\(\int \frac {(c^2-d^2 x^2)^{3/2} (A+B x+C x^2+D x^3)}{(c+d x)^6} \, dx\) [149]

Optimal result

Integrand size = 39, antiderivative size = 247 \[ \int \frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^6} \, dx=-\frac {D \sqrt {c^2-d^2 x^2}}{d^4}+\frac {2 (C d-5 c D) \sqrt {c^2-d^2 x^2}}{d^4 (c+d x)}-\frac {2 (C d-3 c D) \left (c^2-d^2 x^2\right )^{3/2}}{3 d^4 (c+d x)^3}-\frac {\left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \left (c^2-d^2 x^2\right )^{5/2}}{7 c d^4 (c+d x)^6}+\frac {\left (13 c^2 C d-6 B c d^2-A d^3-20 c^3 D\right ) \left (c^2-d^2 x^2\right )^{5/2}}{35 c^2 d^4 (c+d x)^5}+\frac {(C d-6 c D) \arctan \left (\frac {d x}{\sqrt {c^2-d^2 x^2}}\right )}{d^4} \] Output:

-D*(-d^2*x^2+c^2)^(1/2)/d^4+2*(C*d-5*D*c)*(-d^2*x^2+c^2)^(1/2)/d^4/(d*x+c) 
-2/3*(C*d-3*D*c)*(-d^2*x^2+c^2)^(3/2)/d^4/(d*x+c)^3-1/7*(A*d^3-B*c*d^2+C*c 
^2*d-D*c^3)*(-d^2*x^2+c^2)^(5/2)/c/d^4/(d*x+c)^6+1/35*(-A*d^3-6*B*c*d^2+13 
*C*c^2*d-20*D*c^3)*(-d^2*x^2+c^2)^(5/2)/c^2/d^4/(d*x+c)^5+(C*d-6*D*c)*arct 
an(d*x/(-d^2*x^2+c^2)^(1/2))/d^4
 

Mathematica [A] (verified)

Time = 2.05 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.82 \[ \int \frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^6} \, dx=-\frac {\frac {\sqrt {c^2-d^2 x^2} \left (990 c^6 D+3 A d^6 x^3+6 c d^5 x^2 (2 A+3 B x)+c^5 (-164 C d+3330 d D x)+c^4 d^2 (3 B+29 x (-19 C+135 D x))+c^2 d^4 x \left (-33 A+x \left (-33 B-319 C x+105 D x^2\right )\right )+2 c^3 d^3 \left (9 A+x \left (6 B-323 C x+870 D x^2\right )\right )\right )}{c^2 (c+d x)^4}+210 (C d-6 c D) \arctan \left (\frac {d x}{\sqrt {c^2}-\sqrt {c^2-d^2 x^2}}\right )}{105 d^4} \] Input:

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

Output:

-1/105*((Sqrt[c^2 - d^2*x^2]*(990*c^6*D + 3*A*d^6*x^3 + 6*c*d^5*x^2*(2*A + 
 3*B*x) + c^5*(-164*C*d + 3330*d*D*x) + c^4*d^2*(3*B + 29*x*(-19*C + 135*D 
*x)) + c^2*d^4*x*(-33*A + x*(-33*B - 319*C*x + 105*D*x^2)) + 2*c^3*d^3*(9* 
A + x*(6*B - 323*C*x + 870*D*x^2))))/(c^2*(c + d*x)^4) + 210*(C*d - 6*c*D) 
*ArcTan[(d*x)/(Sqrt[c^2] - Sqrt[c^2 - d^2*x^2])])/d^4
 

Rubi [A] (verified)

Time = 1.14 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.20, 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.

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

\(\Big \downarrow \) 2170

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2168

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {d \left (c^2-d^2 x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{35 c^2 (c+d x)^5}-\frac {d \left (c^2-d^2 x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{7 c (c+d x)^6}+d \arctan \left (\frac {d x}{\sqrt {c^2-d^2 x^2}}\right ) (C d-6 c D)+\frac {d \left (c^2-d^2 x^2\right )^{5/2} \left (-B d^2-3 c^2 D+2 c C d\right )}{5 c (c+d x)^5}-\frac {2 d \left (c^2-d^2 x^2\right )^{3/2} (C d-6 c D)}{3 (c+d x)^3}+\frac {2 d \sqrt {c^2-d^2 x^2} (C d-6 c D)}{c+d x}}{d^5}-\frac {D \left (c^2-d^2 x^2\right )^{5/2}}{d^4 (c+d x)^4}\)

Input:

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

Output:

