\(\int \frac {(c^2-d^2 x^2)^{3/2} (A+B x+C x^2+D x^3)}{(c+d x)^{11/2}} \, dx\) [210]

Optimal result

Integrand size = 41, antiderivative size = 332 \[ \int \frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{11/2}} \, dx=\frac {\left (9 c^2 C d-5 B c d^2+A d^3-13 c^3 D\right ) \sqrt {c^2-d^2 x^2}}{4 d^4 (c+d x)^{5/2}}-\frac {\left (73 c^2 C d-21 B c d^2+A d^3-157 c^3 D\right ) \sqrt {c^2-d^2 x^2}}{16 c d^4 (c+d x)^{3/2}}-\frac {2 (3 C d-17 c D) \sqrt {c^2-d^2 x^2}}{3 d^4 \sqrt {c+d x}}-\frac {2 D \sqrt {c+d x} \sqrt {c^2-d^2 x^2}}{3 d^4}-\frac {\left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \left (c^2-d^2 x^2\right )^{3/2}}{3 d^4 (c+d x)^{9/2}}+\frac {\left (119 c^2 C d-11 B c d^2-A d^3-451 c^3 D\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {c+d x}}{\sqrt {c^2-d^2 x^2}}\right )}{16 \sqrt {2} c^{3/2} d^4} \] Output:

1/4*(A*d^3-5*B*c*d^2+9*C*c^2*d-13*D*c^3)*(-d^2*x^2+c^2)^(1/2)/d^4/(d*x+c)^ 
(5/2)-1/16*(A*d^3-21*B*c*d^2+73*C*c^2*d-157*D*c^3)*(-d^2*x^2+c^2)^(1/2)/c/ 
d^4/(d*x+c)^(3/2)-2/3*(3*C*d-17*D*c)*(-d^2*x^2+c^2)^(1/2)/d^4/(d*x+c)^(1/2 
)-2/3*D*(d*x+c)^(1/2)*(-d^2*x^2+c^2)^(1/2)/d^4-1/3*(A*d^3-B*c*d^2+C*c^2*d- 
D*c^3)*(-d^2*x^2+c^2)^(3/2)/d^4/(d*x+c)^(9/2)+1/32*(-A*d^3-11*B*c*d^2+119* 
C*c^2*d-451*D*c^3)*arctanh(2^(1/2)*c^(1/2)*(d*x+c)^(1/2)/(-d^2*x^2+c^2)^(1 
/2))*2^(1/2)/c^(3/2)/d^4
 

Mathematica [A] (verified)

Time = 3.48 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.67 \[ \int \frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{11/2}} \, dx=\frac {\frac {2 \sqrt {c} \sqrt {c^2-d^2 x^2} \left (843 c^5 D-3 A d^5 x^2+c^4 (-223 C d+2274 d D x)+c^3 d^2 (19 B+7 x (-86 C+273 D x))+c d^4 x (22 A+x (63 B-32 x (3 C+D x)))+c^2 d^3 (-7 A+x (50 B+13 x (-39 C+32 D x)))\right )}{(c+d x)^{7/2}}-3 \sqrt {2} \left (-119 c^2 C d+11 B c d^2+A d^3+451 c^3 D\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {c+d x}}{\sqrt {c^2-d^2 x^2}}\right )}{96 c^{3/2} d^4} \] Input:

Integrate[((c^2 - d^2*x^2)^(3/2)*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(11/ 
2),x]
 

Output:

((2*Sqrt[c]*Sqrt[c^2 - d^2*x^2]*(843*c^5*D - 3*A*d^5*x^2 + c^4*(-223*C*d + 
 2274*d*D*x) + c^3*d^2*(19*B + 7*x*(-86*C + 273*D*x)) + c*d^4*x*(22*A + x* 
(63*B - 32*x*(3*C + D*x))) + c^2*d^3*(-7*A + x*(50*B + 13*x*(-39*C + 32*D* 
x)))))/(c + d*x)^(7/2) - 3*Sqrt[2]*(-119*c^2*C*d + 11*B*c*d^2 + A*d^3 + 45 
1*c^3*D)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[c + d*x])/Sqrt[c^2 - d^2*x^2]])/(96 
*c^(3/2)*d^4)
 

Rubi [A] (verified)

Time = 1.20 (sec) , antiderivative size = 298, normalized size of antiderivative = 0.90, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.220, Rules used = {2170, 27, 2170, 27, 671, 465, 465, 471, 221}

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.

