\(\int \frac {A+B x+C x^2+D x^3}{(c+d x)^2 (c^2-d^2 x^2)^{3/2}} \, dx\) [182]

Optimal result

Integrand size = 39, antiderivative size = 181 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^2 \left (c^2-d^2 x^2\right )^{3/2}} \, dx=-\frac {c^2 C d-B c d^2+A d^3-c^3 D}{5 c d^4 (c+d x)^2 \sqrt {c^2-d^2 x^2}}+\frac {7 c^2 C d-2 B c d^2-3 A d^3-12 c^3 D}{15 c^2 d^4 (c+d x) \sqrt {c^2-d^2 x^2}}+\frac {15 c^4 D+d \left (c^2 C d+4 B c d^2+6 A d^3-6 c^3 D\right ) x}{15 c^4 d^4 \sqrt {c^2-d^2 x^2}} \] Output:

-1/5*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)/c/d^4/(d*x+c)^2/(-d^2*x^2+c^2)^(1/2)+1/ 
15*(-3*A*d^3-2*B*c*d^2+7*C*c^2*d-12*D*c^3)/c^2/d^4/(d*x+c)/(-d^2*x^2+c^2)^ 
(1/2)+1/15*(15*c^4*D+d*(6*A*d^3+4*B*c*d^2+C*c^2*d-6*D*c^3)*x)/c^4/d^4/(-d^ 
2*x^2+c^2)^(1/2)
 

Mathematica [A] (verified)

Time = 0.94 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.82 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^2 \left (c^2-d^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {c^2-d^2 x^2} \left (6 c^6 D+6 A d^6 x^3+4 c d^5 x^2 (3 A+B x)+4 c^5 d (C+3 D x)+c^2 d^4 x (3 A+x (8 B+C x))+c^4 d^2 (B+x (8 C+3 D x))+2 c^3 d^3 (-3 A+x (B+x (C-3 D x)))\right )}{15 c^4 d^4 (c-d x) (c+d x)^3} \] Input:

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

Output:

(Sqrt[c^2 - d^2*x^2]*(6*c^6*D + 6*A*d^6*x^3 + 4*c*d^5*x^2*(3*A + B*x) + 4* 
c^5*d*(C + 3*D*x) + c^2*d^4*x*(3*A + x*(8*B + C*x)) + c^4*d^2*(B + x*(8*C 
+ 3*D*x)) + 2*c^3*d^3*(-3*A + x*(B + x*(C - 3*D*x)))))/(15*c^4*d^4*(c - d* 
x)*(c + d*x)^3)
 

Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.22, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2170, 2170, 27, 671, 470, 208}

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.

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

\(\Big \downarrow \) 2170

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

\(\Big \downarrow \) 2170

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

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{2} d^2 \int \frac {-2 D c^3+C d c^2+2 A d^3-d^2 (c C-2 B d) x}{(c+d x)^2 \left (c^2-d^2 x^2\right )^{3/2}}dx+\frac {d (C d-2 c D)}{2 (c+d x) \sqrt {c^2-d^2 x^2}}}{d^5}+\frac {D}{d^4 \sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 671

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

\(\Big \downarrow \) 470

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

\(\Big \downarrow \) 208

\(\displaystyle \frac {\frac {1}{2} d^2 \left (\frac {\left (\frac {2 x}{3 c^3 \sqrt {c^2-d^2 x^2}}-\frac {1}{3 c d (c+d x) \sqrt {c^2-d^2 x^2}}\right ) \left (6 A d^3+4 B c d^2-6 c^3 D+c^2 C d\right )}{5 c}-\frac {2 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 c d (c+d x)^2 \sqrt {c^2-d^2 x^2}}\right )+\frac {d (C d-2 c D)}{2 (c+d x) \sqrt {c^2-d^2 x^2}}}{d^5}+\frac {D}{d^4 \sqrt {c^2-d^2 x^2}}\)

Input:

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

Output:

