\(\int (c^2-d^2 x^2)^{5/2} (A+B x+C x^2+D x^3) \, dx\) [156]

Optimal result

Integrand size = 32, antiderivative size = 231 \[ \int \left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {5}{128} c^4 \left (8 A+\frac {c^2 C}{d^2}\right ) x \sqrt {c^2-d^2 x^2}+\frac {5}{192} c^2 \left (8 A+\frac {c^2 C}{d^2}\right ) x \left (c^2-d^2 x^2\right )^{3/2}+\frac {1}{48} \left (8 A+\frac {c^2 C}{d^2}\right ) x \left (c^2-d^2 x^2\right )^{5/2}-\frac {\left (B d^2+c^2 D\right ) \left (c^2-d^2 x^2\right )^{7/2}}{7 d^4}-\frac {C x \left (c^2-d^2 x^2\right )^{7/2}}{8 d^2}+\frac {D \left (c^2-d^2 x^2\right )^{9/2}}{9 d^4}+\frac {5 c^6 \left (c^2 C+8 A d^2\right ) \arctan \left (\frac {d x}{\sqrt {c^2-d^2 x^2}}\right )}{128 d^3} \] Output:

5/128*c^4*(8*A+c^2*C/d^2)*x*(-d^2*x^2+c^2)^(1/2)+5/192*c^2*(8*A+c^2*C/d^2) 
*x*(-d^2*x^2+c^2)^(3/2)+1/48*(8*A+c^2*C/d^2)*x*(-d^2*x^2+c^2)^(5/2)-1/7*(B 
*d^2+D*c^2)*(-d^2*x^2+c^2)^(7/2)/d^4-1/8*C*x*(-d^2*x^2+c^2)^(7/2)/d^2+1/9* 
D*(-d^2*x^2+c^2)^(9/2)/d^4+5/128*c^6*(8*A*d^2+C*c^2)*arctan(d*x/(-d^2*x^2+ 
c^2)^(1/2))/d^3
 

Mathematica [A] (verified)

Time = 1.18 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.85 \[ \int \left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {\sqrt {c^2-d^2 x^2} \left (-256 c^8 D-c^6 d^2 x (315 C+128 D x)+1152 B d^2 \left (-c^2+d^2 x^2\right )^3+112 d^8 x^5 \left (12 A+9 C x^2+8 D x^3\right )-8 c^2 d^6 x^3 \left (546 A+357 C x^2+304 D x^3\right )+6 c^4 d^4 x \left (924 A+413 C x^2+320 D x^3\right )\right )-630 c^6 d \left (c^2 C+8 A d^2\right ) \arctan \left (\frac {d x}{\sqrt {c^2}-\sqrt {c^2-d^2 x^2}}\right )}{8064 d^4} \] Input:

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

Output:

(Sqrt[c^2 - d^2*x^2]*(-256*c^8*D - c^6*d^2*x*(315*C + 128*D*x) + 1152*B*d^ 
2*(-c^2 + d^2*x^2)^3 + 112*d^8*x^5*(12*A + 9*C*x^2 + 8*D*x^3) - 8*c^2*d^6* 
x^3*(546*A + 357*C*x^2 + 304*D*x^3) + 6*c^4*d^4*x*(924*A + 413*C*x^2 + 320 
*D*x^3)) - 630*c^6*d*(c^2*C + 8*A*d^2)*ArcTan[(d*x)/(Sqrt[c^2] - Sqrt[c^2 
- d^2*x^2])])/(8064*d^4)
 

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.95, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.344, Rules used = {2346, 25, 2346, 25, 27, 455, 211, 211, 211, 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.

