\(\int (c+d x)^{3/2} (c^2-d^2 x^2)^{3/2} (A+B x+C x^2+D x^3) \, dx\) [203]

Optimal result

Integrand size = 41, antiderivative size = 368 \[ \int (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=-\frac {256 c^3 \left (119 c^2 C d+153 B c d^2+663 A d^3+57 c^3 D\right ) \left (c^2-d^2 x^2\right )^{5/2}}{765765 d^4 (c+d x)^{5/2}}-\frac {64 c^2 \left (119 c^2 C d+153 B c d^2+663 A d^3+57 c^3 D\right ) \left (c^2-d^2 x^2\right )^{5/2}}{153153 d^4 (c+d x)^{3/2}}-\frac {8 c \left (119 c^2 C d+153 B c d^2+663 A d^3+57 c^3 D\right ) \left (c^2-d^2 x^2\right )^{5/2}}{21879 d^4 \sqrt {c+d x}}-\frac {2 \left (119 c^2 C d+153 B c d^2+663 A d^3+57 c^3 D\right ) \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{5/2}}{7293 d^4}+\frac {2 \left (34 c C d-51 B d^2-45 c^2 D\right ) (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{5/2}}{663 d^4}-\frac {2 (17 C d-27 c D) (c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{5/2}}{255 d^4}-\frac {2 D (c+d x)^{7/2} \left (c^2-d^2 x^2\right )^{5/2}}{17 d^4} \] Output:

-256/765765*c^3*(663*A*d^3+153*B*c*d^2+119*C*c^2*d+57*D*c^3)*(-d^2*x^2+c^2 
)^(5/2)/d^4/(d*x+c)^(5/2)-64/153153*c^2*(663*A*d^3+153*B*c*d^2+119*C*c^2*d 
+57*D*c^3)*(-d^2*x^2+c^2)^(5/2)/d^4/(d*x+c)^(3/2)-8/21879*c*(663*A*d^3+153 
*B*c*d^2+119*C*c^2*d+57*D*c^3)*(-d^2*x^2+c^2)^(5/2)/d^4/(d*x+c)^(1/2)-2/72 
93*(663*A*d^3+153*B*c*d^2+119*C*c^2*d+57*D*c^3)*(d*x+c)^(1/2)*(-d^2*x^2+c^ 
2)^(5/2)/d^4+2/663*(-51*B*d^2+34*C*c*d-45*D*c^2)*(d*x+c)^(3/2)*(-d^2*x^2+c 
^2)^(5/2)/d^4-2/255*(17*C*d-27*D*c)*(d*x+c)^(5/2)*(-d^2*x^2+c^2)^(5/2)/d^4 
-2/17*D*(d*x+c)^(7/2)*(-d^2*x^2+c^2)^(5/2)/d^4
 

Mathematica [A] (verified)

Time = 2.76 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.55 \[ \int (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=-\frac {2 (c-d x)^2 \sqrt {c^2-d^2 x^2} \left (46320 c^6 D+8 c^5 d (9401 C+14475 D x)+2 c^4 d^2 (70227 B+35 x (2686 C+2895 D x))+21 d^6 x^3 (3315 A+11 x (255 B+13 x (17 C+15 D x)))+21 c d^5 x^2 (14365 A+x (11985 B+11 x (935 C+819 D x)))+15 c^2 d^4 x (33371 A+7 x (4029 B+x (3485 C+3069 D x)))+c^3 d^3 (353379 A+5 x (70227 B+7 x (9401 C+8685 D x)))\right )}{765765 d^4 \sqrt {c+d x}} \] Input:

Integrate[(c + d*x)^(3/2)*(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]*(46320*c^6*D + 8*c^5*d*(9401*C + 14475 
*D*x) + 2*c^4*d^2*(70227*B + 35*x*(2686*C + 2895*D*x)) + 21*d^6*x^3*(3315* 
A + 11*x*(255*B + 13*x*(17*C + 15*D*x))) + 21*c*d^5*x^2*(14365*A + x*(1198 
5*B + 11*x*(935*C + 819*D*x))) + 15*c^2*d^4*x*(33371*A + 7*x*(4029*B + x*( 
3485*C + 3069*D*x))) + c^3*d^3*(353379*A + 5*x*(70227*B + 7*x*(9401*C + 86 
85*D*x)))))/(765765*d^4*Sqrt[c + d*x])
 

Rubi [A] (verified)

Time = 1.22 (sec) , antiderivative size = 312, normalized size of antiderivative = 0.85, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.220, Rules used = {2170, 27, 2170, 27, 672, 459, 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.

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

\(\Big \downarrow \) 2170

\(\displaystyle -\frac {2 \int -\frac {1}{2} (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{3/2} \left ((17 C d-27 c D) x^2 d^4+\left (17 B d^2-3 c^2 D\right ) x d^3+\left (7 D c^3+17 A d^3\right ) d^2\right )dx}{17 d^5}-\frac {2 D \left (c^2-d^2 x^2\right )^{5/2} (c+d x)^{7/2}}{17 d^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{3/2} \left ((17 C d-27 c D) x^2 d^4+\left (17 B d^2-3 c^2 D\right ) x d^3+\left (7 D c^3+17 A d^3\right ) d^2\right )dx}{17 d^5}-\frac {2 D (c+d x)^{7/2} \left (c^2-d^2 x^2\right )^{5/2}}{17 d^4}\)

