\(\int \frac {A+B x+C x^2+D x^3}{(c+d x)^4 \sqrt {c^2-d^2 x^2}} \, dx\) [176]

Optimal result

Integrand size = 39, antiderivative size = 239 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^4 \sqrt {c^2-d^2 x^2}} \, dx=-\frac {\left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \sqrt {c^2-d^2 x^2}}{7 c d^4 (c+d x)^4}+\frac {\left (11 c^2 C d-4 B c d^2-3 A d^3-18 c^3 D\right ) \sqrt {c^2-d^2 x^2}}{35 c^2 d^4 (c+d x)^3}-\frac {\left (13 c^2 C d+8 B c d^2+6 A d^3-69 c^3 D\right ) \sqrt {c^2-d^2 x^2}}{105 c^3 d^4 (c+d x)^2}-\frac {\left (13 c^2 C d+8 B c d^2+6 A d^3+36 c^3 D\right ) \sqrt {c^2-d^2 x^2}}{105 c^4 d^4 (c+d x)} \] Output:

-1/7*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(-d^2*x^2+c^2)^(1/2)/c/d^4/(d*x+c)^4+1/ 
35*(-3*A*d^3-4*B*c*d^2+11*C*c^2*d-18*D*c^3)*(-d^2*x^2+c^2)^(1/2)/c^2/d^4/( 
d*x+c)^3-1/105*(6*A*d^3+8*B*c*d^2+13*C*c^2*d-69*D*c^3)*(-d^2*x^2+c^2)^(1/2 
)/c^3/d^4/(d*x+c)^2-1/105*(6*A*d^3+8*B*c*d^2+13*C*c^2*d+36*D*c^3)*(-d^2*x^ 
2+c^2)^(1/2)/c^4/d^4/(d*x+c)
 

Mathematica [A] (verified)

Time = 1.13 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.62 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^4 \sqrt {c^2-d^2 x^2}} \, dx=-\frac {\sqrt {c^2-d^2 x^2} \left (6 c^6 D+6 A d^6 x^3+8 c d^5 x^2 (3 A+B x)+8 c^5 d (C+3 D x)+c^2 d^4 x (39 A+x (32 B+13 C x))+c^4 d^2 (13 B+x (32 C+39 D x))+4 c^3 d^3 \left (9 A+x \left (13 B+13 C x+9 D x^2\right )\right )\right )}{105 c^4 d^4 (c+d x)^4} \] Input:

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

Output:

-1/105*(Sqrt[c^2 - d^2*x^2]*(6*c^6*D + 6*A*d^6*x^3 + 8*c*d^5*x^2*(3*A + B* 
x) + 8*c^5*d*(C + 3*D*x) + c^2*d^4*x*(39*A + x*(32*B + 13*C*x)) + c^4*d^2* 
(13*B + x*(32*C + 39*D*x)) + 4*c^3*d^3*(9*A + x*(13*B + 13*C*x + 9*D*x^2)) 
))/(c^4*d^4*(c + d*x)^4)
 

Rubi [A] (verified)

Time = 1.03 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.14, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {2170, 2170, 27, 671, 461, 461, 460}

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.

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

\(\Big \downarrow \) 2170

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

\(\Big \downarrow \) 2170

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 671

\(\displaystyle \frac {\frac {1}{2} d^2 \left (\frac {\left (6 A d^3+8 B c d^2+36 c^3 D+13 c^2 C d\right ) \int \frac {1}{(c+d x)^3 \sqrt {c^2-d^2 x^2}}dx}{7 c}-\frac {2 \sqrt {c^2-d^2 x^2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{7 c d (c+d x)^4}\right )+\frac {C d^2 \sqrt {c^2-d^2 x^2}}{2 (c+d x)^3}}{d^5}+\frac {D \sqrt {c^2-d^2 x^2}}{d^4 (c+d x)^2}\)

