\(\int \frac {(c^2-d^2 x^2)^{5/2} (A+B x+C x^2+D x^3)}{(c+d x)^{10}} \, dx\) [166]

Optimal result

Integrand size = 39, antiderivative size = 239 \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{10}} \, dx=-\frac {\left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \left (c^2-d^2 x^2\right )^{7/2}}{13 c d^4 (c+d x)^{10}}+\frac {\left (23 c^2 C d-10 B c d^2-3 A d^3-36 c^3 D\right ) \left (c^2-d^2 x^2\right )^{7/2}}{143 c^2 d^4 (c+d x)^9}-\frac {\left (97 c^2 C d+20 B c d^2+6 A d^3-357 c^3 D\right ) \left (c^2-d^2 x^2\right )^{7/2}}{1287 c^3 d^4 (c+d x)^8}-\frac {\left (97 c^2 C d+20 B c d^2+6 A d^3+930 c^3 D\right ) \left (c^2-d^2 x^2\right )^{7/2}}{9009 c^4 d^4 (c+d x)^7} \] Output:

-1/13*(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)^10+ 
1/143*(-3*A*d^3-10*B*c*d^2+23*C*c^2*d-36*D*c^3)*(-d^2*x^2+c^2)^(7/2)/c^2/d 
^4/(d*x+c)^9-1/1287*(6*A*d^3+20*B*c*d^2+97*C*c^2*d-357*D*c^3)*(-d^2*x^2+c^ 
2)^(7/2)/c^3/d^4/(d*x+c)^8-1/9009*(6*A*d^3+20*B*c*d^2+97*C*c^2*d+930*D*c^3 
)*(-d^2*x^2+c^2)^(7/2)/c^4/d^4/(d*x+c)^7
 

Mathematica [A] (verified)

Time = 2.75 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.65 \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{10}} \, dx=-\frac {(c-d x)^3 \sqrt {c^2-d^2 x^2} \left (6 c^6 D+6 A d^6 x^3+20 c d^5 x^2 (3 A+B x)+20 c^5 d (C+3 D x)+c^2 d^4 x (291 A+x (200 B+97 C x))+c^4 d^2 (97 B+x (200 C+291 D x))+10 c^3 d^3 \left (93 A+x \left (97 B+97 C x+93 D x^2\right )\right )\right )}{9009 c^4 d^4 (c+d x)^7} \] Input:

Integrate[((c^2 - d^2*x^2)^(5/2)*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^10,x 
]
 

Output:

-1/9009*((c - d*x)^3*Sqrt[c^2 - d^2*x^2]*(6*c^6*D + 6*A*d^6*x^3 + 20*c*d^5 
*x^2*(3*A + B*x) + 20*c^5*d*(C + 3*D*x) + c^2*d^4*x*(291*A + x*(200*B + 97 
*C*x)) + c^4*d^2*(97*B + x*(200*C + 291*D*x)) + 10*c^3*d^3*(93*A + x*(97*B 
 + 97*C*x + 93*D*x^2))))/(c^4*d^4*(c + d*x)^7)
 

Rubi [A] (verified)

Time = 1.06 (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.

\(\displaystyle \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{10}} \, dx\)

\(\Big \downarrow \) 2170

\(\displaystyle \frac {\int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left ((C d+6 c D) x^2 d^4+\left (15 D c^2+B d^2\right ) x d^3+\left (8 D c^3+A d^3\right ) d^2\right )}{(c+d x)^{10}}dx}{d^5}+\frac {D \left (c^2-d^2 x^2\right )^{7/2}}{d^4 (c+d x)^8}\)

\(\Big \downarrow \) 2170

\(\displaystyle \frac {\frac {\int \frac {d^6 \left (70 D c^3+9 C d c^2+2 A d^3+d \left (72 D c^2+7 C d c+2 B d^2\right ) x\right ) \left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^{10}}dx}{2 d^4}+\frac {d \left (c^2-d^2 x^2\right )^{7/2} (6 c D+C d)}{2 (c+d x)^9}}{d^5}+\frac {D \left (c^2-d^2 x^2\right )^{7/2}}{d^4 (c+d x)^8}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{2} d^2 \int \frac {\left (70 D c^3+9 C d c^2+2 A d^3+d \left (72 D c^2+7 C d c+2 B d^2\right ) x\right ) \left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^{10}}dx+\frac {d \left (c^2-d^2 x^2\right )^{7/2} (6 c D+C d)}{2 (c+d x)^9}}{d^5}+\frac {D \left (c^2-d^2 x^2\right )^{7/2}}{d^4 (c+d x)^8}\)

