\(\int \frac {(c^2-d^2 x^2)^{5/2} (A+B x+C x^2+D x^3)}{(c+d x)^9} \, dx\) [165]

Optimal result

Integrand size = 39, antiderivative size = 296 \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^9} \, dx=-\frac {2 D \sqrt {c^2-d^2 x^2}}{d^4 (c+d x)}+\frac {2 D \left (c^2-d^2 x^2\right )^{3/2}}{3 d^4 (c+d x)^3}-\frac {2 D \left (c^2-d^2 x^2\right )^{5/2}}{5 d^4 (c+d x)^5}-\frac {\left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \left (c^2-d^2 x^2\right )^{7/2}}{11 c d^4 (c+d x)^9}+\frac {\left (20 c^2 C d-9 B c d^2-2 A d^3-31 c^3 D\right ) \left (c^2-d^2 x^2\right )^{7/2}}{99 c^2 d^4 (c+d x)^8}-\frac {\left (79 c^2 C d+9 B c d^2+2 A d^3-266 c^3 D\right ) \left (c^2-d^2 x^2\right )^{7/2}}{693 c^3 d^4 (c+d x)^7}-\frac {D \arctan \left (\frac {d x}{\sqrt {c^2-d^2 x^2}}\right )}{d^4} \] Output:

-2*D*(-d^2*x^2+c^2)^(1/2)/d^4/(d*x+c)+2/3*D*(-d^2*x^2+c^2)^(3/2)/d^4/(d*x+ 
c)^3-2/5*D*(-d^2*x^2+c^2)^(5/2)/d^4/(d*x+c)^5-1/11*(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)^9+1/99*(-2*A*d^3-9*B*c*d^2+20*C* 
c^2*d-31*D*c^3)*(-d^2*x^2+c^2)^(7/2)/c^2/d^4/(d*x+c)^8-1/693*(2*A*d^3+9*B* 
c*d^2+79*C*c^2*d-266*D*c^3)*(-d^2*x^2+c^2)^(7/2)/c^3/d^4/(d*x+c)^7-D*arcta 
n(d*x/(-d^2*x^2+c^2)^(1/2))/d^4
 

Mathematica [A] (verified)

Time = 2.85 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.84 \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^9} \, dx=-\frac {\sqrt {c^2-d^2 x^2} \left (5446 c^8 D-10 A d^8 x^5-15 c d^7 x^4 (4 A+3 B x)+c^7 d (10 C+29211 D x)-5 c^2 d^6 x^3 (31 A+x (54 B+79 C x))+c^6 d^2 (45 B+x (60 C+63623 D x))+c^3 d^5 x^2 \left (925 A+x \left (1035 B+1095 C x+11956 D x^2\right )\right )+c^4 d^4 x \left (-1095 A+x \left (-1035 B-925 C x+40551 D x^2\right )\right )+c^5 d^3 \left (395 A+x \left (270 B+155 C x+70973 D x^2\right )\right )\right )}{3465 c^3 d^4 (c+d x)^6}+\frac {2 D \arctan \left (\frac {d x}{\sqrt {c^2}-\sqrt {c^2-d^2 x^2}}\right )}{d^4} \] Input:

Integrate[((c^2 - d^2*x^2)^(5/2)*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^9,x]
 

Output:

-1/3465*(Sqrt[c^2 - d^2*x^2]*(5446*c^8*D - 10*A*d^8*x^5 - 15*c*d^7*x^4*(4* 
A + 3*B*x) + c^7*d*(10*C + 29211*D*x) - 5*c^2*d^6*x^3*(31*A + x*(54*B + 79 
*C*x)) + c^6*d^2*(45*B + x*(60*C + 63623*D*x)) + c^3*d^5*x^2*(925*A + x*(1 
035*B + 1095*C*x + 11956*D*x^2)) + c^4*d^4*x*(-1095*A + x*(-1035*B - 925*C 
*x + 40551*D*x^2)) + c^5*d^3*(395*A + x*(270*B + 155*C*x + 70973*D*x^2)))) 
/(c^3*d^4*(c + d*x)^6) + (2*D*ArcTan[(d*x)/(Sqrt[c^2] - Sqrt[c^2 - d^2*x^2 
])])/d^4
 

Rubi [A] (verified)

Time = 1.08 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.47, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {2168, 2009}

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.

