\(\int \sqrt {c+d x} (c^2-d^2 x^2)^{3/2} (A+B x+C x^2+D x^3) \, dx\) [204]

Optimal result

Integrand size = 41, antiderivative size = 306 \[ \int \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=-\frac {64 c^2 \left (23 c^2 C d+13 B c d^2+143 A d^3+5 c^3 D\right ) \left (c^2-d^2 x^2\right )^{5/2}}{45045 d^4 (c+d x)^{5/2}}-\frac {16 c \left (23 c^2 C d+13 B c d^2+143 A d^3+5 c^3 D\right ) \left (c^2-d^2 x^2\right )^{5/2}}{9009 d^4 (c+d x)^{3/2}}-\frac {2 \left (23 c^2 C d+13 B c d^2+143 A d^3+5 c^3 D\right ) \left (c^2-d^2 x^2\right )^{5/2}}{1287 d^4 \sqrt {c+d x}}+\frac {2 \left (30 c C d-39 B d^2-37 c^2 D\right ) \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{5/2}}{429 d^4}-\frac {2 (3 C d-5 c D) (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{5/2}}{39 d^4}-\frac {2 D (c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{5/2}}{15 d^4} \] Output:

-64/45045*c^2*(143*A*d^3+13*B*c*d^2+23*C*c^2*d+5*D*c^3)*(-d^2*x^2+c^2)^(5/ 
2)/d^4/(d*x+c)^(5/2)-16/9009*c*(143*A*d^3+13*B*c*d^2+23*C*c^2*d+5*D*c^3)*( 
-d^2*x^2+c^2)^(5/2)/d^4/(d*x+c)^(3/2)-2/1287*(143*A*d^3+13*B*c*d^2+23*C*c^ 
2*d+5*D*c^3)*(-d^2*x^2+c^2)^(5/2)/d^4/(d*x+c)^(1/2)+2/429*(-39*B*d^2+30*C* 
c*d-37*D*c^2)*(d*x+c)^(1/2)*(-d^2*x^2+c^2)^(5/2)/d^4-2/39*(3*C*d-5*D*c)*(d 
*x+c)^(3/2)*(-d^2*x^2+c^2)^(5/2)/d^4-2/15*D*(d*x+c)^(5/2)*(-d^2*x^2+c^2)^( 
5/2)/d^4
 

Mathematica [A] (verified)

Time = 2.46 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.56 \[ \int \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=-\frac {2 (c-d x)^2 \sqrt {c^2-d^2 x^2} \left (1648 c^5 D+8 c^4 d (347 C+515 D x)+7 d^5 x^2 \left (715 A+585 B x+495 C x^2+429 D x^3\right )+2 c^3 d^2 (2743 B+5 x (694 C+721 D x))+10 c d^4 x (1573 A+7 x (182 B+3 x (51 C+44 D x)))+c^2 d^3 (15301 A+5 x (2743 B+7 x (347 C+309 D x)))\right )}{45045 d^4 \sqrt {c+d x}} \] Input:

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

Output:

(-2*(c - d*x)^2*Sqrt[c^2 - d^2*x^2]*(1648*c^5*D + 8*c^4*d*(347*C + 515*D*x 
) + 7*d^5*x^2*(715*A + 585*B*x + 495*C*x^2 + 429*D*x^3) + 2*c^3*d^2*(2743* 
B + 5*x*(694*C + 721*D*x)) + 10*c*d^4*x*(1573*A + 7*x*(182*B + 3*x*(51*C + 
 44*D*x))) + c^2*d^3*(15301*A + 5*x*(2743*B + 7*x*(347*C + 309*D*x)))))/(4 
5045*d^4*Sqrt[c + d*x])
 

Rubi [A] (verified)

Time = 1.13 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.90, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.195, Rules used = {2170, 27, 2170, 27, 672, 459, 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.

