\(\int \frac {A+B x+C x^2+D x^3}{(c+d x)^4 (c^2-d^2 x^2)^{3/2}} \, dx\) [184]

Optimal result

Integrand size = 39, antiderivative size = 293 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^4 \left (c^2-d^2 x^2\right )^{3/2}} \, dx=\frac {2 \left (11 c^2 C d+16 B c d^2+20 A d^3+4 c^3 D\right ) x}{315 c^6 d^3 \sqrt {c^2-d^2 x^2}}-\frac {c^2 C d-B c d^2+A d^3-c^3 D}{9 c d^4 (c+d x)^4 \sqrt {c^2-d^2 x^2}}+\frac {13 c^2 C d-4 B c d^2-5 A d^3-22 c^3 D}{63 c^2 d^4 (c+d x)^3 \sqrt {c^2-d^2 x^2}}-\frac {11 c^2 C d+16 B c d^2+20 A d^3-101 c^3 D}{315 c^3 d^4 (c+d x)^2 \sqrt {c^2-d^2 x^2}}-\frac {11 c^2 C d+16 B c d^2+20 A d^3+4 c^3 D}{315 c^4 d^4 (c+d x) \sqrt {c^2-d^2 x^2}} \] Output:

2/315*(20*A*d^3+16*B*c*d^2+11*C*c^2*d+4*D*c^3)*x/c^6/d^3/(-d^2*x^2+c^2)^(1 
/2)-1/9*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)/c/d^4/(d*x+c)^4/(-d^2*x^2+c^2)^(1/2) 
+1/63*(-5*A*d^3-4*B*c*d^2+13*C*c^2*d-22*D*c^3)/c^2/d^4/(d*x+c)^3/(-d^2*x^2 
+c^2)^(1/2)-1/315*(20*A*d^3+16*B*c*d^2+11*C*c^2*d-101*D*c^3)/c^3/d^4/(d*x+ 
c)^2/(-d^2*x^2+c^2)^(1/2)-1/315*(20*A*d^3+16*B*c*d^2+11*C*c^2*d+4*D*c^3)/c 
^4/d^4/(d*x+c)/(-d^2*x^2+c^2)^(1/2)
 

Mathematica [A] (verified)

Time = 1.74 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.73 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^4 \left (c^2-d^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {c^2-d^2 x^2} \left (22 c^8 D+40 A d^8 x^5+32 c d^7 x^4 (5 A+B x)+8 c^7 d (C+11 D x)+2 c^2 d^6 x^3 (110 A+x (64 B+11 C x))+c^6 d^2 (-17 B+x (32 C+121 D x))+4 c^5 d^3 (-25 A+x (-17 B+11 x (C+D x)))+8 c^3 d^5 x^2 (10 A+x (22 B+x (11 C+D x)))+c^4 d^4 x (-85 A+x (64 B+x (121 C+32 D x)))\right )}{315 c^6 d^4 (c-d x) (c+d x)^5} \] Input:

Integrate[(A + B*x + C*x^2 + D*x^3)/((c + d*x)^4*(c^2 - d^2*x^2)^(3/2)),x]
 

Output:

(Sqrt[c^2 - d^2*x^2]*(22*c^8*D + 40*A*d^8*x^5 + 32*c*d^7*x^4*(5*A + B*x) + 
 8*c^7*d*(C + 11*D*x) + 2*c^2*d^6*x^3*(110*A + x*(64*B + 11*C*x)) + c^6*d^ 
2*(-17*B + x*(32*C + 121*D*x)) + 4*c^5*d^3*(-25*A + x*(-17*B + 11*x*(C + D 
*x))) + 8*c^3*d^5*x^2*(10*A + x*(22*B + x*(11*C + D*x))) + c^4*d^4*x*(-85* 
A + x*(64*B + x*(121*C + 32*D*x)))))/(315*c^6*d^4*(c - d*x)*(c + d*x)^5)
 

Rubi [A] (verified)

Time = 1.14 (sec) , antiderivative size = 317, 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, 461, 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.

