\(\int \frac {\sqrt {c^2-d^2 x^2} (A+B x+C x^2+D x^3)}{(c+d x)^{13/2}} \, dx\) [202]

Optimal result

Integrand size = 41, antiderivative size = 386 \[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{13/2}} \, dx=-\frac {\left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \sqrt {c^2-d^2 x^2}}{5 d^4 (c+d x)^{11/2}}+\frac {\left (41 c^2 C d-21 B c d^2+A d^3-61 c^3 D\right ) \sqrt {c^2-d^2 x^2}}{80 c d^4 (c+d x)^{9/2}}-\frac {\left (353 c^2 C d-13 B c d^2-7 A d^3-1013 c^3 D\right ) \sqrt {c^2-d^2 x^2}}{960 c^2 d^4 (c+d x)^{7/2}}+\frac {\left (31 c^2 C d+13 B c d^2+7 A d^3-907 c^3 D\right ) \sqrt {c^2-d^2 x^2}}{1536 c^3 d^4 (c+d x)^{5/2}}+\frac {\left (31 c^2 C d+13 B c d^2+7 A d^3+117 c^3 D\right ) \sqrt {c^2-d^2 x^2}}{2048 c^4 d^4 (c+d x)^{3/2}}+\frac {\left (31 c^2 C d+13 B c d^2+7 A d^3+117 c^3 D\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {c+d x}}{\sqrt {c^2-d^2 x^2}}\right )}{2048 \sqrt {2} c^{9/2} d^4} \] Output:

-1/5*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(-d^2*x^2+c^2)^(1/2)/d^4/(d*x+c)^(11/2) 
+1/80*(A*d^3-21*B*c*d^2+41*C*c^2*d-61*D*c^3)*(-d^2*x^2+c^2)^(1/2)/c/d^4/(d 
*x+c)^(9/2)-1/960*(-7*A*d^3-13*B*c*d^2+353*C*c^2*d-1013*D*c^3)*(-d^2*x^2+c 
^2)^(1/2)/c^2/d^4/(d*x+c)^(7/2)+1/1536*(7*A*d^3+13*B*c*d^2+31*C*c^2*d-907* 
D*c^3)*(-d^2*x^2+c^2)^(1/2)/c^3/d^4/(d*x+c)^(5/2)+1/2048*(7*A*d^3+13*B*c*d 
^2+31*C*c^2*d+117*D*c^3)*(-d^2*x^2+c^2)^(1/2)/c^4/d^4/(d*x+c)^(3/2)+1/4096 
*(7*A*d^3+13*B*c*d^2+31*C*c^2*d+117*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^(9/2)/d^4
 

Mathematica [A] (verified)

Time = 1.69 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{13/2}} \, dx=\frac {-\frac {2 \sqrt {c} \sqrt {c^2-d^2 x^2} \left (1249 c^7 D-105 A d^7 x^4-5 c d^6 x^3 (112 A+39 B x)+c^6 d (611 C+5992 D x)-c^2 d^5 x^2 (1274 A+5 x (208 B+93 C x))+c^5 d^2 (1049 B+2 x (1564 C+5737 D x))-c^3 d^4 x (1672 A+x (2366 B+5 x (496 C+351 D x)))+c^4 d^3 (5291 A+2 x (2836 B+x (3323 C+5560 D x)))\right )}{(c+d x)^{11/2}}+15 \sqrt {2} \left (31 c^2 C d+13 B c d^2+7 A d^3+117 c^3 D\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {c+d x}}{\sqrt {c^2-d^2 x^2}}\right )}{61440 c^{9/2} d^4} \] Input:

Integrate[(Sqrt[c^2 - d^2*x^2]*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(13/2) 
,x]
 

Output:

((-2*Sqrt[c]*Sqrt[c^2 - d^2*x^2]*(1249*c^7*D - 105*A*d^7*x^4 - 5*c*d^6*x^3 
*(112*A + 39*B*x) + c^6*d*(611*C + 5992*D*x) - c^2*d^5*x^2*(1274*A + 5*x*( 
208*B + 93*C*x)) + c^5*d^2*(1049*B + 2*x*(1564*C + 5737*D*x)) - c^3*d^4*x* 
(1672*A + x*(2366*B + 5*x*(496*C + 351*D*x))) + c^4*d^3*(5291*A + 2*x*(283 
6*B + x*(3323*C + 5560*D*x)))))/(c + d*x)^(11/2) + 15*Sqrt[2]*(31*c^2*C*d 
+ 13*B*c*d^2 + 7*A*d^3 + 117*c^3*D)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[c + d*x] 
)/Sqrt[c^2 - d^2*x^2]])/(61440*c^(9/2)*d^4)
 

Rubi [A] (verified)

Time = 1.35 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.01, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.268, Rules used = {2170, 27, 2170, 27, 671, 465, 470, 470, 470, 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.