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

\(\Big \downarrow \) 2168

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 \left (c^2-d^2 x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{693 c^3 d^4 (c+d x)^7}-\frac {2 \left (c^2-d^2 x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{99 c^2 d^4 (c+d x)^8}-\frac {\left (c^2-d^2 x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{11 c d^4 (c+d x)^9}-\frac {D \arctan \left (\frac {d x}{\sqrt {c^2-d^2 x^2}}\right )}{d^4}+\frac {\left (c^2-d^2 x^2\right )^{7/2} \left (-B d^2-3 c^2 D+2 c C d\right )}{63 c^2 d^4 (c+d x)^7}+\frac {\left (c^2-d^2 x^2\right )^{7/2} \left (-B d^2-3 c^2 D+2 c C d\right )}{9 c d^4 (c+d x)^8}-\frac {\left (c^2-d^2 x^2\right )^{7/2} (C d-3 c D)}{7 c d^4 (c+d x)^7}-\frac {2 D \left (c^2-d^2 x^2\right )^{5/2}}{5 d^4 (c+d x)^5}+\frac {2 D \left (c^2-d^2 x^2\right )^{3/2}}{3 d^4 (c+d x)^3}-\frac {2 D \sqrt {c^2-d^2 x^2}}{d^4 (c+d x)}\)

Input:

Int[((c^2 - d^2*x^2)^(5/2)*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^9,x]
 

Output:

(-2*D*Sqrt[c^2 - d^2*x^2])/(d^4*(c + d*x)) + (2*D*(c^2 - d^2*x^2)^(3/2))/( 
3*d^4*(c + d*x)^3) - (2*D*(c^2 - d^2*x^2)^(5/2))/(5*d^4*(c + d*x)^5) - ((c 
^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(c^2 - d^2*x^2)^(7/2))/(11*c*d^4*(c + d* 
x)^9) + ((2*c*C*d - B*d^2 - 3*c^2*D)*(c^2 - d^2*x^2)^(7/2))/(9*c*d^4*(c + 
d*x)^8) - (2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(c^2 - d^2*x^2)^(7/2))/(9 
9*c^2*d^4*(c + d*x)^8) - ((C*d - 3*c*D)*(c^2 - d^2*x^2)^(7/2))/(7*c*d^4*(c 
 + d*x)^7) + ((2*c*C*d - B*d^2 - 3*c^2*D)*(c^2 - d^2*x^2)^(7/2))/(63*c^2*d 
^4*(c + d*x)^7) - (2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(c^2 - d^2*x^2)^( 
7/2))/(693*c^3*d^4*(c + d*x)^7) - (D*ArcTan[(d*x)/Sqrt[c^2 - d^2*x^2]])/d^ 
4
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2168

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
 Int[ExpandIntegrand[(a + b*x^2)^p, (d + e*x)^m*Pq, x], x] /; FreeQ[{a, b, 
d, e}, x] && PolyQ[Pq, x] && EqQ[b*d^2 + a*e^2, 0] && EqQ[m + Expon[Pq, x] 
+ 2*p + 1, 0] && ILtQ[m, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(783\) vs. \(2(272)=544\).

Time = 1.52 (sec) , antiderivative size = 784, normalized size of antiderivative = 2.65

method	result	size
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}}}{5 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 {7}{2}}}{3 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 {7}{2}}}{c d \left (x +\frac {c}{d}\right )^{4}}-\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}}}{c d \left (x +\frac {c}{d}\right )^{3}}+\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}}}{3 c d \left (x +\frac {c}{d}\right )^{2}}+\frac {5 d \left (\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {5}{2}}}{5}+c d \left (-\frac {\left (-2 d^{2} \left (x +\frac {c}{d}\right )+2 c d \right ) \left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {3}{2}}}{8 d^{2}}+\frac {3 c^{2} \left (-\frac {\left (-2 d^{2} \left (x +\frac {c}{d}\right )+2 c d \right ) \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}{4 d^{2}}+\frac {c^{2} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}\right )}{2 \sqrt {d^{2}}}\right )}{4}\right )\right )}{3 c}\right )}{c}\right )}{c}\right )}{3 c}\right )}{5 c}\right )}{d^{9}}-\frac {\left (C d -3 D c \right ) \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 (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}}}{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 (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}}}{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}}\)	\(784\)

