\(\int \frac {(c^2-d^2 x^2)^{3/2} (A+B x+C x^2+D x^3)}{(c+d x)^5} \, dx\) [148]

Optimal result

Integrand size = 39, antiderivative size = 252 \[ \int \frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^5} \, dx=-\frac {(C d-5 c D) \sqrt {c^2-d^2 x^2}}{d^4}-\frac {D x \sqrt {c^2-d^2 x^2}}{2 d^3}-\frac {2 \left (4 c C d-B d^2-9 c^2 D\right ) \sqrt {c^2-d^2 x^2}}{d^4 (c+d x)}+\frac {2 \left (2 c C d-B d^2-3 c^2 D\right ) \left (c^2-d^2 x^2\right )^{3/2}}{3 d^4 (c+d x)^3}-\frac {\left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \left (c^2-d^2 x^2\right )^{5/2}}{5 c d^4 (c+d x)^5}-\frac {\left (10 c C d-2 B d^2-27 c^2 D\right ) \arctan \left (\frac {d x}{\sqrt {c^2-d^2 x^2}}\right )}{2 d^4} \] Output:

-(C*d-5*D*c)*(-d^2*x^2+c^2)^(1/2)/d^4-1/2*D*x*(-d^2*x^2+c^2)^(1/2)/d^3-2*( 
-B*d^2+4*C*c*d-9*D*c^2)*(-d^2*x^2+c^2)^(1/2)/d^4/(d*x+c)+2/3*(-B*d^2+2*C*c 
*d-3*D*c^2)*(-d^2*x^2+c^2)^(3/2)/d^4/(d*x+c)^3-1/5*(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)^5-1/2*(-2*B*d^2+10*C*c*d-27*D*c^ 
2)*arctan(d*x/(-d^2*x^2+c^2)^(1/2))/d^4
 

Mathematica [A] (verified)

Time = 2.00 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.76 \[ \int \frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^5} \, dx=\frac {\frac {\sqrt {c^2-d^2 x^2} \left (636 c^5 D-6 A d^5 x^2+c^4 (-236 C d+1503 d D x)+c^3 d^2 (46 B+3 x (-186 C+337 D x))+c d^4 x (12 A+x (86 B-15 x (2 C+D x)))+c^2 d^3 (-6 A+x (108 B+x (-376 C+105 D x)))\right )}{c (c+d x)^3}-30 \left (-10 c C d+2 B d^2+27 c^2 D\right ) \arctan \left (\frac {d x}{\sqrt {c^2}-\sqrt {c^2-d^2 x^2}}\right )}{30 d^4} \] Input:

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

Output:

((Sqrt[c^2 - d^2*x^2]*(636*c^5*D - 6*A*d^5*x^2 + c^4*(-236*C*d + 1503*d*D* 
x) + c^3*d^2*(46*B + 3*x*(-186*C + 337*D*x)) + c*d^4*x*(12*A + x*(86*B - 1 
5*x*(2*C + D*x))) + c^2*d^3*(-6*A + x*(108*B + x*(-376*C + 105*D*x)))))/(c 
*(c + d*x)^3) - 30*(-10*c*C*d + 2*B*d^2 + 27*c^2*D)*ArcTan[(d*x)/(Sqrt[c^2 
] - Sqrt[c^2 - d^2*x^2])])/(30*d^4)
 

Rubi [A] (verified)

Time = 1.08 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.96, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.256, Rules used = {2170, 25, 2170, 27, 671, 465, 463, 25, 224, 216}

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.

