\(\int \sqrt {c+d x} \sqrt {c^2-d^2 x^2} (A+B x+C x^2+D x^3) \, dx\) [195]

Optimal result

Integrand size = 41, antiderivative size = 244 \[ \int \sqrt {c+d x} \sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right ) \, dx=-\frac {8 c \left (55 c^2 C d+33 B c d^2+231 A d^3+17 c^3 D\right ) \left (c^2-d^2 x^2\right )^{3/2}}{3465 d^4 (c+d x)^{3/2}}-\frac {2 \left (55 c^2 C d+33 B c d^2+231 A d^3+17 c^3 D\right ) \left (c^2-d^2 x^2\right )^{3/2}}{1155 d^4 \sqrt {c+d x}}+\frac {2 \left (22 c C d-33 B d^2-31 c^2 D\right ) \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2}}{231 d^4}-\frac {2 (11 C d-17 c D) (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{3/2}}{99 d^4}-\frac {2 D (c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{3/2}}{11 d^4} \] Output:

-8/3465*c*(231*A*d^3+33*B*c*d^2+55*C*c^2*d+17*D*c^3)*(-d^2*x^2+c^2)^(3/2)/ 
d^4/(d*x+c)^(3/2)-2/1155*(231*A*d^3+33*B*c*d^2+55*C*c^2*d+17*D*c^3)*(-d^2* 
x^2+c^2)^(3/2)/d^4/(d*x+c)^(1/2)+2/231*(-33*B*d^2+22*C*c*d-31*D*c^2)*(d*x+ 
c)^(1/2)*(-d^2*x^2+c^2)^(3/2)/d^4-2/99*(11*C*d-17*D*c)*(d*x+c)^(3/2)*(-d^2 
*x^2+c^2)^(3/2)/d^4-2/11*D*(d*x+c)^(5/2)*(-d^2*x^2+c^2)^(3/2)/d^4
 

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.55 \[ \int \sqrt {c+d x} \sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right ) \, dx=-\frac {2 (c-d x) \sqrt {c^2-d^2 x^2} \left (304 c^4 D+8 c^3 d (55 C+57 D x)+6 c^2 d^2 (121 B+5 x (22 C+19 D x))+d^4 x \left (693 A+5 x \left (99 B+77 C x+63 D x^2\right )\right )+c d^3 \left (1617 A+x \left (1089 B+825 C x+665 D x^2\right )\right )\right )}{3465 d^4 \sqrt {c+d x}} \] Input:

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

Output:

(-2*(c - d*x)*Sqrt[c^2 - d^2*x^2]*(304*c^4*D + 8*c^3*d*(55*C + 57*D*x) + 6 
*c^2*d^2*(121*B + 5*x*(22*C + 19*D*x)) + d^4*x*(693*A + 5*x*(99*B + 77*C*x 
 + 63*D*x^2)) + c*d^3*(1617*A + x*(1089*B + 825*C*x + 665*D*x^2))))/(3465* 
d^4*Sqrt[c + d*x])
 

Rubi [A] (verified)

Time = 1.06 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.97, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {2170, 27, 2170, 27, 672, 459, 458}

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.

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

\(\Big \downarrow \) 2170

\(\displaystyle -\frac {2 \int -\frac {1}{2} \sqrt {c+d x} \sqrt {c^2-d^2 x^2} \left ((11 C d-17 c D) x^2 d^4+\left (11 B d^2-c^2 D\right ) x d^3+\left (5 D c^3+11 A d^3\right ) d^2\right )dx}{11 d^5}-\frac {2 D \left (c^2-d^2 x^2\right )^{3/2} (c+d x)^{5/2}}{11 d^4}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2170

\(\displaystyle \frac {-\frac {2 \int -\frac {3}{2} d^6 \sqrt {c+d x} \left (-2 D c^3+11 C d c^2+33 A d^3-d \left (-31 D c^2+22 C d c-33 B d^2\right ) x\right ) \sqrt {c^2-d^2 x^2}dx}{9 d^4}-\frac {2}{9} d (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{3/2} (11 C d-17 c D)}{11 d^5}-\frac {2 D (c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{3/2}}{11 d^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{3} d^2 \int \sqrt {c+d x} \left (-2 D c^3+11 C d c^2+33 A d^3-d \left (-31 D c^2+22 C d c-33 B d^2\right ) x\right ) \sqrt {c^2-d^2 x^2}dx-\frac {2}{9} d (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{3/2} (11 C d-17 c D)}{11 d^5}-\frac {2 D (c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{3/2}}{11 d^4}\)