\(\Big \downarrow \) 461

\(\displaystyle \frac {\frac {1}{2} d^2 \left (\frac {\left (6 A d^3+8 B c d^2+36 c^3 D+13 c^2 C d\right ) \left (\frac {2 \int \frac {1}{(c+d x)^2 \sqrt {c^2-d^2 x^2}}dx}{5 c}-\frac {\sqrt {c^2-d^2 x^2}}{5 c d (c+d x)^3}\right )}{7 c}-\frac {2 \sqrt {c^2-d^2 x^2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{7 c d (c+d x)^4}\right )+\frac {C d^2 \sqrt {c^2-d^2 x^2}}{2 (c+d x)^3}}{d^5}+\frac {D \sqrt {c^2-d^2 x^2}}{d^4 (c+d x)^2}\)

\(\Big \downarrow \) 461

\(\displaystyle \frac {\frac {1}{2} d^2 \left (\frac {\left (6 A d^3+8 B c d^2+36 c^3 D+13 c^2 C d\right ) \left (\frac {2 \left (\frac {\int \frac {1}{(c+d x) \sqrt {c^2-d^2 x^2}}dx}{3 c}-\frac {\sqrt {c^2-d^2 x^2}}{3 c d (c+d x)^2}\right )}{5 c}-\frac {\sqrt {c^2-d^2 x^2}}{5 c d (c+d x)^3}\right )}{7 c}-\frac {2 \sqrt {c^2-d^2 x^2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{7 c d (c+d x)^4}\right )+\frac {C d^2 \sqrt {c^2-d^2 x^2}}{2 (c+d x)^3}}{d^5}+\frac {D \sqrt {c^2-d^2 x^2}}{d^4 (c+d x)^2}\)

\(\Big \downarrow \) 460

\(\displaystyle \frac {\frac {1}{2} d^2 \left (\frac {\left (\frac {2 \left (-\frac {\sqrt {c^2-d^2 x^2}}{3 c^2 d (c+d x)}-\frac {\sqrt {c^2-d^2 x^2}}{3 c d (c+d x)^2}\right )}{5 c}-\frac {\sqrt {c^2-d^2 x^2}}{5 c d (c+d x)^3}\right ) \left (6 A d^3+8 B c d^2+36 c^3 D+13 c^2 C d\right )}{7 c}-\frac {2 \sqrt {c^2-d^2 x^2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{7 c d (c+d x)^4}\right )+\frac {C d^2 \sqrt {c^2-d^2 x^2}}{2 (c+d x)^3}}{d^5}+\frac {D \sqrt {c^2-d^2 x^2}}{d^4 (c+d x)^2}\)

Input:

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

Output:

(D*Sqrt[c^2 - d^2*x^2])/(d^4*(c + d*x)^2) + ((C*d^2*Sqrt[c^2 - d^2*x^2])/( 
2*(c + d*x)^3) + (d^2*((-2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*Sqrt[c^2 - 
d^2*x^2])/(7*c*d*(c + d*x)^4) + ((13*c^2*C*d + 8*B*c*d^2 + 6*A*d^3 + 36*c^ 
3*D)*(-1/5*Sqrt[c^2 - d^2*x^2]/(c*d*(c + d*x)^3) + (2*(-1/3*Sqrt[c^2 - d^2 
*x^2]/(c*d*(c + d*x)^2) - Sqrt[c^2 - d^2*x^2]/(3*c^2*d*(c + d*x))))/(5*c)) 
)/(7*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 460

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(-d)*(c + d*x)^n*((a + b*x^2)^(p + 1)/(b*c*n)), x] /; FreeQ[{a, b, c, d, n, 
 p}, x] && EqQ[b*c^2 + a*d^2, 0] && EqQ[n + 2*p + 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 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 = 0.49 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.79