D/(d^4*Sqrt[c^2 - d^2*x^2]) + ((d*(C*d - 2*c*D))/(2*(c + d*x)*Sqrt[c^2 - d 
^2*x^2]) + (d^2*((-2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(5*c*d*(c + d*x) 
^2*Sqrt[c^2 - d^2*x^2]) + ((c^2*C*d + 4*B*c*d^2 + 6*A*d^3 - 6*c^3*D)*((2*x 
)/(3*c^3*Sqrt[c^2 - d^2*x^2]) - 1/(3*c*d*(c + d*x)*Sqrt[c^2 - d^2*x^2])))/ 
(5*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 208

Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), 
x] /; FreeQ[{a, b}, x]
 

rule 470

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[(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, p}, x] && EqQ[b*c^2 + a*d^2, 0] && LtQ[n, 0] && NeQ[n + 
 p + 1, 0] && IntegerQ[2*p]
 

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 = 0.45 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.08

method	result	size
gosper	\(-\frac {\left (-d x +c \right ) \left (-6 A \,d^{6} x^{3}-4 B c \,d^{5} x^{3}-C \,c^{2} d^{4} x^{3}+6 D c^{3} d^{3} x^{3}-12 A c \,d^{5} x^{2}-8 B \,c^{2} d^{4} x^{2}-2 C \,c^{3} d^{3} x^{2}-3 D c^{4} d^{2} x^{2}-3 A \,c^{2} d^{4} x -2 B \,c^{3} d^{3} x -8 C \,c^{4} d^{2} x -12 D c^{5} d x +6 A \,c^{3} d^{3}-B \,c^{4} d^{2}-4 C \,c^{5} d -6 D c^{6}\right )}{15 \left (d x +c \right ) c^{4} d^{4} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}\)	\(195\)
orering	\(-\frac {\left (-d x +c \right ) \left (-6 A \,d^{6} x^{3}-4 B c \,d^{5} x^{3}-C \,c^{2} d^{4} x^{3}+6 D c^{3} d^{3} x^{3}-12 A c \,d^{5} x^{2}-8 B \,c^{2} d^{4} x^{2}-2 C \,c^{3} d^{3} x^{2}-3 D c^{4} d^{2} x^{2}-3 A \,c^{2} d^{4} x -2 B \,c^{3} d^{3} x -8 C \,c^{4} d^{2} x -12 D c^{5} d x +6 A \,c^{3} d^{3}-B \,c^{4} d^{2}-4 C \,c^{5} d -6 D c^{6}\right )}{15 \left (d x +c \right ) c^{4} d^{4} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}\)	\(195\)
trager	\(-\frac {\left (-6 A \,d^{6} x^{3}-4 B c \,d^{5} x^{3}-C \,c^{2} d^{4} x^{3}+6 D c^{3} d^{3} x^{3}-12 A c \,d^{5} x^{2}-8 B \,c^{2} d^{4} x^{2}-2 C \,c^{3} d^{3} x^{2}-3 D c^{4} d^{2} x^{2}-3 A \,c^{2} d^{4} x -2 B \,c^{3} d^{3} x -8 C \,c^{4} d^{2} x -12 D c^{5} d x +6 A \,c^{3} d^{3}-B \,c^{4} d^{2}-4 C \,c^{5} d -6 D c^{6}\right ) \sqrt {-d^{2} x^{2}+c^{2}}}{15 c^{4} d^{4} \left (d x +c \right )^{3} \left (-d x +c \right )}\)	\(197\)
default	\(\frac {\frac {C d x}{c^{2} \sqrt {-d^{2} x^{2}+c^{2}}}+\frac {D}{d \sqrt {-d^{2} x^{2}+c^{2}}}-\frac {2 D x}{c \sqrt {-d^{2} x^{2}+c^{2}}}}{d^{3}}+\frac {\left (B \,d^{2}-2 C c d +3 D c^{2}\right ) \left (-\frac {1}{3 c d \left (x +\frac {c}{d}\right ) \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}-\frac {-2 d^{2} \left (x +\frac {c}{d}\right )+2 c d}{3 d \,c^{3} \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}\right )}{d^{4}}+\frac {\left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right ) \left (-\frac {1}{5 c d \left (x +\frac {c}{d}\right )^{2} \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}+\frac {3 d \left (-\frac {1}{3 c d \left (x +\frac {c}{d}\right ) \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}-\frac {-2 d^{2} \left (x +\frac {c}{d}\right )+2 c d}{3 d \,c^{3} \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}\right )}{5 c}\right )}{d^{5}}\)	\(368\)

