\(\int \frac {(c^2-d^2 x^2)^{5/2} (A+B x+C x^2+D x^3)}{(c+d x)^8} \, dx\) [164]

Optimal result

Integrand size = 39, antiderivative size = 285 \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^8} \, dx=\frac {D \sqrt {c^2-d^2 x^2}}{d^4}-\frac {2 (C d-7 c D) \sqrt {c^2-d^2 x^2}}{d^4 (c+d x)}+\frac {2 (C d-5 c D) \left (c^2-d^2 x^2\right )^{3/2}}{3 d^4 (c+d x)^3}-\frac {2 (C d-3 c D) \left (c^2-d^2 x^2\right )^{5/2}}{5 d^4 (c+d x)^5}-\frac {\left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \left (c^2-d^2 x^2\right )^{7/2}}{9 c d^4 (c+d x)^8}+\frac {\left (17 c^2 C d-8 B c d^2-A d^3-26 c^3 D\right ) \left (c^2-d^2 x^2\right )^{7/2}}{63 c^2 d^4 (c+d x)^7}-\frac {(C d-8 c D) \arctan \left (\frac {d x}{\sqrt {c^2-d^2 x^2}}\right )}{d^4} \] Output:

D*(-d^2*x^2+c^2)^(1/2)/d^4-2*(C*d-7*D*c)*(-d^2*x^2+c^2)^(1/2)/d^4/(d*x+c)+ 
2/3*(C*d-5*D*c)*(-d^2*x^2+c^2)^(3/2)/d^4/(d*x+c)^3-2/5*(C*d-3*D*c)*(-d^2*x 
^2+c^2)^(5/2)/d^4/(d*x+c)^5-1/9*(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)^8+1/63*(-A*d^3-8*B*c*d^2+17*C*c^2*d-26*D*c^3)*(-d^2 
*x^2+c^2)^(7/2)/c^2/d^4/(d*x+c)^7-(C*d-8*D*c)*arctan(d*x/(-d^2*x^2+c^2)^(1 
/2))/d^4
 

Mathematica [A] (verified)

Time = 2.57 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.81 \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^8} \, dx=\frac {\frac {\sqrt {c^2-d^2 x^2} \left (3958 c^7 D+5 A d^7 x^4+5 c d^6 x^3 (5 A+8 B x)+c^6 (-496 C d+17270 d D x)+c^5 d^2 (-5 B+x (-2165 C+28959 D x))-c^4 d^3 \left (40 A+x \left (25 B+3633 C x-22595 D x^2\right )\right )+c^2 d^5 x^2 \left (-105 A+x \left (-115 B-1051 C x+315 D x^2\right )\right )+c^3 d^4 x \left (115 A+x \left (105 B-2735 C x+7543 D x^2\right )\right )\right )}{c^2 (c+d x)^5}+630 (C d-8 c D) \arctan \left (\frac {d x}{\sqrt {c^2}-\sqrt {c^2-d^2 x^2}}\right )}{315 d^4} \] Input:

Integrate[((c^2 - d^2*x^2)^(5/2)*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^8,x]
 

Output:

((Sqrt[c^2 - d^2*x^2]*(3958*c^7*D + 5*A*d^7*x^4 + 5*c*d^6*x^3*(5*A + 8*B*x 
) + c^6*(-496*C*d + 17270*d*D*x) + c^5*d^2*(-5*B + x*(-2165*C + 28959*D*x) 
) - c^4*d^3*(40*A + x*(25*B + 3633*C*x - 22595*D*x^2)) + c^2*d^5*x^2*(-105 
*A + x*(-115*B - 1051*C*x + 315*D*x^2)) + c^3*d^4*x*(115*A + x*(105*B - 27 
35*C*x + 7543*D*x^2))))/(c^2*(c + d*x)^5) + 630*(C*d - 8*c*D)*ArcTan[(d*x) 
/(Sqrt[c^2] - Sqrt[c^2 - d^2*x^2])])/(315*d^4)
 

Rubi [A] (verified)

Time = 1.20 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.17, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2170, 25, 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.