method	result	size
trager	\(-\frac {\left (6 A \,d^{6} x^{3}+8 B c \,d^{5} x^{3}+13 C \,c^{2} d^{4} x^{3}+36 D c^{3} d^{3} x^{3}+24 A c \,d^{5} x^{2}+32 B \,c^{2} d^{4} x^{2}+52 C \,c^{3} d^{3} x^{2}+39 D c^{4} d^{2} x^{2}+39 A \,c^{2} d^{4} x +52 B \,c^{3} d^{3} x +32 C \,c^{4} d^{2} x +24 D c^{5} d x +36 A \,c^{3} d^{3}+13 B \,c^{4} d^{2}+8 C \,c^{5} d +6 D c^{6}\right ) \sqrt {-d^{2} x^{2}+c^{2}}}{105 c^{4} d^{4} \left (d x +c \right )^{4}}\)	\(189\)
gosper	\(-\frac {\left (-d x +c \right ) \left (6 A \,d^{6} x^{3}+8 B c \,d^{5} x^{3}+13 C \,c^{2} d^{4} x^{3}+36 D c^{3} d^{3} x^{3}+24 A c \,d^{5} x^{2}+32 B \,c^{2} d^{4} x^{2}+52 C \,c^{3} d^{3} x^{2}+39 D c^{4} d^{2} x^{2}+39 A \,c^{2} d^{4} x +52 B \,c^{3} d^{3} x +32 C \,c^{4} d^{2} x +24 D c^{5} d x +36 A \,c^{3} d^{3}+13 B \,c^{4} d^{2}+8 C \,c^{5} d +6 D c^{6}\right )}{105 \left (d x +c \right )^{3} c^{4} d^{4} \sqrt {-d^{2} x^{2}+c^{2}}}\)	\(195\)
orering	\(-\frac {\left (-d x +c \right ) \left (6 A \,d^{6} x^{3}+8 B c \,d^{5} x^{3}+13 C \,c^{2} d^{4} x^{3}+36 D c^{3} d^{3} x^{3}+24 A c \,d^{5} x^{2}+32 B \,c^{2} d^{4} x^{2}+52 C \,c^{3} d^{3} x^{2}+39 D c^{4} d^{2} x^{2}+39 A \,c^{2} d^{4} x +52 B \,c^{3} d^{3} x +32 C \,c^{4} d^{2} x +24 D c^{5} d x +36 A \,c^{3} d^{3}+13 B \,c^{4} d^{2}+8 C \,c^{5} d +6 D c^{6}\right )}{105 \left (d x +c \right )^{3} c^{4} d^{4} \sqrt {-d^{2} x^{2}+c^{2}}}\)	\(195\)
default	\(-\frac {D \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}{d^{5} c \left (x +\frac {c}{d}\right )}+\frac {\left (C d -3 D c \right ) \left (-\frac {\sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}{3 c d \left (x +\frac {c}{d}\right )^{2}}-\frac {\sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}{3 c^{2} \left (x +\frac {c}{d}\right )}\right )}{d^{5}}+\frac {\left (B \,d^{2}-2 C c d +3 D c^{2}\right ) \left (-\frac {\sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}{5 c d \left (x +\frac {c}{d}\right )^{3}}+\frac {2 d \left (-\frac {\sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}{3 c d \left (x +\frac {c}{d}\right )^{2}}-\frac {\sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}{3 c^{2} \left (x +\frac {c}{d}\right )}\right )}{5 c}\right )}{d^{6}}+\frac {\left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right ) \left (-\frac {\sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}{7 c d \left (x +\frac {c}{d}\right )^{4}}+\frac {3 d \left (-\frac {\sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}{5 c d \left (x +\frac {c}{d}\right )^{3}}+\frac {2 d \left (-\frac {\sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}{3 c d \left (x +\frac {c}{d}\right )^{2}}-\frac {\sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}{3 c^{2} \left (x +\frac {c}{d}\right )}\right )}{5 c}\right )}{7 c}\right )}{d^{7}}\)	\(530\)

Input:

int((D*x^3+C*x^2+B*x+A)/(d*x+c)^4/(-d^2*x^2+c^2)^(1/2),x,method=_RETURNVER 
BOSE)
 

Output:

-1/105*(6*A*d^6*x^3+8*B*c*d^5*x^3+13*C*c^2*d^4*x^3+36*D*c^3*d^3*x^3+24*A*c 
*d^5*x^2+32*B*c^2*d^4*x^2+52*C*c^3*d^3*x^2+39*D*c^4*d^2*x^2+39*A*c^2*d^4*x 
+52*B*c^3*d^3*x+32*C*c^4*d^2*x+24*D*c^5*d*x+36*A*c^3*d^3+13*B*c^4*d^2+8*C* 
c^5*d+6*D*c^6)/c^4/d^4/(d*x+c)^4*(-d^2*x^2+c^2)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.67 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^4 \sqrt {c^2-d^2 x^2}} \, dx=-\frac {6 \, D c^{7} + 8 \, C c^{6} d + 13 \, B c^{5} d^{2} + 36 \, A c^{4} d^{3} + {\left (6 \, D c^{3} d^{4} + 8 \, C c^{2} d^{5} + 13 \, B c d^{6} + 36 \, A d^{7}\right )} x^{4} + 4 \, {\left (6 \, D c^{4} d^{3} + 8 \, C c^{3} d^{4} + 13 \, B c^{2} d^{5} + 36 \, A c d^{6}\right )} x^{3} + 6 \, {\left (6 \, D c^{5} d^{2} + 8 \, C c^{4} d^{3} + 13 \, B c^{3} d^{4} + 36 \, A c^{2} d^{5}\right )} x^{2} + 4 \, {\left (6 \, D c^{6} d + 8 \, C c^{5} d^{2} + 13 \, B c^{4} d^{3} + 36 \, A c^{3} d^{4}\right )} x + {\left (6 \, D c^{6} + 8 \, C c^{5} d + 13 \, B c^{4} d^{2} + 36 \, A c^{3} d^{3} + {\left (36 \, D c^{3} d^{3} + 13 \, C c^{2} d^{4} + 8 \, B c d^{5} + 6 \, A d^{6}\right )} x^{3} + {\left (39 \, D c^{4} d^{2} + 52 \, C c^{3} d^{3} + 32 \, B c^{2} d^{4} + 24 \, A c d^{5}\right )} x^{2} + {\left (24 \, D c^{5} d + 32 \, C c^{4} d^{2} + 52 \, B c^{3} d^{3} + 39 \, A c^{2} d^{4}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{105 \, {\left (c^{4} d^{8} x^{4} + 4 \, c^{5} d^{7} x^{3} + 6 \, c^{6} d^{6} x^{2} + 4 \, c^{7} d^{5} x + c^{8} d^{4}\right )}} \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(d*x+c)^4/(-d^2*x^2+c^2)^(1/2),x, algorithm= 
"fricas")
 

Output:

-1/105*(6*D*c^7 + 8*C*c^6*d + 13*B*c^5*d^2 + 36*A*c^4*d^3 + (6*D*c^3*d^4 + 
 8*C*c^2*d^5 + 13*B*c*d^6 + 36*A*d^7)*x^4 + 4*(6*D*c^4*d^3 + 8*C*c^3*d^4 + 
 13*B*c^2*d^5 + 36*A*c*d^6)*x^3 + 6*(6*D*c^5*d^2 + 8*C*c^4*d^3 + 13*B*c^3* 
d^4 + 36*A*c^2*d^5)*x^2 + 4*(6*D*c^6*d + 8*C*c^5*d^2 + 13*B*c^4*d^3 + 36*A 
*c^3*d^4)*x + (6*D*c^6 + 8*C*c^5*d + 13*B*c^4*d^2 + 36*A*c^3*d^3 + (36*D*c 
^3*d^3 + 13*C*c^2*d^4 + 8*B*c*d^5 + 6*A*d^6)*x^3 + (39*D*c^4*d^2 + 52*C*c^ 
3*d^3 + 32*B*c^2*d^4 + 24*A*c*d^5)*x^2 + (24*D*c^5*d + 32*C*c^4*d^2 + 52*B 
*c^3*d^3 + 39*A*c^2*d^4)*x)*sqrt(-d^2*x^2 + c^2))/(c^4*d^8*x^4 + 4*c^5*d^7 
*x^3 + 6*c^6*d^6*x^2 + 4*c^7*d^5*x + c^8*d^4)
 

Sympy [F]

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

integrate((D*x**3+C*x**2+B*x+A)/(d*x+c)**4/(-d**2*x**2+c**2)**(1/2),x)
 

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1448 vs. \(2 (223) = 446\).

Time = 0.13 (sec) , antiderivative size = 1448, normalized size of antiderivative = 6.06 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^4 \sqrt {c^2-d^2 x^2}} \, dx=\text {Too large to display} \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(d*x+c)^4/(-d^2*x^2+c^2)^(1/2),x, algorithm= 
"maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 660 vs. \(2 (223) = 446\).

