\(\int \frac {A+B x+C x^2+D x^3}{(c+d x)^3 (c^2-d^2 x^2)^{5/2}} \, dx\) [192]

Optimal result

Integrand size = 39, antiderivative size = 286 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^3 \left (c^2-d^2 x^2\right )^{5/2}} \, dx=\frac {\left (4 c^2 C d+20 B c d^2+40 A d^3-7 c^3 D\right ) x}{315 c^5 d^3 \left (c^2-d^2 x^2\right )^{3/2}}-\frac {c^2 C d-B c d^2+A d^3-c^3 D}{9 c d^4 (c+d x)^3 \left (c^2-d^2 x^2\right )^{3/2}}+\frac {4 c^2 C d-B c d^2-2 A d^3-7 c^3 D}{21 c^2 d^4 (c+d x)^2 \left (c^2-d^2 x^2\right )^{3/2}}-\frac {c^2 C d+5 B c d^2+10 A d^3-28 c^3 D}{105 c^3 d^4 (c+d x) \left (c^2-d^2 x^2\right )^{3/2}}+\frac {2 \left (4 c^2 C d+20 B c d^2+40 A d^3-7 c^3 D\right ) x}{315 c^7 d^3 \sqrt {c^2-d^2 x^2}} \] Output:

1/315*(40*A*d^3+20*B*c*d^2+4*C*c^2*d-7*D*c^3)*x/c^5/d^3/(-d^2*x^2+c^2)^(3/ 
2)-1/9*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)/c/d^4/(d*x+c)^3/(-d^2*x^2+c^2)^(3/2)+ 
1/21*(-2*A*d^3-B*c*d^2+4*C*c^2*d-7*D*c^3)/c^2/d^4/(d*x+c)^2/(-d^2*x^2+c^2) 
^(3/2)-1/105*(10*A*d^3+5*B*c*d^2+C*c^2*d-28*D*c^3)/c^3/d^4/(d*x+c)/(-d^2*x 
^2+c^2)^(3/2)+2/315*(40*A*d^3+20*B*c*d^2+4*C*c^2*d-7*D*c^3)*x/c^7/d^3/(-d^ 
2*x^2+c^2)^(1/2)
 

Mathematica [A] (verified)

Time = 1.83 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.88 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^3 \left (c^2-d^2 x^2\right )^{5/2}} \, dx=\frac {\sqrt {c^2-d^2 x^2} \left (14 c^9 D-80 A d^9 x^6-40 c d^8 x^5 (6 A+B x)+c^8 (22 C d+42 d D x)-8 c^2 d^7 x^4 (15 A+x (15 B+C x))+c^7 d^2 (5 B+3 x (22 C+7 D x))+c^6 d^3 (-95 A+x (15 B+x (33 C-49 D x)))+c^5 d^4 x (30 A+x (165 B+7 x (4 C+3 D x)))+2 c^3 d^6 x^3 (140 A+x (-30 B+x (-12 C+7 D x)))+2 c^4 d^5 x^2 (165 A+x (70 B+3 x (-2 C+7 D x)))\right )}{315 c^7 d^4 (c-d x)^2 (c+d x)^5} \] Input:

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

Output:

(Sqrt[c^2 - d^2*x^2]*(14*c^9*D - 80*A*d^9*x^6 - 40*c*d^8*x^5*(6*A + B*x) + 
 c^8*(22*C*d + 42*d*D*x) - 8*c^2*d^7*x^4*(15*A + x*(15*B + C*x)) + c^7*d^2 
*(5*B + 3*x*(22*C + 7*D*x)) + c^6*d^3*(-95*A + x*(15*B + x*(33*C - 49*D*x) 
)) + c^5*d^4*x*(30*A + x*(165*B + 7*x*(4*C + 3*D*x))) + 2*c^3*d^6*x^3*(140 
*A + x*(-30*B + x*(-12*C + 7*D*x))) + 2*c^4*d^5*x^2*(165*A + x*(70*B + 3*x 
*(-2*C + 7*D*x)))))/(315*c^7*d^4*(c - d*x)^2*(c + d*x)^5)
 

Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.205, Rules used = {2170, 2170, 27, 671, 461, 470, 209, 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.