-((D*(c^2 - d^2*x^2)^(5/2))/(d^4*(c + d*x)^4)) + ((2*d*(C*d - 6*c*D)*Sqrt[ 
c^2 - d^2*x^2])/(c + d*x) - (2*d*(C*d - 6*c*D)*(c^2 - d^2*x^2)^(3/2))/(3*( 
c + d*x)^3) - (d*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(c^2 - d^2*x^2)^(5/2) 
)/(7*c*(c + d*x)^6) + (d*(2*c*C*d - B*d^2 - 3*c^2*D)*(c^2 - d^2*x^2)^(5/2) 
)/(5*c*(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)^(5/2))/(35*c^2*(c + d*x)^5) + d*(C*d - 6*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. \(717\) vs. \(2(229)=458\).

Time = 0.58 (sec) , antiderivative size = 718, normalized size of antiderivative = 2.91

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 {5}{2}}}{c d \left (x +\frac {c}{d}\right )^{3}}-\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 {5}{2}}}{c d \left (x +\frac {c}{d}\right )^{2}}+\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 {3}{2}}}{3}+c d \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 )\right )}{c}\right )}{c}\right )}{d^{6}}+\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 {5}{2}}}{3 c d \left (x +\frac {c}{d}\right )^{4}}-\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 {5}{2}}}{c d \left (x +\frac {c}{d}\right )^{3}}-\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 {5}{2}}}{c d \left (x +\frac {c}{d}\right )^{2}}+\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 {3}{2}}}{3}+c d \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 )\right )}{c}\right )}{c}\right )}{3 c}\right )}{d^{7}}-\frac {\left (B \,d^{2}-2 C c d +3 D c^{2}\right ) \left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {5}{2}}}{5 d^{9} c \left (x +\frac {c}{d}\right )^{5}}+\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 {5}{2}}}{7 c d \left (x +\frac {c}{d}\right )^{6}}-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {5}{2}}}{35 c^{2} \left (x +\frac {c}{d}\right )^{5}}\right )}{d^{9}}\)	\(718\)

Input:

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

Output:

D/d^6*(-1/c/d/(x+c/d)^3*(-d^2*(x+c/d)^2+2*c*d*(x+c/d))^(5/2)-2*d/c*(1/c/d/ 
(x+c/d)^2*(-d^2*(x+c/d)^2+2*c*d*(x+c/d))^(5/2)+3*d/c*(1/3*(-d^2*(x+c/d)^2+ 
2*c*d*(x+c/d))^(3/2)+c*d*(-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^7*(-1/3/c/d/(x+c/d)^4*(-d^2*(x 
+c/d)^2+2*c*d*(x+c/d))^(5/2)-1/3*d/c*(-1/c/d/(x+c/d)^3*(-d^2*(x+c/d)^2+2*c 
*d*(x+c/d))^(5/2)-2*d/c*(1/c/d/(x+c/d)^2*(-d^2*(x+c/d)^2+2*c*d*(x+c/d))^(5 
/2)+3*d/c*(1/3*(-d^2*(x+c/d)^2+2*c*d*(x+c/d))^(3/2)+c*d*(-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)*ar 
ctan((d^2)^(1/2)*x/(-d^2*(x+c/d)^2+2*c*d*(x+c/d))^(1/2)))))))-1/5*(B*d^2-2 
*C*c*d+3*D*c^2)/d^9/c/(x+c/d)^5*(-d^2*(x+c/d)^2+2*c*d*(x+c/d))^(5/2)+(A*d^ 
3-B*c*d^2+C*c^2*d-D*c^3)/d^9*(-1/7/c/d/(x+c/d)^6*(-d^2*(x+c/d)^2+2*c*d*(x+ 
c/d))^(5/2)-1/35/c^2/(x+c/d)^5*(-d^2*(x+c/d)^2+2*c*d*(x+c/d))^(5/2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 546 vs. \(2 (232) = 464\).