\(\displaystyle \int \frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{11/2}} \, dx\)

\(\Big \downarrow \) 2170

\(\displaystyle -\frac {2 \int -\frac {\left (c^2-d^2 x^2\right )^{3/2} \left ((3 C d-13 c D) x^2 d^4+\left (3 B d^2-17 c^2 D\right ) x d^3+\left (3 A d^3-7 c^3 D\right ) d^2\right )}{2 (c+d x)^{11/2}}dx}{3 d^5}-\frac {2 D \left (c^2-d^2 x^2\right )^{5/2}}{3 d^4 (c+d x)^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\left (c^2-d^2 x^2\right )^{3/2} \left ((3 C d-13 c D) x^2 d^4+\left (3 B d^2-17 c^2 D\right ) x d^3+\left (3 A d^3-7 c^3 D\right ) d^2\right )}{(c+d x)^{11/2}}dx}{3 d^5}-\frac {2 D \left (c^2-d^2 x^2\right )^{5/2}}{3 d^4 (c+d x)^{7/2}}\)

\(\Big \downarrow \) 2170

\(\displaystyle \frac {-\frac {2 \int \frac {d^6 \left (-110 D c^3+27 C d c^2-3 A d^3+d \left (-113 D c^2+30 C d c-3 B d^2\right ) x\right ) \left (c^2-d^2 x^2\right )^{3/2}}{2 (c+d x)^{11/2}}dx}{d^4}-\frac {2 d \left (c^2-d^2 x^2\right )^{5/2} (3 C d-13 c D)}{(c+d x)^{9/2}}}{3 d^5}-\frac {2 D \left (c^2-d^2 x^2\right )^{5/2}}{3 d^4 (c+d x)^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-d^2 \int \frac {\left (-110 D c^3+27 C d c^2-3 A d^3+d \left (-113 D c^2+30 C d c-3 B d^2\right ) x\right ) \left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^{11/2}}dx-\frac {2 d \left (c^2-d^2 x^2\right )^{5/2} (3 C d-13 c D)}{(c+d x)^{9/2}}}{3 d^5}-\frac {2 D \left (c^2-d^2 x^2\right )^{5/2}}{3 d^4 (c+d x)^{7/2}}\)

\(\Big \downarrow \) 671

\(\displaystyle \frac {-d^2 \left (\frac {\left (-A d^3-11 B c d^2-451 c^3 D+119 c^2 C d\right ) \int \frac {\left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^{9/2}}dx}{4 c}+\frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 c d (c+d x)^{11/2}}\right )-\frac {2 d \left (c^2-d^2 x^2\right )^{5/2} (3 C d-13 c D)}{(c+d x)^{9/2}}}{3 d^5}-\frac {2 D \left (c^2-d^2 x^2\right )^{5/2}}{3 d^4 (c+d x)^{7/2}}\)

\(\Big \downarrow \) 465

\(\displaystyle \frac {-d^2 \left (\frac {\left (-A d^3-11 B c d^2-451 c^3 D+119 c^2 C d\right ) \left (-\frac {3}{4} \int \frac {\sqrt {c^2-d^2 x^2}}{(c+d x)^{5/2}}dx-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{2 d (c+d x)^{7/2}}\right )}{4 c}+\frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 c d (c+d x)^{11/2}}\right )-\frac {2 d \left (c^2-d^2 x^2\right )^{5/2} (3 C d-13 c D)}{(c+d x)^{9/2}}}{3 d^5}-\frac {2 D \left (c^2-d^2 x^2\right )^{5/2}}{3 d^4 (c+d x)^{7/2}}\)