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

\(\Big \downarrow \) 2170

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

\(\Big \downarrow \) 2170

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 671

\(\displaystyle \frac {\frac {1}{4} d^2 \left (\frac {\left (20 A d^3+16 B c d^2+4 c^3 D+11 c^2 C d\right ) \int \frac {1}{(c+d x)^3 \left (c^2-d^2 x^2\right )^{3/2}}dx}{3 c}-\frac {4 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 c d (c+d x)^4 \sqrt {c^2-d^2 x^2}}\right )+\frac {d (3 C d-4 c D)}{4 (c+d x)^3 \sqrt {c^2-d^2 x^2}}}{3 d^5}+\frac {D}{3 d^4 (c+d x)^2 \sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 461

\(\displaystyle \frac {\frac {1}{4} d^2 \left (\frac {\left (20 A d^3+16 B c d^2+4 c^3 D+11 c^2 C d\right ) \left (\frac {4 \int \frac {1}{(c+d x)^2 \left (c^2-d^2 x^2\right )^{3/2}}dx}{7 c}-\frac {1}{7 c d (c+d x)^3 \sqrt {c^2-d^2 x^2}}\right )}{3 c}-\frac {4 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 c d (c+d x)^4 \sqrt {c^2-d^2 x^2}}\right )+\frac {d (3 C d-4 c D)}{4 (c+d x)^3 \sqrt {c^2-d^2 x^2}}}{3 d^5}+\frac {D}{3 d^4 (c+d x)^2 \sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 461

\(\displaystyle \frac {\frac {1}{4} d^2 \left (\frac {\left (20 A d^3+16 B c d^2+4 c^3 D+11 c^2 C d\right ) \left (\frac {4 \left (\frac {3 \int \frac {1}{(c+d x) \left (c^2-d^2 x^2\right )^{3/2}}dx}{5 c}-\frac {1}{5 c d (c+d x)^2 \sqrt {c^2-d^2 x^2}}\right )}{7 c}-\frac {1}{7 c d (c+d x)^3 \sqrt {c^2-d^2 x^2}}\right )}{3 c}-\frac {4 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 c d (c+d x)^4 \sqrt {c^2-d^2 x^2}}\right )+\frac {d (3 C d-4 c D)}{4 (c+d x)^3 \sqrt {c^2-d^2 x^2}}}{3 d^5}+\frac {D}{3 d^4 (c+d x)^2 \sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 470

\(\displaystyle \frac {\frac {1}{4} d^2 \left (\frac {\left (20 A d^3+16 B c d^2+4 c^3 D+11 c^2 C d\right ) \left (\frac {4 \left (\frac {3 \left (\frac {2 \int \frac {1}{\left (c^2-d^2 x^2\right )^{3/2}}dx}{3 c}-\frac {1}{3 c d (c+d x) \sqrt {c^2-d^2 x^2}}\right )}{5 c}-\frac {1}{5 c d (c+d x)^2 \sqrt {c^2-d^2 x^2}}\right )}{7 c}-\frac {1}{7 c d (c+d x)^3 \sqrt {c^2-d^2 x^2}}\right )}{3 c}-\frac {4 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 c d (c+d x)^4 \sqrt {c^2-d^2 x^2}}\right )+\frac {d (3 C d-4 c D)}{4 (c+d x)^3 \sqrt {c^2-d^2 x^2}}}{3 d^5}+\frac {D}{3 d^4 (c+d x)^2 \sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 208