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

\(\Big \downarrow \) 2170

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2170

\(\displaystyle \frac {-\frac {\int \frac {d^6 \left (-25 D c^3+8 C d c^2-2 A d^3+d \left (-27 D c^2+10 C d c-2 B d^2\right ) x\right ) \left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^5}dx}{d^4}-\frac {d \left (c^2-d^2 x^2\right )^{5/2} (2 C d-7 c D)}{(c+d x)^4}}{2 d^5}-\frac {D \left (c^2-d^2 x^2\right )^{5/2}}{2 d^4 (c+d x)^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-d^2 \int \frac {\left (-25 D c^3+8 C d c^2-2 A d^3+d \left (-27 D c^2+10 C d c-2 B d^2\right ) x\right ) \left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^5}dx-\frac {d \left (c^2-d^2 x^2\right )^{5/2} (2 C d-7 c D)}{(c+d x)^4}}{2 d^5}-\frac {D \left (c^2-d^2 x^2\right )^{5/2}}{2 d^4 (c+d x)^3}\)

\(\Big \downarrow \) 671

\(\displaystyle \frac {-d^2 \left (\left (-2 B d^2-27 c^2 D+10 c C d\right ) \int \frac {\left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^4}dx+\frac {2 \left (c^2-d^2 x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 c d (c+d x)^5}\right )-\frac {d \left (c^2-d^2 x^2\right )^{5/2} (2 C d-7 c D)}{(c+d x)^4}}{2 d^5}-\frac {D \left (c^2-d^2 x^2\right )^{5/2}}{2 d^4 (c+d x)^3}\)

\(\Big \downarrow \) 465

\(\displaystyle \frac {-d^2 \left (\left (-2 B d^2-27 c^2 D+10 c C d\right ) \left (-\int \frac {\sqrt {c^2-d^2 x^2}}{(c+d x)^2}dx-\frac {2 \left (c^2-d^2 x^2\right )^{3/2}}{3 d (c+d x)^3}\right )+\frac {2 \left (c^2-d^2 x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 c d (c+d x)^5}\right )-\frac {d \left (c^2-d^2 x^2\right )^{5/2} (2 C d-7 c D)}{(c+d x)^4}}{2 d^5}-\frac {D \left (c^2-d^2 x^2\right )^{5/2}}{2 d^4 (c+d x)^3}\)

\(\Big \downarrow \) 463

\(\displaystyle \frac {-d^2 \left (\left (-2 B d^2-27 c^2 D+10 c C d\right ) \left (-\int -\frac {1}{\sqrt {c^2-d^2 x^2}}dx-\frac {2 \left (c^2-d^2 x^2\right )^{3/2}}{3 d (c+d x)^3}+\frac {2 \sqrt {c^2-d^2 x^2}}{d (c+d x)}\right )+\frac {2 \left (c^2-d^2 x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 c d (c+d x)^5}\right )-\frac {d \left (c^2-d^2 x^2\right )^{5/2} (2 C d-7 c D)}{(c+d x)^4}}{2 d^5}-\frac {D \left (c^2-d^2 x^2\right )^{5/2}}{2 d^4 (c+d x)^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {-d^2 \left (\left (-2 B d^2-27 c^2 D+10 c C d\right ) \left (\int \frac {1}{\sqrt {c^2-d^2 x^2}}dx-\frac {2 \left (c^2-d^2 x^2\right )^{3/2}}{3 d (c+d x)^3}+\frac {2 \sqrt {c^2-d^2 x^2}}{d (c+d x)}\right )+\frac {2 \left (c^2-d^2 x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 c d (c+d x)^5}\right )-\frac {d \left (c^2-d^2 x^2\right )^{5/2} (2 C d-7 c D)}{(c+d x)^4}}{2 d^5}-\frac {D \left (c^2-d^2 x^2\right )^{5/2}}{2 d^4 (c+d x)^3}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {-d^2 \left (\left (-2 B d^2-27 c^2 D+10 c C d\right ) \left (\int \frac {1}{\frac {d^2 x^2}{c^2-d^2 x^2}+1}d\frac {x}{\sqrt {c^2-d^2 x^2}}-\frac {2 \left (c^2-d^2 x^2\right )^{3/2}}{3 d (c+d x)^3}+\frac {2 \sqrt {c^2-d^2 x^2}}{d (c+d x)}\right )+\frac {2 \left (c^2-d^2 x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 c d (c+d x)^5}\right )-\frac {d \left (c^2-d^2 x^2\right )^{5/2} (2 C d-7 c D)}{(c+d x)^4}}{2 d^5}-\frac {D \left (c^2-d^2 x^2\right )^{5/2}}{2 d^4 (c+d x)^3}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {-d^2 \left (\frac {2 \left (c^2-d^2 x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 c d (c+d x)^5}+\left (\frac {\arctan \left (\frac {d x}{\sqrt {c^2-d^2 x^2}}\right )}{d}-\frac {2 \left (c^2-d^2 x^2\right )^{3/2}}{3 d (c+d x)^3}+\frac {2 \sqrt {c^2-d^2 x^2}}{d (c+d x)}\right ) \left (-2 B d^2-27 c^2 D+10 c C d\right )\right )-\frac {d \left (c^2-d^2 x^2\right )^{5/2} (2 C d-7 c D)}{(c+d x)^4}}{2 d^5}-\frac {D \left (c^2-d^2 x^2\right )^{5/2}}{2 d^4 (c+d x)^3}\)