\(\Big \downarrow \) 671

\(\displaystyle \frac {\frac {1}{2} d^2 \left (\frac {\left (6 A d^3+20 B c d^2+930 c^3 D+97 c^2 C d\right ) \int \frac {\left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^9}dx}{13 c}-\frac {2 \left (c^2-d^2 x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{13 c d (c+d x)^{10}}\right )+\frac {d \left (c^2-d^2 x^2\right )^{7/2} (6 c D+C d)}{2 (c+d x)^9}}{d^5}+\frac {D \left (c^2-d^2 x^2\right )^{7/2}}{d^4 (c+d x)^8}\)

\(\Big \downarrow \) 461

\(\displaystyle \frac {\frac {1}{2} d^2 \left (\frac {\left (6 A d^3+20 B c d^2+930 c^3 D+97 c^2 C d\right ) \left (\frac {2 \int \frac {\left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^8}dx}{11 c}-\frac {\left (c^2-d^2 x^2\right )^{7/2}}{11 c d (c+d x)^9}\right )}{13 c}-\frac {2 \left (c^2-d^2 x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{13 c d (c+d x)^{10}}\right )+\frac {d \left (c^2-d^2 x^2\right )^{7/2} (6 c D+C d)}{2 (c+d x)^9}}{d^5}+\frac {D \left (c^2-d^2 x^2\right )^{7/2}}{d^4 (c+d x)^8}\)

\(\Big \downarrow \) 461

\(\displaystyle \frac {\frac {1}{2} d^2 \left (\frac {\left (6 A d^3+20 B c d^2+930 c^3 D+97 c^2 C d\right ) \left (\frac {2 \left (\frac {\int \frac {\left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^7}dx}{9 c}-\frac {\left (c^2-d^2 x^2\right )^{7/2}}{9 c d (c+d x)^8}\right )}{11 c}-\frac {\left (c^2-d^2 x^2\right )^{7/2}}{11 c d (c+d x)^9}\right )}{13 c}-\frac {2 \left (c^2-d^2 x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{13 c d (c+d x)^{10}}\right )+\frac {d \left (c^2-d^2 x^2\right )^{7/2} (6 c D+C d)}{2 (c+d x)^9}}{d^5}+\frac {D \left (c^2-d^2 x^2\right )^{7/2}}{d^4 (c+d x)^8}\)

\(\Big \downarrow \) 460

\(\displaystyle \frac {\frac {1}{2} d^2 \left (\frac {\left (\frac {2 \left (-\frac {\left (c^2-d^2 x^2\right )^{7/2}}{63 c^2 d (c+d x)^7}-\frac {\left (c^2-d^2 x^2\right )^{7/2}}{9 c d (c+d x)^8}\right )}{11 c}-\frac {\left (c^2-d^2 x^2\right )^{7/2}}{11 c d (c+d x)^9}\right ) \left (6 A d^3+20 B c d^2+930 c^3 D+97 c^2 C d\right )}{13 c}-\frac {2 \left (c^2-d^2 x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{13 c d (c+d x)^{10}}\right )+\frac {d \left (c^2-d^2 x^2\right )^{7/2} (6 c D+C d)}{2 (c+d x)^9}}{d^5}+\frac {D \left (c^2-d^2 x^2\right )^{7/2}}{d^4 (c+d x)^8}\)

Input:

Int[((c^2 - d^2*x^2)^(5/2)*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^10,x]
 

Output:

(D*(c^2 - d^2*x^2)^(7/2))/(d^4*(c + d*x)^8) + ((d*(C*d + 6*c*D)*(c^2 - d^2 
*x^2)^(7/2))/(2*(c + d*x)^9) + (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))/(13*c*d*(c + d*x)^10) + ((97*c^2*C*d + 20*B*c*d^ 
2 + 6*A*d^3 + 930*c^3*D)*(-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)))/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 = 2.11 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.82