\(\Big \downarrow \) 672

\(\displaystyle \frac {\frac {1}{3} d^2 \left (\frac {1}{7} \left (231 A d^3+33 B c d^2+17 c^3 D+55 c^2 C d\right ) \int \sqrt {c+d x} \sqrt {c^2-d^2 x^2}dx+\frac {2 \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2} \left (-33 B d^2-31 c^2 D+22 c C d\right )}{7 d}\right )-\frac {2}{9} d (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{3/2} (11 C d-17 c D)}{11 d^5}-\frac {2 D (c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{3/2}}{11 d^4}\)

\(\Big \downarrow \) 459

\(\displaystyle \frac {\frac {1}{3} d^2 \left (\frac {1}{7} \left (231 A d^3+33 B c d^2+17 c^3 D+55 c^2 C d\right ) \left (\frac {4}{5} c \int \frac {\sqrt {c^2-d^2 x^2}}{\sqrt {c+d x}}dx-\frac {2 \left (c^2-d^2 x^2\right )^{3/2}}{5 d \sqrt {c+d x}}\right )+\frac {2 \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2} \left (-33 B d^2-31 c^2 D+22 c C d\right )}{7 d}\right )-\frac {2}{9} d (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{3/2} (11 C d-17 c D)}{11 d^5}-\frac {2 D (c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{3/2}}{11 d^4}\)

\(\Big \downarrow \) 458

\(\displaystyle \frac {\frac {1}{3} d^2 \left (\frac {1}{7} \left (-\frac {2 \left (c^2-d^2 x^2\right )^{3/2}}{5 d \sqrt {c+d x}}-\frac {8 c \left (c^2-d^2 x^2\right )^{3/2}}{15 d (c+d x)^{3/2}}\right ) \left (231 A d^3+33 B c d^2+17 c^3 D+55 c^2 C d\right )+\frac {2 \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2} \left (-33 B d^2-31 c^2 D+22 c C d\right )}{7 d}\right )-\frac {2}{9} d (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{3/2} (11 C d-17 c D)}{11 d^5}-\frac {2 D (c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{3/2}}{11 d^4}\)

Input:

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

Output:

(-2*D*(c + d*x)^(5/2)*(c^2 - d^2*x^2)^(3/2))/(11*d^4) + ((-2*d*(11*C*d - 1 
7*c*D)*(c + d*x)^(3/2)*(c^2 - d^2*x^2)^(3/2))/9 + (d^2*((2*(22*c*C*d - 33* 
B*d^2 - 31*c^2*D)*Sqrt[c + d*x]*(c^2 - d^2*x^2)^(3/2))/(7*d) + ((55*c^2*C* 
d + 33*B*c*d^2 + 231*A*d^3 + 17*c^3*D)*((-8*c*(c^2 - d^2*x^2)^(3/2))/(15*d 
*(c + d*x)^(3/2)) - (2*(c^2 - d^2*x^2)^(3/2))/(5*d*Sqrt[c + d*x])))/7))/3) 
/(11*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 458

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

rule 459

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

rule 672

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_ 
), x_Symbol] :> Simp[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), 
 x] + Simp[(m*(d*g + e*f) + 2*e*f*(p + 1))/(e*(m + 2*p + 2))   Int[(d + e*x 
)^m*(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] && NeQ[m + 2*p + 2, 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 [A] (verified)