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

\(\Big \downarrow \) 2170

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2168

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {d \left (c^2-d^2 x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{63 c^2 (c+d x)^7}-\frac {d \left (c^2-d^2 x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{9 c (c+d x)^8}-d \arctan \left (\frac {d x}{\sqrt {c^2-d^2 x^2}}\right ) (C d-8 c D)+\frac {d \left (c^2-d^2 x^2\right )^{7/2} \left (-B d^2-3 c^2 D+2 c C d\right )}{7 c (c+d x)^7}-\frac {2 d \left (c^2-d^2 x^2\right )^{5/2} (C d-8 c D)}{5 (c+d x)^5}+\frac {2 d \left (c^2-d^2 x^2\right )^{3/2} (C d-8 c D)}{3 (c+d x)^3}-\frac {2 d \sqrt {c^2-d^2 x^2} (C d-8 c D)}{c+d x}}{d^5}-\frac {D \left (c^2-d^2 x^2\right )^{7/2}}{d^4 (c+d x)^6}\)

Input:

Int[((c^2 - d^2*x^2)^(5/2)*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^8,x]
 

Output:

-((D*(c^2 - d^2*x^2)^(7/2))/(d^4*(c + d*x)^6)) + ((-2*d*(C*d - 8*c*D)*Sqrt 
[c^2 - d^2*x^2])/(c + d*x) + (2*d*(C*d - 8*c*D)*(c^2 - d^2*x^2)^(3/2))/(3* 
(c + d*x)^3) - (2*d*(C*d - 8*c*D)*(c^2 - d^2*x^2)^(5/2))/(5*(c + d*x)^5) - 
 (d*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(c^2 - d^2*x^2)^(7/2))/(9*c*(c + d 
*x)^8) + (d*(2*c*C*d - B*d^2 - 3*c^2*D)*(c^2 - d^2*x^2)^(7/2))/(7*c*(c + d 
*x)^7) - (d*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(c^2 - d^2*x^2)^(7/2))/(63 
*c^2*(c + d*x)^7) - d*(C*d - 8*c*D)*ArcTan[(d*x)/Sqrt[c^2 - d^2*x^2]])/d^5
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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]
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1037\) vs. \(2(263)=526\).

Time = 1.12 (sec) , antiderivative size = 1038, normalized size of antiderivative = 3.64

method	result	size
default	\(\text {Expression too large to display}\)	\(1038\)

Input:

int((-d^2*x^2+c^2)^(5/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^8,x,method=_RETURNVER 
BOSE)
 

Output:

D/d^8*(-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)))))))))+(C*d-3*D*c)/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*(B*d^2-2*C*c*d+3*D*c^2)/d^11/c/( 
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^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))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 667 vs. \(2 (265) = 530\).