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

\(\Big \downarrow \) 2170

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

\(\Big \downarrow \) 2170

\(\displaystyle \frac {\frac {\int \frac {d^6 \left (-9 D c^3+8 C d c^2+20 A d^3-d \left (-11 D c^2+12 C d c-20 B d^2\right ) x\right )}{(c+d x)^3 \left (c^2-d^2 x^2\right )^{5/2}}dx}{5 d^4}+\frac {d (4 C d-7 c D)}{5 (c+d x)^2 \left (c^2-d^2 x^2\right )^{3/2}}}{4 d^5}+\frac {D}{4 d^4 (c+d x) \left (c^2-d^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{5} d^2 \int \frac {-9 D c^3+8 C d c^2+20 A d^3-d \left (-11 D c^2+12 C d c-20 B d^2\right ) x}{(c+d x)^3 \left (c^2-d^2 x^2\right )^{5/2}}dx+\frac {d (4 C d-7 c D)}{5 (c+d x)^2 \left (c^2-d^2 x^2\right )^{3/2}}}{4 d^5}+\frac {D}{4 d^4 (c+d x) \left (c^2-d^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 671

\(\displaystyle \frac {\frac {1}{5} d^2 \left (\frac {\left (40 A d^3+20 B c d^2-7 c^3 D+4 c^2 C d\right ) \int \frac {1}{(c+d x)^2 \left (c^2-d^2 x^2\right )^{5/2}}dx}{3 c}-\frac {20 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{9 c d (c+d x)^3 \left (c^2-d^2 x^2\right )^{3/2}}\right )+\frac {d (4 C d-7 c D)}{5 (c+d x)^2 \left (c^2-d^2 x^2\right )^{3/2}}}{4 d^5}+\frac {D}{4 d^4 (c+d x) \left (c^2-d^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 461

\(\displaystyle \frac {\frac {1}{5} d^2 \left (\frac {\left (40 A d^3+20 B c d^2-7 c^3 D+4 c^2 C d\right ) \left (\frac {5 \int \frac {1}{(c+d x) \left (c^2-d^2 x^2\right )^{5/2}}dx}{7 c}-\frac {1}{7 c d (c+d x)^2 \left (c^2-d^2 x^2\right )^{3/2}}\right )}{3 c}-\frac {20 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{9 c d (c+d x)^3 \left (c^2-d^2 x^2\right )^{3/2}}\right )+\frac {d (4 C d-7 c D)}{5 (c+d x)^2 \left (c^2-d^2 x^2\right )^{3/2}}}{4 d^5}+\frac {D}{4 d^4 (c+d x) \left (c^2-d^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 470

\(\displaystyle \frac {\frac {1}{5} d^2 \left (\frac {\left (40 A d^3+20 B c d^2-7 c^3 D+4 c^2 C d\right ) \left (\frac {5 \left (\frac {4 \int \frac {1}{\left (c^2-d^2 x^2\right )^{5/2}}dx}{5 c}-\frac {1}{5 c d (c+d x) \left (c^2-d^2 x^2\right )^{3/2}}\right )}{7 c}-\frac {1}{7 c d (c+d x)^2 \left (c^2-d^2 x^2\right )^{3/2}}\right )}{3 c}-\frac {20 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{9 c d (c+d x)^3 \left (c^2-d^2 x^2\right )^{3/2}}\right )+\frac {d (4 C d-7 c D)}{5 (c+d x)^2 \left (c^2-d^2 x^2\right )^{3/2}}}{4 d^5}+\frac {D}{4 d^4 (c+d x) \left (c^2-d^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 209

\(\displaystyle \frac {\frac {1}{5} d^2 \left (\frac {\left (40 A d^3+20 B c d^2-7 c^3 D+4 c^2 C d\right ) \left (\frac {5 \left (\frac {4 \left (\frac {2 \int \frac {1}{\left (c^2-d^2 x^2\right )^{3/2}}dx}{3 c^2}+\frac {x}{3 c^2 \left (c^2-d^2 x^2\right )^{3/2}}\right )}{5 c}-\frac {1}{5 c d (c+d x) \left (c^2-d^2 x^2\right )^{3/2}}\right )}{7 c}-\frac {1}{7 c d (c+d x)^2 \left (c^2-d^2 x^2\right )^{3/2}}\right )}{3 c}-\frac {20 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{9 c d (c+d x)^3 \left (c^2-d^2 x^2\right )^{3/2}}\right )+\frac {d (4 C d-7 c D)}{5 (c+d x)^2 \left (c^2-d^2 x^2\right )^{3/2}}}{4 d^5}+\frac {D}{4 d^4 (c+d x) \left (c^2-d^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 208