Time = 0.36 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.64

method	result	size
gosper	\(-\frac {2 \left (-d x +c \right ) \left (315 D x^{4} d^{4}+385 C \,d^{4} x^{3}+665 D c \,d^{3} x^{3}+495 B \,d^{4} x^{2}+825 C c \,d^{3} x^{2}+570 D c^{2} d^{2} x^{2}+693 A \,d^{4} x +1089 B c \,d^{3} x +660 C \,c^{2} d^{2} x +456 D c^{3} d x +1617 A c \,d^{3}+726 B \,c^{2} d^{2}+440 C \,c^{3} d +304 c^{4} D\right ) \sqrt {-d^{2} x^{2}+c^{2}}}{3465 d^{4} \sqrt {d x +c}}\)	\(155\)
default	\(-\frac {2 \left (-d x +c \right ) \left (315 D x^{4} d^{4}+385 C \,d^{4} x^{3}+665 D c \,d^{3} x^{3}+495 B \,d^{4} x^{2}+825 C c \,d^{3} x^{2}+570 D c^{2} d^{2} x^{2}+693 A \,d^{4} x +1089 B c \,d^{3} x +660 C \,c^{2} d^{2} x +456 D c^{3} d x +1617 A c \,d^{3}+726 B \,c^{2} d^{2}+440 C \,c^{3} d +304 c^{4} D\right ) \sqrt {-d^{2} x^{2}+c^{2}}}{3465 d^{4} \sqrt {d x +c}}\)	\(155\)
orering	\(-\frac {2 \left (-d x +c \right ) \left (315 D x^{4} d^{4}+385 C \,d^{4} x^{3}+665 D c \,d^{3} x^{3}+495 B \,d^{4} x^{2}+825 C c \,d^{3} x^{2}+570 D c^{2} d^{2} x^{2}+693 A \,d^{4} x +1089 B c \,d^{3} x +660 C \,c^{2} d^{2} x +456 D c^{3} d x +1617 A c \,d^{3}+726 B \,c^{2} d^{2}+440 C \,c^{3} d +304 c^{4} D\right ) \sqrt {-d^{2} x^{2}+c^{2}}}{3465 d^{4} \sqrt {d x +c}}\)	\(155\)

Input:

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

Output:

-2/3465*(-d*x+c)*(315*D*d^4*x^4+385*C*d^4*x^3+665*D*c*d^3*x^3+495*B*d^4*x^ 
2+825*C*c*d^3*x^2+570*D*c^2*d^2*x^2+693*A*d^4*x+1089*B*c*d^3*x+660*C*c^2*d 
^2*x+456*D*c^3*d*x+1617*A*c*d^3+726*B*c^2*d^2+440*C*c^3*d+304*D*c^4)*(-d^2 
*x^2+c^2)^(1/2)/d^4/(d*x+c)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.81 \[ \int \sqrt {c+d x} \sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {2 \, {\left (315 \, D d^{5} x^{5} - 304 \, D c^{5} - 440 \, C c^{4} d - 726 \, B c^{3} d^{2} - 1617 \, A c^{2} d^{3} + 35 \, {\left (10 \, D c d^{4} + 11 \, C d^{5}\right )} x^{4} - 5 \, {\left (19 \, D c^{2} d^{3} - 88 \, C c d^{4} - 99 \, B d^{5}\right )} x^{3} - 3 \, {\left (38 \, D c^{3} d^{2} + 55 \, C c^{2} d^{3} - 198 \, B c d^{4} - 231 \, A d^{5}\right )} x^{2} - {\left (152 \, D c^{4} d + 220 \, C c^{3} d^{2} + 363 \, B c^{2} d^{3} - 924 \, A c d^{4}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}} \sqrt {d x + c}}{3465 \, {\left (d^{5} x + c d^{4}\right )}} \] Input:

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

Output:

2/3465*(315*D*d^5*x^5 - 304*D*c^5 - 440*C*c^4*d - 726*B*c^3*d^2 - 1617*A*c 
^2*d^3 + 35*(10*D*c*d^4 + 11*C*d^5)*x^4 - 5*(19*D*c^2*d^3 - 88*C*c*d^4 - 9 
9*B*d^5)*x^3 - 3*(38*D*c^3*d^2 + 55*C*c^2*d^3 - 198*B*c*d^4 - 231*A*d^5)*x 
^2 - (152*D*c^4*d + 220*C*c^3*d^2 + 363*B*c^2*d^3 - 924*A*c*d^4)*x)*sqrt(- 
d^2*x^2 + c^2)*sqrt(d*x + c)/(d^5*x + c*d^4)
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.05 \[ \int \sqrt {c+d x} \sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {2 \, {\left (3 \, d^{2} x^{2} + 4 \, c d x - 7 \, c^{2}\right )} {\left (d x + c\right )} \sqrt {-d x + c} A}{15 \, {\left (d^{2} x + c d\right )}} + \frac {2 \, {\left (15 \, d^{3} x^{3} + 18 \, c d^{2} x^{2} - 11 \, c^{2} d x - 22 \, c^{3}\right )} {\left (d x + c\right )} \sqrt {-d x + c} B}{105 \, {\left (d^{3} x + c d^{2}\right )}} + \frac {2 \, {\left (7 \, d^{4} x^{4} + 8 \, c d^{3} x^{3} - 3 \, c^{2} d^{2} x^{2} - 4 \, c^{3} d x - 8 \, c^{4}\right )} {\left (d x + c\right )} \sqrt {-d x + c} C}{63 \, {\left (d^{4} x + c d^{3}\right )}} + \frac {2 \, {\left (315 \, d^{5} x^{5} + 350 \, c d^{4} x^{4} - 95 \, c^{2} d^{3} x^{3} - 114 \, c^{3} d^{2} x^{2} - 152 \, c^{4} d x - 304 \, c^{5}\right )} {\left (d x + c\right )} \sqrt {-d x + c} D}{3465 \, {\left (d^{5} x + c d^{4}\right )}} \] Input:

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

Output:

2/15*(3*d^2*x^2 + 4*c*d*x - 7*c^2)*(d*x + c)*sqrt(-d*x + c)*A/(d^2*x + c*d 
) + 2/105*(15*d^3*x^3 + 18*c*d^2*x^2 - 11*c^2*d*x - 22*c^3)*(d*x + c)*sqrt 
(-d*x + c)*B/(d^3*x + c*d^2) + 2/63*(7*d^4*x^4 + 8*c*d^3*x^3 - 3*c^2*d^2*x 
^2 - 4*c^3*d*x - 8*c^4)*(d*x + c)*sqrt(-d*x + c)*C/(d^4*x + c*d^3) + 2/346 
5*(315*d^5*x^5 + 350*c*d^4*x^4 - 95*c^2*d^3*x^3 - 114*c^3*d^2*x^2 - 152*c^ 
4*d*x - 304*c^5)*(d*x + c)*sqrt(-d*x + c)*D/(d^5*x + c*d^4)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 502 vs. \(2 (214) = 428\).

Time = 0.12 (sec) , antiderivative size = 502, normalized size of antiderivative = 2.06 \[ \int \sqrt {c+d x} \sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right ) \, dx=-\frac {2 \, {\left (3465 \, \sqrt {-d x + c} A c^{2} d^{3} - 1155 \, {\left ({\left (-d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {-d x + c} c\right )} B c^{2} d^{2} + 231 \, {\left (3 \, {\left (d x - c\right )}^{2} \sqrt {-d x + c} - 10 \, {\left (-d x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {-d x + c} c^{2}\right )} C c^{2} d - 231 \, {\left (3 \, {\left (d x - c\right )}^{2} \sqrt {-d x + c} - 10 \, {\left (-d x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {-d x + c} c^{2}\right )} A d^{3} + 99 \, {\left (5 \, {\left (d x - c\right )}^{3} \sqrt {-d x + c} + 21 \, {\left (d x - c\right )}^{2} \sqrt {-d x + c} c - 35 \, {\left (-d x + c\right )}^{\frac {3}{2}} c^{2} + 35 \, \sqrt {-d x + c} c^{3}\right )} D c^{2} - 99 \, {\left (5 \, {\left (d x - c\right )}^{3} \sqrt {-d x + c} + 21 \, {\left (d x - c\right )}^{2} \sqrt {-d x + c} c - 35 \, {\left (-d x + c\right )}^{\frac {3}{2}} c^{2} + 35 \, \sqrt {-d x + c} c^{3}\right )} B d^{2} - 11 \, {\left (35 \, {\left (d x - c\right )}^{4} \sqrt {-d x + c} + 180 \, {\left (d x - c\right )}^{3} \sqrt {-d x + c} c + 378 \, {\left (d x - c\right )}^{2} \sqrt {-d x + c} c^{2} - 420 \, {\left (-d x + c\right )}^{\frac {3}{2}} c^{3} + 315 \, \sqrt {-d x + c} c^{4}\right )} C d - 5 \, {\left (63 \, {\left (d x - c\right )}^{5} \sqrt {-d x + c} + 385 \, {\left (d x - c\right )}^{4} \sqrt {-d x + c} c + 990 \, {\left (d x - c\right )}^{3} \sqrt {-d x + c} c^{2} + 1386 \, {\left (d x - c\right )}^{2} \sqrt {-d x + c} c^{3} - 1155 \, {\left (-d x + c\right )}^{\frac {3}{2}} c^{4} + 693 \, \sqrt {-d x + c} c^{5}\right )} D\right )}}{3465 \, d^{4}} \] Input:

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

Output:

-2/3465*(3465*sqrt(-d*x + c)*A*c^2*d^3 - 1155*((-d*x + c)^(3/2) - 3*sqrt(- 
d*x + c)*c)*B*c^2*d^2 + 231*(3*(d*x - c)^2*sqrt(-d*x + c) - 10*(-d*x + c)^ 
(3/2)*c + 15*sqrt(-d*x + c)*c^2)*C*c^2*d - 231*(3*(d*x - c)^2*sqrt(-d*x + 
c) - 10*(-d*x + c)^(3/2)*c + 15*sqrt(-d*x + c)*c^2)*A*d^3 + 99*(5*(d*x - c 
)^3*sqrt(-d*x + c) + 21*(d*x - c)^2*sqrt(-d*x + c)*c - 35*(-d*x + c)^(3/2) 
*c^2 + 35*sqrt(-d*x + c)*c^3)*D*c^2 - 99*(5*(d*x - c)^3*sqrt(-d*x + c) + 2 
1*(d*x - c)^2*sqrt(-d*x + c)*c - 35*(-d*x + c)^(3/2)*c^2 + 35*sqrt(-d*x + 
c)*c^3)*B*d^2 - 11*(35*(d*x - c)^4*sqrt(-d*x + c) + 180*(d*x - c)^3*sqrt(- 
d*x + c)*c + 378*(d*x - c)^2*sqrt(-d*x + c)*c^2 - 420*(-d*x + c)^(3/2)*c^3 
 + 315*sqrt(-d*x + c)*c^4)*C*d - 5*(63*(d*x - c)^5*sqrt(-d*x + c) + 385*(d 
*x - c)^4*sqrt(-d*x + c)*c + 990*(d*x - c)^3*sqrt(-d*x + c)*c^2 + 1386*(d* 
x - c)^2*sqrt(-d*x + c)*c^3 - 1155*(-d*x + c)^(3/2)*c^4 + 693*sqrt(-d*x + 
c)*c^5)*D)/d^4
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.52 \[ \int \sqrt {c+d x} \sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {2 \sqrt {-d x +c}\, \left (105 d^{5} x^{5}+245 c \,d^{4} x^{4}+165 b \,d^{4} x^{3}+115 c^{2} d^{3} x^{3}+231 a \,d^{4} x^{2}+198 b c \,d^{3} x^{2}-93 c^{3} d^{2} x^{2}+308 a c \,d^{3} x -121 b \,c^{2} d^{2} x -124 c^{4} d x -539 a \,c^{2} d^{2}-242 b \,c^{3} d -248 c^{5}\right )}{1155 d^{3}} \] Input:

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

Output:

(2*sqrt(c - d*x)*( - 539*a*c**2*d**2 + 308*a*c*d**3*x + 231*a*d**4*x**2 - 
242*b*c**3*d - 121*b*c**2*d**2*x + 198*b*c*d**3*x**2 + 165*b*d**4*x**3 - 2 
48*c**5 - 124*c**4*d*x - 93*c**3*d**2*x**2 + 115*c**2*d**3*x**3 + 245*c*d* 
*4*x**4 + 105*d**5*x**5))/(1155*d**3)