Time = 0.31 (sec) , antiderivative size = 667, normalized size of antiderivative = 2.34 \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^8} \, dx=\frac {3958 \, D c^{8} - 496 \, C c^{7} d - 5 \, B c^{6} d^{2} - 40 \, A c^{5} d^{3} + {\left (3958 \, D c^{3} d^{5} - 496 \, C c^{2} d^{6} - 5 \, B c d^{7} - 40 \, A d^{8}\right )} x^{5} + 5 \, {\left (3958 \, D c^{4} d^{4} - 496 \, C c^{3} d^{5} - 5 \, B c^{2} d^{6} - 40 \, A c d^{7}\right )} x^{4} + 10 \, {\left (3958 \, D c^{5} d^{3} - 496 \, C c^{4} d^{4} - 5 \, B c^{3} d^{5} - 40 \, A c^{2} d^{6}\right )} x^{3} + 10 \, {\left (3958 \, D c^{6} d^{2} - 496 \, C c^{5} d^{3} - 5 \, B c^{4} d^{4} - 40 \, A c^{3} d^{5}\right )} x^{2} + 5 \, {\left (3958 \, D c^{7} d - 496 \, C c^{6} d^{2} - 5 \, B c^{5} d^{3} - 40 \, A c^{4} d^{4}\right )} x - 630 \, {\left (8 \, D c^{8} - C c^{7} d + {\left (8 \, D c^{3} d^{5} - C c^{2} d^{6}\right )} x^{5} + 5 \, {\left (8 \, D c^{4} d^{4} - C c^{3} d^{5}\right )} x^{4} + 10 \, {\left (8 \, D c^{5} d^{3} - C c^{4} d^{4}\right )} x^{3} + 10 \, {\left (8 \, D c^{6} d^{2} - C c^{5} d^{3}\right )} x^{2} + 5 \, {\left (8 \, D c^{7} d - C c^{6} d^{2}\right )} x\right )} \arctan \left (-\frac {c - \sqrt {-d^{2} x^{2} + c^{2}}}{d x}\right ) + {\left (315 \, D c^{2} d^{5} x^{5} + 3958 \, D c^{7} - 496 \, C c^{6} d - 5 \, B c^{5} d^{2} - 40 \, A c^{4} d^{3} + {\left (7543 \, D c^{3} d^{4} - 1051 \, C c^{2} d^{5} + 40 \, B c d^{6} + 5 \, A d^{7}\right )} x^{4} + 5 \, {\left (4519 \, D c^{4} d^{3} - 547 \, C c^{3} d^{4} - 23 \, B c^{2} d^{5} + 5 \, A c d^{6}\right )} x^{3} + 21 \, {\left (1379 \, D c^{5} d^{2} - 173 \, C c^{4} d^{3} + 5 \, B c^{3} d^{4} - 5 \, A c^{2} d^{5}\right )} x^{2} + 5 \, {\left (3454 \, D c^{6} d - 433 \, C c^{5} d^{2} - 5 \, B c^{4} d^{3} + 23 \, A c^{3} d^{4}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{315 \, {\left (c^{2} d^{9} x^{5} + 5 \, c^{3} d^{8} x^{4} + 10 \, c^{4} d^{7} x^{3} + 10 \, c^{5} d^{6} x^{2} + 5 \, c^{6} d^{5} x + c^{7} d^{4}\right )}} \] Input:

integrate((-d^2*x^2+c^2)^(5/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^8,x, algorithm= 
"fricas")
 

Output:

1/315*(3958*D*c^8 - 496*C*c^7*d - 5*B*c^6*d^2 - 40*A*c^5*d^3 + (3958*D*c^3 
*d^5 - 496*C*c^2*d^6 - 5*B*c*d^7 - 40*A*d^8)*x^5 + 5*(3958*D*c^4*d^4 - 496 
*C*c^3*d^5 - 5*B*c^2*d^6 - 40*A*c*d^7)*x^4 + 10*(3958*D*c^5*d^3 - 496*C*c^ 
4*d^4 - 5*B*c^3*d^5 - 40*A*c^2*d^6)*x^3 + 10*(3958*D*c^6*d^2 - 496*C*c^5*d 
^3 - 5*B*c^4*d^4 - 40*A*c^3*d^5)*x^2 + 5*(3958*D*c^7*d - 496*C*c^6*d^2 - 5 
*B*c^5*d^3 - 40*A*c^4*d^4)*x - 630*(8*D*c^8 - C*c^7*d + (8*D*c^3*d^5 - C*c 
^2*d^6)*x^5 + 5*(8*D*c^4*d^4 - C*c^3*d^5)*x^4 + 10*(8*D*c^5*d^3 - C*c^4*d^ 
4)*x^3 + 10*(8*D*c^6*d^2 - C*c^5*d^3)*x^2 + 5*(8*D*c^7*d - C*c^6*d^2)*x)*a 
rctan(-(c - sqrt(-d^2*x^2 + c^2))/(d*x)) + (315*D*c^2*d^5*x^5 + 3958*D*c^7 
 - 496*C*c^6*d - 5*B*c^5*d^2 - 40*A*c^4*d^3 + (7543*D*c^3*d^4 - 1051*C*c^2 
*d^5 + 40*B*c*d^6 + 5*A*d^7)*x^4 + 5*(4519*D*c^4*d^3 - 547*C*c^3*d^4 - 23* 
B*c^2*d^5 + 5*A*c*d^6)*x^3 + 21*(1379*D*c^5*d^2 - 173*C*c^4*d^3 + 5*B*c^3* 
d^4 - 5*A*c^2*d^5)*x^2 + 5*(3454*D*c^6*d - 433*C*c^5*d^2 - 5*B*c^4*d^3 + 2 
3*A*c^3*d^4)*x)*sqrt(-d^2*x^2 + c^2))/(c^2*d^9*x^5 + 5*c^3*d^8*x^4 + 10*c^ 
4*d^7*x^3 + 10*c^5*d^6*x^2 + 5*c^6*d^5*x + c^7*d^4)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^8} \, dx=\text {Timed out} \] Input:

integrate((-d**2*x**2+c**2)**(5/2)*(D*x**3+C*x**2+B*x+A)/(d*x+c)**8,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3095 vs. \(2 (265) = 530\).

