**2)*b*c**2*d**4*x**4 - 105*sqrt 
(c**2 - d**2*x**2)*b*c*d**5*x**5 + 44*sqrt(c**2 - d**2*x**2)*c**7*x + 242* 
sqrt(c**2 - d**2*x**2)*c**6*d*x**2 - 583*sqrt(c**2 - d**2*x**2)*c**5*d**2* 
x**3 - 946*sqrt(c**2 - d**2*x**2)*c**4*d**3*x**4 - 165*sqrt(c**2 - d**2*x* 
*2)*c**3*d**4*x**5 - 630*a*c**6*d + 232*a*c**5*d**2*x - 1244*a*c**4*d**3*x 
**2 - 1720*a*c**3*d**4*x**3 - 1321*a*c**2*d**5*x**4 - 538*a*c*d**6*x**5 - 
91*a*d**7*x**6 + 91*b*c**6*d*x + 2233*b*c**5*d**2*x**2 + 1715*b*c**4*d**3* 
x**3 + 1232*b*c**3*d**4*x**4 + 476*b*c**2*d**5*x**5 + 77*b*c*d**6*x**6 + 4 
4*c**8*x + 242*c**7*d*x**2 + 1705*c**6*d**2*x**3 + 1243*c**5*d**3*x**4 - 3 
41*c**4*d**4*x**5 - 77*c**3*d**5*x**6)/(3465*c**5*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))