Time = 0.14 (sec) , antiderivative size = 660, normalized size of antiderivative = 2.76 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^4 \sqrt {c^2-d^2 x^2}} \, dx =\text {Too large to display} \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(d*x+c)^4/(-d^2*x^2+c^2)^(1/2),x, algorithm= 
"giac")
 

Output:

2/105*(6*D*c^3 + 8*C*c^2*d + 13*B*c*d^2 + 36*A*d^3 + 91*(c*d + sqrt(-d^2*x 
^2 + c^2)*abs(d))*B*c/x + 42*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))*D*c^3/(d^ 
2*x) + 56*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))*C*c^2/(d*x) + 147*(c*d + sqr 
t(-d^2*x^2 + c^2)*abs(d))*A*d/x + 126*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^ 
2*D*c^3/(d^4*x^2) + 168*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^2*C*c^2/(d^3*x 
^2) + 168*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^2*B*c/(d^2*x^2) + 441*(c*d + 
 sqrt(-d^2*x^2 + c^2)*abs(d))^2*A/(d*x^2) + 210*(c*d + sqrt(-d^2*x^2 + c^2 
)*abs(d))^3*D*c^3/(d^6*x^3) + 140*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^3*C* 
c^2/(d^5*x^3) + 280*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^3*B*c/(d^4*x^3) + 
630*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^3*A/(d^3*x^3) + 140*(c*d + sqrt(-d 
^2*x^2 + c^2)*abs(d))^4*C*c^2/(d^7*x^4) + 175*(c*d + sqrt(-d^2*x^2 + c^2)* 
abs(d))^4*B*c/(d^6*x^4) + 630*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^4*A/(d^5 
*x^4) + 105*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^5*B*c/(d^8*x^5) + 315*(c*d 
 + sqrt(-d^2*x^2 + c^2)*abs(d))^5*A/(d^7*x^5) + 105*(c*d + sqrt(-d^2*x^2 + 
 c^2)*abs(d))^6*A/(d^9*x^6))/(c^4*d^3*((c*d + sqrt(-d^2*x^2 + c^2)*abs(d)) 
/(d^2*x) + 1)^7*abs(d))
 

Mupad [F(-1)]

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

int((A + B*x + C*x^2 + x^3*D)/((c^2 - d^2*x^2)^(1/2)*(c + d*x)^4),x)
 

Output:

int((A + B*x + C*x^2 + x^3*D)/((c^2 - d^2*x^2)^(1/2)*(c + d*x)^4), x)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.72 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^4 \sqrt {c^2-d^2 x^2}} \, dx=\frac {\sqrt {-d^{2} x^{2}+c^{2}}\, \left (-6 a \,d^{5} x^{3}-8 b c \,d^{4} x^{3}-49 c^{3} d^{3} x^{3}-24 a c \,d^{4} x^{2}-32 b \,c^{2} d^{3} x^{2}-91 c^{4} d^{2} x^{2}-39 a \,c^{2} d^{3} x -52 b \,c^{3} d^{2} x -56 c^{5} d x -36 a \,c^{3} d^{2}-13 b \,c^{4} d -14 c^{6}\right )}{105 c^{4} d^{3} \left (d^{4} x^{4}+4 c \,d^{3} x^{3}+6 c^{2} d^{2} x^{2}+4 c^{3} d x +c^{4}\right )} \] Input:

int((D*x^3+C*x^2+B*x+A)/(d*x+c)^4/(-d^2*x^2+c^2)^(1/2),x)
 

Output:

(sqrt(c**2 - d**2*x**2)*( - 36*a*c**3*d**2 - 39*a*c**2*d**3*x - 24*a*c*d** 
4*x**2 - 6*a*d**5*x**3 - 13*b*c**4*d - 52*b*c**3*d**2*x - 32*b*c**2*d**3*x 
**2 - 8*b*c*d**4*x**3 - 14*c**6 - 56*c**5*d*x - 91*c**4*d**2*x**2 - 49*c** 
3*d**3*x**3))/(105*c**4*d**3*(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d 
**3*x**3 + d**4*x**4))