\(\displaystyle \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{13/2}} \, dx\)

\(\Big \downarrow \) 2170

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2170

\(\displaystyle \frac {\frac {2 \int \frac {d^6 \left (26 D c^3+11 C d c^2+5 A d^3+d \left (31 D c^2+6 C d c+5 B d^2\right ) x\right ) \sqrt {c^2-d^2 x^2}}{2 (c+d x)^{13/2}}dx}{5 d^4}+\frac {2 d \left (c^2-d^2 x^2\right )^{3/2} (c D+C d)}{5 (c+d x)^{11/2}}}{d^5}+\frac {2 D \left (c^2-d^2 x^2\right )^{3/2}}{3 d^4 (c+d x)^{9/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{5} d^2 \int \frac {\left (26 D c^3+11 C d c^2+5 A d^3+d \left (31 D c^2+6 C d c+5 B d^2\right ) x\right ) \sqrt {c^2-d^2 x^2}}{(c+d x)^{13/2}}dx+\frac {2 d \left (c^2-d^2 x^2\right )^{3/2} (c D+C d)}{5 (c+d x)^{11/2}}}{d^5}+\frac {2 D \left (c^2-d^2 x^2\right )^{3/2}}{3 d^4 (c+d x)^{9/2}}\)

\(\Big \downarrow \) 671

\(\displaystyle \frac {\frac {1}{5} d^2 \left (\frac {\left (7 A d^3+13 B c d^2+117 c^3 D+31 c^2 C d\right ) \int \frac {\sqrt {c^2-d^2 x^2}}{(c+d x)^{11/2}}dx}{4 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 c d (c+d x)^{13/2}}\right )+\frac {2 d \left (c^2-d^2 x^2\right )^{3/2} (c D+C d)}{5 (c+d x)^{11/2}}}{d^5}+\frac {2 D \left (c^2-d^2 x^2\right )^{3/2}}{3 d^4 (c+d x)^{9/2}}\)

\(\Big \downarrow \) 465

\(\displaystyle \frac {\frac {1}{5} d^2 \left (\frac {\left (7 A d^3+13 B c d^2+117 c^3 D+31 c^2 C d\right ) \left (-\frac {1}{8} \int \frac {1}{(c+d x)^{7/2} \sqrt {c^2-d^2 x^2}}dx-\frac {\sqrt {c^2-d^2 x^2}}{4 d (c+d x)^{9/2}}\right )}{4 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 c d (c+d x)^{13/2}}\right )+\frac {2 d \left (c^2-d^2 x^2\right )^{3/2} (c D+C d)}{5 (c+d x)^{11/2}}}{d^5}+\frac {2 D \left (c^2-d^2 x^2\right )^{3/2}}{3 d^4 (c+d x)^{9/2}}\)

\(\Big \downarrow \) 470

\(\displaystyle \frac {\frac {1}{5} d^2 \left (\frac {\left (7 A d^3+13 B c d^2+117 c^3 D+31 c^2 C d\right ) \left (\frac {1}{8} \left (\frac {\sqrt {c^2-d^2 x^2}}{6 c d (c+d x)^{7/2}}-\frac {5 \int \frac {1}{(c+d x)^{5/2} \sqrt {c^2-d^2 x^2}}dx}{12 c}\right )-\frac {\sqrt {c^2-d^2 x^2}}{4 d (c+d x)^{9/2}}\right )}{4 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 c d (c+d x)^{13/2}}\right )+\frac {2 d \left (c^2-d^2 x^2\right )^{3/2} (c D+C d)}{5 (c+d x)^{11/2}}}{d^5}+\frac {2 D \left (c^2-d^2 x^2\right )^{3/2}}{3 d^4 (c+d x)^{9/2}}\)