\(\displaystyle \frac {\frac {1}{5} d^2 \left (\frac {\left (\frac {5 \left (\frac {4 \left (\frac {x}{3 c^2 \left (c^2-d^2 x^2\right )^{3/2}}+\frac {2 x}{3 c^4 \sqrt {c^2-d^2 x^2}}\right )}{5 c}-\frac {1}{5 c d (c+d x) \left (c^2-d^2 x^2\right )^{3/2}}\right )}{7 c}-\frac {1}{7 c d (c+d x)^2 \left (c^2-d^2 x^2\right )^{3/2}}\right ) \left (40 A d^3+20 B c d^2-7 c^3 D+4 c^2 C d\right )}{3 c}-\frac {20 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{9 c d (c+d x)^3 \left (c^2-d^2 x^2\right )^{3/2}}\right )+\frac {d (4 C d-7 c D)}{5 (c+d x)^2 \left (c^2-d^2 x^2\right )^{3/2}}}{4 d^5}+\frac {D}{4 d^4 (c+d x) \left (c^2-d^2 x^2\right )^{3/2}}\)

Input:

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

Output:

D/(4*d^4*(c + d*x)*(c^2 - d^2*x^2)^(3/2)) + ((d*(4*C*d - 7*c*D))/(5*(c + d 
*x)^2*(c^2 - d^2*x^2)^(3/2)) + (d^2*((-20*(c^2*C*d - B*c*d^2 + A*d^3 - c^3 
*D))/(9*c*d*(c + d*x)^3*(c^2 - d^2*x^2)^(3/2)) + ((4*c^2*C*d + 20*B*c*d^2 
+ 40*A*d^3 - 7*c^3*D)*(-1/7*1/(c*d*(c + d*x)^2*(c^2 - d^2*x^2)^(3/2)) + (5 
*(-1/5*1/(c*d*(c + d*x)*(c^2 - d^2*x^2)^(3/2)) + (4*(x/(3*c^2*(c^2 - d^2*x 
^2)^(3/2)) + (2*x)/(3*c^4*Sqrt[c^2 - d^2*x^2])))/(5*c)))/(7*c)))/(3*c)))/5 
)/(4*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 209

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && ILtQ[p + 3/2, 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 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 = 339, normalized size of antiderivative = 1.19