\(\Big \downarrow \) 465

\(\displaystyle \frac {-d^2 \left (\frac {\left (-A d^3-11 B c d^2-451 c^3 D+119 c^2 C d\right ) \left (-\frac {3}{4} \left (-\frac {1}{2} \int \frac {1}{\sqrt {c+d x} \sqrt {c^2-d^2 x^2}}dx-\frac {\sqrt {c^2-d^2 x^2}}{d (c+d x)^{3/2}}\right )-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{2 d (c+d x)^{7/2}}\right )}{4 c}+\frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 c d (c+d x)^{11/2}}\right )-\frac {2 d \left (c^2-d^2 x^2\right )^{5/2} (3 C d-13 c D)}{(c+d x)^{9/2}}}{3 d^5}-\frac {2 D \left (c^2-d^2 x^2\right )^{5/2}}{3 d^4 (c+d x)^{7/2}}\)

\(\Big \downarrow \) 471

\(\displaystyle \frac {-d^2 \left (\frac {\left (-A d^3-11 B c d^2-451 c^3 D+119 c^2 C d\right ) \left (-\frac {3}{4} \left (-d \int \frac {1}{\frac {d^2 \left (c^2-d^2 x^2\right )}{c+d x}-2 c d^2}d\frac {\sqrt {c^2-d^2 x^2}}{\sqrt {c+d x}}-\frac {\sqrt {c^2-d^2 x^2}}{d (c+d x)^{3/2}}\right )-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{2 d (c+d x)^{7/2}}\right )}{4 c}+\frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 c d (c+d x)^{11/2}}\right )-\frac {2 d \left (c^2-d^2 x^2\right )^{5/2} (3 C d-13 c D)}{(c+d x)^{9/2}}}{3 d^5}-\frac {2 D \left (c^2-d^2 x^2\right )^{5/2}}{3 d^4 (c+d x)^{7/2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {-d^2 \left (\frac {\left (-\frac {3}{4} \left (\frac {\text {arctanh}\left (\frac {\sqrt {c^2-d^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {c+d x}}\right )}{\sqrt {2} \sqrt {c} d}-\frac {\sqrt {c^2-d^2 x^2}}{d (c+d x)^{3/2}}\right )-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{2 d (c+d x)^{7/2}}\right ) \left (-A d^3-11 B c d^2-451 c^3 D+119 c^2 C d\right )}{4 c}+\frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 c d (c+d x)^{11/2}}\right )-\frac {2 d \left (c^2-d^2 x^2\right )^{5/2} (3 C d-13 c D)}{(c+d x)^{9/2}}}{3 d^5}-\frac {2 D \left (c^2-d^2 x^2\right )^{5/2}}{3 d^4 (c+d x)^{7/2}}\)

Input:

Int[((c^2 - d^2*x^2)^(3/2)*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(11/2),x]
 

Output:

(-2*D*(c^2 - d^2*x^2)^(5/2))/(3*d^4*(c + d*x)^(7/2)) + ((-2*d*(3*C*d - 13* 
c*D)*(c^2 - d^2*x^2)^(5/2))/(c + d*x)^(9/2) - d^2*(((c^2*C*d - B*c*d^2 + A 
*d^3 - c^3*D)*(c^2 - d^2*x^2)^(5/2))/(2*c*d*(c + d*x)^(11/2)) + ((119*c^2* 
C*d - 11*B*c*d^2 - A*d^3 - 451*c^3*D)*(-1/2*(c^2 - d^2*x^2)^(3/2)/(d*(c + 
d*x)^(7/2)) - (3*(-(Sqrt[c^2 - d^2*x^2]/(d*(c + d*x)^(3/2))) + ArcTanh[Sqr 
t[c^2 - d^2*x^2]/(Sqrt[2]*Sqrt[c]*Sqrt[c + d*x])]/(Sqrt[2]*Sqrt[c]*d)))/4) 
)/(4*c)))/(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 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

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 471

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim 
p[2*d   Subst[Int[1/(2*b*c + d^2*x^2), x], x, Sqrt[a + b*x^2]/Sqrt[c + d*x] 
], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(789\) vs. \(2(288)=576\).