\(\Big \downarrow \) 470

\(\displaystyle \frac {\frac {1}{5} d^2 \left (\frac {\left (7 A d^3+13 B c d^2+117 c^3 D+31 c^2 C d\right ) \left (\frac {1}{8} \left (\frac {\sqrt {c^2-d^2 x^2}}{6 c d (c+d x)^{7/2}}-\frac {5 \left (\frac {3 \int \frac {1}{(c+d x)^{3/2} \sqrt {c^2-d^2 x^2}}dx}{8 c}-\frac {\sqrt {c^2-d^2 x^2}}{4 c d (c+d x)^{5/2}}\right )}{12 c}\right )-\frac {\sqrt {c^2-d^2 x^2}}{4 d (c+d x)^{9/2}}\right )}{4 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 c d (c+d x)^{13/2}}\right )+\frac {2 d \left (c^2-d^2 x^2\right )^{3/2} (c D+C d)}{5 (c+d x)^{11/2}}}{d^5}+\frac {2 D \left (c^2-d^2 x^2\right )^{3/2}}{3 d^4 (c+d x)^{9/2}}\)

\(\Big \downarrow \) 470

\(\displaystyle \frac {\frac {1}{5} d^2 \left (\frac {\left (7 A d^3+13 B c d^2+117 c^3 D+31 c^2 C d\right ) \left (\frac {1}{8} \left (\frac {\sqrt {c^2-d^2 x^2}}{6 c d (c+d x)^{7/2}}-\frac {5 \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {c+d x} \sqrt {c^2-d^2 x^2}}dx}{4 c}-\frac {\sqrt {c^2-d^2 x^2}}{2 c d (c+d x)^{3/2}}\right )}{8 c}-\frac {\sqrt {c^2-d^2 x^2}}{4 c d (c+d x)^{5/2}}\right )}{12 c}\right )-\frac {\sqrt {c^2-d^2 x^2}}{4 d (c+d x)^{9/2}}\right )}{4 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 c d (c+d x)^{13/2}}\right )+\frac {2 d \left (c^2-d^2 x^2\right )^{3/2} (c D+C d)}{5 (c+d x)^{11/2}}}{d^5}+\frac {2 D \left (c^2-d^2 x^2\right )^{3/2}}{3 d^4 (c+d x)^{9/2}}\)

\(\Big \downarrow \) 471

\(\displaystyle \frac {\frac {1}{5} d^2 \left (\frac {\left (7 A d^3+13 B c d^2+117 c^3 D+31 c^2 C d\right ) \left (\frac {1}{8} \left (\frac {\sqrt {c^2-d^2 x^2}}{6 c d (c+d x)^{7/2}}-\frac {5 \left (\frac {3 \left (\frac {d \int \frac {1}{\frac {d^2 \left (c^2-d^2 x^2\right )}{c+d x}-2 c d^2}d\frac {\sqrt {c^2-d^2 x^2}}{\sqrt {c+d x}}}{2 c}-\frac {\sqrt {c^2-d^2 x^2}}{2 c d (c+d x)^{3/2}}\right )}{8 c}-\frac {\sqrt {c^2-d^2 x^2}}{4 c d (c+d x)^{5/2}}\right )}{12 c}\right )-\frac {\sqrt {c^2-d^2 x^2}}{4 d (c+d x)^{9/2}}\right )}{4 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 c d (c+d x)^{13/2}}\right )+\frac {2 d \left (c^2-d^2 x^2\right )^{3/2} (c D+C d)}{5 (c+d x)^{11/2}}}{d^5}+\frac {2 D \left (c^2-d^2 x^2\right )^{3/2}}{3 d^4 (c+d x)^{9/2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {1}{5} d^2 \left (\frac {\left (\frac {1}{8} \left (\frac {\sqrt {c^2-d^2 x^2}}{6 c d (c+d x)^{7/2}}-\frac {5 \left (\frac {3 \left (-\frac {\text {arctanh}\left (\frac {\sqrt {c^2-d^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {c+d x}}\right )}{2 \sqrt {2} c^{3/2} d}-\frac {\sqrt {c^2-d^2 x^2}}{2 c d (c+d x)^{3/2}}\right )}{8 c}-\frac {\sqrt {c^2-d^2 x^2}}{4 c d (c+d x)^{5/2}}\right )}{12 c}\right )-\frac {\sqrt {c^2-d^2 x^2}}{4 d (c+d x)^{9/2}}\right ) \left (7 A d^3+13 B c d^2+117 c^3 D+31 c^2 C d\right )}{4 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 c d (c+d x)^{13/2}}\right )+\frac {2 d \left (c^2-d^2 x^2\right )^{3/2} (c D+C d)}{5 (c+d x)^{11/2}}}{d^5}+\frac {2 D \left (c^2-d^2 x^2\right )^{3/2}}{3 d^4 (c+d x)^{9/2}}\)