Input:

int((-d^2*x^2+c^2)^(5/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^9,x,method=_RETURNVER 
BOSE)
 

Output:

D/d^9*(-1/5/c/d/(x+c/d)^6*(-d^2*(x+c/d)^2+2*c*d*(x+c/d))^(7/2)-1/5*d/c*(-1 
/3/c/d/(x+c/d)^5*(-d^2*(x+c/d)^2+2*c*d*(x+c/d))^(7/2)-2/3*d/c*(-1/c/d/(x+c 
/d)^4*(-d^2*(x+c/d)^2+2*c*d*(x+c/d))^(7/2)-3*d/c*(1/c/d/(x+c/d)^3*(-d^2*(x 
+c/d)^2+2*c*d*(x+c/d))^(7/2)+4*d/c*(1/3/c/d/(x+c/d)^2*(-d^2*(x+c/d)^2+2*c* 
d*(x+c/d))^(7/2)+5/3*d/c*(1/5*(-d^2*(x+c/d)^2+2*c*d*(x+c/d))^(5/2)+c*d*(-1 
/8*(-2*d^2*(x+c/d)+2*c*d)/d^2*(-d^2*(x+c/d)^2+2*c*d*(x+c/d))^(3/2)+3/4*c^2 
*(-1/4*(-2*d^2*(x+c/d)+2*c*d)/d^2*(-d^2*(x+c/d)^2+2*c*d*(x+c/d))^(1/2)+1/2 
*c^2/(d^2)^(1/2)*arctan((d^2)^(1/2)*x/(-d^2*(x+c/d)^2+2*c*d*(x+c/d))^(1/2) 
)))))))))-1/7*(C*d-3*D*c)/d^11/c/(x+c/d)^7*(-d^2*(x+c/d)^2+2*c*d*(x+c/d))^ 
(7/2)+(B*d^2-2*C*c*d+3*D*c^2)/d^11*(-1/9/c/d/(x+c/d)^8*(-d^2*(x+c/d)^2+2*c 
*d*(x+c/d))^(7/2)-1/63/c^2/(x+c/d)^7*(-d^2*(x+c/d)^2+2*c*d*(x+c/d))^(7/2)) 
+(A*d^3-B*c*d^2+C*c^2*d-D*c^3)/d^12*(-1/11/c/d/(x+c/d)^9*(-d^2*(x+c/d)^2+2 
*c*d*(x+c/d))^(7/2)+2/11*d/c*(-1/9/c/d/(x+c/d)^8*(-d^2*(x+c/d)^2+2*c*d*(x+ 
c/d))^(7/2)-1/63/c^2/(x+c/d)^7*(-d^2*(x+c/d)^2+2*c*d*(x+c/d))^(7/2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 692 vs. \(2 (272) = 544\).