\(\displaystyle \frac {\frac {1}{4} d^2 \left (\frac {\left (\frac {4 \left (\frac {3 \left (\frac {2 x}{3 c^3 \sqrt {c^2-d^2 x^2}}-\frac {1}{3 c d (c+d x) \sqrt {c^2-d^2 x^2}}\right )}{5 c}-\frac {1}{5 c d (c+d x)^2 \sqrt {c^2-d^2 x^2}}\right )}{7 c}-\frac {1}{7 c d (c+d x)^3 \sqrt {c^2-d^2 x^2}}\right ) \left (20 A d^3+16 B c d^2+4 c^3 D+11 c^2 C d\right )}{3 c}-\frac {4 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 c d (c+d x)^4 \sqrt {c^2-d^2 x^2}}\right )+\frac {d (3 C d-4 c D)}{4 (c+d x)^3 \sqrt {c^2-d^2 x^2}}}{3 d^5}+\frac {D}{3 d^4 (c+d x)^2 \sqrt {c^2-d^2 x^2}}\)

Input:

Int[(A + B*x + C*x^2 + D*x^3)/((c + d*x)^4*(c^2 - d^2*x^2)^(3/2)),x]
 

Output:

D/(3*d^4*(c + d*x)^2*Sqrt[c^2 - d^2*x^2]) + ((d*(3*C*d - 4*c*D))/(4*(c + d 
*x)^3*Sqrt[c^2 - d^2*x^2]) + (d^2*((-4*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D) 
)/(3*c*d*(c + d*x)^4*Sqrt[c^2 - d^2*x^2]) + ((11*c^2*C*d + 16*B*c*d^2 + 20 
*A*d^3 + 4*c^3*D)*(-1/7*1/(c*d*(c + d*x)^3*Sqrt[c^2 - d^2*x^2]) + (4*(-1/5 
*1/(c*d*(c + d*x)^2*Sqrt[c^2 - d^2*x^2]) + (3*((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)))/(7*c)))/(3*c))) 
/4)/(3*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 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 = 0.50 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.99

method	result	size
gosper	\(-\frac {\left (-d x +c \right ) \left (-40 A \,d^{8} x^{5}-32 B c \,d^{7} x^{5}-22 C \,c^{2} d^{6} x^{5}-8 D c^{3} d^{5} x^{5}-160 A c \,d^{7} x^{4}-128 B \,c^{2} d^{6} x^{4}-88 C \,c^{3} d^{5} x^{4}-32 D c^{4} d^{4} x^{4}-220 A \,c^{2} d^{6} x^{3}-176 B \,c^{3} d^{5} x^{3}-121 C \,c^{4} d^{4} x^{3}-44 D c^{5} d^{3} x^{3}-80 A \,c^{3} d^{5} x^{2}-64 B \,c^{4} d^{4} x^{2}-44 C \,c^{5} d^{3} x^{2}-121 D c^{6} d^{2} x^{2}+85 A \,c^{4} d^{4} x +68 B \,c^{5} d^{3} x -32 C \,c^{6} d^{2} x -88 D c^{7} d x +100 A \,c^{5} d^{3}+17 B \,c^{6} d^{2}-8 C \,c^{7} d -22 D c^{8}\right )}{315 \left (d x +c \right )^{3} c^{6} d^{4} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}\)	\(291\)