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

\(\Big \downarrow \) 2346

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2346

\(\displaystyle \frac {-\frac {\int -d^2 \left (9 \left (C c^2+8 A d^2\right )+8 \left (2 D c^2+9 B d^2\right ) x\right ) \left (c^2-d^2 x^2\right )^{5/2}dx}{8 d^2}-\frac {9}{8} C x \left (c^2-d^2 x^2\right )^{7/2}}{9 d^2}-\frac {D x^2 \left (c^2-d^2 x^2\right )^{7/2}}{9 d^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int d^2 \left (9 \left (C c^2+8 A d^2\right )+8 \left (2 D c^2+9 B d^2\right ) x\right ) \left (c^2-d^2 x^2\right )^{5/2}dx}{8 d^2}-\frac {9}{8} C x \left (c^2-d^2 x^2\right )^{7/2}}{9 d^2}-\frac {D x^2 \left (c^2-d^2 x^2\right )^{7/2}}{9 d^2}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 455

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 216

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

Input:

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

Output:

-1/9*(D*x^2*(c^2 - d^2*x^2)^(7/2))/d^2 + ((-9*C*x*(c^2 - d^2*x^2)^(7/2))/8 
 + ((-8*(9*B + (2*c^2*D)/d^2)*(c^2 - d^2*x^2)^(7/2))/7 + 9*(c^2*C + 8*A*d^ 
2)*((x*(c^2 - d^2*x^2)^(5/2))/6 + (5*c^2*((x*(c^2 - d^2*x^2)^(3/2))/4 + (3 
*c^2*((x*Sqrt[c^2 - d^2*x^2])/2 + (c^2*ArcTan[(d*x)/Sqrt[c^2 - d^2*x^2]])/ 
(2*d)))/4))/6))/8)/(9*d^2)
 

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 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

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 455

Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( 
a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c   Int[(a + b*x^2)^p, x], x] 
/; FreeQ[{a, b, c, d, p}, x] &&  !LeQ[p, -1]
 

rule 2346

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], 
e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x^2)^(p + 1)/(b*( 
q + 2*p + 1))), x] + Simp[1/(b*(q + 2*p + 1))   Int[(a + b*x^2)^p*ExpandToS 
um[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], 
x], x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] &&  !LeQ[p, -1]
 

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.27

method	result	size
default	\(A \left (\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 c^{2} \left (\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 c^{2} \left (\frac {x \sqrt {-d^{2} x^{2}+c^{2}}}{2}+\frac {c^{2} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right )}{2 \sqrt {d^{2}}}\right )}{4}\right )}{6}\right )-\frac {B \left (-d^{2} x^{2}+c^{2}\right )^{\frac {7}{2}}}{7 d^{2}}+C \left (-\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {7}{2}}}{8 d^{2}}+\frac {c^{2} \left (\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 c^{2} \left (\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 c^{2} \left (\frac {x \sqrt {-d^{2} x^{2}+c^{2}}}{2}+\frac {c^{2} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right )}{2 \sqrt {d^{2}}}\right )}{4}\right )}{6}\right )}{8 d^{2}}\right )+D \left (-\frac {x^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {7}{2}}}{9 d^{2}}-\frac {2 c^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {7}{2}}}{63 d^{4}}\right )\)	\(294\)

Input:

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

Output:

A*(1/6*x*(-d^2*x^2+c^2)^(5/2)+5/6*c^2*(1/4*x*(-d^2*x^2+c^2)^(3/2)+3/4*c^2* 
(1/2*x*(-d^2*x^2+c^2)^(1/2)+1/2*c^2/(d^2)^(1/2)*arctan((d^2)^(1/2)*x/(-d^2 
*x^2+c^2)^(1/2)))))-1/7*B*(-d^2*x^2+c^2)^(7/2)/d^2+C*(-1/8*x*(-d^2*x^2+c^2 
)^(7/2)/d^2+1/8*c^2/d^2*(1/6*x*(-d^2*x^2+c^2)^(5/2)+5/6*c^2*(1/4*x*(-d^2*x 
^2+c^2)^(3/2)+3/4*c^2*(1/2*x*(-d^2*x^2+c^2)^(1/2)+1/2*c^2/(d^2)^(1/2)*arct 
an((d^2)^(1/2)*x/(-d^2*x^2+c^2)^(1/2))))))+D*(-1/9*x^2*(-d^2*x^2+c^2)^(7/2 
)/d^2-2/63*c^2*(-d^2*x^2+c^2)^(7/2)/d^4)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.02 \[ \int \left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right ) \, dx=-\frac {630 \, {\left (C c^{8} d + 8 \, A c^{6} d^{3}\right )} \arctan \left (-\frac {c - \sqrt {-d^{2} x^{2} + c^{2}}}{d x}\right ) - {\left (896 \, D d^{8} x^{8} + 1008 \, C d^{8} x^{7} - 256 \, D c^{8} - 1152 \, B c^{6} d^{2} - 128 \, {\left (19 \, D c^{2} d^{6} - 9 \, B d^{8}\right )} x^{6} - 168 \, {\left (17 \, C c^{2} d^{6} - 8 \, A d^{8}\right )} x^{5} + 384 \, {\left (5 \, D c^{4} d^{4} - 9 \, B c^{2} d^{6}\right )} x^{4} + 42 \, {\left (59 \, C c^{4} d^{4} - 104 \, A c^{2} d^{6}\right )} x^{3} - 128 \, {\left (D c^{6} d^{2} - 27 \, B c^{4} d^{4}\right )} x^{2} - 63 \, {\left (5 \, C c^{6} d^{2} - 88 \, A c^{4} d^{4}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{8064 \, d^{4}} \] Input:

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

Output:

-1/8064*(630*(C*c^8*d + 8*A*c^6*d^3)*arctan(-(c - sqrt(-d^2*x^2 + c^2))/(d 
*x)) - (896*D*d^8*x^8 + 1008*C*d^8*x^7 - 256*D*c^8 - 1152*B*c^6*d^2 - 128* 
(19*D*c^2*d^6 - 9*B*d^8)*x^6 - 168*(17*C*c^2*d^6 - 8*A*d^8)*x^5 + 384*(5*D 
*c^4*d^4 - 9*B*c^2*d^6)*x^4 + 42*(59*C*c^4*d^4 - 104*A*c^2*d^6)*x^3 - 128* 
(D*c^6*d^2 - 27*B*c^4*d^4)*x^2 - 63*(5*C*c^6*d^2 - 88*A*c^4*d^4)*x)*sqrt(- 
d^2*x^2 + c^2))/d^4
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 627 vs. \(2 (209) = 418\).