Input:

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

Output:

-1/2*(D*(c^2 - d^2*x^2)^(5/2))/(d^4*(c + d*x)^3) + (-((d*(2*C*d - 7*c*D)*( 
c^2 - d^2*x^2)^(5/2))/(c + d*x)^4) - d^2*((2*(c^2*C*d - B*c*d^2 + A*d^3 - 
c^3*D)*(c^2 - d^2*x^2)^(5/2))/(5*c*d*(c + d*x)^5) + (10*c*C*d - 2*B*d^2 - 
27*c^2*D)*((2*Sqrt[c^2 - d^2*x^2])/(d*(c + d*x)) - (2*(c^2 - d^2*x^2)^(3/2 
))/(3*d*(c + d*x)^3) + ArcTan[(d*x)/Sqrt[c^2 - d^2*x^2]]/d)))/(2*d^5)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 463

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(-(-c)^(-n - 2))*d^(2*n + 3)*(Sqrt[a + b*x^2]/(2^(n + 1)*b^(n + 2)*(c + d*x 
))), x] - Simp[d^(2*n + 2)/b^(n + 1)   Int[(1/Sqrt[a + b*x^2])*ExpandToSum[ 
(2^(-n - 1)*(-c)^(-n - 1) - (-c + d*x)^(-n - 1))/(c + d*x), x], x], x] /; F 
reeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0] && ILtQ[n, 0] && EqQ[n + p, 
-3/2]
 

rule 465

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(c + d*x)^(n + 1)*((a + b*x^2)^p/(d*(n + p + 1))), x] - Simp[b*(p/(d^2*(n + 
 p + 1)))   Int[(c + d*x)^(n + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, 
b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0] && GtQ[p, 0] && (LtQ[n, -2] || EqQ[n 
+ 2*p + 1, 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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(811\) vs. \(2(232)=464\).

Time = 0.52 (sec) , antiderivative size = 812, normalized size of antiderivative = 3.22

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 )^{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 )}{d^{5}}+\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}}}{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 (B \,d^{2}-2 C c d +3 D c^{2}\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 (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\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}}\)	\(812\)

Input:

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

Output:

D/d^5*(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^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))))))+(B*d^2-2*C*c*d+3*D*c^2)/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)*arctan((d^2) 
^(1/2)*x/(-d^2*(x+c/d)^2+2*c*d*(x+c/d))^(1/2)))))))-1/5*(A*d^3-B*c*d^2+C*c 
^2*d-D*c^3)/d^9/c/(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. 484 vs. \(2 (230) = 460\).