Input:

Int[(Sqrt[c^2 - d^2*x^2]*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(13/2),x]
 

Output:

(2*D*(c^2 - d^2*x^2)^(3/2))/(3*d^4*(c + d*x)^(9/2)) + ((2*d*(C*d + c*D)*(c 
^2 - d^2*x^2)^(3/2))/(5*(c + d*x)^(11/2)) + (d^2*(-1/2*((c^2*C*d - B*c*d^2 
 + A*d^3 - c^3*D)*(c^2 - d^2*x^2)^(3/2))/(c*d*(c + d*x)^(13/2)) + ((31*c^2 
*C*d + 13*B*c*d^2 + 7*A*d^3 + 117*c^3*D)*(-1/4*Sqrt[c^2 - d^2*x^2]/(d*(c + 
 d*x)^(9/2)) + (Sqrt[c^2 - d^2*x^2]/(6*c*d*(c + d*x)^(7/2)) - (5*(-1/4*Sqr 
t[c^2 - d^2*x^2]/(c*d*(c + d*x)^(5/2)) + (3*(-1/2*Sqrt[c^2 - d^2*x^2]/(c*d 
*(c + d*x)^(3/2)) - ArcTanh[Sqrt[c^2 - d^2*x^2]/(Sqrt[2]*Sqrt[c]*Sqrt[c + 
d*x])]/(2*Sqrt[2]*c^(3/2)*d)))/(8*c)))/(12*c))/8))/(4*c)))/5)/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 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 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. \(1145\) vs. \(2(342)=684\).

Time = 0.37 (sec) , antiderivative size = 1146, normalized size of antiderivative = 2.97

method	result	size
default	\(\text {Expression too large to display}\)	\(1146\)

Input:

int((-d^2*x^2+c^2)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(13/2),x,method=_RETU 
RNVERBOSE)
 

Output:

1/61440*(-d^2*x^2+c^2)^(1/2)/c^(9/2)*(4650*C*2^(1/2)*arctanh(1/2*(-d*x+c)^ 
(1/2)*2^(1/2)/c^(1/2))*c^4*d^4*x^3-22948*D*(-d*x+c)^(1/2)*c^(11/2)*d^2*x^2 
+17550*D*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^5*d^3*x^3+5 
25*A*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^4*d^4*x+975*B*2 
^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^5*d^3*x+2325*C*2^(1/2 
)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^6*d^2*x+525*A*2^(1/2)*arct 
anh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c*d^7*x^4-11984*D*(-d*x+c)^(1/2)*c 
^(13/2)*d*x+1050*A*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^3 
*d^5*x^2+4650*C*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^5*d^ 
3*x^2+105*A*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^5*d^3+19 
5*B*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^6*d^2+465*C*2^(1 
/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^7*d+1950*B*2^(1/2)*arcta 
nh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^4*d^4*x^2+8775*D*2^(1/2)*arctanh( 
1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^7*d*x-10582*A*(-d*x+c)^(1/2)*c^(9/2) 
*d^3-2098*B*(-d*x+c)^(1/2)*c^(11/2)*d^2-1222*C*(-d*x+c)^(1/2)*c^(13/2)*d+1 
7550*D*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^6*d^2*x^2-222 
40*D*(-d*x+c)^(1/2)*c^(9/2)*d^3*x^3+1050*A*2^(1/2)*arctanh(1/2*(-d*x+c)^(1 
/2)*2^(1/2)/c^(1/2))*c^2*d^6*x^3+1950*B*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2) 
*2^(1/2)/c^(1/2))*c^3*d^5*x^3+930*C*c^(5/2)*d^5*x^4*(-d*x+c)^(1/2)+3510*D* 
c^(7/2)*d^4*x^4*(-d*x+c)^(1/2)+1120*A*c^(3/2)*d^6*x^3*(-d*x+c)^(1/2)+25...
 