method	result	size
gosper	\(-\frac {\left (-d x +c \right ) \left (80 A \,d^{9} x^{6}+40 B c \,d^{8} x^{6}+8 C \,c^{2} d^{7} x^{6}-14 D c^{3} d^{6} x^{6}+240 A c \,d^{8} x^{5}+120 B \,c^{2} d^{7} x^{5}+24 C \,c^{3} d^{6} x^{5}-42 D c^{4} d^{5} x^{5}+120 A \,c^{2} d^{7} x^{4}+60 B \,c^{3} d^{6} x^{4}+12 C \,c^{4} d^{5} x^{4}-21 D c^{5} d^{4} x^{4}-280 A \,c^{3} d^{6} x^{3}-140 B \,c^{4} d^{5} x^{3}-28 C \,c^{5} d^{4} x^{3}+49 D c^{6} d^{3} x^{3}-330 A \,c^{4} d^{5} x^{2}-165 B \,c^{5} d^{4} x^{2}-33 C \,c^{6} d^{3} x^{2}-21 D c^{7} d^{2} x^{2}-30 A \,c^{5} d^{4} x -15 B \,c^{6} d^{3} x -66 C \,c^{7} d^{2} x -42 D c^{8} d x +95 A \,c^{6} d^{3}-5 B \,c^{7} d^{2}-22 C \,c^{8} d -14 D c^{9}\right )}{315 \left (d x +c \right )^{2} c^{7} d^{4} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}\)	\(339\)
orering	\(-\frac {\left (-d x +c \right ) \left (80 A \,d^{9} x^{6}+40 B c \,d^{8} x^{6}+8 C \,c^{2} d^{7} x^{6}-14 D c^{3} d^{6} x^{6}+240 A c \,d^{8} x^{5}+120 B \,c^{2} d^{7} x^{5}+24 C \,c^{3} d^{6} x^{5}-42 D c^{4} d^{5} x^{5}+120 A \,c^{2} d^{7} x^{4}+60 B \,c^{3} d^{6} x^{4}+12 C \,c^{4} d^{5} x^{4}-21 D c^{5} d^{4} x^{4}-280 A \,c^{3} d^{6} x^{3}-140 B \,c^{4} d^{5} x^{3}-28 C \,c^{5} d^{4} x^{3}+49 D c^{6} d^{3} x^{3}-330 A \,c^{4} d^{5} x^{2}-165 B \,c^{5} d^{4} x^{2}-33 C \,c^{6} d^{3} x^{2}-21 D c^{7} d^{2} x^{2}-30 A \,c^{5} d^{4} x -15 B \,c^{6} d^{3} x -66 C \,c^{7} d^{2} x -42 D c^{8} d x +95 A \,c^{6} d^{3}-5 B \,c^{7} d^{2}-22 C \,c^{8} d -14 D c^{9}\right )}{315 \left (d x +c \right )^{2} c^{7} d^{4} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}\)	\(339\)
trager	\(-\frac {\left (80 A \,d^{9} x^{6}+40 B c \,d^{8} x^{6}+8 C \,c^{2} d^{7} x^{6}-14 D c^{3} d^{6} x^{6}+240 A c \,d^{8} x^{5}+120 B \,c^{2} d^{7} x^{5}+24 C \,c^{3} d^{6} x^{5}-42 D c^{4} d^{5} x^{5}+120 A \,c^{2} d^{7} x^{4}+60 B \,c^{3} d^{6} x^{4}+12 C \,c^{4} d^{5} x^{4}-21 D c^{5} d^{4} x^{4}-280 A \,c^{3} d^{6} x^{3}-140 B \,c^{4} d^{5} x^{3}-28 C \,c^{5} d^{4} x^{3}+49 D c^{6} d^{3} x^{3}-330 A \,c^{4} d^{5} x^{2}-165 B \,c^{5} d^{4} x^{2}-33 C \,c^{6} d^{3} x^{2}-21 D c^{7} d^{2} x^{2}-30 A \,c^{5} d^{4} x -15 B \,c^{6} d^{3} x -66 C \,c^{7} d^{2} x -42 D c^{8} d x +95 A \,c^{6} d^{3}-5 B \,c^{7} d^{2}-22 C \,c^{8} d -14 D c^{9}\right ) \sqrt {-d^{2} x^{2}+c^{2}}}{315 c^{7} \left (d x +c \right )^{5} d^{4} \left (-d x +c \right )^{2}}\)	\(341\)
default	\(\frac {D \left (\frac {x}{3 c^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}+\frac {2 x}{3 c^{4} \sqrt {-d^{2} x^{2}+c^{2}}}\right )}{d^{3}}+\frac {\left (C d -3 D c \right ) \left (-\frac {1}{5 c d \left (x +\frac {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}}}+\frac {4 d \left (-\frac {-2 d^{2} \left (x +\frac {c}{d}\right )+2 c d}{6 c^{2} d^{2} \left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {3}{2}}}-\frac {-2 d^{2} \left (x +\frac {c}{d}\right )+2 c d}{3 d^{2} c^{4} \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}\right )}{5 c}\right )}{d^{4}}+\frac {\left (B \,d^{2}-2 C c d +3 D c^{2}\right ) \left (-\frac {1}{7 c d \left (x +\frac {c}{d}\right )^{2} \left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {3}{2}}}+\frac {5 d \left (-\frac {1}{5 c d \left (x +\frac {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}}}+\frac {4 d \left (-\frac {-2 d^{2} \left (x +\frac {c}{d}\right )+2 c d}{6 c^{2} d^{2} \left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {3}{2}}}-\frac {-2 d^{2} \left (x +\frac {c}{d}\right )+2 c d}{3 d^{2} c^{4} \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}\right )}{5 c}\right )}{7 c}\right )}{d^{5}}+\frac {\left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right ) \left (-\frac {1}{9 c d \left (x +\frac {c}{d}\right )^{3} \left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {3}{2}}}+\frac {2 d \left (-\frac {1}{7 c d \left (x +\frac {c}{d}\right )^{2} \left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {3}{2}}}+\frac {5 d \left (-\frac {1}{5 c d \left (x +\frac {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}}}+\frac {4 d \left (-\frac {-2 d^{2} \left (x +\frac {c}{d}\right )+2 c d}{6 c^{2} d^{2} \left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {3}{2}}}-\frac {-2 d^{2} \left (x +\frac {c}{d}\right )+2 c d}{3 d^{2} c^{4} \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}\right )}{5 c}\right )}{7 c}\right )}{3 c}\right )}{d^{6}}\)	\(743\)