orering	\(-\frac {\left (-d x +c \right ) \left (-40 A \,d^{8} x^{5}-32 B c \,d^{7} x^{5}-22 C \,c^{2} d^{6} x^{5}-8 D c^{3} d^{5} x^{5}-160 A c \,d^{7} x^{4}-128 B \,c^{2} d^{6} x^{4}-88 C \,c^{3} d^{5} x^{4}-32 D c^{4} d^{4} x^{4}-220 A \,c^{2} d^{6} x^{3}-176 B \,c^{3} d^{5} x^{3}-121 C \,c^{4} d^{4} x^{3}-44 D c^{5} d^{3} x^{3}-80 A \,c^{3} d^{5} x^{2}-64 B \,c^{4} d^{4} x^{2}-44 C \,c^{5} d^{3} x^{2}-121 D c^{6} d^{2} x^{2}+85 A \,c^{4} d^{4} x +68 B \,c^{5} d^{3} x -32 C \,c^{6} d^{2} x -88 D c^{7} d x +100 A \,c^{5} d^{3}+17 B \,c^{6} d^{2}-8 C \,c^{7} d -22 D c^{8}\right )}{315 \left (d x +c \right )^{3} c^{6} d^{4} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}\)	\(291\)
trager	\(-\frac {\left (-40 A \,d^{8} x^{5}-32 B c \,d^{7} x^{5}-22 C \,c^{2} d^{6} x^{5}-8 D c^{3} d^{5} x^{5}-160 A c \,d^{7} x^{4}-128 B \,c^{2} d^{6} x^{4}-88 C \,c^{3} d^{5} x^{4}-32 D c^{4} d^{4} x^{4}-220 A \,c^{2} d^{6} x^{3}-176 B \,c^{3} d^{5} x^{3}-121 C \,c^{4} d^{4} x^{3}-44 D c^{5} d^{3} x^{3}-80 A \,c^{3} d^{5} x^{2}-64 B \,c^{4} d^{4} x^{2}-44 C \,c^{5} d^{3} x^{2}-121 D c^{6} d^{2} x^{2}+85 A \,c^{4} d^{4} x +68 B \,c^{5} d^{3} x -32 C \,c^{6} d^{2} x -88 D c^{7} d x +100 A \,c^{5} d^{3}+17 B \,c^{6} d^{2}-8 C \,c^{7} d -22 D c^{8}\right ) \sqrt {-d^{2} x^{2}+c^{2}}}{315 c^{6} \left (d x +c \right )^{5} d^{4} \left (-d x +c \right )}\)	\(293\)
default	\(\frac {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 )}{d^{4}}+\frac {\left (C d -3 D c \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}}+\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 )^{3} \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}+\frac {4 d \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 )}{7 c}\right )}{d^{6}}+\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 )^{4} \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}+\frac {5 d \left (-\frac {1}{7 c d \left (x +\frac {c}{d}\right )^{3} \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}+\frac {4 d \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 )}{7 c}\right )}{9 c}\right )}{d^{7}}\)	\(777\)