Fricas [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 1259, normalized size of antiderivative = 3.26 \[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{13/2}} \, dx=\text {Too large to display} \] Input:

integrate((-d^2*x^2+c^2)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(13/2),x, algor 
ithm="fricas")
 

Output:

[1/122880*(15*sqrt(2)*(117*D*c^9 + 31*C*c^8*d + 13*B*c^7*d^2 + 7*A*c^6*d^3 
 + (117*D*c^3*d^6 + 31*C*c^2*d^7 + 13*B*c*d^8 + 7*A*d^9)*x^6 + 6*(117*D*c^ 
4*d^5 + 31*C*c^3*d^6 + 13*B*c^2*d^7 + 7*A*c*d^8)*x^5 + 15*(117*D*c^5*d^4 + 
 31*C*c^4*d^5 + 13*B*c^3*d^6 + 7*A*c^2*d^7)*x^4 + 20*(117*D*c^6*d^3 + 31*C 
*c^5*d^4 + 13*B*c^4*d^5 + 7*A*c^3*d^6)*x^3 + 15*(117*D*c^7*d^2 + 31*C*c^6* 
d^3 + 13*B*c^5*d^4 + 7*A*c^4*d^5)*x^2 + 6*(117*D*c^8*d + 31*C*c^7*d^2 + 13 
*B*c^6*d^3 + 7*A*c^5*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*(1249*D*c^8 + 611*C*c^7*d + 1049*B*c^6*d^2 + 5291*A*c^5*d^3 - 15*( 
117*D*c^4*d^4 + 31*C*c^3*d^5 + 13*B*c^2*d^6 + 7*A*c*d^7)*x^4 + 80*(139*D*c 
^5*d^3 - 31*C*c^4*d^4 - 13*B*c^3*d^5 - 7*A*c^2*d^6)*x^3 + 2*(5737*D*c^6*d^ 
2 + 3323*C*c^5*d^3 - 1183*B*c^4*d^4 - 637*A*c^3*d^5)*x^2 + 8*(749*D*c^7*d 
+ 391*C*c^6*d^2 + 709*B*c^5*d^3 - 209*A*c^4*d^4)*x)*sqrt(-d^2*x^2 + c^2)*s 
qrt(d*x + c))/(c^5*d^10*x^6 + 6*c^6*d^9*x^5 + 15*c^7*d^8*x^4 + 20*c^8*d^7* 
x^3 + 15*c^9*d^6*x^2 + 6*c^10*d^5*x + c^11*d^4), -1/61440*(15*sqrt(2)*(117 
*D*c^9 + 31*C*c^8*d + 13*B*c^7*d^2 + 7*A*c^6*d^3 + (117*D*c^3*d^6 + 31*C*c 
^2*d^7 + 13*B*c*d^8 + 7*A*d^9)*x^6 + 6*(117*D*c^4*d^5 + 31*C*c^3*d^6 + 13* 
B*c^2*d^7 + 7*A*c*d^8)*x^5 + 15*(117*D*c^5*d^4 + 31*C*c^4*d^5 + 13*B*c^3*d 
^6 + 7*A*c^2*d^7)*x^4 + 20*(117*D*c^6*d^3 + 31*C*c^5*d^4 + 13*B*c^4*d^5 + 
7*A*c^3*d^6)*x^3 + 15*(117*D*c^7*d^2 + 31*C*c^6*d^3 + 13*B*c^5*d^4 + 7*...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{13/2}} \, dx=\text {Timed out} \] Input:

integrate((-d**2*x**2+c**2)**(1/2)*(D*x**3+C*x**2+B*x+A)/(d*x+c)**(13/2),x 
)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{13/2}} \, dx=\int { \frac {\sqrt {-d^{2} x^{2} + c^{2}} {\left (D x^{3} + C x^{2} + B x + A\right )}}{{\left (d x + c\right )}^{\frac {13}{2}}} \,d x } \] Input:

integrate((-d^2*x^2+c^2)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(13/2),x, algor 
ithm="maxima")
 