Input:

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

Output:

-1/15*(-d*x+c)*(-6*A*d^6*x^3-4*B*c*d^5*x^3-C*c^2*d^4*x^3+6*D*c^3*d^3*x^3-1 
2*A*c*d^5*x^2-8*B*c^2*d^4*x^2-2*C*c^3*d^3*x^2-3*D*c^4*d^2*x^2-3*A*c^2*d^4* 
x-2*B*c^3*d^3*x-8*C*c^4*d^2*x-12*D*c^5*d*x+6*A*c^3*d^3-B*c^4*d^2-4*C*c^5*d 
-6*D*c^6)/(d*x+c)/c^4/d^4/(-d^2*x^2+c^2)^(3/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 345 vs. \(2 (171) = 342\).

Time = 0.11 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.91 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^2 \left (c^2-d^2 x^2\right )^{3/2}} \, dx=-\frac {6 \, D c^{7} + 4 \, C c^{6} d + B c^{5} d^{2} - 6 \, A c^{4} d^{3} - {\left (6 \, D c^{3} d^{4} + 4 \, C c^{2} d^{5} + B c d^{6} - 6 \, A d^{7}\right )} x^{4} - 2 \, {\left (6 \, D c^{4} d^{3} + 4 \, C c^{3} d^{4} + B c^{2} d^{5} - 6 \, A c d^{6}\right )} x^{3} + 2 \, {\left (6 \, D c^{6} d + 4 \, C c^{5} d^{2} + B c^{4} d^{3} - 6 \, A c^{3} d^{4}\right )} x + {\left (6 \, D c^{6} + 4 \, C c^{5} d + B c^{4} d^{2} - 6 \, A c^{3} d^{3} - {\left (6 \, D c^{3} d^{3} - C c^{2} d^{4} - 4 \, B c d^{5} - 6 \, A d^{6}\right )} x^{3} + {\left (3 \, D c^{4} d^{2} + 2 \, C c^{3} d^{3} + 8 \, B c^{2} d^{4} + 12 \, A c d^{5}\right )} x^{2} + {\left (12 \, D c^{5} d + 8 \, C c^{4} d^{2} + 2 \, B c^{3} d^{3} + 3 \, A c^{2} d^{4}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{15 \, {\left (c^{4} d^{8} x^{4} + 2 \, c^{5} d^{7} x^{3} - 2 \, c^{7} d^{5} x - c^{8} d^{4}\right )}} \] Input:

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

Output:

-1/15*(6*D*c^7 + 4*C*c^6*d + B*c^5*d^2 - 6*A*c^4*d^3 - (6*D*c^3*d^4 + 4*C* 
c^2*d^5 + B*c*d^6 - 6*A*d^7)*x^4 - 2*(6*D*c^4*d^3 + 4*C*c^3*d^4 + B*c^2*d^ 
5 - 6*A*c*d^6)*x^3 + 2*(6*D*c^6*d + 4*C*c^5*d^2 + B*c^4*d^3 - 6*A*c^3*d^4) 
*x + (6*D*c^6 + 4*C*c^5*d + B*c^4*d^2 - 6*A*c^3*d^3 - (6*D*c^3*d^3 - C*c^2 
*d^4 - 4*B*c*d^5 - 6*A*d^6)*x^3 + (3*D*c^4*d^2 + 2*C*c^3*d^3 + 8*B*c^2*d^4 
 + 12*A*c*d^5)*x^2 + (12*D*c^5*d + 8*C*c^4*d^2 + 2*B*c^3*d^3 + 3*A*c^2*d^4 
)*x)*sqrt(-d^2*x^2 + c^2))/(c^4*d^8*x^4 + 2*c^5*d^7*x^3 - 2*c^7*d^5*x - c^ 
8*d^4)
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 752 vs. \(2 (171) = 342\).