Input:

int((D*x^3+C*x^2+B*x+A)/(d*x+c)^4/(-d^2*x^2+c^2)^(3/2),x,method=_RETURNVER 
BOSE)
 

Output:

-1/315*(-d*x+c)*(-40*A*d^8*x^5-32*B*c*d^7*x^5-22*C*c^2*d^6*x^5-8*D*c^3*d^5 
*x^5-160*A*c*d^7*x^4-128*B*c^2*d^6*x^4-88*C*c^3*d^5*x^4-32*D*c^4*d^4*x^4-2 
20*A*c^2*d^6*x^3-176*B*c^3*d^5*x^3-121*C*c^4*d^4*x^3-44*D*c^5*d^3*x^3-80*A 
*c^3*d^5*x^2-64*B*c^4*d^4*x^2-44*C*c^5*d^3*x^2-121*D*c^6*d^2*x^2+85*A*c^4* 
d^4*x+68*B*c^5*d^3*x-32*C*c^6*d^2*x-88*D*c^7*d*x+100*A*c^5*d^3+17*B*c^6*d^ 
2-8*C*c^7*d-22*D*c^8)/(d*x+c)^3/c^6/d^4/(-d^2*x^2+c^2)^(3/2)
 

Fricas [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 540, normalized size of antiderivative = 1.84 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^4 \left (c^2-d^2 x^2\right )^{3/2}} \, dx=-\frac {22 \, D c^{9} + 8 \, C c^{8} d - 17 \, B c^{7} d^{2} - 100 \, A c^{6} d^{3} - {\left (22 \, D c^{3} d^{6} + 8 \, C c^{2} d^{7} - 17 \, B c d^{8} - 100 \, A d^{9}\right )} x^{6} - 4 \, {\left (22 \, D c^{4} d^{5} + 8 \, C c^{3} d^{6} - 17 \, B c^{2} d^{7} - 100 \, A c d^{8}\right )} x^{5} - 5 \, {\left (22 \, D c^{5} d^{4} + 8 \, C c^{4} d^{5} - 17 \, B c^{3} d^{6} - 100 \, A c^{2} d^{7}\right )} x^{4} + 5 \, {\left (22 \, D c^{7} d^{2} + 8 \, C c^{6} d^{3} - 17 \, B c^{5} d^{4} - 100 \, A c^{4} d^{5}\right )} x^{2} + 4 \, {\left (22 \, D c^{8} d + 8 \, C c^{7} d^{2} - 17 \, B c^{6} d^{3} - 100 \, A c^{5} d^{4}\right )} x + {\left (22 \, D c^{8} + 8 \, C c^{7} d - 17 \, B c^{6} d^{2} - 100 \, A c^{5} d^{3} + 2 \, {\left (4 \, D c^{3} d^{5} + 11 \, C c^{2} d^{6} + 16 \, B c d^{7} + 20 \, A d^{8}\right )} x^{5} + 8 \, {\left (4 \, D c^{4} d^{4} + 11 \, C c^{3} d^{5} + 16 \, B c^{2} d^{6} + 20 \, A c d^{7}\right )} x^{4} + 11 \, {\left (4 \, D c^{5} d^{3} + 11 \, C c^{4} d^{4} + 16 \, B c^{3} d^{5} + 20 \, A c^{2} d^{6}\right )} x^{3} + {\left (121 \, D c^{6} d^{2} + 44 \, C c^{5} d^{3} + 64 \, B c^{4} d^{4} + 80 \, A c^{3} d^{5}\right )} x^{2} + {\left (88 \, D c^{7} d + 32 \, C c^{6} d^{2} - 68 \, B c^{5} d^{3} - 85 \, A c^{4} d^{4}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{315 \, {\left (c^{6} d^{10} x^{6} + 4 \, c^{7} d^{9} x^{5} + 5 \, c^{8} d^{8} x^{4} - 5 \, c^{10} d^{6} x^{2} - 4 \, c^{11} d^{5} x - c^{12} d^{4}\right )}} \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(d*x+c)^4/(-d^2*x^2+c^2)^(3/2),x, algorithm= 
"fricas")
 

Output:

-1/315*(22*D*c^9 + 8*C*c^8*d - 17*B*c^7*d^2 - 100*A*c^6*d^3 - (22*D*c^3*d^ 
6 + 8*C*c^2*d^7 - 17*B*c*d^8 - 100*A*d^9)*x^6 - 4*(22*D*c^4*d^5 + 8*C*c^3* 
d^6 - 17*B*c^2*d^7 - 100*A*c*d^8)*x^5 - 5*(22*D*c^5*d^4 + 8*C*c^4*d^5 - 17 
*B*c^3*d^6 - 100*A*c^2*d^7)*x^4 + 5*(22*D*c^7*d^2 + 8*C*c^6*d^3 - 17*B*c^5 
*d^4 - 100*A*c^4*d^5)*x^2 + 4*(22*D*c^8*d + 8*C*c^7*d^2 - 17*B*c^6*d^3 - 1 
00*A*c^5*d^4)*x + (22*D*c^8 + 8*C*c^7*d - 17*B*c^6*d^2 - 100*A*c^5*d^3 + 2 
*(4*D*c^3*d^5 + 11*C*c^2*d^6 + 16*B*c*d^7 + 20*A*d^8)*x^5 + 8*(4*D*c^4*d^4 
 + 11*C*c^3*d^5 + 16*B*c^2*d^6 + 20*A*c*d^7)*x^4 + 11*(4*D*c^5*d^3 + 11*C* 
c^4*d^4 + 16*B*c^3*d^5 + 20*A*c^2*d^6)*x^3 + (121*D*c^6*d^2 + 44*C*c^5*d^3 
 + 64*B*c^4*d^4 + 80*A*c^3*d^5)*x^2 + (88*D*c^7*d + 32*C*c^6*d^2 - 68*B*c^ 
5*d^3 - 85*A*c^4*d^4)*x)*sqrt(-d^2*x^2 + c^2))/(c^6*d^10*x^6 + 4*c^7*d^9*x 
^5 + 5*c^8*d^8*x^4 - 5*c^10*d^6*x^2 - 4*c^11*d^5*x - c^12*d^4)
 

