\(\int (c^2-d^2 x^2)^{3/2} (A+B x+C x^2+D x^3) \, dx\) [143]

Optimal result

Integrand size = 32, antiderivative size = 195 \[ \int \left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {1}{16} c^2 \left (6 A+\frac {c^2 C}{d^2}\right ) x \sqrt {c^2-d^2 x^2}+\frac {1}{24} \left (6 A+\frac {c^2 C}{d^2}\right ) x \left (c^2-d^2 x^2\right )^{3/2}-\frac {\left (B d^2+c^2 D\right ) \left (c^2-d^2 x^2\right )^{5/2}}{5 d^4}-\frac {C x \left (c^2-d^2 x^2\right )^{5/2}}{6 d^2}+\frac {D \left (c^2-d^2 x^2\right )^{7/2}}{7 d^4}+\frac {c^4 \left (c^2 C+6 A d^2\right ) \arctan \left (\frac {d x}{\sqrt {c^2-d^2 x^2}}\right )}{16 d^3} \] Output:

1/16*c^2*(6*A+c^2*C/d^2)*x*(-d^2*x^2+c^2)^(1/2)+1/24*(6*A+c^2*C/d^2)*x*(-d 
^2*x^2+c^2)^(3/2)-1/5*(B*d^2+D*c^2)*(-d^2*x^2+c^2)^(5/2)/d^4-1/6*C*x*(-d^2 
*x^2+c^2)^(5/2)/d^2+1/7*D*(-d^2*x^2+c^2)^(7/2)/d^4+1/16*c^4*(6*A*d^2+C*c^2 
)*arctan(d*x/(-d^2*x^2+c^2)^(1/2))/d^3
 

Mathematica [A] (verified)

Time = 0.99 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.87 \[ \int \left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=-\frac {\sqrt {c^2-d^2 x^2} \left (96 c^6 D+3 c^4 d^2 x (35 C+16 D x)+336 B d^2 \left (c^2-d^2 x^2\right )^2+20 d^6 x^3 \left (21 A+2 x^2 (7 C+6 D x)\right )-2 c^2 d^4 x \left (525 A+x^2 (245 C+192 D x)\right )\right )+210 c^4 d \left (c^2 C+6 A d^2\right ) \arctan \left (\frac {d x}{\sqrt {c^2}-\sqrt {c^2-d^2 x^2}}\right )}{1680 d^4} \] Input:

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

Output:

-1/1680*(Sqrt[c^2 - d^2*x^2]*(96*c^6*D + 3*c^4*d^2*x*(35*C + 16*D*x) + 336 
*B*d^2*(c^2 - d^2*x^2)^2 + 20*d^6*x^3*(21*A + 2*x^2*(7*C + 6*D*x)) - 2*c^2 
*d^4*x*(525*A + x^2*(245*C + 192*D*x))) + 210*c^4*d*(c^2*C + 6*A*d^2)*ArcT 
an[(d*x)/(Sqrt[c^2] - Sqrt[c^2 - d^2*x^2])])/d^4
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.98, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {2346, 25, 2346, 25, 27, 455, 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.

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

\(\Big \downarrow \) 2346

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2346

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 455

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 216

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

Input:

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

Output:

-1/7*(D*x^2*(c^2 - d^2*x^2)^(5/2))/d^2 + ((-7*C*x*(c^2 - d^2*x^2)^(5/2))/6 
 + ((-6*(7*B + (2*c^2*D)/d^2)*(c^2 - d^2*x^2)^(5/2))/5 + 7*(c^2*C + 6*A*d^ 
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)/(7*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 = 0.35 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.27

method	result	size
default	\(A \left (\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 c^{2} \left (\frac {x \sqrt {-d^{2} x^{2}+c^{2}}}{2}+\frac {c^{2} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right )}{2 \sqrt {d^{2}}}\right )}{4}\right )-\frac {B \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{5 d^{2}}+C \left (-\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{6 d^{2}}+\frac {c^{2} \left (\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 c^{2} \left (\frac {x \sqrt {-d^{2} x^{2}+c^{2}}}{2}+\frac {c^{2} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right )}{2 \sqrt {d^{2}}}\right )}{4}\right )}{6 d^{2}}\right )+D \left (-\frac {x^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{7 d^{2}}-\frac {2 c^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{35 d^{4}}\right )\)	\(248\)

Input:

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

Output:

A*(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))))-1/5*B*(-d^2*x^2+c 
^2)^(5/2)/d^2+C*(-1/6*x*(-d^2*x^2+c^2)^(5/2)/d^2+1/6*c^2/d^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)*arc 
tan((d^2)^(1/2)*x/(-d^2*x^2+c^2)^(1/2)))))+D*(-1/7*x^2*(-d^2*x^2+c^2)^(5/2 
)/d^2-2/35*c^2*(-d^2*x^2+c^2)^(5/2)/d^4)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.95 \[ \int \left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=-\frac {210 \, {\left (C c^{6} d + 6 \, A c^{4} d^{3}\right )} \arctan \left (-\frac {c - \sqrt {-d^{2} x^{2} + c^{2}}}{d x}\right ) + {\left (240 \, D d^{6} x^{6} + 280 \, C d^{6} x^{5} + 96 \, D c^{6} + 336 \, B c^{4} d^{2} - 48 \, {\left (8 \, D c^{2} d^{4} - 7 \, B d^{6}\right )} x^{4} - 70 \, {\left (7 \, C c^{2} d^{4} - 6 \, A d^{6}\right )} x^{3} + 48 \, {\left (D c^{4} d^{2} - 14 \, B c^{2} d^{4}\right )} x^{2} + 105 \, {\left (C c^{4} d^{2} - 10 \, A c^{2} d^{4}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{1680 \, d^{4}} \] Input:

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

Output:

-1/1680*(210*(C*c^6*d + 6*A*c^4*d^3)*arctan(-(c - sqrt(-d^2*x^2 + c^2))/(d 
*x)) + (240*D*d^6*x^6 + 280*C*d^6*x^5 + 96*D*c^6 + 336*B*c^4*d^2 - 48*(8*D 
*c^2*d^4 - 7*B*d^6)*x^4 - 70*(7*C*c^2*d^4 - 6*A*d^6)*x^3 + 48*(D*c^4*d^2 - 
 14*B*c^2*d^4)*x^2 + 105*(C*c^4*d^2 - 10*A*c^2*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. 394 vs. \(2 (172) = 344\).

Time = 0.52 (sec) , antiderivative size = 394, normalized size of antiderivative = 2.02 \[ \int \left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\begin {cases} \sqrt {c^{2} - d^{2} x^{2}} \left (- \frac {C d^{2} x^{5}}{6} - \frac {D d^{2} x^{6}}{7} - \frac {x^{4} \left (B d^{4} - \frac {8 D c^{2} d^{2}}{7}\right )}{5 d^{2}} - \frac {x^{3} \left (A d^{4} - \frac {7 C c^{2} d^{2}}{6}\right )}{4 d^{2}} - \frac {x^{2} \left (- 2 B c^{2} d^{2} + D c^{4} + \frac {4 c^{2} \left (B d^{4} - \frac {8 D c^{2} d^{2}}{7}\right )}{5 d^{2}}\right )}{3 d^{2}} - \frac {x \left (- 2 A c^{2} d^{2} + C c^{4} + \frac {3 c^{2} \left (A d^{4} - \frac {7 C c^{2} d^{2}}{6}\right )}{4 d^{2}}\right )}{2 d^{2}} - \frac {B c^{4} + \frac {2 c^{2} \left (- 2 B c^{2} d^{2} + D c^{4} + \frac {4 c^{2} \left (B d^{4} - \frac {8 D c^{2} d^{2}}{7}\right )}{5 d^{2}}\right )}{3 d^{2}}}{d^{2}}\right ) + \left (A c^{4} + \frac {c^{2} \left (- 2 A c^{2} d^{2} + C c^{4} + \frac {3 c^{2} \left (A d^{4} - \frac {7 C c^{2} d^{2}}{6}\right )}{4 d^{2}}\right )}{2 d^{2}}\right ) \left (\begin {cases} \frac {\log {\left (- 2 d^{2} x + 2 \sqrt {- d^{2}} \sqrt {c^{2} - d^{2} x^{2}} \right )}}{\sqrt {- d^{2}}} & \text {for}\: c^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- d^{2} x^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: d^{2} \neq 0 \\\left (A x + \frac {B x^{2}}{2} + \frac {C x^{3}}{3} + \frac {D x^{4}}{4}\right ) \left (c^{2}\right )^{\frac {3}{2}} & \text {otherwise} \end {cases} \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.06 \[ \int \left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {C c^{6} \arcsin \left (\frac {d x}{c}\right )}{16 \, d^{3}} + \frac {3 \, A c^{4} \arcsin \left (\frac {d x}{c}\right )}{8 \, d} + \frac {3}{8} \, \sqrt {-d^{2} x^{2} + c^{2}} A c^{2} x + \frac {\sqrt {-d^{2} x^{2} + c^{2}} C c^{4} x}{16 \, d^{2}} + \frac {1}{4} \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} A x + \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} C c^{2} x}{24 \, d^{2}} - \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} D x^{2}}{7 \, d^{2}} - \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} C x}{6 \, d^{2}} - \frac {2 \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} D c^{2}}{35 \, d^{4}} - \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} B}{5 \, d^{2}} \] Input:

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

Output:

1/16*C*c^6*arcsin(d*x/c)/d^3 + 3/8*A*c^4*arcsin(d*x/c)/d + 3/8*sqrt(-d^2*x 
^2 + c^2)*A*c^2*x + 1/16*sqrt(-d^2*x^2 + c^2)*C*c^4*x/d^2 + 1/4*(-d^2*x^2 
+ c^2)^(3/2)*A*x + 1/24*(-d^2*x^2 + c^2)^(3/2)*C*c^2*x/d^2 - 1/7*(-d^2*x^2 
 + c^2)^(5/2)*D*x^2/d^2 - 1/6*(-d^2*x^2 + c^2)^(5/2)*C*x/d^2 - 2/35*(-d^2* 
x^2 + c^2)^(5/2)*D*c^2/d^4 - 1/5*(-d^2*x^2 + c^2)^(5/2)*B/d^2
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.01 \[ \int \left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=-\frac {1}{1680} \, \sqrt {-d^{2} x^{2} + c^{2}} {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (6 \, D d^{2} x + 7 \, C d^{2}\right )} x - \frac {6 \, {\left (8 \, D c^{2} d^{10} - 7 \, B d^{12}\right )}}{d^{10}}\right )} x - \frac {35 \, {\left (7 \, C c^{2} d^{10} - 6 \, A d^{12}\right )}}{d^{10}}\right )} x + \frac {24 \, {\left (D c^{4} d^{8} - 14 \, B c^{2} d^{10}\right )}}{d^{10}}\right )} x + \frac {105 \, {\left (C c^{4} d^{8} - 10 \, A c^{2} d^{10}\right )}}{d^{10}}\right )} x + \frac {48 \, {\left (2 \, D c^{6} d^{6} + 7 \, B c^{4} d^{8}\right )}}{d^{10}}\right )} + \frac {{\left (C c^{6} + 6 \, A c^{4} d^{2}\right )} \arcsin \left (\frac {d x}{c}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right )}{16 \, d^{2} {\left | d \right |}} \] Input:

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

Output:

-1/1680*sqrt(-d^2*x^2 + c^2)*((2*((4*(5*(6*D*d^2*x + 7*C*d^2)*x - 6*(8*D*c 
^2*d^10 - 7*B*d^12)/d^10)*x - 35*(7*C*c^2*d^10 - 6*A*d^12)/d^10)*x + 24*(D 
*c^4*d^8 - 14*B*c^2*d^10)/d^10)*x + 105*(C*c^4*d^8 - 10*A*c^2*d^10)/d^10)* 
x + 48*(2*D*c^6*d^6 + 7*B*c^4*d^8)/d^10) + 1/16*(C*c^6 + 6*A*c^4*d^2)*arcs 
in(d*x/c)*sgn(c)*sgn(d)/(d^2*abs(d))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.59 \[ \int \left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {630 \mathit {asin} \left (\frac {d x}{c}\right ) a \,c^{4} d^{2}+105 \mathit {asin} \left (\frac {d x}{c}\right ) c^{7}+1050 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{2} d^{3} x -420 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,d^{5} x^{3}-336 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{4} d +672 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{2} d^{3} x^{2}-336 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,d^{5} x^{4}-96 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{6}-105 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{5} d x -48 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{4} d^{2} x^{2}+490 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{3} d^{3} x^{3}+384 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{2} d^{4} x^{4}-280 \sqrt {-d^{2} x^{2}+c^{2}}\, c \,d^{5} x^{5}-240 \sqrt {-d^{2} x^{2}+c^{2}}\, d^{6} x^{6}+336 b \,c^{5} d +96 c^{7}}{1680 d^{3}} \] Input:

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

Output:

(630*asin((d*x)/c)*a*c**4*d**2 + 105*asin((d*x)/c)*c**7 + 1050*sqrt(c**2 - 
 d**2*x**2)*a*c**2*d**3*x - 420*sqrt(c**2 - d**2*x**2)*a*d**5*x**3 - 336*s 
qrt(c**2 - d**2*x**2)*b*c**4*d + 672*sqrt(c**2 - d**2*x**2)*b*c**2*d**3*x* 
*2 - 336*sqrt(c**2 - d**2*x**2)*b*d**5*x**4 - 96*sqrt(c**2 - d**2*x**2)*c* 
*6 - 105*sqrt(c**2 - d**2*x**2)*c**5*d*x - 48*sqrt(c**2 - d**2*x**2)*c**4* 
d**2*x**2 + 490*sqrt(c**2 - d**2*x**2)*c**3*d**3*x**3 + 384*sqrt(c**2 - d* 
*2*x**2)*c**2*d**4*x**4 - 280*sqrt(c**2 - d**2*x**2)*c*d**5*x**5 - 240*sqr 
t(c**2 - d**2*x**2)*d**6*x**6 + 336*b*c**5*d + 96*c**7)/(1680*d**3)