method	result	size
gosper	\(-\frac {\left (-d x +c \right ) \left (6 A \,d^{6} x^{3}+20 B c \,d^{5} x^{3}+97 C \,c^{2} d^{4} x^{3}+930 D c^{3} d^{3} x^{3}+60 A c \,d^{5} x^{2}+200 B \,c^{2} d^{4} x^{2}+970 C \,c^{3} d^{3} x^{2}+291 D c^{4} d^{2} x^{2}+291 A \,c^{2} d^{4} x +970 B \,c^{3} d^{3} x +200 C \,c^{4} d^{2} x +60 D c^{5} d x +930 A \,c^{3} d^{3}+97 B \,c^{4} d^{2}+20 C \,c^{5} d +6 D c^{6}\right ) \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{9009 \left (d x +c \right )^{9} c^{4} d^{4}}\)	\(195\)
orering	\(-\frac {\left (-d x +c \right ) \left (6 A \,d^{6} x^{3}+20 B c \,d^{5} x^{3}+97 C \,c^{2} d^{4} x^{3}+930 D c^{3} d^{3} x^{3}+60 A c \,d^{5} x^{2}+200 B \,c^{2} d^{4} x^{2}+970 C \,c^{3} d^{3} x^{2}+291 D c^{4} d^{2} x^{2}+291 A \,c^{2} d^{4} x +970 B \,c^{3} d^{3} x +200 C \,c^{4} d^{2} x +60 D c^{5} d x +930 A \,c^{3} d^{3}+97 B \,c^{4} d^{2}+20 C \,c^{5} d +6 D c^{6}\right ) \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{9009 \left (d x +c \right )^{9} c^{4} d^{4}}\)	\(195\)
trager	\(-\frac {\left (-6 A \,d^{9} x^{6}-20 B c \,d^{8} x^{6}-97 C \,c^{2} d^{7} x^{6}-930 D c^{3} d^{6} x^{6}-42 A c \,d^{8} x^{5}-140 B \,c^{2} d^{7} x^{5}-679 C \,c^{3} d^{6} x^{5}+2499 D c^{4} d^{5} x^{5}-129 A \,c^{2} d^{7} x^{4}-430 B \,c^{3} d^{6} x^{4}+2419 C \,c^{4} d^{5} x^{4}-1977 D c^{5} d^{4} x^{4}-231 A \,c^{3} d^{6} x^{3}+2233 B \,c^{4} d^{5} x^{3}-2233 C \,c^{5} d^{4} x^{3}+231 D c^{6} d^{3} x^{3}+1977 A \,c^{4} d^{5} x^{2}-2419 B \,c^{5} d^{4} x^{2}+430 C \,c^{6} d^{3} x^{2}+129 D c^{7} d^{2} x^{2}-2499 A \,c^{5} d^{4} x +679 B \,c^{6} d^{3} x +140 C \,c^{7} d^{2} x +42 D c^{8} d x +930 A \,c^{6} d^{3}+97 B \,c^{7} d^{2}+20 C \,c^{8} d +6 D c^{9}\right ) \sqrt {-d^{2} x^{2}+c^{2}}}{9009 c^{4} \left (d x +c \right )^{7} d^{4}}\)	\(333\)
default	\(-\frac {D \left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {7}{2}}}{7 d^{11} c \left (x +\frac {c}{d}\right )^{7}}+\frac {\left (C d -3 D c \right ) \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {7}{2}}}{9 c d \left (x +\frac {c}{d}\right )^{8}}-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {7}{2}}}{63 c^{2} \left (x +\frac {c}{d}\right )^{7}}\right )}{d^{11}}+\frac {\left (B \,d^{2}-2 C c d +3 D c^{2}\right ) \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {7}{2}}}{11 c d \left (x +\frac {c}{d}\right )^{9}}+\frac {2 d \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {7}{2}}}{9 c d \left (x +\frac {c}{d}\right )^{8}}-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {7}{2}}}{63 c^{2} \left (x +\frac {c}{d}\right )^{7}}\right )}{11 c}\right )}{d^{12}}+\frac {\left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right ) \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {7}{2}}}{13 c d \left (x +\frac {c}{d}\right )^{10}}+\frac {3 d \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {7}{2}}}{11 c d \left (x +\frac {c}{d}\right )^{9}}+\frac {2 d \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {7}{2}}}{9 c d \left (x +\frac {c}{d}\right )^{8}}-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {7}{2}}}{63 c^{2} \left (x +\frac {c}{d}\right )^{7}}\right )}{11 c}\right )}{13 c}\right )}{d^{13}}\)	\(530\)