Time = 0.06 (sec) , antiderivative size = 752, normalized size of antiderivative = 4.15 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^2 \left (c^2-d^2 x^2\right )^{3/2}} \, dx=\frac {D c^{3}}{5 \, {\left (\sqrt {-d^{2} x^{2} + c^{2}} c d^{6} x^{2} + 2 \, \sqrt {-d^{2} x^{2} + c^{2}} c^{2} d^{5} x + \sqrt {-d^{2} x^{2} + c^{2}} c^{3} d^{4}\right )}} + \frac {D c^{3}}{5 \, {\left (\sqrt {-d^{2} x^{2} + c^{2}} c^{2} d^{5} x + \sqrt {-d^{2} x^{2} + c^{2}} c^{3} d^{4}\right )}} - \frac {C c^{2}}{5 \, {\left (\sqrt {-d^{2} x^{2} + c^{2}} c d^{5} x^{2} + 2 \, \sqrt {-d^{2} x^{2} + c^{2}} c^{2} d^{4} x + \sqrt {-d^{2} x^{2} + c^{2}} c^{3} d^{3}\right )}} - \frac {C c^{2}}{5 \, {\left (\sqrt {-d^{2} x^{2} + c^{2}} c^{2} d^{4} x + \sqrt {-d^{2} x^{2} + c^{2}} c^{3} d^{3}\right )}} - \frac {D c^{2}}{\sqrt {-d^{2} x^{2} + c^{2}} c d^{5} x + \sqrt {-d^{2} x^{2} + c^{2}} c^{2} d^{4}} + \frac {B c}{5 \, {\left (\sqrt {-d^{2} x^{2} + c^{2}} c d^{4} x^{2} + 2 \, \sqrt {-d^{2} x^{2} + c^{2}} c^{2} d^{3} x + \sqrt {-d^{2} x^{2} + c^{2}} c^{3} d^{2}\right )}} + \frac {B c}{5 \, {\left (\sqrt {-d^{2} x^{2} + c^{2}} c^{2} d^{3} x + \sqrt {-d^{2} x^{2} + c^{2}} c^{3} d^{2}\right )}} + \frac {2 \, C c}{3 \, {\left (\sqrt {-d^{2} x^{2} + c^{2}} c d^{4} x + \sqrt {-d^{2} x^{2} + c^{2}} c^{2} d^{3}\right )}} - \frac {A}{5 \, {\left (\sqrt {-d^{2} x^{2} + c^{2}} c d^{3} x^{2} + 2 \, \sqrt {-d^{2} x^{2} + c^{2}} c^{2} d^{2} x + \sqrt {-d^{2} x^{2} + c^{2}} c^{3} d\right )}} - \frac {A}{5 \, {\left (\sqrt {-d^{2} x^{2} + c^{2}} c^{2} d^{2} x + \sqrt {-d^{2} x^{2} + c^{2}} c^{3} d\right )}} - \frac {B}{3 \, {\left (\sqrt {-d^{2} x^{2} + c^{2}} c d^{3} x + \sqrt {-d^{2} x^{2} + c^{2}} c^{2} d^{2}\right )}} + \frac {2 \, A x}{5 \, \sqrt {-d^{2} x^{2} + c^{2}} c^{4}} - \frac {2 \, D x}{5 \, \sqrt {-d^{2} x^{2} + c^{2}} c d^{3}} + \frac {C x}{15 \, \sqrt {-d^{2} x^{2} + c^{2}} c^{2} d^{2}} + \frac {4 \, B x}{15 \, \sqrt {-d^{2} x^{2} + c^{2}} c^{3} d} + \frac {D}{\sqrt {-d^{2} x^{2} + c^{2}} d^{4}} \] Input:

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

Output:

1/5*D*c^3/(sqrt(-d^2*x^2 + c^2)*c*d^6*x^2 + 2*sqrt(-d^2*x^2 + c^2)*c^2*d^5 
*x + sqrt(-d^2*x^2 + c^2)*c^3*d^4) + 1/5*D*c^3/(sqrt(-d^2*x^2 + c^2)*c^2*d 
^5*x + sqrt(-d^2*x^2 + c^2)*c^3*d^4) - 1/5*C*c^2/(sqrt(-d^2*x^2 + c^2)*c*d 
^5*x^2 + 2*sqrt(-d^2*x^2 + c^2)*c^2*d^4*x + sqrt(-d^2*x^2 + c^2)*c^3*d^3) 
- 1/5*C*c^2/(sqrt(-d^2*x^2 + c^2)*c^2*d^4*x + sqrt(-d^2*x^2 + c^2)*c^3*d^3 
) - D*c^2/(sqrt(-d^2*x^2 + c^2)*c*d^5*x + sqrt(-d^2*x^2 + c^2)*c^2*d^4) + 
1/5*B*c/(sqrt(-d^2*x^2 + c^2)*c*d^4*x^2 + 2*sqrt(-d^2*x^2 + c^2)*c^2*d^3*x 
 + sqrt(-d^2*x^2 + c^2)*c^3*d^2) + 1/5*B*c/(sqrt(-d^2*x^2 + c^2)*c^2*d^3*x 
 + sqrt(-d^2*x^2 + c^2)*c^3*d^2) + 2/3*C*c/(sqrt(-d^2*x^2 + c^2)*c*d^4*x + 
 sqrt(-d^2*x^2 + c^2)*c^2*d^3) - 1/5*A/(sqrt(-d^2*x^2 + c^2)*c*d^3*x^2 + 2 
*sqrt(-d^2*x^2 + c^2)*c^2*d^2*x + sqrt(-d^2*x^2 + c^2)*c^3*d) - 1/5*A/(sqr 
t(-d^2*x^2 + c^2)*c^2*d^2*x + sqrt(-d^2*x^2 + c^2)*c^3*d) - 1/3*B/(sqrt(-d 
^2*x^2 + c^2)*c*d^3*x + sqrt(-d^2*x^2 + c^2)*c^2*d^2) + 2/5*A*x/(sqrt(-d^2 
*x^2 + c^2)*c^4) - 2/5*D*x/(sqrt(-d^2*x^2 + c^2)*c*d^3) + 1/15*C*x/(sqrt(- 
d^2*x^2 + c^2)*c^2*d^2) + 4/15*B*x/(sqrt(-d^2*x^2 + c^2)*c^3*d) + D/(sqrt( 
-d^2*x^2 + c^2)*d^4)
                                                                                    
                                                                                    
 

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.17 (sec) , antiderivative size = 577, normalized size of antiderivative = 3.19 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^2 \left (c^2-d^2 x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