Input:

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

Output:

-1/315*(-d*x+c)*(80*A*d^9*x^6+40*B*c*d^8*x^6+8*C*c^2*d^7*x^6-14*D*c^3*d^6* 
x^6+240*A*c*d^8*x^5+120*B*c^2*d^7*x^5+24*C*c^3*d^6*x^5-42*D*c^4*d^5*x^5+12 
0*A*c^2*d^7*x^4+60*B*c^3*d^6*x^4+12*C*c^4*d^5*x^4-21*D*c^5*d^4*x^4-280*A*c 
^3*d^6*x^3-140*B*c^4*d^5*x^3-28*C*c^5*d^4*x^3+49*D*c^6*d^3*x^3-330*A*c^4*d 
^5*x^2-165*B*c^5*d^4*x^2-33*C*c^6*d^3*x^2-21*D*c^7*d^2*x^2-30*A*c^5*d^4*x- 
15*B*c^6*d^3*x-66*C*c^7*d^2*x-42*D*c^8*d*x+95*A*c^6*d^3-5*B*c^7*d^2-22*C*c 
^8*d-14*D*c^9)/(d*x+c)^2/c^7/d^4/(-d^2*x^2+c^2)^(5/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 684 vs. \(2 (266) = 532\).

Time = 0.37 (sec) , antiderivative size = 684, normalized size of antiderivative = 2.39 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^3 \left (c^2-d^2 x^2\right )^{5/2}} \, dx=\frac {14 \, D c^{10} + 22 \, C c^{9} d + 5 \, B c^{8} d^{2} - 95 \, A c^{7} d^{3} + {\left (14 \, D c^{3} d^{7} + 22 \, C c^{2} d^{8} + 5 \, B c d^{9} - 95 \, A d^{10}\right )} x^{7} + 3 \, {\left (14 \, D c^{4} d^{6} + 22 \, C c^{3} d^{7} + 5 \, B c^{2} d^{8} - 95 \, A c d^{9}\right )} x^{6} + {\left (14 \, D c^{5} d^{5} + 22 \, C c^{4} d^{6} + 5 \, B c^{3} d^{7} - 95 \, A c^{2} d^{8}\right )} x^{5} - 5 \, {\left (14 \, D c^{6} d^{4} + 22 \, C c^{5} d^{5} + 5 \, B c^{4} d^{6} - 95 \, A c^{3} d^{7}\right )} x^{4} - 5 \, {\left (14 \, D c^{7} d^{3} + 22 \, C c^{6} d^{4} + 5 \, B c^{5} d^{5} - 95 \, A c^{4} d^{6}\right )} x^{3} + {\left (14 \, D c^{8} d^{2} + 22 \, C c^{7} d^{3} + 5 \, B c^{6} d^{4} - 95 \, A c^{5} d^{5}\right )} x^{2} + 3 \, {\left (14 \, D c^{9} d + 22 \, C c^{8} d^{2} + 5 \, B c^{7} d^{3} - 95 \, A c^{6} d^{4}\right )} x + {\left (14 \, D c^{9} + 22 \, C c^{8} d + 5 \, B c^{7} d^{2} - 95 \, A c^{6} d^{3} + 2 \, {\left (7 \, D c^{3} d^{6} - 4 \, C c^{2} d^{7} - 20 \, B c d^{8} - 40 \, A d^{9}\right )} x^{6} + 6 \, {\left (7 \, D c^{4} d^{5} - 4 \, C c^{3} d^{6} - 20 \, B c^{2} d^{7} - 40 \, A c d^{8}\right )} x^{5} + 3 \, {\left (7 \, D c^{5} d^{4} - 4 \, C c^{4} d^{5} - 20 \, B c^{3} d^{6} - 40 \, A c^{2} d^{7}\right )} x^{4} - 7 \, {\left (7 \, D c^{6} d^{3} - 4 \, C c^{5} d^{4} - 20 \, B c^{4} d^{5} - 40 \, A c^{3} d^{6}\right )} x^{3} + 3 \, {\left (7 \, D c^{7} d^{2} + 11 \, C c^{6} d^{3} + 55 \, B c^{5} d^{4} + 110 \, A c^{4} d^{5}\right )} x^{2} + 3 \, {\left (14 \, D c^{8} d + 22 \, C c^{7} d^{2} + 5 \, B c^{6} d^{3} + 10 \, A c^{5} d^{4}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{315 \, {\left (c^{7} d^{11} x^{7} + 3 \, c^{8} d^{10} x^{6} + c^{9} d^{9} x^{5} - 5 \, c^{10} d^{8} x^{4} - 5 \, c^{11} d^{7} x^{3} + c^{12} d^{6} x^{2} + 3 \, c^{13} d^{5} x + c^{14} d^{4}\right )}} \] Input:

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

Output:

1/315*(14*D*c^10 + 22*C*c^9*d + 5*B*c^8*d^2 - 95*A*c^7*d^3 + (14*D*c^3*d^7 
 + 22*C*c^2*d^8 + 5*B*c*d^9 - 95*A*d^10)*x^7 + 3*(14*D*c^4*d^6 + 22*C*c^3* 
d^7 + 5*B*c^2*d^8 - 95*A*c*d^9)*x^6 + (14*D*c^5*d^5 + 22*C*c^4*d^6 + 5*B*c 
^3*d^7 - 95*A*c^2*d^8)*x^5 - 5*(14*D*c^6*d^4 + 22*C*c^5*d^5 + 5*B*c^4*d^6 
- 95*A*c^3*d^7)*x^4 - 5*(14*D*c^7*d^3 + 22*C*c^6*d^4 + 5*B*c^5*d^5 - 95*A* 
c^4*d^6)*x^3 + (14*D*c^8*d^2 + 22*C*c^7*d^3 + 5*B*c^6*d^4 - 95*A*c^5*d^5)* 
x^2 + 3*(14*D*c^9*d + 22*C*c^8*d^2 + 5*B*c^7*d^3 - 95*A*c^6*d^4)*x + (14*D 
*c^9 + 22*C*c^8*d + 5*B*c^7*d^2 - 95*A*c^6*d^3 + 2*(7*D*c^3*d^6 - 4*C*c^2* 
d^7 - 20*B*c*d^8 - 40*A*d^9)*x^6 + 6*(7*D*c^4*d^5 - 4*C*c^3*d^6 - 20*B*c^2 
*d^7 - 40*A*c*d^8)*x^5 + 3*(7*D*c^5*d^4 - 4*C*c^4*d^5 - 20*B*c^3*d^6 - 40* 
A*c^2*d^7)*x^4 - 7*(7*D*c^6*d^3 - 4*C*c^5*d^4 - 20*B*c^4*d^5 - 40*A*c^3*d^ 
6)*x^3 + 3*(7*D*c^7*d^2 + 11*C*c^6*d^3 + 55*B*c^5*d^4 + 110*A*c^4*d^5)*x^2 
 + 3*(14*D*c^8*d + 22*C*c^7*d^2 + 5*B*c^6*d^3 + 10*A*c^5*d^4)*x)*sqrt(-d^2 
*x^2 + c^2))/(c^7*d^11*x^7 + 3*c^8*d^10*x^6 + c^9*d^9*x^5 - 5*c^10*d^8*x^4 
 - 5*c^11*d^7*x^3 + c^12*d^6*x^2 + 3*c^13*d^5*x + c^14*d^4)
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1548 vs. \(2 (266) = 532\).