Input:

int((-d^2*x^2+c^2)^(5/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^10,x,method=_RETURNVE 
RBOSE)
 

Output:

-1/9009*(-d*x+c)*(6*A*d^6*x^3+20*B*c*d^5*x^3+97*C*c^2*d^4*x^3+930*D*c^3*d^ 
3*x^3+60*A*c*d^5*x^2+200*B*c^2*d^4*x^2+970*C*c^3*d^3*x^2+291*D*c^4*d^2*x^2 
+291*A*c^2*d^4*x+970*B*c^3*d^3*x+200*C*c^4*d^2*x+60*D*c^5*d*x+930*A*c^3*d^ 
3+97*B*c^4*d^2+20*C*c^5*d+6*D*c^6)*(-d^2*x^2+c^2)^(5/2)/(d*x+c)^9/c^4/d^4
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 687 vs. \(2 (223) = 446\).

Time = 0.49 (sec) , antiderivative size = 687, normalized size of antiderivative = 2.87 \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{10}} \, dx=-\frac {6 \, D c^{10} + 20 \, C c^{9} d + 97 \, B c^{8} d^{2} + 930 \, A c^{7} d^{3} + {\left (6 \, D c^{3} d^{7} + 20 \, C c^{2} d^{8} + 97 \, B c d^{9} + 930 \, A d^{10}\right )} x^{7} + 7 \, {\left (6 \, D c^{4} d^{6} + 20 \, C c^{3} d^{7} + 97 \, B c^{2} d^{8} + 930 \, A c d^{9}\right )} x^{6} + 21 \, {\left (6 \, D c^{5} d^{5} + 20 \, C c^{4} d^{6} + 97 \, B c^{3} d^{7} + 930 \, A c^{2} d^{8}\right )} x^{5} + 35 \, {\left (6 \, D c^{6} d^{4} + 20 \, C c^{5} d^{5} + 97 \, B c^{4} d^{6} + 930 \, A c^{3} d^{7}\right )} x^{4} + 35 \, {\left (6 \, D c^{7} d^{3} + 20 \, C c^{6} d^{4} + 97 \, B c^{5} d^{5} + 930 \, A c^{4} d^{6}\right )} x^{3} + 21 \, {\left (6 \, D c^{8} d^{2} + 20 \, C c^{7} d^{3} + 97 \, B c^{6} d^{4} + 930 \, A c^{5} d^{5}\right )} x^{2} + 7 \, {\left (6 \, D c^{9} d + 20 \, C c^{8} d^{2} + 97 \, B c^{7} d^{3} + 930 \, A c^{6} d^{4}\right )} x + {\left (6 \, D c^{9} + 20 \, C c^{8} d + 97 \, B c^{7} d^{2} + 930 \, A c^{6} d^{3} - {\left (930 \, D c^{3} d^{6} + 97 \, C c^{2} d^{7} + 20 \, B c d^{8} + 6 \, A d^{9}\right )} x^{6} + 7 \, {\left (357 \, D c^{4} d^{5} - 97 \, C c^{3} d^{6} - 20 \, B c^{2} d^{7} - 6 \, A c d^{8}\right )} x^{5} - {\left (1977 \, D c^{5} d^{4} - 2419 \, C c^{4} d^{5} + 430 \, B c^{3} d^{6} + 129 \, A c^{2} d^{7}\right )} x^{4} + 77 \, {\left (3 \, D c^{6} d^{3} - 29 \, C c^{5} d^{4} + 29 \, B c^{4} d^{5} - 3 \, A c^{3} d^{6}\right )} x^{3} + {\left (129 \, D c^{7} d^{2} + 430 \, C c^{6} d^{3} - 2419 \, B c^{5} d^{4} + 1977 \, A c^{4} d^{5}\right )} x^{2} + 7 \, {\left (6 \, D c^{8} d + 20 \, C c^{7} d^{2} + 97 \, B c^{6} d^{3} - 357 \, A c^{5} d^{4}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{9009 \, {\left (c^{4} d^{11} x^{7} + 7 \, c^{5} d^{10} x^{6} + 21 \, c^{6} d^{9} x^{5} + 35 \, c^{7} d^{8} x^{4} + 35 \, c^{8} d^{7} x^{3} + 21 \, c^{9} d^{6} x^{2} + 7 \, c^{10} d^{5} x + c^{11} d^{4}\right )}} \] Input:

integrate((-d^2*x^2+c^2)^(5/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^10,x, algorithm 
="fricas")
 