Sympy [F]

\[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^4 \left (c^2-d^2 x^2\right )^{3/2}} \, dx=\int \frac {A + B x + C x^{2} + D x^{3}}{\left (- \left (- c + d x\right ) \left (c + d x\right )\right )^{\frac {3}{2}} \left (c + d x\right )^{4}}\, dx \] Input:

integrate((D*x**3+C*x**2+B*x+A)/(d*x+c)**4/(-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)**4), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2453 vs. \(2 (273) = 546\).

Time = 0.09 (sec) , antiderivative size = 2453, normalized size of antiderivative = 8.37 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^4 \left (c^2-d^2 x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(d*x+c)^4/(-d^2*x^2+c^2)^(3/2),x, algorithm= 
"maxima")
                                                                                    
                                                                                    
 

Output:

1/9*D*c^3/(sqrt(-d^2*x^2 + c^2)*c*d^8*x^4 + 4*sqrt(-d^2*x^2 + c^2)*c^2*d^7 
*x^3 + 6*sqrt(-d^2*x^2 + c^2)*c^3*d^6*x^2 + 4*sqrt(-d^2*x^2 + c^2)*c^4*d^5 
*x + sqrt(-d^2*x^2 + c^2)*c^5*d^4) + 5/63*D*c^3/(sqrt(-d^2*x^2 + c^2)*c^2* 
d^7*x^3 + 3*sqrt(-d^2*x^2 + c^2)*c^3*d^6*x^2 + 3*sqrt(-d^2*x^2 + c^2)*c^4* 
d^5*x + sqrt(-d^2*x^2 + c^2)*c^5*d^4) + 4/63*D*c^3/(sqrt(-d^2*x^2 + c^2)*c 
^3*d^6*x^2 + 2*sqrt(-d^2*x^2 + c^2)*c^4*d^5*x + sqrt(-d^2*x^2 + c^2)*c^5*d 
^4) + 4/63*D*c^3/(sqrt(-d^2*x^2 + c^2)*c^4*d^5*x + sqrt(-d^2*x^2 + c^2)*c^ 
5*d^4) - 1/9*C*c^2/(sqrt(-d^2*x^2 + c^2)*c*d^7*x^4 + 4*sqrt(-d^2*x^2 + c^2 
)*c^2*d^6*x^3 + 6*sqrt(-d^2*x^2 + c^2)*c^3*d^5*x^2 + 4*sqrt(-d^2*x^2 + c^2 
)*c^4*d^4*x + sqrt(-d^2*x^2 + c^2)*c^5*d^3) - 5/63*C*c^2/(sqrt(-d^2*x^2 + 
c^2)*c^2*d^6*x^3 + 3*sqrt(-d^2*x^2 + c^2)*c^3*d^5*x^2 + 3*sqrt(-d^2*x^2 + 
c^2)*c^4*d^4*x + sqrt(-d^2*x^2 + c^2)*c^5*d^3) - 4/63*C*c^2/(sqrt(-d^2*x^2 
 + c^2)*c^3*d^5*x^2 + 2*sqrt(-d^2*x^2 + c^2)*c^4*d^4*x + sqrt(-d^2*x^2 + c 
^2)*c^5*d^3) - 4/63*C*c^2/(sqrt(-d^2*x^2 + c^2)*c^4*d^4*x + sqrt(-d^2*x^2 
+ c^2)*c^5*d^3) - 3/7*D*c^2/(sqrt(-d^2*x^2 + c^2)*c*d^7*x^3 + 3*sqrt(-d^2* 
x^2 + c^2)*c^2*d^6*x^2 + 3*sqrt(-d^2*x^2 + c^2)*c^3*d^5*x + sqrt(-d^2*x^2 
+ c^2)*c^4*d^4) - 12/35*D*c^2/(sqrt(-d^2*x^2 + c^2)*c^2*d^6*x^2 + 2*sqrt(- 
d^2*x^2 + c^2)*c^3*d^5*x + sqrt(-d^2*x^2 + c^2)*c^4*d^4) - 12/35*D*c^2/(sq 
rt(-d^2*x^2 + c^2)*c^3*d^5*x + sqrt(-d^2*x^2 + c^2)*c^4*d^4) + 1/9*B*c/(sq 
rt(-d^2*x^2 + c^2)*c*d^6*x^4 + 4*sqrt(-d^2*x^2 + c^2)*c^2*d^5*x^3 + 6*s...
 