Time = 0.09 (sec) , antiderivative size = 1548, normalized size of antiderivative = 5.41 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^3 \left (c^2-d^2 x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

1/9*D*c^3/((-d^2*x^2 + c^2)^(3/2)*c*d^7*x^3 + 3*(-d^2*x^2 + c^2)^(3/2)*c^2 
*d^6*x^2 + 3*(-d^2*x^2 + c^2)^(3/2)*c^3*d^5*x + (-d^2*x^2 + c^2)^(3/2)*c^4 
*d^4) + 2/21*D*c^3/((-d^2*x^2 + c^2)^(3/2)*c^2*d^6*x^2 + 2*(-d^2*x^2 + c^2 
)^(3/2)*c^3*d^5*x + (-d^2*x^2 + c^2)^(3/2)*c^4*d^4) + 2/21*D*c^3/((-d^2*x^ 
2 + c^2)^(3/2)*c^3*d^5*x + (-d^2*x^2 + c^2)^(3/2)*c^4*d^4) - 1/9*C*c^2/((- 
d^2*x^2 + c^2)^(3/2)*c*d^6*x^3 + 3*(-d^2*x^2 + c^2)^(3/2)*c^2*d^5*x^2 + 3* 
(-d^2*x^2 + c^2)^(3/2)*c^3*d^4*x + (-d^2*x^2 + c^2)^(3/2)*c^4*d^3) - 2/21* 
C*c^2/((-d^2*x^2 + c^2)^(3/2)*c^2*d^5*x^2 + 2*(-d^2*x^2 + c^2)^(3/2)*c^3*d 
^4*x + (-d^2*x^2 + c^2)^(3/2)*c^4*d^3) - 2/21*C*c^2/((-d^2*x^2 + c^2)^(3/2 
)*c^3*d^4*x + (-d^2*x^2 + c^2)^(3/2)*c^4*d^3) - 3/7*D*c^2/((-d^2*x^2 + c^2 
)^(3/2)*c*d^6*x^2 + 2*(-d^2*x^2 + c^2)^(3/2)*c^2*d^5*x + (-d^2*x^2 + c^2)^ 
(3/2)*c^3*d^4) - 3/7*D*c^2/((-d^2*x^2 + c^2)^(3/2)*c^2*d^5*x + (-d^2*x^2 + 
 c^2)^(3/2)*c^3*d^4) + 1/9*B*c/((-d^2*x^2 + c^2)^(3/2)*c*d^5*x^3 + 3*(-d^2 
*x^2 + c^2)^(3/2)*c^2*d^4*x^2 + 3*(-d^2*x^2 + c^2)^(3/2)*c^3*d^3*x + (-d^2 
*x^2 + c^2)^(3/2)*c^4*d^2) + 2/21*B*c/((-d^2*x^2 + c^2)^(3/2)*c^2*d^4*x^2 
+ 2*(-d^2*x^2 + c^2)^(3/2)*c^3*d^3*x + (-d^2*x^2 + c^2)^(3/2)*c^4*d^2) + 2 
/21*B*c/((-d^2*x^2 + c^2)^(3/2)*c^3*d^3*x + (-d^2*x^2 + c^2)^(3/2)*c^4*d^2 
) + 2/7*C*c/((-d^2*x^2 + c^2)^(3/2)*c*d^5*x^2 + 2*(-d^2*x^2 + c^2)^(3/2)*c 
^2*d^4*x + (-d^2*x^2 + c^2)^(3/2)*c^3*d^3) + 2/7*C*c/((-d^2*x^2 + c^2)^(3/ 
2)*c^2*d^4*x + (-d^2*x^2 + c^2)^(3/2)*c^3*d^3) + 3/5*D*c/((-d^2*x^2 + c...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 706, normalized size of antiderivative = 2.47 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^3 \left (c^2-d^2 x^2\right )^{5/2}} \, dx=\frac {30 a \,c^{5} d^{3} x +330 a \,c^{4} d^{4} x^{2}+280 a \,c^{3} d^{5} x^{3}-120 a \,c^{2} d^{6} x^{4}-240 a c \,d^{7} x^{5}+15 b \,c^{6} d^{2} x +165 b \,c^{5} d^{3} x^{2}+140 b \,c^{4} d^{4} x^{3}-60 b \,c^{3} d^{5} x^{4}-120 b \,c^{2} d^{6} x^{5}-80 a \,d^{8} x^{6}+108 c^{8} d x +54 c^{7} d^{2} x^{2}-21 c^{6} d^{3} x^{3}+9 c^{5} d^{4} x^{4}+18 c^{4} d^{5} x^{5}+6 c^{3} d^{6} x^{6}-95 a \,c^{6} d^{2}+5 b \,c^{7} d -40 b c \,d^{7} x^{6}+\sqrt {-d^{2} x^{2}+c^{2}}\, c^{8}+36 c^{9}+30 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{4} d^{3} x +20 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{3} d^{4} x^{2}-20 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{2} d^{5} x^{3}-30 \sqrt {-d^{2} x^{2}+c^{2}}\, a c \,d^{6} x^{4}+15 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{5} d^{2} x +10 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{4} d^{3} x^{2}-10 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{3} d^{4} x^{3}-15 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{2} d^{5} x^{4}-5 \sqrt {-d^{2} x^{2}+c^{2}}\, b c \,d^{6} x^{5}+10 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{5} d^{2}-10 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,d^{7} x^{5}+5 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{6} d +3 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{7} d x +2 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{6} d^{2} x^{2}-2 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{5} d^{3} x^{3}-3 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{4} d^{4} x^{4}-\sqrt {-d^{2} x^{2}+c^{2}}\, c^{3} d^{5} x^{5}}{315 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{7} d^{3} \left (-d^{5} x^{5}-3 c \,d^{4} x^{4}-2 c^{2} d^{3} x^{3}+2 c^{3} d^{2} x^{2}+3 c^{4} d x +c^{5}\right )} \] Input:

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

Output:

(10*sqrt(c**2 - d**2*x**2)*a*c**5*d**2 + 30*sqrt(c**2 - d**2*x**2)*a*c**4* 
d**3*x + 20*sqrt(c**2 - d**2*x**2)*a*c**3*d**4*x**2 - 20*sqrt(c**2 - d**2* 
x**2)*a*c**2*d**5*x**3 - 30*sqrt(c**2 - d**2*x**2)*a*c*d**6*x**4 - 10*sqrt 
(c**2 - d**2*x**2)*a*d**7*x**5 + 5*sqrt(c**2 - d**2*x**2)*b*c**6*d + 15*sq 
rt(c**2 - d**2*x**2)*b*c**5*d**2*x + 10*sqrt(c**2 - d**2*x**2)*b*c**4*d**3 
*x**2 - 10*sqrt(c**2 - d**2*x**2)*b*c**3*d**4*x**3 - 15*sqrt(c**2 - d**2*x 
**2)*b*c**2*d**5*x**4 - 5*sqrt(c**2 - d**2*x**2)*b*c*d**6*x**5 + sqrt(c**2 
 - d**2*x**2)*c**8 + 3*sqrt(c**2 - d**2*x**2)*c**7*d*x + 2*sqrt(c**2 - d** 
2*x**2)*c**6*d**2*x**2 - 2*sqrt(c**2 - d**2*x**2)*c**5*d**3*x**3 - 3*sqrt( 
c**2 - d**2*x**2)*c**4*d**4*x**4 - sqrt(c**2 - d**2*x**2)*c**3*d**5*x**5 - 
 95*a*c**6*d**2 + 30*a*c**5*d**3*x + 330*a*c**4*d**4*x**2 + 280*a*c**3*d** 
5*x**3 - 120*a*c**2*d**6*x**4 - 240*a*c*d**7*x**5 - 80*a*d**8*x**6 + 5*b*c 
**7*d + 15*b*c**6*d**2*x + 165*b*c**5*d**3*x**2 + 140*b*c**4*d**4*x**3 - 6 
0*b*c**3*d**5*x**4 - 120*b*c**2*d**6*x**5 - 40*b*c*d**7*x**6 + 36*c**9 + 1 
08*c**8*d*x + 54*c**7*d**2*x**2 - 21*c**6*d**3*x**3 + 9*c**5*d**4*x**4 + 1 
8*c**4*d**5*x**5 + 6*c**3*d**6*x**6)/(315*sqrt(c**2 - d**2*x**2)*c**7*d**3 
*(c**5 + 3*c**4*d*x + 2*c**3*d**2*x**2 - 2*c**2*d**3*x**3 - 3*c*d**4*x**4 
- d**5*x**5))