Time = 0.36 (sec) , antiderivative size = 790, normalized size of antiderivative = 2.38

method	result	size
default	\(-\frac {\sqrt {-d^{2} x^{2}+c^{2}}\, \left (33 B \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c \,d^{5} x^{3}-357 C \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{2} d^{4} x^{3}+1353 D \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{3} d^{3} x^{3}-1071 C \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{4} d^{2} x -4548 D \sqrt {-d x +c}\, c^{\frac {9}{2}} d x +4059 D \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{5} d x +33 B \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{4} d^{2}+446 C \sqrt {-d x +c}\, c^{\frac {9}{2}} d -357 C \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{5} d +4059 D \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{4} d^{2} x^{2}+9 A \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{2} d^{4} x +99 B \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{3} d^{3} x -1686 D \sqrt {-d x +c}\, c^{\frac {11}{2}}+99 B \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{2} d^{4} x^{2}+9 A \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c \,d^{5} x^{2}-126 B \,c^{\frac {3}{2}} d^{4} x^{2} \sqrt {-d x +c}-44 A \,c^{\frac {3}{2}} d^{4} x \sqrt {-d x +c}+1204 C \sqrt {-d x +c}\, c^{\frac {7}{2}} d^{2} x +1014 C \sqrt {-d x +c}\, c^{\frac {5}{2}} d^{3} x^{2}-100 B \,c^{\frac {5}{2}} d^{3} x \sqrt {-d x +c}+3 A \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) d^{6} x^{3}+14 A \sqrt {-d x +c}\, c^{\frac {5}{2}} d^{3}+3 A \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{3} d^{3}-38 B \sqrt {-d x +c}\, c^{\frac {7}{2}} d^{2}-3822 D \sqrt {-d x +c}\, c^{\frac {7}{2}} d^{2} x^{2}-1071 C \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{3} d^{3} x^{2}+1353 D \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{6}+6 A \,d^{5} x^{2} \sqrt {-d x +c}\, \sqrt {c}-832 D \sqrt {-d x +c}\, c^{\frac {5}{2}} d^{3} x^{3}+64 D c^{\frac {3}{2}} d^{4} x^{4} \sqrt {-d x +c}+192 C \,c^{\frac {3}{2}} d^{4} x^{3} \sqrt {-d x +c}\right )}{96 c^{\frac {3}{2}} \left (d x +c \right )^{\frac {7}{2}} \sqrt {-d x +c}\, d^{4}}\)	\(790\)

Input:

int((-d^2*x^2+c^2)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(11/2),x,method=_RETU 
RNVERBOSE)
 

Output:

-1/96*(-d^2*x^2+c^2)^(1/2)/c^(3/2)*(33*B*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2 
)*2^(1/2)/c^(1/2))*c*d^5*x^3-357*C*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1 
/2)/c^(1/2))*c^2*d^4*x^3+1353*D*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2) 
/c^(1/2))*c^3*d^3*x^3-1071*C*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^ 
(1/2))*c^4*d^2*x-4548*D*(-d*x+c)^(1/2)*c^(9/2)*d*x+4059*D*2^(1/2)*arctanh( 
1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^5*d*x+33*B*2^(1/2)*arctanh(1/2*(-d*x 
+c)^(1/2)*2^(1/2)/c^(1/2))*c^4*d^2+446*C*(-d*x+c)^(1/2)*c^(9/2)*d-357*C*2^ 
(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^5*d+4059*D*2^(1/2)*arc 
tanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^4*d^2*x^2+9*A*2^(1/2)*arctanh(1 
/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^2*d^4*x+99*B*2^(1/2)*arctanh(1/2*(-d* 
x+c)^(1/2)*2^(1/2)/c^(1/2))*c^3*d^3*x-1686*D*(-d*x+c)^(1/2)*c^(11/2)+99*B* 
2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^2*d^4*x^2+9*A*2^(1/2 
)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c*d^5*x^2-126*B*c^(3/2)*d^4* 
x^2*(-d*x+c)^(1/2)-44*A*c^(3/2)*d^4*x*(-d*x+c)^(1/2)+1204*C*(-d*x+c)^(1/2) 
*c^(7/2)*d^2*x+1014*C*(-d*x+c)^(1/2)*c^(5/2)*d^3*x^2-100*B*c^(5/2)*d^3*x*( 
-d*x+c)^(1/2)+3*A*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*d^6* 
x^3+14*A*(-d*x+c)^(1/2)*c^(5/2)*d^3+3*A*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2) 
*2^(1/2)/c^(1/2))*c^3*d^3-38*B*(-d*x+c)^(1/2)*c^(7/2)*d^2-3822*D*(-d*x+c)^ 
(1/2)*c^(7/2)*d^2*x^2-1071*C*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^ 
(1/2))*c^3*d^3*x^2+1353*D*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^...
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 941, normalized size of antiderivative = 2.83 \[ \int \frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{11/2}} \, dx =\text {Too large to display} \] Input:

integrate((-d^2*x^2+c^2)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(11/2),x, algor 
ithm="fricas")
 

Output:

[1/192*(3*sqrt(2)*(451*D*c^7 - 119*C*c^6*d + 11*B*c^5*d^2 + A*c^4*d^3 + (4 
51*D*c^3*d^4 - 119*C*c^2*d^5 + 11*B*c*d^6 + A*d^7)*x^4 + 4*(451*D*c^4*d^3 
- 119*C*c^3*d^4 + 11*B*c^2*d^5 + A*c*d^6)*x^3 + 6*(451*D*c^5*d^2 - 119*C*c 
^4*d^3 + 11*B*c^3*d^4 + A*c^2*d^5)*x^2 + 4*(451*D*c^6*d - 119*C*c^5*d^2 + 
11*B*c^4*d^3 + A*c^3*d^4)*x)*sqrt(c)*log(-(d^2*x^2 - 2*c*d*x + 2*sqrt(2)*s 
qrt(-d^2*x^2 + c^2)*sqrt(d*x + c)*sqrt(c) - 3*c^2)/(d^2*x^2 + 2*c*d*x + c^ 
2)) - 4*(32*D*c^2*d^4*x^4 - 843*D*c^6 + 223*C*c^5*d - 19*B*c^4*d^2 + 7*A*c 
^3*d^3 - 32*(13*D*c^3*d^3 - 3*C*c^2*d^4)*x^3 - 3*(637*D*c^4*d^2 - 169*C*c^ 
3*d^3 + 21*B*c^2*d^4 - A*c*d^5)*x^2 - 2*(1137*D*c^5*d - 301*C*c^4*d^2 + 25 
*B*c^3*d^3 + 11*A*c^2*d^4)*x)*sqrt(-d^2*x^2 + c^2)*sqrt(d*x + c))/(c^2*d^8 
*x^4 + 4*c^3*d^7*x^3 + 6*c^4*d^6*x^2 + 4*c^5*d^5*x + c^6*d^4), 1/96*(3*sqr 
t(2)*(451*D*c^7 - 119*C*c^6*d + 11*B*c^5*d^2 + A*c^4*d^3 + (451*D*c^3*d^4 
- 119*C*c^2*d^5 + 11*B*c*d^6 + A*d^7)*x^4 + 4*(451*D*c^4*d^3 - 119*C*c^3*d 
^4 + 11*B*c^2*d^5 + A*c*d^6)*x^3 + 6*(451*D*c^5*d^2 - 119*C*c^4*d^3 + 11*B 
*c^3*d^4 + A*c^2*d^5)*x^2 + 4*(451*D*c^6*d - 119*C*c^5*d^2 + 11*B*c^4*d^3 
+ A*c^3*d^4)*x)*sqrt(-c)*arctan(1/2*sqrt(2)*sqrt(-d^2*x^2 + c^2)*sqrt(d*x 
+ c)*sqrt(-c)/(c*d*x + c^2)) - 2*(32*D*c^2*d^4*x^4 - 843*D*c^6 + 223*C*c^5 
*d - 19*B*c^4*d^2 + 7*A*c^3*d^3 - 32*(13*D*c^3*d^3 - 3*C*c^2*d^4)*x^3 - 3* 
(637*D*c^4*d^2 - 169*C*c^3*d^3 + 21*B*c^2*d^4 - A*c*d^5)*x^2 - 2*(1137*D*c 
^5*d - 301*C*c^4*d^2 + 25*B*c^3*d^3 + 11*A*c^2*d^4)*x)*sqrt(-d^2*x^2 + ...
 