Time = 0.17 (sec) , antiderivative size = 546, normalized size of antiderivative = 2.21 \[ \int \frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^6} \, dx=-\frac {990 \, D c^{7} - 164 \, C c^{6} d + 3 \, B c^{5} d^{2} + 18 \, A c^{4} d^{3} + {\left (990 \, D c^{3} d^{4} - 164 \, C c^{2} d^{5} + 3 \, B c d^{6} + 18 \, A d^{7}\right )} x^{4} + 4 \, {\left (990 \, D c^{4} d^{3} - 164 \, C c^{3} d^{4} + 3 \, B c^{2} d^{5} + 18 \, A c d^{6}\right )} x^{3} + 6 \, {\left (990 \, D c^{5} d^{2} - 164 \, C c^{4} d^{3} + 3 \, B c^{3} d^{4} + 18 \, A c^{2} d^{5}\right )} x^{2} + 4 \, {\left (990 \, D c^{6} d - 164 \, C c^{5} d^{2} + 3 \, B c^{4} d^{3} + 18 \, A c^{3} d^{4}\right )} x - 210 \, {\left (6 \, D c^{7} - C c^{6} d + {\left (6 \, D c^{3} d^{4} - C c^{2} d^{5}\right )} x^{4} + 4 \, {\left (6 \, D c^{4} d^{3} - C c^{3} d^{4}\right )} x^{3} + 6 \, {\left (6 \, D c^{5} d^{2} - C c^{4} d^{3}\right )} x^{2} + 4 \, {\left (6 \, D c^{6} d - C c^{5} d^{2}\right )} x\right )} \arctan \left (-\frac {c - \sqrt {-d^{2} x^{2} + c^{2}}}{d x}\right ) + {\left (105 \, D c^{2} d^{4} x^{4} + 990 \, D c^{6} - 164 \, C c^{5} d + 3 \, B c^{4} d^{2} + 18 \, A c^{3} d^{3} + {\left (1740 \, D c^{3} d^{3} - 319 \, C c^{2} d^{4} + 18 \, B c d^{5} + 3 \, A d^{6}\right )} x^{3} + {\left (3915 \, D c^{4} d^{2} - 646 \, C c^{3} d^{3} - 33 \, B c^{2} d^{4} + 12 \, A c d^{5}\right )} x^{2} + {\left (3330 \, D c^{5} d - 551 \, C c^{4} d^{2} + 12 \, B c^{3} d^{3} - 33 \, A c^{2} d^{4}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{105 \, {\left (c^{2} d^{8} x^{4} + 4 \, c^{3} d^{7} x^{3} + 6 \, c^{4} d^{6} x^{2} + 4 \, c^{5} d^{5} x + c^{6} d^{4}\right )}} \] Input:

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

Output:

-1/105*(990*D*c^7 - 164*C*c^6*d + 3*B*c^5*d^2 + 18*A*c^4*d^3 + (990*D*c^3* 
d^4 - 164*C*c^2*d^5 + 3*B*c*d^6 + 18*A*d^7)*x^4 + 4*(990*D*c^4*d^3 - 164*C 
*c^3*d^4 + 3*B*c^2*d^5 + 18*A*c*d^6)*x^3 + 6*(990*D*c^5*d^2 - 164*C*c^4*d^ 
3 + 3*B*c^3*d^4 + 18*A*c^2*d^5)*x^2 + 4*(990*D*c^6*d - 164*C*c^5*d^2 + 3*B 
*c^4*d^3 + 18*A*c^3*d^4)*x - 210*(6*D*c^7 - C*c^6*d + (6*D*c^3*d^4 - C*c^2 
*d^5)*x^4 + 4*(6*D*c^4*d^3 - C*c^3*d^4)*x^3 + 6*(6*D*c^5*d^2 - C*c^4*d^3)* 
x^2 + 4*(6*D*c^6*d - C*c^5*d^2)*x)*arctan(-(c - sqrt(-d^2*x^2 + c^2))/(d*x 
)) + (105*D*c^2*d^4*x^4 + 990*D*c^6 - 164*C*c^5*d + 3*B*c^4*d^2 + 18*A*c^3 
*d^3 + (1740*D*c^3*d^3 - 319*C*c^2*d^4 + 18*B*c*d^5 + 3*A*d^6)*x^3 + (3915 
*D*c^4*d^2 - 646*C*c^3*d^3 - 33*B*c^2*d^4 + 12*A*c*d^5)*x^2 + (3330*D*c^5* 
d - 551*C*c^4*d^2 + 12*B*c^3*d^3 - 33*A*c^2*d^4)*x)*sqrt(-d^2*x^2 + c^2))/ 
(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)
 

Sympy [F]

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

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

Output:

Integral((-(-c + d*x)*(c + d*x))**(3/2)*(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. 1755 vs. \(2 (232) = 464\).