Output:

integrate(sqrt(-d^2*x^2 + c^2)*(D*x^3 + C*x^2 + B*x + A)/(d*x + c)^(13/2), 
 x)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 494, normalized size of antiderivative = 1.28 \[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{13/2}} \, dx=-\frac {\frac {15 \, \sqrt {2} {\left (117 \, D c^{3} + 31 \, C c^{2} d + 13 \, B c d^{2} + 7 \, A d^{3}\right )} \arctan \left (\frac {\sqrt {2} \sqrt {-d x + c}}{2 \, \sqrt {-c}}\right )}{\sqrt {-c} c^{4}} - \frac {2 \, {\left (1755 \, {\left (d x - c\right )}^{4} \sqrt {-d x + c} D c^{3} - 4100 \, {\left (d x - c\right )}^{3} \sqrt {-d x + c} D c^{4} - 34304 \, {\left (d x - c\right )}^{2} \sqrt {-d x + c} D c^{5} + 55280 \, {\left (-d x + c\right )}^{\frac {3}{2}} D c^{6} - 28080 \, \sqrt {-d x + c} D c^{7} + 465 \, {\left (d x - c\right )}^{4} \sqrt {-d x + c} C c^{2} d + 4340 \, {\left (d x - c\right )}^{3} \sqrt {-d x + c} C c^{3} d + 3584 \, {\left (d x - c\right )}^{2} \sqrt {-d x + c} C c^{4} d + 7120 \, {\left (-d x + c\right )}^{\frac {3}{2}} C c^{5} d - 7440 \, \sqrt {-d x + c} C c^{6} d + 195 \, {\left (d x - c\right )}^{4} \sqrt {-d x + c} B c d^{2} + 1820 \, {\left (d x - c\right )}^{3} \sqrt {-d x + c} B c^{2} d^{2} + 6656 \, {\left (d x - c\right )}^{2} \sqrt {-d x + c} B c^{3} d^{2} - 2960 \, {\left (-d x + c\right )}^{\frac {3}{2}} B c^{4} d^{2} - 3120 \, \sqrt {-d x + c} B c^{5} d^{2} + 105 \, {\left (d x - c\right )}^{4} \sqrt {-d x + c} A d^{3} + 980 \, {\left (d x - c\right )}^{3} \sqrt {-d x + c} A c d^{3} + 3584 \, {\left (d x - c\right )}^{2} \sqrt {-d x + c} A c^{2} d^{3} - 6320 \, {\left (-d x + c\right )}^{\frac {3}{2}} A c^{3} d^{3} - 1680 \, \sqrt {-d x + c} A c^{4} d^{3}\right )}}{{\left (d x + c\right )}^{5} c^{4}}}{61440 \, d^{4}} \] Input:

integrate((-d^2*x^2+c^2)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(13/2),x, algor 
ithm="giac")
 

Output:

-1/61440*(15*sqrt(2)*(117*D*c^3 + 31*C*c^2*d + 13*B*c*d^2 + 7*A*d^3)*arcta 
n(1/2*sqrt(2)*sqrt(-d*x + c)/sqrt(-c))/(sqrt(-c)*c^4) - 2*(1755*(d*x - c)^ 
4*sqrt(-d*x + c)*D*c^3 - 4100*(d*x - c)^3*sqrt(-d*x + c)*D*c^4 - 34304*(d* 
x - c)^2*sqrt(-d*x + c)*D*c^5 + 55280*(-d*x + c)^(3/2)*D*c^6 - 28080*sqrt( 
-d*x + c)*D*c^7 + 465*(d*x - c)^4*sqrt(-d*x + c)*C*c^2*d + 4340*(d*x - c)^ 
3*sqrt(-d*x + c)*C*c^3*d + 3584*(d*x - c)^2*sqrt(-d*x + c)*C*c^4*d + 7120* 
(-d*x + c)^(3/2)*C*c^5*d - 7440*sqrt(-d*x + c)*C*c^6*d + 195*(d*x - c)^4*s 
qrt(-d*x + c)*B*c*d^2 + 1820*(d*x - c)^3*sqrt(-d*x + c)*B*c^2*d^2 + 6656*( 
d*x - c)^2*sqrt(-d*x + c)*B*c^3*d^2 - 2960*(-d*x + c)^(3/2)*B*c^4*d^2 - 31 
20*sqrt(-d*x + c)*B*c^5*d^2 + 105*(d*x - c)^4*sqrt(-d*x + c)*A*d^3 + 980*( 
d*x - c)^3*sqrt(-d*x + c)*A*c*d^3 + 3584*(d*x - c)^2*sqrt(-d*x + c)*A*c^2* 
d^3 - 6320*(-d*x + c)^(3/2)*A*c^3*d^3 - 1680*sqrt(-d*x + c)*A*c^4*d^3)/((d 
*x + c)^5*c^4))/d^4
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{13/2}} \, dx=\int \frac {\sqrt {c^2-d^2\,x^2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^{13/2}} \,d x \] Input:

int(((c^2 - d^2*x^2)^(1/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(13/2),x)
 

Output:

int(((c^2 - d^2*x^2)^(1/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(13/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 929, normalized size of antiderivative = 2.41 \[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{13/2}} \, dx =\text {Too large to display} \] Input:

int((-d^2*x^2+c^2)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(13/2),x)
 

Output:

( - 10582*sqrt(c - d*x)*a*c**5*d**2 + 3344*sqrt(c - d*x)*a*c**4*d**3*x + 2 
548*sqrt(c - d*x)*a*c**3*d**4*x**2 + 1120*sqrt(c - d*x)*a*c**2*d**5*x**3 + 
 210*sqrt(c - d*x)*a*c*d**6*x**4 - 2098*sqrt(c - d*x)*b*c**6*d - 11344*sqr 
t(c - d*x)*b*c**5*d**2*x + 4732*sqrt(c - d*x)*b*c**4*d**3*x**2 + 2080*sqrt 
(c - d*x)*b*c**3*d**4*x**3 + 390*sqrt(c - d*x)*b*c**2*d**5*x**4 - 3720*sqr 
t(c - d*x)*c**8 - 18240*sqrt(c - d*x)*c**7*d*x - 36240*sqrt(c - d*x)*c**6* 
d**2*x**2 - 17280*sqrt(c - d*x)*c**5*d**3*x**3 + 4440*sqrt(c - d*x)*c**4*d 
**4*x**4 - 105*sqrt(c)*sqrt(2)*log(tan(asin(sqrt(c + d*x)/(sqrt(c)*sqrt(2) 
))/2))*a*c**5*d**2 - 525*sqrt(c)*sqrt(2)*log(tan(asin(sqrt(c + d*x)/(sqrt( 
c)*sqrt(2)))/2))*a*c**4*d**3*x - 1050*sqrt(c)*sqrt(2)*log(tan(asin(sqrt(c 
+ d*x)/(sqrt(c)*sqrt(2)))/2))*a*c**3*d**4*x**2 - 1050*sqrt(c)*sqrt(2)*log( 
tan(asin(sqrt(c + d*x)/(sqrt(c)*sqrt(2)))/2))*a*c**2*d**5*x**3 - 525*sqrt( 
c)*sqrt(2)*log(tan(asin(sqrt(c + d*x)/(sqrt(c)*sqrt(2)))/2))*a*c*d**6*x**4 
 - 105*sqrt(c)*sqrt(2)*log(tan(asin(sqrt(c + d*x)/(sqrt(c)*sqrt(2)))/2))*a 
*d**7*x**5 - 195*sqrt(c)*sqrt(2)*log(tan(asin(sqrt(c + d*x)/(sqrt(c)*sqrt( 
2)))/2))*b*c**6*d - 975*sqrt(c)*sqrt(2)*log(tan(asin(sqrt(c + d*x)/(sqrt(c 
)*sqrt(2)))/2))*b*c**5*d**2*x - 1950*sqrt(c)*sqrt(2)*log(tan(asin(sqrt(c + 
 d*x)/(sqrt(c)*sqrt(2)))/2))*b*c**4*d**3*x**2 - 1950*sqrt(c)*sqrt(2)*log(t 
an(asin(sqrt(c + d*x)/(sqrt(c)*sqrt(2)))/2))*b*c**3*d**4*x**3 - 975*sqrt(c 
)*sqrt(2)*log(tan(asin(sqrt(c + d*x)/(sqrt(c)*sqrt(2)))/2))*b*c**2*d**5...