Giac [F]

\[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^4 \left (c^2-d^2 x^2\right )^{3/2}} \, dx=\int { \frac {D x^{3} + C x^{2} + B x + A}{{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{4}} \,d x } \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(d*x+c)^4/(-d^2*x^2+c^2)^(3/2),x, algorithm= 
"giac")
 

Output:

integrate((D*x^3 + C*x^2 + B*x + A)/((-d^2*x^2 + c^2)^(3/2)*(d*x + c)^4), 
x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^4 \left (c^2-d^2 x^2\right )^{3/2}} \, dx=\int \frac {A+B\,x+C\,x^2+x^3\,D}{{\left (c^2-d^2\,x^2\right )}^{3/2}\,{\left (c+d\,x\right )}^4} \,d x \] Input:

int((A + B*x + C*x^2 + x^3*D)/((c^2 - d^2*x^2)^(3/2)*(c + d*x)^4),x)
 

Output:

int((A + B*x + C*x^2 + x^3*D)/((c^2 - d^2*x^2)^(3/2)*(c + d*x)^4), x)
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.87 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^4 \left (c^2-d^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {-d^{2} x^{2}+c^{2}}\, \left (40 a \,d^{7} x^{5}+32 b c \,d^{6} x^{5}+30 c^{3} d^{5} x^{5}+160 a c \,d^{6} x^{4}+128 b \,c^{2} d^{5} x^{4}+120 c^{4} d^{4} x^{4}+220 a \,c^{2} d^{5} x^{3}+176 b \,c^{3} d^{4} x^{3}+165 c^{5} d^{3} x^{3}+80 a \,c^{3} d^{4} x^{2}+64 b \,c^{4} d^{3} x^{2}+165 c^{6} d^{2} x^{2}-85 a \,c^{4} d^{3} x -68 b \,c^{5} d^{2} x +120 c^{7} d x -100 a \,c^{5} d^{2}-17 b \,c^{6} d +30 c^{8}\right )}{315 c^{6} d^{3} \left (-d^{6} x^{6}-4 c \,d^{5} x^{5}-5 c^{2} d^{4} x^{4}+5 c^{4} d^{2} x^{2}+4 c^{5} d x +c^{6}\right )} \] Input:

int((D*x^3+C*x^2+B*x+A)/(d*x+c)^4/(-d^2*x^2+c^2)^(3/2),x)
 

Output:

(sqrt(c**2 - d**2*x**2)*( - 100*a*c**5*d**2 - 85*a*c**4*d**3*x + 80*a*c**3 
*d**4*x**2 + 220*a*c**2*d**5*x**3 + 160*a*c*d**6*x**4 + 40*a*d**7*x**5 - 1 
7*b*c**6*d - 68*b*c**5*d**2*x + 64*b*c**4*d**3*x**2 + 176*b*c**3*d**4*x**3 
 + 128*b*c**2*d**5*x**4 + 32*b*c*d**6*x**5 + 30*c**8 + 120*c**7*d*x + 165* 
c**6*d**2*x**2 + 165*c**5*d**3*x**3 + 120*c**4*d**4*x**4 + 30*c**3*d**5*x* 
*5))/(315*c**6*d**3*(c**6 + 4*c**5*d*x + 5*c**4*d**2*x**2 - 5*c**2*d**4*x* 
*4 - 4*c*d**5*x**5 - d**6*x**6))