1/120*(8*(-6*I*D*c^3 + I*C*c^2*d + 4*I*B*c*d^2 + 6*I*A*d^3)*sgn(1/(d*x + c 
))*sgn(d)/(c^4*d^3) + 15*(D*c^3 + C*c^2*d + B*c*d^2 + A*d^3)/(c^4*d^3*sqrt 
(2*c/(d*x + c) - 1)*sgn(1/(d*x + c))*sgn(d)) + (3*D*c^19*d^12*(2*c/(d*x + 
c) - 1)^(5/2)*sgn(1/(d*x + c))^4*sgn(d)^4 - 3*C*c^18*d^13*(2*c/(d*x + c) - 
 1)^(5/2)*sgn(1/(d*x + c))^4*sgn(d)^4 + 3*B*c^17*d^14*(2*c/(d*x + c) - 1)^ 
(5/2)*sgn(1/(d*x + c))^4*sgn(d)^4 - 3*A*c^16*d^15*(2*c/(d*x + c) - 1)^(5/2 
)*sgn(1/(d*x + c))^4*sgn(d)^4 - 15*D*c^19*d^12*(2*c/(d*x + c) - 1)^(3/2)*s 
gn(1/(d*x + c))^4*sgn(d)^4 + 5*C*c^18*d^13*(2*c/(d*x + c) - 1)^(3/2)*sgn(1 
/(d*x + c))^4*sgn(d)^4 + 5*B*c^17*d^14*(2*c/(d*x + c) - 1)^(3/2)*sgn(1/(d* 
x + c))^4*sgn(d)^4 - 15*A*c^16*d^15*(2*c/(d*x + c) - 1)^(3/2)*sgn(1/(d*x + 
 c))^4*sgn(d)^4 + 45*D*c^19*d^12*sqrt(2*c/(d*x + c) - 1)*sgn(1/(d*x + c))^ 
4*sgn(d)^4 + 15*C*c^18*d^13*sqrt(2*c/(d*x + c) - 1)*sgn(1/(d*x + c))^4*sgn 
(d)^4 - 15*B*c^17*d^14*sqrt(2*c/(d*x + c) - 1)*sgn(1/(d*x + c))^4*sgn(d)^4 
 - 45*A*c^16*d^15*sqrt(2*c/(d*x + c) - 1)*sgn(1/(d*x + c))^4*sgn(d)^4)/(c^ 
20*d^15*sgn(1/(d*x + c))^5*sgn(d)^5))/abs(d)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.91 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^2 \left (c^2-d^2 x^2\right )^{3/2}} \, dx=\frac {3 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{2} d^{2}+6 \sqrt {-d^{2} x^{2}+c^{2}}\, a c \,d^{3} x +3 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,d^{4} x^{2}+2 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{3} d +4 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{2} d^{2} x +2 \sqrt {-d^{2} x^{2}+c^{2}}\, b c \,d^{3} x^{2}-10 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{5}-20 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{4} d x -10 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{3} d^{2} x^{2}-12 a \,c^{3} d^{2}+6 a \,c^{2} d^{3} x +24 a c \,d^{4} x^{2}+12 a \,d^{5} x^{3}+2 b \,c^{4} d +4 b \,c^{3} d^{2} x +16 b \,c^{2} d^{3} x^{2}+8 b c \,d^{4} x^{3}+20 c^{6}+40 c^{5} d x +10 c^{4} d^{2} x^{2}-10 c^{3} d^{3} x^{3}}{30 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{4} d^{3} \left (d^{2} x^{2}+2 c d x +c^{2}\right )} \] Input:

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

Output:

(3*sqrt(c**2 - d**2*x**2)*a*c**2*d**2 + 6*sqrt(c**2 - d**2*x**2)*a*c*d**3* 
x + 3*sqrt(c**2 - d**2*x**2)*a*d**4*x**2 + 2*sqrt(c**2 - d**2*x**2)*b*c**3 
*d + 4*sqrt(c**2 - d**2*x**2)*b*c**2*d**2*x + 2*sqrt(c**2 - d**2*x**2)*b*c 
*d**3*x**2 - 10*sqrt(c**2 - d**2*x**2)*c**5 - 20*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 - 12*a*c**3*d**2 + 6*a*c 
**2*d**3*x + 24*a*c*d**4*x**2 + 12*a*d**5*x**3 + 2*b*c**4*d + 4*b*c**3*d** 
2*x + 16*b*c**2*d**3*x**2 + 8*b*c*d**4*x**3 + 20*c**6 + 40*c**5*d*x + 10*c 
**4*d**2*x**2 - 10*c**3*d**3*x**3)/(30*sqrt(c**2 - d**2*x**2)*c**4*d**3*(c 
**2 + 2*c*d*x + d**2*x**2))