Time = 0.59 (sec) , antiderivative size = 627, normalized size of antiderivative = 2.71 \[ \int \left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\begin {cases} \sqrt {c^{2} - d^{2} x^{2}} \left (\frac {C d^{4} x^{7}}{8} + \frac {D d^{4} x^{8}}{9} - \frac {x^{6} \left (- B d^{6} + \frac {19 D c^{2} d^{4}}{9}\right )}{7 d^{2}} - \frac {x^{5} \left (- A d^{6} + \frac {17 C c^{2} d^{4}}{8}\right )}{6 d^{2}} - \frac {x^{4} \cdot \left (3 B c^{2} d^{4} - 3 D c^{4} d^{2} + \frac {6 c^{2} \left (- B d^{6} + \frac {19 D c^{2} d^{4}}{9}\right )}{7 d^{2}}\right )}{5 d^{2}} - \frac {x^{3} \cdot \left (3 A c^{2} d^{4} - 3 C c^{4} d^{2} + \frac {5 c^{2} \left (- A d^{6} + \frac {17 C c^{2} d^{4}}{8}\right )}{6 d^{2}}\right )}{4 d^{2}} - \frac {x^{2} \left (- 3 B c^{4} d^{2} + D c^{6} + \frac {4 c^{2} \cdot \left (3 B c^{2} d^{4} - 3 D c^{4} d^{2} + \frac {6 c^{2} \left (- B d^{6} + \frac {19 D c^{2} d^{4}}{9}\right )}{7 d^{2}}\right )}{5 d^{2}}\right )}{3 d^{2}} - \frac {x \left (- 3 A c^{4} d^{2} + C c^{6} + \frac {3 c^{2} \cdot \left (3 A c^{2} d^{4} - 3 C c^{4} d^{2} + \frac {5 c^{2} \left (- A d^{6} + \frac {17 C c^{2} d^{4}}{8}\right )}{6 d^{2}}\right )}{4 d^{2}}\right )}{2 d^{2}} - \frac {B c^{6} + \frac {2 c^{2} \left (- 3 B c^{4} d^{2} + D c^{6} + \frac {4 c^{2} \cdot \left (3 B c^{2} d^{4} - 3 D c^{4} d^{2} + \frac {6 c^{2} \left (- B d^{6} + \frac {19 D c^{2} d^{4}}{9}\right )}{7 d^{2}}\right )}{5 d^{2}}\right )}{3 d^{2}}}{d^{2}}\right ) + \left (A c^{6} + \frac {c^{2} \left (- 3 A c^{4} d^{2} + C c^{6} + \frac {3 c^{2} \cdot \left (3 A c^{2} d^{4} - 3 C c^{4} d^{2} + \frac {5 c^{2} \left (- A d^{6} + \frac {17 C c^{2} d^{4}}{8}\right )}{6 d^{2}}\right )}{4 d^{2}}\right )}{2 d^{2}}\right ) \left (\begin {cases} \frac {\log {\left (- 2 d^{2} x + 2 \sqrt {- d^{2}} \sqrt {c^{2} - d^{2} x^{2}} \right )}}{\sqrt {- d^{2}}} & \text {for}\: c^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- d^{2} x^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: d^{2} \neq 0 \\\left (A x + \frac {B x^{2}}{2} + \frac {C x^{3}}{3} + \frac {D x^{4}}{4}\right ) \left (c^{2}\right )^{\frac {5}{2}} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((sqrt(c**2 - d**2*x**2)*(C*d**4*x**7/8 + D*d**4*x**8/9 - x**6*(- 
B*d**6 + 19*D*c**2*d**4/9)/(7*d**2) - x**5*(-A*d**6 + 17*C*c**2*d**4/8)/(6 
*d**2) - x**4*(3*B*c**2*d**4 - 3*D*c**4*d**2 + 6*c**2*(-B*d**6 + 19*D*c**2 
*d**4/9)/(7*d**2))/(5*d**2) - x**3*(3*A*c**2*d**4 - 3*C*c**4*d**2 + 5*c**2 
*(-A*d**6 + 17*C*c**2*d**4/8)/(6*d**2))/(4*d**2) - x**2*(-3*B*c**4*d**2 + 
D*c**6 + 4*c**2*(3*B*c**2*d**4 - 3*D*c**4*d**2 + 6*c**2*(-B*d**6 + 19*D*c* 
*2*d**4/9)/(7*d**2))/(5*d**2))/(3*d**2) - x*(-3*A*c**4*d**2 + C*c**6 + 3*c 
**2*(3*A*c**2*d**4 - 3*C*c**4*d**2 + 5*c**2*(-A*d**6 + 17*C*c**2*d**4/8)/( 
6*d**2))/(4*d**2))/(2*d**2) - (B*c**6 + 2*c**2*(-3*B*c**4*d**2 + D*c**6 + 
4*c**2*(3*B*c**2*d**4 - 3*D*c**4*d**2 + 6*c**2*(-B*d**6 + 19*D*c**2*d**4/9 
)/(7*d**2))/(5*d**2))/(3*d**2))/d**2) + (A*c**6 + c**2*(-3*A*c**4*d**2 + C 
*c**6 + 3*c**2*(3*A*c**2*d**4 - 3*C*c**4*d**2 + 5*c**2*(-A*d**6 + 17*C*c** 
2*d**4/8)/(6*d**2))/(4*d**2))/(2*d**2))*Piecewise((log(-2*d**2*x + 2*sqrt( 
-d**2)*sqrt(c**2 - d**2*x**2))/sqrt(-d**2), Ne(c**2, 0)), (x*log(x)/sqrt(- 
d**2*x**2), True)), Ne(d**2, 0)), ((A*x + B*x**2/2 + C*x**3/3 + D*x**4/4)* 
(c**2)**(5/2), True))
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.09 \[ \int \left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {5 \, C c^{8} \arcsin \left (\frac {d x}{c}\right )}{128 \, d^{3}} + \frac {5 \, A c^{6} \arcsin \left (\frac {d x}{c}\right )}{16 \, d} + \frac {5}{16} \, \sqrt {-d^{2} x^{2} + c^{2}} A c^{4} x + \frac {5 \, \sqrt {-d^{2} x^{2} + c^{2}} C c^{6} x}{128 \, d^{2}} + \frac {5}{24} \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} A c^{2} x + \frac {5 \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} C c^{4} x}{192 \, d^{2}} + \frac {1}{6} \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} A x + \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} C c^{2} x}{48 \, d^{2}} - \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {7}{2}} D x^{2}}{9 \, d^{2}} - \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {7}{2}} C x}{8 \, d^{2}} - \frac {2 \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {7}{2}} D c^{2}}{63 \, d^{4}} - \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {7}{2}} B}{7 \, d^{2}} \] Input:

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

Output:

5/128*C*c^8*arcsin(d*x/c)/d^3 + 5/16*A*c^6*arcsin(d*x/c)/d + 5/16*sqrt(-d^ 
2*x^2 + c^2)*A*c^4*x + 5/128*sqrt(-d^2*x^2 + c^2)*C*c^6*x/d^2 + 5/24*(-d^2 
*x^2 + c^2)^(3/2)*A*c^2*x + 5/192*(-d^2*x^2 + c^2)^(3/2)*C*c^4*x/d^2 + 1/6 
*(-d^2*x^2 + c^2)^(5/2)*A*x + 1/48*(-d^2*x^2 + c^2)^(5/2)*C*c^2*x/d^2 - 1/ 
9*(-d^2*x^2 + c^2)^(7/2)*D*x^2/d^2 - 1/8*(-d^2*x^2 + c^2)^(7/2)*C*x/d^2 - 
2/63*(-d^2*x^2 + c^2)^(7/2)*D*c^2/d^4 - 1/7*(-d^2*x^2 + c^2)^(7/2)*B/d^2
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.09 \[ \int \left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {1}{8064} \, \sqrt {-d^{2} x^{2} + c^{2}} {\left ({\left (2 \, {\left ({\left (4 \, {\left ({\left (2 \, {\left (7 \, {\left (8 \, D d^{4} x + 9 \, C d^{4}\right )} x - \frac {8 \, {\left (19 \, D c^{2} d^{16} - 9 \, B d^{18}\right )}}{d^{14}}\right )} x - \frac {21 \, {\left (17 \, C c^{2} d^{16} - 8 \, A d^{18}\right )}}{d^{14}}\right )} x + \frac {48 \, {\left (5 \, D c^{4} d^{14} - 9 \, B c^{2} d^{16}\right )}}{d^{14}}\right )} x + \frac {21 \, {\left (59 \, C c^{4} d^{14} - 104 \, A c^{2} d^{16}\right )}}{d^{14}}\right )} x - \frac {64 \, {\left (D c^{6} d^{12} - 27 \, B c^{4} d^{14}\right )}}{d^{14}}\right )} x - \frac {63 \, {\left (5 \, C c^{6} d^{12} - 88 \, A c^{4} d^{14}\right )}}{d^{14}}\right )} x - \frac {128 \, {\left (2 \, D c^{8} d^{10} + 9 \, B c^{6} d^{12}\right )}}{d^{14}}\right )} + \frac {5 \, {\left (C c^{8} + 8 \, A c^{6} d^{2}\right )} \arcsin \left (\frac {d x}{c}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right )}{128 \, d^{2} {\left | d \right |}} \] Input:

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

Output:

1/8064*sqrt(-d^2*x^2 + c^2)*((2*((4*((2*(7*(8*D*d^4*x + 9*C*d^4)*x - 8*(19 
*D*c^2*d^16 - 9*B*d^18)/d^14)*x - 21*(17*C*c^2*d^16 - 8*A*d^18)/d^14)*x + 
48*(5*D*c^4*d^14 - 9*B*c^2*d^16)/d^14)*x + 21*(59*C*c^4*d^14 - 104*A*c^2*d 
^16)/d^14)*x - 64*(D*c^6*d^12 - 27*B*c^4*d^14)/d^14)*x - 63*(5*C*c^6*d^12 
- 88*A*c^4*d^14)/d^14)*x - 128*(2*D*c^8*d^10 + 9*B*c^6*d^12)/d^14) + 5/128 
*(C*c^8 + 8*A*c^6*d^2)*arcsin(d*x/c)*sgn(c)*sgn(d)/(d^2*abs(d))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.77 \[ \int \left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {2520 \mathit {asin} \left (\frac {d x}{c}\right ) a \,c^{6} d^{2}+315 \mathit {asin} \left (\frac {d x}{c}\right ) c^{9}+5544 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{4} d^{3} x -4368 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{2} d^{5} x^{3}+1344 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,d^{7} x^{5}-1152 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{6} d +3456 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{4} d^{3} x^{2}-3456 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{2} d^{5} x^{4}+1152 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,d^{7} x^{6}-256 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{8}-315 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{7} d x -128 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{6} d^{2} x^{2}+2478 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{5} d^{3} x^{3}+1920 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{4} d^{4} x^{4}-2856 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{3} d^{5} x^{5}-2432 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{2} d^{6} x^{6}+1008 \sqrt {-d^{2} x^{2}+c^{2}}\, c \,d^{7} x^{7}+896 \sqrt {-d^{2} x^{2}+c^{2}}\, d^{8} x^{8}+1152 b \,c^{7} d +256 c^{9}}{8064 d^{3}} \] Input:

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

Output:

(2520*asin((d*x)/c)*a*c**6*d**2 + 315*asin((d*x)/c)*c**9 + 5544*sqrt(c**2 
- d**2*x**2)*a*c**4*d**3*x - 4368*sqrt(c**2 - d**2*x**2)*a*c**2*d**5*x**3 
+ 1344*sqrt(c**2 - d**2*x**2)*a*d**7*x**5 - 1152*sqrt(c**2 - d**2*x**2)*b* 
c**6*d + 3456*sqrt(c**2 - d**2*x**2)*b*c**4*d**3*x**2 - 3456*sqrt(c**2 - d 
**2*x**2)*b*c**2*d**5*x**4 + 1152*sqrt(c**2 - d**2*x**2)*b*d**7*x**6 - 256 
*sqrt(c**2 - d**2*x**2)*c**8 - 315*sqrt(c**2 - d**2*x**2)*c**7*d*x - 128*s 
qrt(c**2 - d**2*x**2)*c**6*d**2*x**2 + 2478*sqrt(c**2 - d**2*x**2)*c**5*d* 
*3*x**3 + 1920*sqrt(c**2 - d**2*x**2)*c**4*d**4*x**4 - 2856*sqrt(c**2 - d* 
*2*x**2)*c**3*d**5*x**5 - 2432*sqrt(c**2 - d**2*x**2)*c**2*d**6*x**6 + 100 
8*sqrt(c**2 - d**2*x**2)*c*d**7*x**7 + 896*sqrt(c**2 - d**2*x**2)*d**8*x** 
8 + 1152*b*c**7*d + 256*c**9)/(8064*d**3)