Output:

-1/9009*(6*D*c^10 + 20*C*c^9*d + 97*B*c^8*d^2 + 930*A*c^7*d^3 + (6*D*c^3*d 
^7 + 20*C*c^2*d^8 + 97*B*c*d^9 + 930*A*d^10)*x^7 + 7*(6*D*c^4*d^6 + 20*C*c 
^3*d^7 + 97*B*c^2*d^8 + 930*A*c*d^9)*x^6 + 21*(6*D*c^5*d^5 + 20*C*c^4*d^6 
+ 97*B*c^3*d^7 + 930*A*c^2*d^8)*x^5 + 35*(6*D*c^6*d^4 + 20*C*c^5*d^5 + 97* 
B*c^4*d^6 + 930*A*c^3*d^7)*x^4 + 35*(6*D*c^7*d^3 + 20*C*c^6*d^4 + 97*B*c^5 
*d^5 + 930*A*c^4*d^6)*x^3 + 21*(6*D*c^8*d^2 + 20*C*c^7*d^3 + 97*B*c^6*d^4 
+ 930*A*c^5*d^5)*x^2 + 7*(6*D*c^9*d + 20*C*c^8*d^2 + 97*B*c^7*d^3 + 930*A* 
c^6*d^4)*x + (6*D*c^9 + 20*C*c^8*d + 97*B*c^7*d^2 + 930*A*c^6*d^3 - (930*D 
*c^3*d^6 + 97*C*c^2*d^7 + 20*B*c*d^8 + 6*A*d^9)*x^6 + 7*(357*D*c^4*d^5 - 9 
7*C*c^3*d^6 - 20*B*c^2*d^7 - 6*A*c*d^8)*x^5 - (1977*D*c^5*d^4 - 2419*C*c^4 
*d^5 + 430*B*c^3*d^6 + 129*A*c^2*d^7)*x^4 + 77*(3*D*c^6*d^3 - 29*C*c^5*d^4 
 + 29*B*c^4*d^5 - 3*A*c^3*d^6)*x^3 + (129*D*c^7*d^2 + 430*C*c^6*d^3 - 2419 
*B*c^5*d^4 + 1977*A*c^4*d^5)*x^2 + 7*(6*D*c^8*d + 20*C*c^7*d^2 + 97*B*c^6* 
d^3 - 357*A*c^5*d^4)*x)*sqrt(-d^2*x^2 + c^2))/(c^4*d^11*x^7 + 7*c^5*d^10*x 
^6 + 21*c^6*d^9*x^5 + 35*c^7*d^8*x^4 + 35*c^8*d^7*x^3 + 21*c^9*d^6*x^2 + 7 
*c^10*d^5*x + c^11*d^4)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{10}} \, dx=\text {Timed out} \] Input:

integrate((-d**2*x**2+c**2)**(5/2)*(D*x**3+C*x**2+B*x+A)/(d*x+c)**10,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4892 vs. \(2 (223) = 446\).

Time = 0.14 (sec) , antiderivative size = 4892, normalized size of antiderivative = 20.47 \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{10}} \, dx=\text {Too large to display} \] Input:

integrate((-d^2*x^2+c^2)^(5/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^10,x, algorithm 
="maxima")
 

Output:

1/4*(-d^2*x^2 + c^2)^(5/2)*D*c^3/(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) - 1/4*(-d^2*x^2 + c^2)^(3/2)*D*c^ 
4/(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) + 3/26*sqr 
t(-d^2*x^2 + c^2)*D*c^5/(d^11*x^7 + 7*c*d^10*x^6 + 21*c^2*d^9*x^5 + 35*c^3 
*d^8*x^4 + 35*c^4*d^7*x^3 + 21*c^5*d^6*x^2 + 7*c^6*d^5*x + c^7*d^4) - 1/4* 
(-d^2*x^2 + c^2)^(5/2)*C*c^2/(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) - (-d^2*x^2 + c^2)^(5/2)*D*c^2/(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) + 1/4*(-d^2*x^2 + c 
^2)^(3/2)*C*c^3/(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*d^5*x^2 + 8*c^7*d^4*x + c^8*d^ 
3) + 5/4*(-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) - 3/26*sqrt(-d^2*x^2 + c^2)*C*c^4/(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) - 393/572*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^...
                                                                                    
                                                                                    
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1470 vs. \(2 (223) = 446\).