Time = 0.15 (sec) , antiderivative size = 1755, normalized size of antiderivative = 7.11 \[ \int \frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^6} \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*(-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) - 3/7*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)^(3/2)*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) - 3*(-d^2*x^2 + c^2)^(3/2)*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) + 3/7*sqrt(-d^2*x^ 
2 + c^2)*C*c^3/(d^7*x^4 + 4*c*d^6*x^3 + 6*c^2*d^5*x^2 + 4*c^3*d^4*x + c^4* 
d^3) + 51/14*sqrt(-d^2*x^2 + c^2)*D*c^3/(d^7*x^3 + 3*c*d^6*x^2 + 3*c^2*d^5 
*x + c^3*d^4) + 1/35*sqrt(-d^2*x^2 + c^2)*D*c^3/(c*d^6*x^2 + 2*c^2*d^5*x + 
 c^3*d^4) + 1/35*sqrt(-d^2*x^2 + c^2)*D*c^3/(c^2*d^5*x + c^3*d^4) + 1/2*(- 
d^2*x^2 + c^2)^(3/2)*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) + 2*(-d^2*x^2 + c^2)^(3/2)*C*c/(d^7*x^4 + 
 4*c*d^6*x^3 + 6*c^2*d^5*x^2 + 4*c^3*d^4*x + c^4*d^3) + (-d^2*x^2 + c^2)^( 
3/2)*D*c/(d^7*x^3 + 3*c*d^6*x^2 + 3*c^2*d^5*x + c^3*d^4) - 3/7*sqrt(-d^2*x 
^2 + c^2)*B*c^2/(d^6*x^4 + 4*c*d^5*x^3 + 6*c^2*d^4*x^2 + 4*c^3*d^3*x + c^4 
*d^2) - 171/70*sqrt(-d^2*x^2 + c^2)*C*c^2/(d^6*x^3 + 3*c*d^5*x^2 + 3*c^2*d 
^4*x + c^3*d^3) - 1/35*sqrt(-d^2*x^2 + c^2)*C*c^2/(c*d^5*x^2 + 2*c^2*d^4*x 
 + c^3*d^3) - 1/35*sqrt(-d^2*x^2 + c^2)*C*c^2/(c^2*d^4*x + c^3*d^3) + 7/5* 
sqrt(-d^2*x^2 + c^2)*D*c^2/(d^6*x^2 + 2*c*d^5*x + c^2*d^4) - 3/5*sqrt(-d^2 
*x^2 + c^2)*D*c^2/(c*d^5*x + c^2*d^4) - 1/2*(-d^2*x^2 + c^2)^(3/2)*A/(d...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 885 vs. \(2 (232) = 464\).

Time = 0.15 (sec) , antiderivative size = 885, normalized size of antiderivative = 3.58 \[ \int \frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^6} \, dx =\text {Too large to display} \] Input:

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

Output:

-(6*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 + 2/105*(885*D*c^3 - 164*C*c^2*d + 3*B*c*d^2 + 18*A*d^3 + 21*(c 
*d + sqrt(-d^2*x^2 + c^2)*abs(d))*B*c/x + 5565*(c*d + sqrt(-d^2*x^2 + c^2) 
*abs(d))*D*c^3/(d^2*x) - 1043*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))*C*c^2/(d 
*x) + 21*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))*A*d/x + 14280*(c*d + sqrt(-d^ 
2*x^2 + c^2)*abs(d))^2*D*c^3/(d^4*x^2) - 2709*(c*d + sqrt(-d^2*x^2 + c^2)* 
abs(d))^2*C*c^2/(d^3*x^2) - 42*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^2*B*c/( 
d^2*x^2) + 273*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^2*A/(d*x^2) + 18690*(c* 
d + sqrt(-d^2*x^2 + c^2)*abs(d))^3*D*c^3/(d^6*x^3) - 3710*(c*d + sqrt(-d^2 
*x^2 + c^2)*abs(d))^3*C*c^2/(d^5*x^3) + 210*(c*d + sqrt(-d^2*x^2 + c^2)*ab 
s(d))^3*B*c/(d^4*x^3) + 210*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^3*A/(d^3*x 
^3) + 12285*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^4*D*c^3/(d^8*x^4) - 2030*( 
c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^4*C*c^2/(d^7*x^4) - 105*(c*d + sqrt(-d^ 
2*x^2 + c^2)*abs(d))^4*B*c/(d^6*x^4) + 420*(c*d + sqrt(-d^2*x^2 + c^2)*abs 
(d))^4*A/(d^5*x^4) + 4305*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^5*D*c^3/(d^1 
0*x^5) - 735*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^5*C*c^2/(d^9*x^5) + 105*( 
c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^5*B*c/(d^8*x^5) + 105*(c*d + sqrt(-d^2* 
x^2 + c^2)*abs(d))^5*A/(d^7*x^5) + 630*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d)) 
^6*D*c^3/(d^12*x^6) - 105*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^6*C*c^2/(d^1 
1*x^6) + 105*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^6*A/(d^9*x^6))/(c^2*d^...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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