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

\(\Big \downarrow \) 2170

\(\displaystyle -\frac {2 \int -\frac {5}{2} \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2} \left ((3 C d-5 c D) x^2 d^4+\left (3 B d^2-c^2 D\right ) x d^3+\left (D c^3+3 A d^3\right ) d^2\right )dx}{15 d^5}-\frac {2 D (c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{5/2}}{15 d^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2} \left ((3 C d-5 c D) x^2 d^4+\left (3 B d^2-c^2 D\right ) x d^3+\left (D c^3+3 A d^3\right ) d^2\right )dx}{3 d^5}-\frac {2 D (c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{5/2}}{15 d^4}\)

\(\Big \downarrow \) 2170

\(\displaystyle \frac {-\frac {2 \int -\frac {1}{2} d^6 \sqrt {c+d x} \left (-2 D c^3+9 C d c^2+39 A d^3-d \left (-37 D c^2+30 C d c-39 B d^2\right ) x\right ) \left (c^2-d^2 x^2\right )^{3/2}dx}{13 d^4}-\frac {2}{13} d (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{5/2} (3 C d-5 c D)}{3 d^5}-\frac {2 D (c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{5/2}}{15 d^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{13} d^2 \int \sqrt {c+d x} \left (-2 D c^3+9 C d c^2+39 A d^3-d \left (-37 D c^2+30 C d c-39 B d^2\right ) x\right ) \left (c^2-d^2 x^2\right )^{3/2}dx-\frac {2}{13} d (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{5/2} (3 C d-5 c D)}{3 d^5}-\frac {2 D (c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{5/2}}{15 d^4}\)

\(\Big \downarrow \) 672

\(\displaystyle \frac {\frac {1}{13} d^2 \left (\frac {3}{11} \left (143 A d^3+13 B c d^2+5 c^3 D+23 c^2 C d\right ) \int \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2}dx+\frac {2 \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{5/2} \left (-39 B d^2-37 c^2 D+30 c C d\right )}{11 d}\right )-\frac {2}{13} d (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{5/2} (3 C d-5 c D)}{3 d^5}-\frac {2 D (c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{5/2}}{15 d^4}\)

\(\Big \downarrow \) 459

\(\displaystyle \frac {\frac {1}{13} d^2 \left (\frac {3}{11} \left (143 A d^3+13 B c d^2+5 c^3 D+23 c^2 C d\right ) \left (\frac {8}{9} c \int \frac {\left (c^2-d^2 x^2\right )^{3/2}}{\sqrt {c+d x}}dx-\frac {2 \left (c^2-d^2 x^2\right )^{5/2}}{9 d \sqrt {c+d x}}\right )+\frac {2 \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{5/2} \left (-39 B d^2-37 c^2 D+30 c C d\right )}{11 d}\right )-\frac {2}{13} d (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{5/2} (3 C d-5 c D)}{3 d^5}-\frac {2 D (c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{5/2}}{15 d^4}\)

\(\Big \downarrow \) 459

\(\displaystyle \frac {\frac {1}{13} d^2 \left (\frac {3}{11} \left (143 A d^3+13 B c d^2+5 c^3 D+23 c^2 C d\right ) \left (\frac {8}{9} c \left (\frac {4}{7} c \int \frac {\left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^{3/2}}dx-\frac {2 \left (c^2-d^2 x^2\right )^{5/2}}{7 d (c+d x)^{3/2}}\right )-\frac {2 \left (c^2-d^2 x^2\right )^{5/2}}{9 d \sqrt {c+d x}}\right )+\frac {2 \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{5/2} \left (-39 B d^2-37 c^2 D+30 c C d\right )}{11 d}\right )-\frac {2}{13} d (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{5/2} (3 C d-5 c D)}{3 d^5}-\frac {2 D (c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{5/2}}{15 d^4}\)

\(\Big \downarrow \) 458

\(\displaystyle \frac {\frac {1}{13} d^2 \left (\frac {3}{11} \left (\frac {8}{9} c \left (-\frac {2 \left (c^2-d^2 x^2\right )^{5/2}}{7 d (c+d x)^{3/2}}-\frac {8 c \left (c^2-d^2 x^2\right )^{5/2}}{35 d (c+d x)^{5/2}}\right )-\frac {2 \left (c^2-d^2 x^2\right )^{5/2}}{9 d \sqrt {c+d x}}\right ) \left (143 A d^3+13 B c d^2+5 c^3 D+23 c^2 C d\right )+\frac {2 \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{5/2} \left (-39 B d^2-37 c^2 D+30 c C d\right )}{11 d}\right )-\frac {2}{13} d (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{5/2} (3 C d-5 c D)}{3 d^5}-\frac {2 D (c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{5/2}}{15 d^4}\)

Input:

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

Output:

(-2*D*(c + d*x)^(5/2)*(c^2 - d^2*x^2)^(5/2))/(15*d^4) + ((-2*d*(3*C*d - 5* 
c*D)*(c + d*x)^(3/2)*(c^2 - d^2*x^2)^(5/2))/13 + (d^2*((2*(30*c*C*d - 39*B 
*d^2 - 37*c^2*D)*Sqrt[c + d*x]*(c^2 - d^2*x^2)^(5/2))/(11*d) + (3*(23*c^2* 
C*d + 13*B*c*d^2 + 143*A*d^3 + 5*c^3*D)*((-2*(c^2 - d^2*x^2)^(5/2))/(9*d*S 
qrt[c + d*x]) + (8*c*((-8*c*(c^2 - d^2*x^2)^(5/2))/(35*d*(c + d*x)^(5/2)) 
- (2*(c^2 - d^2*x^2)^(5/2))/(7*d*(c + d*x)^(3/2))))/9))/11))/13)/(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 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 = 0.37 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.66

method	result	size
gosper	\(-\frac {2 \left (-d x +c \right ) \left (3003 D d^{5} x^{5}+3465 C \,d^{5} x^{4}+9240 D c \,d^{4} x^{4}+4095 B \,d^{5} x^{3}+10710 C c \,d^{4} x^{3}+10815 D c^{2} d^{3} x^{3}+5005 A \,d^{5} x^{2}+12740 B c \,d^{4} x^{2}+12145 C \,c^{2} d^{3} x^{2}+7210 D c^{3} d^{2} x^{2}+15730 A c \,d^{4} x +13715 B \,c^{2} d^{3} x +6940 C \,c^{3} d^{2} x +4120 D c^{4} d x +15301 A \,c^{2} d^{3}+5486 B \,c^{3} d^{2}+2776 C \,c^{4} d +1648 D c^{5}\right ) \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{45045 d^{4} \left (d x +c \right )^{\frac {3}{2}}}\)	\(203\)
orering	\(-\frac {2 \left (-d x +c \right ) \left (3003 D d^{5} x^{5}+3465 C \,d^{5} x^{4}+9240 D c \,d^{4} x^{4}+4095 B \,d^{5} x^{3}+10710 C c \,d^{4} x^{3}+10815 D c^{2} d^{3} x^{3}+5005 A \,d^{5} x^{2}+12740 B c \,d^{4} x^{2}+12145 C \,c^{2} d^{3} x^{2}+7210 D c^{3} d^{2} x^{2}+15730 A c \,d^{4} x +13715 B \,c^{2} d^{3} x +6940 C \,c^{3} d^{2} x +4120 D c^{4} d x +15301 A \,c^{2} d^{3}+5486 B \,c^{3} d^{2}+2776 C \,c^{4} d +1648 D c^{5}\right ) \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{45045 d^{4} \left (d x +c \right )^{\frac {3}{2}}}\)	\(203\)
default	\(-\frac {2 \sqrt {-d^{2} x^{2}+c^{2}}\, \left (-d x +c \right )^{2} \left (3003 D d^{5} x^{5}+3465 C \,d^{5} x^{4}+9240 D c \,d^{4} x^{4}+4095 B \,d^{5} x^{3}+10710 C c \,d^{4} x^{3}+10815 D c^{2} d^{3} x^{3}+5005 A \,d^{5} x^{2}+12740 B c \,d^{4} x^{2}+12145 C \,c^{2} d^{3} x^{2}+7210 D c^{3} d^{2} x^{2}+15730 A c \,d^{4} x +13715 B \,c^{2} d^{3} x +6940 C \,c^{3} d^{2} x +4120 D c^{4} d x +15301 A \,c^{2} d^{3}+5486 B \,c^{3} d^{2}+2776 C \,c^{4} d +1648 D c^{5}\right )}{45045 \sqrt {d x +c}\, d^{4}}\)	\(205\)

Input:

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

Output:

-2/45045*(-d*x+c)*(3003*D*d^5*x^5+3465*C*d^5*x^4+9240*D*c*d^4*x^4+4095*B*d 
^5*x^3+10710*C*c*d^4*x^3+10815*D*c^2*d^3*x^3+5005*A*d^5*x^2+12740*B*c*d^4* 
x^2+12145*C*c^2*d^3*x^2+7210*D*c^3*d^2*x^2+15730*A*c*d^4*x+13715*B*c^2*d^3 
*x+6940*C*c^3*d^2*x+4120*D*c^4*d*x+15301*A*c^2*d^3+5486*B*c^3*d^2+2776*C*c 
^4*d+1648*D*c^5)*(-d^2*x^2+c^2)^(3/2)/d^4/(d*x+c)^(3/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 280, normalized size of antiderivative = 0.92 \[ \int \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=-\frac {2 \, {\left (3003 \, D d^{7} x^{7} + 1648 \, D c^{7} + 2776 \, C c^{6} d + 5486 \, B c^{5} d^{2} + 15301 \, A c^{4} d^{3} + 231 \, {\left (14 \, D c d^{6} + 15 \, C d^{7}\right )} x^{6} - 63 \, {\left (74 \, D c^{2} d^{5} - 60 \, C c d^{6} - 65 \, B d^{7}\right )} x^{5} - 35 \, {\left (148 \, D c^{3} d^{4} + 166 \, C c^{2} d^{5} - 130 \, B c d^{6} - 143 \, A d^{7}\right )} x^{4} + 5 \, {\left (103 \, D c^{4} d^{3} - 1328 \, C c^{3} d^{4} - 1534 \, B c^{2} d^{5} + 1144 \, A c d^{6}\right )} x^{3} + 3 \, {\left (206 \, D c^{5} d^{2} + 347 \, C c^{4} d^{3} - 3068 \, B c^{3} d^{4} - 3718 \, A c^{2} d^{5}\right )} x^{2} + {\left (824 \, D c^{6} d + 1388 \, C c^{5} d^{2} + 2743 \, B c^{4} d^{3} - 14872 \, A c^{3} d^{4}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}} \sqrt {d x + c}}{45045 \, {\left (d^{5} x + c d^{4}\right )}} \] Input:

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

Output:

-2/45045*(3003*D*d^7*x^7 + 1648*D*c^7 + 2776*C*c^6*d + 5486*B*c^5*d^2 + 15 
301*A*c^4*d^3 + 231*(14*D*c*d^6 + 15*C*d^7)*x^6 - 63*(74*D*c^2*d^5 - 60*C* 
c*d^6 - 65*B*d^7)*x^5 - 35*(148*D*c^3*d^4 + 166*C*c^2*d^5 - 130*B*c*d^6 - 
143*A*d^7)*x^4 + 5*(103*D*c^4*d^3 - 1328*C*c^3*d^4 - 1534*B*c^2*d^5 + 1144 
*A*c*d^6)*x^3 + 3*(206*D*c^5*d^2 + 347*C*c^4*d^3 - 3068*B*c^3*d^4 - 3718*A 
*c^2*d^5)*x^2 + (824*D*c^6*d + 1388*C*c^5*d^2 + 2743*B*c^4*d^3 - 14872*A*c 
^3*d^4)*x)*sqrt(-d^2*x^2 + c^2)*sqrt(d*x + c)/(d^5*x + c*d^4)
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.13 \[ \int \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=-\frac {2 \, {\left (35 \, d^{4} x^{4} + 40 \, c d^{3} x^{3} - 78 \, c^{2} d^{2} x^{2} - 104 \, c^{3} d x + 107 \, c^{4}\right )} {\left (d x + c\right )} \sqrt {-d x + c} A}{315 \, {\left (d^{2} x + c d\right )}} - \frac {2 \, {\left (315 \, d^{5} x^{5} + 350 \, c d^{4} x^{4} - 590 \, c^{2} d^{3} x^{3} - 708 \, c^{3} d^{2} x^{2} + 211 \, c^{4} d x + 422 \, c^{5}\right )} {\left (d x + c\right )} \sqrt {-d x + c} B}{3465 \, {\left (d^{3} x + c d^{2}\right )}} - \frac {2 \, {\left (3465 \, d^{6} x^{6} + 3780 \, c d^{5} x^{5} - 5810 \, c^{2} d^{4} x^{4} - 6640 \, c^{3} d^{3} x^{3} + 1041 \, c^{4} d^{2} x^{2} + 1388 \, c^{5} d x + 2776 \, c^{6}\right )} {\left (d x + c\right )} \sqrt {-d x + c} C}{45045 \, {\left (d^{4} x + c d^{3}\right )}} - \frac {2 \, {\left (3003 \, d^{7} x^{7} + 3234 \, c d^{6} x^{6} - 4662 \, c^{2} d^{5} x^{5} - 5180 \, c^{3} d^{4} x^{4} + 515 \, c^{4} d^{3} x^{3} + 618 \, c^{5} d^{2} x^{2} + 824 \, c^{6} d x + 1648 \, c^{7}\right )} {\left (d x + c\right )} \sqrt {-d x + c} D}{45045 \, {\left (d^{5} x + c d^{4}\right )}} \] Input:

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

Output:

-2/315*(35*d^4*x^4 + 40*c*d^3*x^3 - 78*c^2*d^2*x^2 - 104*c^3*d*x + 107*c^4 
)*(d*x + c)*sqrt(-d*x + c)*A/(d^2*x + c*d) - 2/3465*(315*d^5*x^5 + 350*c*d 
^4*x^4 - 590*c^2*d^3*x^3 - 708*c^3*d^2*x^2 + 211*c^4*d*x + 422*c^5)*(d*x + 
 c)*sqrt(-d*x + c)*B/(d^3*x + c*d^2) - 2/45045*(3465*d^6*x^6 + 3780*c*d^5* 
x^5 - 5810*c^2*d^4*x^4 - 6640*c^3*d^3*x^3 + 1041*c^4*d^2*x^2 + 1388*c^5*d* 
x + 2776*c^6)*(d*x + c)*sqrt(-d*x + c)*C/(d^4*x + c*d^3) - 2/45045*(3003*d 
^7*x^7 + 3234*c*d^6*x^6 - 4662*c^2*d^5*x^5 - 5180*c^3*d^4*x^4 + 515*c^4*d^ 
3*x^3 + 618*c^5*d^2*x^2 + 824*c^6*d*x + 1648*c^7)*(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. 1017 vs. \(2 (270) = 540\).

Time = 0.12 (sec) , antiderivative size = 1017, normalized size of antiderivative = 3.32 \[ \int \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

-2/45045*(45045*sqrt(-d*x + c)*A*c^4*d^3 - 15015*((-d*x + c)^(3/2) - 3*sqr 
t(-d*x + c)*c)*B*c^4*d^2 + 3003*(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^4*d - 6006*(3*(d*x - c)^2*sqrt(-d 
*x + c) - 10*(-d*x + c)^(3/2)*c + 15*sqrt(-d*x + c)*c^2)*A*c^2*d^3 + 1287* 
(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^4 - 2574*(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)*B*c^2*d^2 - 286*(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*c^2*d + 143*(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)*A*d^3 - 
 130*(63*(d*x - c)^5*sqrt(-d*x + c) + 385*(d*x - c)^4*sqrt(-d*x + c)*c + 9 
90*(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*c^2 + 65*(63*(d*x - 
c)^5*sqrt(-d*x + c) + 385*(d*x - c)^4*sqrt(-d*x + c)*c + 990*(d*x - c)^3*s 
qrt(-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)*B*d^2 + 15*(231*(d*x - c)^6*sqrt(-d*x 
+ c) + 1638*(d*x - c)^5*sqrt(-d*x + c)*c + 5005*(d*x - c)^4*sqrt(-d*x + c) 
*c^2 + 8580*(d*x - c)^3*sqrt(-d*x + c)*c^3 + 9009*(d*x - c)^2*sqrt(-d*x...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.64 \[ \int \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {2 \sqrt {-d x +c}\, \left (-3003 d^{7} x^{7}-6699 c \,d^{6} x^{6}-4095 b \,d^{6} x^{5}+882 c^{2} d^{5} x^{5}-5005 a \,d^{6} x^{4}-4550 b c \,d^{5} x^{4}+10990 c^{3} d^{4} x^{4}-5720 a c \,d^{5} x^{3}+7670 b \,c^{2} d^{4} x^{3}+6125 c^{4} d^{3} x^{3}+11154 a \,c^{2} d^{4} x^{2}+9204 b \,c^{3} d^{3} x^{2}-1659 c^{5} d^{2} x^{2}+14872 a \,c^{3} d^{3} x -2743 b \,c^{4} d^{2} x -2212 c^{6} d x -15301 a \,c^{4} d^{2}-5486 b \,c^{5} d -4424 c^{7}\right )}{45045 d^{3}} \] Input:

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

Output:

(2*sqrt(c - d*x)*( - 15301*a*c**4*d**2 + 14872*a*c**3*d**3*x + 11154*a*c** 
2*d**4*x**2 - 5720*a*c*d**5*x**3 - 5005*a*d**6*x**4 - 5486*b*c**5*d - 2743 
*b*c**4*d**2*x + 9204*b*c**3*d**3*x**2 + 7670*b*c**2*d**4*x**3 - 4550*b*c* 
d**5*x**4 - 4095*b*d**6*x**5 - 4424*c**7 - 2212*c**6*d*x - 1659*c**5*d**2* 
x**2 + 6125*c**4*d**3*x**3 + 10990*c**3*d**4*x**4 + 882*c**2*d**5*x**5 - 6 
699*c*d**6*x**6 - 3003*d**7*x**7))/(45045*d**3)