Sympy [F]

\[ \int \frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{11/2}} \, dx=\int \frac {\left (- \left (- c + d x\right ) \left (c + d x\right )\right )^{\frac {3}{2}} \left (A + B x + C x^{2} + D x^{3}\right )}{\left (c + d x\right )^{\frac {11}{2}}}\, dx \] Input:

integrate((-d**2*x**2+c**2)**(3/2)*(D*x**3+C*x**2+B*x+A)/(d*x+c)**(11/2),x 
)
 

Output:

Integral((-(-c + d*x)*(c + d*x))**(3/2)*(A + B*x + C*x**2 + D*x**3)/(c + d 
*x)**(11/2), x)
 

Maxima [F]

\[ \int \frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{11/2}} \, dx=\int { \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} {\left (D x^{3} + C x^{2} + B x + A\right )}}{{\left (d x + c\right )}^{\frac {11}{2}}} \,d x } \] Input:

integrate((-d^2*x^2+c^2)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(11/2),x, algor 
ithm="maxima")
 

Output:

integrate((-d^2*x^2 + c^2)^(3/2)*(D*x^3 + C*x^2 + B*x + A)/(d*x + c)^(11/2 
), x)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 330, normalized size of antiderivative = 0.99 \[ \int \frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{11/2}} \, dx=\frac {64 \, {\left (-d x + c\right )}^{\frac {3}{2}} D + 960 \, \sqrt {-d x + c} D c - 192 \, \sqrt {-d x + c} C d + \frac {3 \, \sqrt {2} {\left (451 \, D c^{3} - 119 \, C c^{2} d + 11 \, B c d^{2} + A d^{3}\right )} \arctan \left (\frac {\sqrt {2} \sqrt {-d x + c}}{2 \, \sqrt {-c}}\right )}{\sqrt {-c} c} + \frac {2 \, {\left (471 \, {\left (d x - c\right )}^{2} \sqrt {-d x + c} D c^{3} - 1712 \, {\left (-d x + c\right )}^{\frac {3}{2}} D c^{4} + 1572 \, \sqrt {-d x + c} D c^{5} - 219 \, {\left (d x - c\right )}^{2} \sqrt {-d x + c} C c^{2} d + 752 \, {\left (-d x + c\right )}^{\frac {3}{2}} C c^{3} d - 660 \, \sqrt {-d x + c} C c^{4} d + 63 \, {\left (d x - c\right )}^{2} \sqrt {-d x + c} B c d^{2} - 176 \, {\left (-d x + c\right )}^{\frac {3}{2}} B c^{2} d^{2} + 132 \, \sqrt {-d x + c} B c^{3} d^{2} - 3 \, {\left (d x - c\right )}^{2} \sqrt {-d x + c} A d^{3} - 16 \, {\left (-d x + c\right )}^{\frac {3}{2}} A c d^{3} + 12 \, \sqrt {-d x + c} A c^{2} d^{3}\right )}}{{\left (d x + c\right )}^{3} c}}{96 \, d^{4}} \] Input:

integrate((-d^2*x^2+c^2)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(11/2),x, algor 
ithm="giac")
 

Output:

1/96*(64*(-d*x + c)^(3/2)*D + 960*sqrt(-d*x + c)*D*c - 192*sqrt(-d*x + c)* 
C*d + 3*sqrt(2)*(451*D*c^3 - 119*C*c^2*d + 11*B*c*d^2 + A*d^3)*arctan(1/2* 
sqrt(2)*sqrt(-d*x + c)/sqrt(-c))/(sqrt(-c)*c) + 2*(471*(d*x - c)^2*sqrt(-d 
*x + c)*D*c^3 - 1712*(-d*x + c)^(3/2)*D*c^4 + 1572*sqrt(-d*x + c)*D*c^5 - 
219*(d*x - c)^2*sqrt(-d*x + c)*C*c^2*d + 752*(-d*x + c)^(3/2)*C*c^3*d - 66 
0*sqrt(-d*x + c)*C*c^4*d + 63*(d*x - c)^2*sqrt(-d*x + c)*B*c*d^2 - 176*(-d 
*x + c)^(3/2)*B*c^2*d^2 + 132*sqrt(-d*x + c)*B*c^3*d^2 - 3*(d*x - c)^2*sqr 
t(-d*x + c)*A*d^3 - 16*(-d*x + c)^(3/2)*A*c*d^3 + 12*sqrt(-d*x + c)*A*c^2* 
d^3)/((d*x + c)^3*c))/d^4
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{11/2}} \, dx=\int \frac {{\left (c^2-d^2\,x^2\right )}^{3/2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^{11/2}} \,d x \] Input:

int(((c^2 - d^2*x^2)^(3/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(11/2),x)
 

Output:

int(((c^2 - d^2*x^2)^(3/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(11/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 776, normalized size of antiderivative = 2.34 \[ \int \frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{11/2}} \, dx =\text {Too large to display} \] Input:

int((-d^2*x^2+c^2)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(11/2),x)
 

Output:

( - 112*sqrt(c - d*x)*a*c**3*d**2 + 352*sqrt(c - d*x)*a*c**2*d**3*x - 48*s 
qrt(c - d*x)*a*c*d**4*x**2 + 304*sqrt(c - d*x)*b*c**4*d + 800*sqrt(c - d*x 
)*b*c**3*d**2*x + 1008*sqrt(c - d*x)*b*c**2*d**3*x**2 + 9920*sqrt(c - d*x) 
*c**6 + 26752*sqrt(c - d*x)*c**5*d*x + 22464*sqrt(c - d*x)*c**4*d**2*x**2 
+ 5120*sqrt(c - d*x)*c**3*d**3*x**3 - 512*sqrt(c - d*x)*c**2*d**4*x**4 + 2 
4*sqrt(c)*sqrt(2)*log(tan(asin(sqrt(c + d*x)/(sqrt(c)*sqrt(2)))/2))*a*c**3 
*d**2 + 72*sqrt(c)*sqrt(2)*log(tan(asin(sqrt(c + d*x)/(sqrt(c)*sqrt(2)))/2 
))*a*c**2*d**3*x + 72*sqrt(c)*sqrt(2)*log(tan(asin(sqrt(c + d*x)/(sqrt(c)* 
sqrt(2)))/2))*a*c*d**4*x**2 + 24*sqrt(c)*sqrt(2)*log(tan(asin(sqrt(c + d*x 
)/(sqrt(c)*sqrt(2)))/2))*a*d**5*x**3 + 264*sqrt(c)*sqrt(2)*log(tan(asin(sq 
rt(c + d*x)/(sqrt(c)*sqrt(2)))/2))*b*c**4*d + 792*sqrt(c)*sqrt(2)*log(tan( 
asin(sqrt(c + d*x)/(sqrt(c)*sqrt(2)))/2))*b*c**3*d**2*x + 792*sqrt(c)*sqrt 
(2)*log(tan(asin(sqrt(c + d*x)/(sqrt(c)*sqrt(2)))/2))*b*c**2*d**3*x**2 + 2 
64*sqrt(c)*sqrt(2)*log(tan(asin(sqrt(c + d*x)/(sqrt(c)*sqrt(2)))/2))*b*c*d 
**4*x**3 + 7968*sqrt(c)*sqrt(2)*log(tan(asin(sqrt(c + d*x)/(sqrt(c)*sqrt(2 
)))/2))*c**6 + 23904*sqrt(c)*sqrt(2)*log(tan(asin(sqrt(c + d*x)/(sqrt(c)*s 
qrt(2)))/2))*c**5*d*x + 23904*sqrt(c)*sqrt(2)*log(tan(asin(sqrt(c + d*x)/( 
sqrt(c)*sqrt(2)))/2))*c**4*d**2*x**2 + 7968*sqrt(c)*sqrt(2)*log(tan(asin(s 
qrt(c + d*x)/(sqrt(c)*sqrt(2)))/2))*c**3*d**3*x**3 - 17*sqrt(c)*sqrt(2)*a* 
c**3*d**2 - 51*sqrt(c)*sqrt(2)*a*c**2*d**3*x - 51*sqrt(c)*sqrt(2)*a*c*d...