Time = 0.11 (sec) , antiderivative size = 484, normalized size of antiderivative = 1.92 \[ \int \frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^5} \, dx=\frac {636 \, D c^{6} - 236 \, C c^{5} d + 46 \, B c^{4} d^{2} - 6 \, A c^{3} d^{3} + 2 \, {\left (318 \, D c^{3} d^{3} - 118 \, C c^{2} d^{4} + 23 \, B c d^{5} - 3 \, A d^{6}\right )} x^{3} + 6 \, {\left (318 \, D c^{4} d^{2} - 118 \, C c^{3} d^{3} + 23 \, B c^{2} d^{4} - 3 \, A c d^{5}\right )} x^{2} + 6 \, {\left (318 \, D c^{5} d - 118 \, C c^{4} d^{2} + 23 \, B c^{3} d^{3} - 3 \, A c^{2} d^{4}\right )} x - 30 \, {\left (27 \, D c^{6} - 10 \, C c^{5} d + 2 \, B c^{4} d^{2} + {\left (27 \, D c^{3} d^{3} - 10 \, C c^{2} d^{4} + 2 \, B c d^{5}\right )} x^{3} + 3 \, {\left (27 \, D c^{4} d^{2} - 10 \, C c^{3} d^{3} + 2 \, B c^{2} d^{4}\right )} x^{2} + 3 \, {\left (27 \, D c^{5} d - 10 \, C c^{4} d^{2} + 2 \, B c^{3} d^{3}\right )} x\right )} \arctan \left (-\frac {c - \sqrt {-d^{2} x^{2} + c^{2}}}{d x}\right ) - {\left (15 \, D c d^{4} x^{4} - 636 \, D c^{5} + 236 \, C c^{4} d - 46 \, B c^{3} d^{2} + 6 \, A c^{2} d^{3} - 15 \, {\left (7 \, D c^{2} d^{3} - 2 \, C c d^{4}\right )} x^{3} - {\left (1011 \, D c^{3} d^{2} - 376 \, C c^{2} d^{3} + 86 \, B c d^{4} - 6 \, A d^{5}\right )} x^{2} - 3 \, {\left (501 \, D c^{4} d - 186 \, C c^{3} d^{2} + 36 \, B c^{2} d^{3} + 4 \, A c d^{4}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{30 \, {\left (c d^{7} x^{3} + 3 \, c^{2} d^{6} x^{2} + 3 \, c^{3} d^{5} x + c^{4} d^{4}\right )}} \] Input:

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

Output:

1/30*(636*D*c^6 - 236*C*c^5*d + 46*B*c^4*d^2 - 6*A*c^3*d^3 + 2*(318*D*c^3* 
d^3 - 118*C*c^2*d^4 + 23*B*c*d^5 - 3*A*d^6)*x^3 + 6*(318*D*c^4*d^2 - 118*C 
*c^3*d^3 + 23*B*c^2*d^4 - 3*A*c*d^5)*x^2 + 6*(318*D*c^5*d - 118*C*c^4*d^2 
+ 23*B*c^3*d^3 - 3*A*c^2*d^4)*x - 30*(27*D*c^6 - 10*C*c^5*d + 2*B*c^4*d^2 
+ (27*D*c^3*d^3 - 10*C*c^2*d^4 + 2*B*c*d^5)*x^3 + 3*(27*D*c^4*d^2 - 10*C*c 
^3*d^3 + 2*B*c^2*d^4)*x^2 + 3*(27*D*c^5*d - 10*C*c^4*d^2 + 2*B*c^3*d^3)*x) 
*arctan(-(c - sqrt(-d^2*x^2 + c^2))/(d*x)) - (15*D*c*d^4*x^4 - 636*D*c^5 + 
 236*C*c^4*d - 46*B*c^3*d^2 + 6*A*c^2*d^3 - 15*(7*D*c^2*d^3 - 2*C*c*d^4)*x 
^3 - (1011*D*c^3*d^2 - 376*C*c^2*d^3 + 86*B*c*d^4 - 6*A*d^5)*x^2 - 3*(501* 
D*c^4*d - 186*C*c^3*d^2 + 36*B*c^2*d^3 + 4*A*c*d^4)*x)*sqrt(-d^2*x^2 + c^2 
))/(c*d^7*x^3 + 3*c^2*d^6*x^2 + 3*c^3*d^5*x + c^4*d^4)
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1207 vs. \(2 (230) = 460\).