Time = 0.21 (sec) , antiderivative size = 3095, normalized size of antiderivative = 10.86 \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^8} \, dx=\text {Too large to display} \] Input:

integrate((-d^2*x^2+c^2)^(5/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^8,x, algorithm= 
"maxima")
 

Output:

1/2*(-d^2*x^2 + c^2)^(5/2)*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) - 5/6*(-d^2*x^2 + c^2)^(3/2)*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^4) + 5/9*sqrt 
(-d^2*x^2 + c^2)*D*c^5/(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) - 1/2*(-d^2*x^2 + c^2)^(5/2)*C*c^2/(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) - 3*(-d^2*x^2 + c^2)^(5/2)*D*c^2/(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) + 5/6*(-d^2*x^2 + c^2)^(3/2)*C*c^3/(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) + 15/2*(-d^2*x^2 + c^2)^(3/2)*D*c^3/(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) - 5/9*sqrt(-d^2*x^2 + c^ 
2)*C*c^4/(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) - 815/126*sqrt(-d^2*x^2 + c^2)*D*c^4/(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) + 1/2*(-d^2*x^2 + c^2)^(5/2)*B 
*c/(d^9*x^7 + 7*c*d^8*x^6 + 21*c^2*d^7*x^5 + 35*c^3*d^6*x^4 + 35*c^4*d^5*x 
^3 + 21*c^5*d^4*x^2 + 7*c^6*d^3*x + c^7*d^2) + 2*(-d^2*x^2 + c^2)^(5/2)*C* 
c/(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) + 3/5*(-d^2*x^2 + c^2)^(5/2)*D*c/(d^9*x^5 + ...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1153 vs. \(2 (265) = 530\).

Time = 0.16 (sec) , antiderivative size = 1153, normalized size of antiderivative = 4.05 \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^8} \, dx=\text {Too large to display} \] Input:

integrate((-d^2*x^2+c^2)^(5/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^8,x, algorithm= 
"giac")
 

Output:

(8*D*c - C*d)*arcsin(d*x/c)*sgn(c)*sgn(d)/(d^3*abs(d)) + sqrt(-d^2*x^2 + c 
^2)*D/d^4 - 2/315*(3643*D*c^3 - 496*C*c^2*d - 5*B*c*d^2 - 40*A*d^3 - 45*(c 
*d + sqrt(-d^2*x^2 + c^2)*abs(d))*B*c/x + 30267*(c*d + sqrt(-d^2*x^2 + c^2 
)*abs(d))*D*c^3/(d^2*x) - 4149*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))*C*c^2/( 
d*x) - 45*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))*A*d/x + 108783*(c*d + sqrt(- 
d^2*x^2 + c^2)*abs(d))^2*D*c^3/(d^4*x^2) - 15021*(c*d + sqrt(-d^2*x^2 + c^ 
2)*abs(d))^2*C*c^2/(d^3*x^2) + 135*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^2*B 
*c/(d^2*x^2) - 1125*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^2*A/(d*x^2) + 2189 
67*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^3*D*c^3/(d^6*x^3) - 30009*(c*d + sq 
rt(-d^2*x^2 + c^2)*abs(d))^3*C*c^2/(d^5*x^3) - 945*(c*d + sqrt(-d^2*x^2 + 
c^2)*abs(d))^3*B*c/(d^4*x^3) - 945*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^3*A 
/(d^3*x^3) + 266553*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^4*D*c^3/(d^8*x^4) 
- 38241*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^4*C*c^2/(d^7*x^4) + 945*(c*d + 
 sqrt(-d^2*x^2 + c^2)*abs(d))^4*B*c/(d^6*x^4) - 3465*(c*d + sqrt(-d^2*x^2 
+ c^2)*abs(d))^4*A/(d^5*x^4) + 192465*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^ 
5*D*c^3/(d^10*x^5) - 24255*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^5*C*c^2/(d^ 
9*x^5) - 1575*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^5*B*c/(d^8*x^5) - 1575*( 
c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^5*A/(d^7*x^5) + 87045*(c*d + sqrt(-d^2* 
x^2 + c^2)*abs(d))^6*D*c^3/(d^12*x^6) - 11655*(c*d + sqrt(-d^2*x^2 + c^2)* 
abs(d))^6*C*c^2/(d^11*x^6) + 525*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^6*...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^8} \, dx=\int \frac {{\left (c^2-d^2\,x^2\right )}^{5/2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^8} \,d x \] Input:

int(((c^2 - d^2*x^2)^(5/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^8,x)
 

Output:

int(((c^2 - d^2*x^2)^(5/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^8, x)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 939, normalized size of antiderivative = 3.29 \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^8} \, dx =\text {Too large to display} \] Input:

int((-d^2*x^2+c^2)^(5/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^8,x)
 

Output:

(2205*sqrt(c**2 - d**2*x**2)*asin((d*x)/c)*c**7 + 8820*sqrt(c**2 - d**2*x* 
*2)*asin((d*x)/c)*c**6*d*x + 13230*sqrt(c**2 - d**2*x**2)*asin((d*x)/c)*c* 
*5*d**2*x**2 + 8820*sqrt(c**2 - d**2*x**2)*asin((d*x)/c)*c**4*d**3*x**3 + 
2205*sqrt(c**2 - d**2*x**2)*asin((d*x)/c)*c**3*d**4*x**4 - 2205*asin((d*x) 
/c)*c**8 - 11025*asin((d*x)/c)*c**7*d*x - 22050*asin((d*x)/c)*c**6*d**2*x* 
*2 - 22050*asin((d*x)/c)*c**5*d**3*x**3 - 11025*asin((d*x)/c)*c**4*d**4*x* 
*4 - 2205*asin((d*x)/c)*c**3*d**5*x**5 + 70*sqrt(c**2 - d**2*x**2)*a*c**4* 
d**2 + 5*sqrt(c**2 - d**2*x**2)*a*c**3*d**3*x + 285*sqrt(c**2 - d**2*x**2) 
*a*c**2*d**4*x**2 + 95*sqrt(c**2 - d**2*x**2)*a*c*d**5*x**3 + 25*sqrt(c**2 
 - d**2*x**2)*a*d**6*x**4 + 5*sqrt(c**2 - d**2*x**2)*b*c**4*d**2*x - 135*s 
qrt(c**2 - d**2*x**2)*b*c**3*d**3*x**2 + 95*sqrt(c**2 - d**2*x**2)*b*c**2* 
d**4*x**3 - 45*sqrt(c**2 - d**2*x**2)*b*c*d**5*x**4 - 490*sqrt(c**2 - d**2 
*x**2)*c**7 - 3217*sqrt(c**2 - d**2*x**2)*c**6*d*x - 7494*sqrt(c**2 - d**2 
*x**2)*c**5*d**2*x**2 - 7972*sqrt(c**2 - d**2*x**2)*c**4*d**3*x**3 - 3520* 
sqrt(c**2 - d**2*x**2)*c**3*d**4*x**4 - 315*sqrt(c**2 - d**2*x**2)*c**2*d* 
*5*x**5 - 70*a*c**5*d**2 + 5*a*c**4*d**3*x - 520*a*c**3*d**4*x**2 - 170*a* 
c**2*d**5*x**3 - 170*a*c*d**6*x**4 - 35*a*d**7*x**5 + 5*b*c**5*d**2*x + 18 
0*b*c**4*d**3*x**2 - 170*b*c**3*d**4*x**3 + 180*b*c**2*d**5*x**4 - 35*b*c* 
d**6*x**5 + 490*c**8 - 3217*c**7*d*x - 19499*c**6*d**2*x**2 - 35186*c**5*d 
**3*x**3 - 28228*c**4*d**4*x**4 - 9149*c**3*d**5*x**5 - 315*c**2*d**6*x...