Time = 0.18 (sec) , antiderivative size = 1470, normalized size of antiderivative = 6.15 \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{10}} \, dx=\text {Too large to display} \] Input:

integrate((-d^2*x^2+c^2)^(5/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^10,x, algorithm 
="giac")
 

Output:

2/9009*(6*D*c^3 + 20*C*c^2*d + 97*B*c*d^2 + 930*A*d^3 + 1261*(c*d + sqrt(- 
d^2*x^2 + c^2)*abs(d))*B*c/x + 78*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))*D*c^ 
3/(d^2*x) + 260*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))*C*c^2/(d*x) + 3081*(c* 
d + sqrt(-d^2*x^2 + c^2)*abs(d))*A*d/x + 468*(c*d + sqrt(-d^2*x^2 + c^2)*a 
bs(d))^2*D*c^3/(d^4*x^2) + 1560*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^2*C*c^ 
2/(d^3*x^2) - 1443*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^2*B*c/(d^2*x^2) + 4 
5513*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^2*A/(d*x^2) + 1716*(c*d + sqrt(-d 
^2*x^2 + c^2)*abs(d))^3*D*c^3/(d^6*x^3) - 6292*(c*d + sqrt(-d^2*x^2 + c^2) 
*abs(d))^3*C*c^2/(d^5*x^3) + 30745*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^3*B 
*c/(d^4*x^3) + 112827*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^3*A/(d^3*x^3) - 
13728*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^4*D*c^3/(d^8*x^4) + 38324*(c*d + 
 sqrt(-d^2*x^2 + c^2)*abs(d))^4*C*c^2/(d^7*x^4) + 286*(c*d + sqrt(-d^2*x^2 
 + c^2)*abs(d))^4*B*c/(d^6*x^4) + 367653*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d 
))^4*A/(d^5*x^4) + 61776*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^5*D*c^3/(d^10 
*x^5) - 46332*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^5*C*c^2/(d^9*x^5) + 1158 
30*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^5*B*c/(d^8*x^5) + 548262*(c*d + sqr 
t(-d^2*x^2 + c^2)*abs(d))^5*A/(d^7*x^5) - 97812*(c*d + sqrt(-d^2*x^2 + c^2 
)*abs(d))^6*D*c^3/(d^12*x^6) + 94380*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^6 
*C*c^2/(d^11*x^6) + 16302*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^6*B*c/(d^10* 
x^6) + 857142*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^6*A/(d^9*x^6) + 10810...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{10}} \, dx=\int \frac {{\left (c^2-d^2\,x^2\right )}^{5/2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^{10}} \,d x \] Input:

int(((c^2 - d^2*x^2)^(5/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^10,x)
 

Output:

int(((c^2 - d^2*x^2)^(5/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^10, x)
 

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 912, normalized size of antiderivative = 3.82 \[ \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)^{10}} \, dx=\frac {-1386 a \,c^{7} d +26 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{8} x -462 a \,d^{8} x^{7}+169 c^{8} d \,x^{2}+3471 c^{7} d^{2} x^{3}-1534 c^{6} d^{3} x^{4}-832 c^{5} d^{4} x^{5}+3029 c^{4} d^{5} x^{6}-117 \sqrt {-d^{2} x^{2}+c^{2}}\, b c \,d^{6} x^{6}-1885 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{3} d^{4} x^{4}-722 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{2} d^{5} x^{5}+293 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{4} d^{3} x^{3}+2694 \sqrt {-d^{2} x^{2}+c^{2}}\, a c \,d^{6} x^{5}+97 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{6} d x -3874 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{5} d^{2} x^{2}+237 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{5} d^{2} x +8817 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{4} d^{3} x^{2}+8889 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{3} d^{4} x^{3}+6711 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{2} d^{5} x^{4}+5135 b \,c^{6} d^{2} x^{2}-1257 b \,c^{5} d^{3} x^{3}+6058 b \,c^{4} d^{4} x^{4}+1747 b \,c^{3} d^{5} x^{5}+559 b \,c^{2} d^{6} x^{6}+77 b c \,d^{7} x^{7}+1386 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{6} d +450 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,d^{7} x^{6}-1001 c^{3} d^{6} x^{7}+169 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{7} d \,x^{2}-2522 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{6} d^{2} x^{3}+52 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{5} d^{3} x^{4}+1664 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{4} d^{4} x^{5}-1053 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{3} d^{5} x^{6}+237 a \,c^{6} d^{2} x -14052 a \,c^{5} d^{3} x^{2}-13752 a \,c^{4} d^{4} x^{3}-16062 a \,c^{3} d^{5} x^{4}-9663 a \,c^{2} d^{6} x^{5}-3228 a c \,d^{7} x^{6}+97 b \,c^{7} d x +26 c^{9} x}{9009 c^{4} d^{2} \left (\sqrt {-d^{2} x^{2}+c^{2}}\, c^{6}+6 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{5} d x +15 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{4} d^{2} x^{2}+20 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{3} d^{3} x^{3}+15 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{2} d^{4} x^{4}+6 \sqrt {-d^{2} x^{2}+c^{2}}\, c \,d^{5} x^{5}+\sqrt {-d^{2} x^{2}+c^{2}}\, d^{6} x^{6}-c^{7}-7 c^{6} d x -21 c^{5} d^{2} x^{2}-35 c^{4} d^{3} x^{3}-35 c^{3} d^{4} x^{4}-21 c^{2} d^{5} x^{5}-7 c \,d^{6} x^{6}-d^{7} x^{7}\right )} \] Input:

int((-d^2*x^2+c^2)^(5/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^10,x)
 

Output:

(1386*sqrt(c**2 - d**2*x**2)*a*c**6*d + 237*sqrt(c**2 - d**2*x**2)*a*c**5* 
d**2*x + 8817*sqrt(c**2 - d**2*x**2)*a*c**4*d**3*x**2 + 8889*sqrt(c**2 - d 
**2*x**2)*a*c**3*d**4*x**3 + 6711*sqrt(c**2 - d**2*x**2)*a*c**2*d**5*x**4 
+ 2694*sqrt(c**2 - d**2*x**2)*a*c*d**6*x**5 + 450*sqrt(c**2 - d**2*x**2)*a 
*d**7*x**6 + 97*sqrt(c**2 - d**2*x**2)*b*c**6*d*x - 3874*sqrt(c**2 - d**2* 
x**2)*b*c**5*d**2*x**2 + 293*sqrt(c**2 - d**2*x**2)*b*c**4*d**3*x**3 - 188 
5*sqrt(c**2 - d**2*x**2)*b*c**3*d**4*x**4 - 722*sqrt(c**2 - d**2*x**2)*b*c 
**2*d**5*x**5 - 117*sqrt(c**2 - d**2*x**2)*b*c*d**6*x**6 + 26*sqrt(c**2 - 
d**2*x**2)*c**8*x + 169*sqrt(c**2 - d**2*x**2)*c**7*d*x**2 - 2522*sqrt(c** 
2 - d**2*x**2)*c**6*d**2*x**3 + 52*sqrt(c**2 - d**2*x**2)*c**5*d**3*x**4 + 
 1664*sqrt(c**2 - d**2*x**2)*c**4*d**4*x**5 - 1053*sqrt(c**2 - d**2*x**2)* 
c**3*d**5*x**6 - 1386*a*c**7*d + 237*a*c**6*d**2*x - 14052*a*c**5*d**3*x** 
2 - 13752*a*c**4*d**4*x**3 - 16062*a*c**3*d**5*x**4 - 9663*a*c**2*d**6*x** 
5 - 3228*a*c*d**7*x**6 - 462*a*d**8*x**7 + 97*b*c**7*d*x + 5135*b*c**6*d** 
2*x**2 - 1257*b*c**5*d**3*x**3 + 6058*b*c**4*d**4*x**4 + 1747*b*c**3*d**5* 
x**5 + 559*b*c**2*d**6*x**6 + 77*b*c*d**7*x**7 + 26*c**9*x + 169*c**8*d*x* 
*2 + 3471*c**7*d**2*x**3 - 1534*c**6*d**3*x**4 - 832*c**5*d**4*x**5 + 3029 
*c**4*d**5*x**6 - 1001*c**3*d**6*x**7)/(9009*c**4*d**2*(sqrt(c**2 - d**2*x 
**2)*c**6 + 6*sqrt(c**2 - d**2*x**2)*c**5*d*x + 15*sqrt(c**2 - d**2*x**2)* 
c**4*d**2*x**2 + 20*sqrt(c**2 - d**2*x**2)*c**3*d**3*x**3 + 15*sqrt(c**...