Time = 0.34 (sec) , antiderivative size = 692, normalized size of antiderivative = 2.34 \[ \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)^9} \, dx=-\frac {5446 \, D c^{9} + 10 \, C c^{8} d + 45 \, B c^{7} d^{2} + 395 \, A c^{6} d^{3} + {\left (5446 \, D c^{3} d^{6} + 10 \, C c^{2} d^{7} + 45 \, B c d^{8} + 395 \, A d^{9}\right )} x^{6} + 6 \, {\left (5446 \, D c^{4} d^{5} + 10 \, C c^{3} d^{6} + 45 \, B c^{2} d^{7} + 395 \, A c d^{8}\right )} x^{5} + 15 \, {\left (5446 \, D c^{5} d^{4} + 10 \, C c^{4} d^{5} + 45 \, B c^{3} d^{6} + 395 \, A c^{2} d^{7}\right )} x^{4} + 20 \, {\left (5446 \, D c^{6} d^{3} + 10 \, C c^{5} d^{4} + 45 \, B c^{4} d^{5} + 395 \, A c^{3} d^{6}\right )} x^{3} + 15 \, {\left (5446 \, D c^{7} d^{2} + 10 \, C c^{6} d^{3} + 45 \, B c^{5} d^{4} + 395 \, A c^{4} d^{5}\right )} x^{2} + 6 \, {\left (5446 \, D c^{8} d + 10 \, C c^{7} d^{2} + 45 \, B c^{6} d^{3} + 395 \, A c^{5} d^{4}\right )} x - 6930 \, {\left (D c^{3} d^{6} x^{6} + 6 \, D c^{4} d^{5} x^{5} + 15 \, D c^{5} d^{4} x^{4} + 20 \, D c^{6} d^{3} x^{3} + 15 \, D c^{7} d^{2} x^{2} + 6 \, D c^{8} d x + D c^{9}\right )} \arctan \left (-\frac {c - \sqrt {-d^{2} x^{2} + c^{2}}}{d x}\right ) + {\left (5446 \, D c^{8} + 10 \, C c^{7} d + 45 \, B c^{6} d^{2} + 395 \, A c^{5} d^{3} + {\left (11956 \, D c^{3} d^{5} - 395 \, C c^{2} d^{6} - 45 \, B c d^{7} - 10 \, A d^{8}\right )} x^{5} + 3 \, {\left (13517 \, D c^{4} d^{4} + 365 \, C c^{3} d^{5} - 90 \, B c^{2} d^{6} - 20 \, A c d^{7}\right )} x^{4} + {\left (70973 \, D c^{5} d^{3} - 925 \, C c^{4} d^{4} + 1035 \, B c^{3} d^{5} - 155 \, A c^{2} d^{6}\right )} x^{3} + {\left (63623 \, D c^{6} d^{2} + 155 \, C c^{5} d^{3} - 1035 \, B c^{4} d^{4} + 925 \, A c^{3} d^{5}\right )} x^{2} + 3 \, {\left (9737 \, D c^{7} d + 20 \, C c^{6} d^{2} + 90 \, B c^{5} d^{3} - 365 \, A c^{4} d^{4}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{3465 \, {\left (c^{3} d^{10} x^{6} + 6 \, c^{4} d^{9} x^{5} + 15 \, c^{5} d^{8} x^{4} + 20 \, c^{6} d^{7} x^{3} + 15 \, c^{7} d^{6} x^{2} + 6 \, c^{8} d^{5} x + c^{9} 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)^9,x, algorithm= 
"fricas")
 

Output:

-1/3465*(5446*D*c^9 + 10*C*c^8*d + 45*B*c^7*d^2 + 395*A*c^6*d^3 + (5446*D* 
c^3*d^6 + 10*C*c^2*d^7 + 45*B*c*d^8 + 395*A*d^9)*x^6 + 6*(5446*D*c^4*d^5 + 
 10*C*c^3*d^6 + 45*B*c^2*d^7 + 395*A*c*d^8)*x^5 + 15*(5446*D*c^5*d^4 + 10* 
C*c^4*d^5 + 45*B*c^3*d^6 + 395*A*c^2*d^7)*x^4 + 20*(5446*D*c^6*d^3 + 10*C* 
c^5*d^4 + 45*B*c^4*d^5 + 395*A*c^3*d^6)*x^3 + 15*(5446*D*c^7*d^2 + 10*C*c^ 
6*d^3 + 45*B*c^5*d^4 + 395*A*c^4*d^5)*x^2 + 6*(5446*D*c^8*d + 10*C*c^7*d^2 
 + 45*B*c^6*d^3 + 395*A*c^5*d^4)*x - 6930*(D*c^3*d^6*x^6 + 6*D*c^4*d^5*x^5 
 + 15*D*c^5*d^4*x^4 + 20*D*c^6*d^3*x^3 + 15*D*c^7*d^2*x^2 + 6*D*c^8*d*x + 
D*c^9)*arctan(-(c - sqrt(-d^2*x^2 + c^2))/(d*x)) + (5446*D*c^8 + 10*C*c^7* 
d + 45*B*c^6*d^2 + 395*A*c^5*d^3 + (11956*D*c^3*d^5 - 395*C*c^2*d^6 - 45*B 
*c*d^7 - 10*A*d^8)*x^5 + 3*(13517*D*c^4*d^4 + 365*C*c^3*d^5 - 90*B*c^2*d^6 
 - 20*A*c*d^7)*x^4 + (70973*D*c^5*d^3 - 925*C*c^4*d^4 + 1035*B*c^3*d^5 - 1 
55*A*c^2*d^6)*x^3 + (63623*D*c^6*d^2 + 155*C*c^5*d^3 - 1035*B*c^4*d^4 + 92 
5*A*c^3*d^5)*x^2 + 3*(9737*D*c^7*d + 20*C*c^6*d^2 + 90*B*c^5*d^3 - 365*A*c 
^4*d^4)*x)*sqrt(-d^2*x^2 + c^2))/(c^3*d^10*x^6 + 6*c^4*d^9*x^5 + 15*c^5*d^ 
8*x^4 + 20*c^6*d^7*x^3 + 15*c^7*d^6*x^2 + 6*c^8*d^5*x + c^9*d^4)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^9} \, dx=\text {Timed out} \] Input:

integrate((-d**2*x**2+c**2)**(5/2)*(D*x**3+C*x**2+B*x+A)/(d*x+c)**9,x)
                                                                                    
                                                                                    
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3960 vs. \(2 (272) = 544\).

Time = 0.22 (sec) , antiderivative size = 3960, normalized size of antiderivative = 13.38 \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^9} \, dx=\text {Too large to display} \] Input:

integrate((-d^2*x^2+c^2)^(5/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^9,x, algorithm= 
"maxima")
 

Output:

1/3*(-d^2*x^2 + c^2)^(5/2)*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) - 5/12*(-d^2*x^2 + c^2)^(3/2)*D*c^4/(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) + 5/22*sqrt(-d^2*x^2 + c^2)*D*c^5/(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) - 1/3*(-d^2*x^2 + c^2)^(5/2)*C*c^2/(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) - 3/2*(-d^2*x^2 + c^2)^(5/2)*D*c^2/(d^1 
1*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) + 5/12*(-d^2*x^2 + c^2)^(3/2)*C*c^ 
3/(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) + 5/2*(-d^2*x^2 + c^2)^(3/2)* 
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) - 5/22*sqrt(-d^2*x^2 + c^2)*C*c^4/(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) - 665/396*sqrt(-d^2*x^2 + c^2)*D*c^4/(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) + 1/3*(-d^2* 
x^2 + c^2)^(5/2)*B*c/(d^10*x^8 + 8*c*d^9*x^7 + 28*c^2*d^8*x^6 + 56*c^3*d^7 
*x^5 + 70*c^4*d^6*x^4 + 56*c^5*d^5*x^3 + 28*c^6*d^4*x^2 + 8*c^7*d^3*x +...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1327 vs. \(2 (272) = 544\).

Time = 0.16 (sec) , antiderivative size = 1327, normalized size of antiderivative = 4.48 \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^9} \, dx=\text {Too large to display} \] Input:

integrate((-d^2*x^2+c^2)^(5/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^9,x, algorithm= 
"giac")
 

Output:

-D*arcsin(d*x/c)*sgn(c)*sgn(d)/(d^3*abs(d)) + 2/3465*(5446*D*c^3 + 10*C*c^ 
2*d + 45*B*c*d^2 + 395*A*d^3 + 495*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))*B*c 
/x + 56441*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))*D*c^3/(d^2*x) + 110*(c*d + 
sqrt(-d^2*x^2 + c^2)*abs(d))*C*c^2/(d*x) + 880*(c*d + sqrt(-d^2*x^2 + c^2) 
*abs(d))*A*d/x + 261415*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^2*D*c^3/(d^4*x 
^2) + 550*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^2*C*c^2/(d^3*x^2) - 990*(c*d 
 + sqrt(-d^2*x^2 + c^2)*abs(d))^2*B*c/(d^2*x^2) + 14795*(c*d + sqrt(-d^2*x 
^2 + c^2)*abs(d))^2*A/(d*x^2) + 709170*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d)) 
^3*D*c^3/(d^6*x^3) - 2970*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^3*C*c^2/(d^5 
*x^3) + 10890*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^3*B*c/(d^4*x^3) + 24750* 
(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^3*A/(d^3*x^3) + 1231230*(c*d + sqrt(-d 
^2*x^2 + c^2)*abs(d))^4*D*c^3/(d^8*x^4) + 14850*(c*d + sqrt(-d^2*x^2 + c^2 
)*abs(d))^4*C*c^2/(d^7*x^4) - 5940*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^4*B 
*c/(d^6*x^4) + 77220*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^4*A/(d^5*x^4) + 1 
458996*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^5*D*c^3/(d^10*x^5) - 20790*(c*d 
 + sqrt(-d^2*x^2 + c^2)*abs(d))^5*C*c^2/(d^9*x^5) + 27720*(c*d + sqrt(-d^2 
*x^2 + c^2)*abs(d))^5*B*c/(d^8*x^5) + 76230*(c*d + sqrt(-d^2*x^2 + c^2)*ab 
s(d))^5*A/(d^7*x^5) + 1057056*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^6*D*c^3/ 
(d^12*x^6) + 25410*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^6*C*c^2/(d^11*x^6) 
- 6930*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^6*B*c/(d^10*x^6) + 106260*(c...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^9} \, dx=\int \frac {{\left (c^2-d^2\,x^2\right )}^{5/2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^9} \,d x \] Input:

int(((c^2 - d^2*x^2)^(5/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^9,x)
 

Output:

int(((c^2 - d^2*x^2)^(5/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^9, x)
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 1087, normalized size of antiderivative = 3.67 \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^9} \, dx =\text {Too large to display} \] Input:

int((-d^2*x^2+c^2)^(5/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^9,x)
 

Output:

( - 3465*sqrt(c**2 - d**2*x**2)*asin((d*x)/c)*c**8 - 17325*sqrt(c**2 - d** 
2*x**2)*asin((d*x)/c)*c**7*d*x - 34650*sqrt(c**2 - d**2*x**2)*asin((d*x)/c 
)*c**6*d**2*x**2 - 34650*sqrt(c**2 - d**2*x**2)*asin((d*x)/c)*c**5*d**3*x* 
*3 - 17325*sqrt(c**2 - d**2*x**2)*asin((d*x)/c)*c**4*d**4*x**4 - 3465*sqrt 
(c**2 - d**2*x**2)*asin((d*x)/c)*c**3*d**5*x**5 + 3465*asin((d*x)/c)*c**9 
+ 20790*asin((d*x)/c)*c**8*d*x + 51975*asin((d*x)/c)*c**7*d**2*x**2 + 6930 
0*asin((d*x)/c)*c**6*d**3*x**3 + 51975*asin((d*x)/c)*c**5*d**4*x**4 + 2079 
0*asin((d*x)/c)*c**4*d**5*x**5 + 3465*asin((d*x)/c)*c**3*d**6*x**6 + 630*s 
qrt(c**2 - d**2*x**2)*a*c**5*d**2 + 80*sqrt(c**2 - d**2*x**2)*a*c**4*d**3* 
x + 3275*sqrt(c**2 - d**2*x**2)*a*c**3*d**4*x**2 + 2195*sqrt(c**2 - d**2*x 
**2)*a*c**2*d**5*x**3 + 1115*sqrt(c**2 - d**2*x**2)*a*c*d**6*x**4 + 225*sq 
rt(c**2 - d**2*x**2)*a*d**7*x**5 + 45*sqrt(c**2 - d**2*x**2)*b*c**5*d**2*x 
 - 1485*sqrt(c**2 - d**2*x**2)*b*c**4*d**3*x**2 + 585*sqrt(c**2 - d**2*x** 
2)*b*c**3*d**4*x**3 - 495*sqrt(c**2 - d**2*x**2)*b*c**2*d**5*x**4 - 90*sqr 
t(c**2 - d**2*x**2)*b*c*d**6*x**5 + 630*sqrt(c**2 - d**2*x**2)*c**8 + 5141 
*sqrt(c**2 - d**2*x**2)*c**7*d*x + 15518*sqrt(c**2 - d**2*x**2)*c**6*d**2* 
x**2 + 21788*sqrt(c**2 - d**2*x**2)*c**5*d**3*x**3 + 17516*sqrt(c**2 - d** 
2*x**2)*c**4*d**4*x**4 + 6735*sqrt(c**2 - d**2*x**2)*c**3*d**5*x**5 - 630* 
a*c**6*d**2 + 80*a*c**5*d**3*x - 5545*a*c**4*d**4*x**2 - 3620*a*c**3*d**5* 
x**3 - 3620*a*c**2*d**6*x**4 - 1460*a*c*d**7*x**5 - 245*a*d**8*x**6 + 4...