Time = 0.14 (sec) , antiderivative size = 1207, normalized size of antiderivative = 4.79 \[ \int \frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^5} \, dx=\text {Too large to display} \] Input:

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

Output:

(-d^2*x^2 + c^2)^(3/2)*D*c^3/(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) - 6/5*sqrt(-d^2*x^2 + c^2)*D*c^4/(d^7*x^3 + 3*c*d^6*x^2 
 + 3*c^2*d^5*x + c^3*d^4) - (-d^2*x^2 + c^2)^(3/2)*C*c^2/(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^2/(d^7*x^3 + 3*c*d^6*x^2 + 3*c^2*d^5*x + c^3*d^4) + 6/5*sqrt(-d^2*x^2 + 
c^2)*C*c^3/(d^6*x^3 + 3*c*d^5*x^2 + 3*c^2*d^4*x + c^3*d^3) - 9/5*sqrt(-d^2 
*x^2 + c^2)*D*c^3/(d^6*x^2 + 2*c*d^5*x + c^2*d^4) + 1/5*sqrt(-d^2*x^2 + c^ 
2)*D*c^3/(c*d^5*x + c^2*d^4) + (-d^2*x^2 + c^2)^(3/2)*B*c/(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) + 2/3*(-d^2*x^2 + c^2)^(3/ 
2)*C*c/(d^6*x^3 + 3*c*d^5*x^2 + 3*c^2*d^4*x + c^3*d^3) - 3*(-d^2*x^2 + c^2 
)^(3/2)*D*c/(d^6*x^2 + 2*c*d^5*x + c^2*d^4) - 6/5*sqrt(-d^2*x^2 + c^2)*B*c 
^2/(d^5*x^3 + 3*c*d^4*x^2 + 3*c^2*d^3*x + c^3*d^2) + 17/15*sqrt(-d^2*x^2 + 
 c^2)*C*c^2/(d^5*x^2 + 2*c*d^4*x + c^2*d^3) - 1/5*sqrt(-d^2*x^2 + c^2)*C*c 
^2/(c*d^4*x + c^2*d^3) + 25*sqrt(-d^2*x^2 + c^2)*D*c^2/(d^5*x + c*d^4) - ( 
-d^2*x^2 + c^2)^(3/2)*A/(d^5*x^4 + 4*c*d^4*x^3 + 6*c^2*d^3*x^2 + 4*c^3*d^2 
*x + c^4*d) - 1/3*(-d^2*x^2 + c^2)^(3/2)*B/(d^5*x^3 + 3*c*d^4*x^2 + 3*c^2* 
d^3*x + c^3*d^2) + (-d^2*x^2 + c^2)^(3/2)*C/(d^5*x^2 + 2*c*d^4*x + c^2*d^3 
) + 1/2*(-d^2*x^2 + c^2)^(3/2)*D/(d^5*x + c*d^4) + 6/5*sqrt(-d^2*x^2 + c^2 
)*A*c/(d^4*x^3 + 3*c*d^3*x^2 + 3*c^2*d^2*x + c^3*d) - 7/15*sqrt(-d^2*x^2 + 
 c^2)*B*c/(d^4*x^2 + 2*c*d^3*x + c^2*d^2) + 1/5*sqrt(-d^2*x^2 + c^2)*B*...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 550 vs. \(2 (230) = 460\).

Time = 0.19 (sec) , antiderivative size = 550, normalized size of antiderivative = 2.18 \[ \int \frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^5} \, dx =\text {Too large to display} \] Input:

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

Output:

1/60*(12*D*c^3*d^3*(2*c/(d*x + c) - 1)^(5/2)*sgn(1/(d*x + c))*sgn(d) - 12* 
C*c^2*d^4*(2*c/(d*x + c) - 1)^(5/2)*sgn(1/(d*x + c))*sgn(d) + 12*B*c*d^5*( 
2*c/(d*x + c) - 1)^(5/2)*sgn(1/(d*x + c))*sgn(d) - 12*A*d^6*(2*c/(d*x + c) 
 - 1)^(5/2)*sgn(1/(d*x + c))*sgn(d) - 120*D*c^3*d^3*(2*c/(d*x + c) - 1)^(3 
/2)*sgn(1/(d*x + c))*sgn(d) + 80*C*c^2*d^4*(2*c/(d*x + c) - 1)^(3/2)*sgn(1 
/(d*x + c))*sgn(d) - 40*B*c*d^5*(2*c/(d*x + c) - 1)^(3/2)*sgn(1/(d*x + c)) 
*sgn(d) + 1080*D*c^3*d^3*sqrt(2*c/(d*x + c) - 1)*sgn(1/(d*x + c))*sgn(d) - 
 480*C*c^2*d^4*sqrt(2*c/(d*x + c) - 1)*sgn(1/(d*x + c))*sgn(d) + 120*B*c*d 
^5*sqrt(2*c/(d*x + c) - 1)*sgn(1/(d*x + c))*sgn(d) - 60*(27*D*c^3*d^3*sgn( 
1/(d*x + c))*sgn(d) - 10*C*c^2*d^4*sgn(1/(d*x + c))*sgn(d) + 2*B*c*d^5*sgn 
(1/(d*x + c))*sgn(d))*arctan(sqrt(2*c/(d*x + c) - 1)) + 15*(11*D*c^3*d^3*( 
2*c/(d*x + c) - 1)^(3/2)*sgn(1/(d*x + c))*sgn(d) - 2*C*c^2*d^4*(2*c/(d*x + 
 c) - 1)^(3/2)*sgn(1/(d*x + c))*sgn(d) + 9*D*c^3*d^3*sqrt(2*c/(d*x + c) - 
1)*sgn(1/(d*x + c))*sgn(d) - 2*C*c^2*d^4*sqrt(2*c/(d*x + c) - 1)*sgn(1/(d* 
x + c))*sgn(d))*(d*x + c)^2/c^2)*abs(d)/(c*d^8)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 822, normalized size of antiderivative = 3.26 \[ \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)^5} \, dx=\frac {-30 \mathit {atan} \left (\frac {\sqrt {-d^{2} x^{2}+c^{2}}\, c^{2}-2 \sqrt {-d^{2} x^{2}+c^{2}}\, d^{2} x^{2}}{-2 d^{3} x^{3}+2 c^{2} d x}\right ) b \,c^{4} d -90 \mathit {atan} \left (\frac {\sqrt {-d^{2} x^{2}+c^{2}}\, c^{2}-2 \sqrt {-d^{2} x^{2}+c^{2}}\, d^{2} x^{2}}{-2 d^{3} x^{3}+2 c^{2} d x}\right ) b \,c^{3} d^{2} x -90 \mathit {atan} \left (\frac {\sqrt {-d^{2} x^{2}+c^{2}}\, c^{2}-2 \sqrt {-d^{2} x^{2}+c^{2}}\, d^{2} x^{2}}{-2 d^{3} x^{3}+2 c^{2} d x}\right ) b \,c^{2} d^{3} x^{2}-30 \mathit {atan} \left (\frac {\sqrt {-d^{2} x^{2}+c^{2}}\, c^{2}-2 \sqrt {-d^{2} x^{2}+c^{2}}\, d^{2} x^{2}}{-2 d^{3} x^{3}+2 c^{2} d x}\right ) b c \,d^{4} x^{3}-255 \mathit {atan} \left (\frac {\sqrt {-d^{2} x^{2}+c^{2}}\, c^{2}-2 \sqrt {-d^{2} x^{2}+c^{2}}\, d^{2} x^{2}}{-2 d^{3} x^{3}+2 c^{2} d x}\right ) c^{6}-765 \mathit {atan} \left (\frac {\sqrt {-d^{2} x^{2}+c^{2}}\, c^{2}-2 \sqrt {-d^{2} x^{2}+c^{2}}\, d^{2} x^{2}}{-2 d^{3} x^{3}+2 c^{2} d x}\right ) c^{5} d x -765 \mathit {atan} \left (\frac {\sqrt {-d^{2} x^{2}+c^{2}}\, c^{2}-2 \sqrt {-d^{2} x^{2}+c^{2}}\, d^{2} x^{2}}{-2 d^{3} x^{3}+2 c^{2} d x}\right ) c^{4} d^{2} x^{2}-255 \mathit {atan} \left (\frac {\sqrt {-d^{2} x^{2}+c^{2}}\, c^{2}-2 \sqrt {-d^{2} x^{2}+c^{2}}\, d^{2} x^{2}}{-2 d^{3} x^{3}+2 c^{2} d x}\right ) c^{3} d^{3} x^{3}-12 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{2} d^{2}+24 \sqrt {-d^{2} x^{2}+c^{2}}\, a c \,d^{3} x -12 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,d^{4} x^{2}+92 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{3} d +216 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{2} d^{2} x +172 \sqrt {-d^{2} x^{2}+c^{2}}\, b c \,d^{3} x^{2}+800 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{5}+1890 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{4} d x +1270 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{3} d^{2} x^{2}+150 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{2} d^{3} x^{3}-30 \sqrt {-d^{2} x^{2}+c^{2}}\, c \,d^{4} x^{4}}{60 c \,d^{3} \left (d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}\right )} \] Input:

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

Output:

( - 30*atan((sqrt(c**2 - d**2*x**2)*c**2 - 2*sqrt(c**2 - d**2*x**2)*d**2*x 
**2)/(2*c**2*d*x - 2*d**3*x**3))*b*c**4*d - 90*atan((sqrt(c**2 - d**2*x**2 
)*c**2 - 2*sqrt(c**2 - d**2*x**2)*d**2*x**2)/(2*c**2*d*x - 2*d**3*x**3))*b 
*c**3*d**2*x - 90*atan((sqrt(c**2 - d**2*x**2)*c**2 - 2*sqrt(c**2 - d**2*x 
**2)*d**2*x**2)/(2*c**2*d*x - 2*d**3*x**3))*b*c**2*d**3*x**2 - 30*atan((sq 
rt(c**2 - d**2*x**2)*c**2 - 2*sqrt(c**2 - d**2*x**2)*d**2*x**2)/(2*c**2*d* 
x - 2*d**3*x**3))*b*c*d**4*x**3 - 255*atan((sqrt(c**2 - d**2*x**2)*c**2 - 
2*sqrt(c**2 - d**2*x**2)*d**2*x**2)/(2*c**2*d*x - 2*d**3*x**3))*c**6 - 765 
*atan((sqrt(c**2 - d**2*x**2)*c**2 - 2*sqrt(c**2 - d**2*x**2)*d**2*x**2)/( 
2*c**2*d*x - 2*d**3*x**3))*c**5*d*x - 765*atan((sqrt(c**2 - d**2*x**2)*c** 
2 - 2*sqrt(c**2 - d**2*x**2)*d**2*x**2)/(2*c**2*d*x - 2*d**3*x**3))*c**4*d 
**2*x**2 - 255*atan((sqrt(c**2 - d**2*x**2)*c**2 - 2*sqrt(c**2 - d**2*x**2 
)*d**2*x**2)/(2*c**2*d*x - 2*d**3*x**3))*c**3*d**3*x**3 - 12*sqrt(c**2 - d 
**2*x**2)*a*c**2*d**2 + 24*sqrt(c**2 - d**2*x**2)*a*c*d**3*x - 12*sqrt(c** 
2 - d**2*x**2)*a*d**4*x**2 + 92*sqrt(c**2 - d**2*x**2)*b*c**3*d + 216*sqrt 
(c**2 - d**2*x**2)*b*c**2*d**2*x + 172*sqrt(c**2 - d**2*x**2)*b*c*d**3*x** 
2 + 800*sqrt(c**2 - d**2*x**2)*c**5 + 1890*sqrt(c**2 - d**2*x**2)*c**4*d*x 
 + 1270*sqrt(c**2 - d**2*x**2)*c**3*d**2*x**2 + 150*sqrt(c**2 - d**2*x**2) 
*c**2*d**3*x**3 - 30*sqrt(c**2 - d**2*x**2)*c*d**4*x**4)/(60*c*d**3*(c**3 
+ 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3))