\(\Big \downarrow \) 2170

\(\displaystyle \frac {-\frac {2 \int -\frac {5}{2} d^6 (c+d x)^{3/2} \left (-6 D c^3+17 C d c^2+51 A d^3-d \left (-45 D c^2+34 C d c-51 B d^2\right ) x\right ) \left (c^2-d^2 x^2\right )^{3/2}dx}{15 d^4}-\frac {2}{15} d (c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{5/2} (17 C d-27 c D)}{17 d^5}-\frac {2 D (c+d x)^{7/2} \left (c^2-d^2 x^2\right )^{5/2}}{17 d^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{3} d^2 \int (c+d x)^{3/2} \left (-6 D c^3+17 C d c^2+51 A d^3-d \left (-45 D c^2+34 C d c-51 B d^2\right ) x\right ) \left (c^2-d^2 x^2\right )^{3/2}dx-\frac {2}{15} d (c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{5/2} (17 C d-27 c D)}{17 d^5}-\frac {2 D (c+d x)^{7/2} \left (c^2-d^2 x^2\right )^{5/2}}{17 d^4}\)

\(\Big \downarrow \) 672

\(\displaystyle \frac {\frac {1}{3} d^2 \left (\frac {1}{13} \left (663 A d^3+153 B c d^2+57 c^3 D+119 c^2 C d\right ) \int (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{3/2}dx+\frac {2 (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{5/2} \left (-51 B d^2-45 c^2 D+34 c C d\right )}{13 d}\right )-\frac {2}{15} d (c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{5/2} (17 C d-27 c D)}{17 d^5}-\frac {2 D (c+d x)^{7/2} \left (c^2-d^2 x^2\right )^{5/2}}{17 d^4}\)

\(\Big \downarrow \) 459

\(\displaystyle \frac {\frac {1}{3} d^2 \left (\frac {1}{13} \left (663 A d^3+153 B c d^2+57 c^3 D+119 c^2 C d\right ) \left (\frac {12}{11} c \int \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2}dx-\frac {2 \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{5/2}}{11 d}\right )+\frac {2 (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{5/2} \left (-51 B d^2-45 c^2 D+34 c C d\right )}{13 d}\right )-\frac {2}{15} d (c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{5/2} (17 C d-27 c D)}{17 d^5}-\frac {2 D (c+d x)^{7/2} \left (c^2-d^2 x^2\right )^{5/2}}{17 d^4}\)

\(\Big \downarrow \) 459

\(\displaystyle \frac {\frac {1}{3} d^2 \left (\frac {1}{13} \left (663 A d^3+153 B c d^2+57 c^3 D+119 c^2 C d\right ) \left (\frac {12}{11} c \left (\frac {8}{9} c \int \frac {\left (c^2-d^2 x^2\right )^{3/2}}{\sqrt {c+d x}}dx-\frac {2 \left (c^2-d^2 x^2\right )^{5/2}}{9 d \sqrt {c+d x}}\right )-\frac {2 \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{5/2}}{11 d}\right )+\frac {2 (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{5/2} \left (-51 B d^2-45 c^2 D+34 c C d\right )}{13 d}\right )-\frac {2}{15} d (c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{5/2} (17 C d-27 c D)}{17 d^5}-\frac {2 D (c+d x)^{7/2} \left (c^2-d^2 x^2\right )^{5/2}}{17 d^4}\)

\(\Big \downarrow \) 459

\(\displaystyle \frac {\frac {1}{3} d^2 \left (\frac {1}{13} \left (663 A d^3+153 B c d^2+57 c^3 D+119 c^2 C d\right ) \left (\frac {12}{11} c \left (\frac {8}{9} c \left (\frac {4}{7} c \int \frac {\left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^{3/2}}dx-\frac {2 \left (c^2-d^2 x^2\right )^{5/2}}{7 d (c+d x)^{3/2}}\right )-\frac {2 \left (c^2-d^2 x^2\right )^{5/2}}{9 d \sqrt {c+d x}}\right )-\frac {2 \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{5/2}}{11 d}\right )+\frac {2 (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{5/2} \left (-51 B d^2-45 c^2 D+34 c C d\right )}{13 d}\right )-\frac {2}{15} d (c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{5/2} (17 C d-27 c D)}{17 d^5}-\frac {2 D (c+d x)^{7/2} \left (c^2-d^2 x^2\right )^{5/2}}{17 d^4}\)

\(\Big \downarrow \) 458

\(\displaystyle \frac {\frac {1}{3} d^2 \left (\frac {1}{13} \left (\frac {12}{11} c \left (\frac {8}{9} c \left (-\frac {2 \left (c^2-d^2 x^2\right )^{5/2}}{7 d (c+d x)^{3/2}}-\frac {8 c \left (c^2-d^2 x^2\right )^{5/2}}{35 d (c+d x)^{5/2}}\right )-\frac {2 \left (c^2-d^2 x^2\right )^{5/2}}{9 d \sqrt {c+d x}}\right )-\frac {2 \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{5/2}}{11 d}\right ) \left (663 A d^3+153 B c d^2+57 c^3 D+119 c^2 C d\right )+\frac {2 (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{5/2} \left (-51 B d^2-45 c^2 D+34 c C d\right )}{13 d}\right )-\frac {2}{15} d (c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{5/2} (17 C d-27 c D)}{17 d^5}-\frac {2 D (c+d x)^{7/2} \left (c^2-d^2 x^2\right )^{5/2}}{17 d^4}\)

Input:

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

Output:

(-2*D*(c + d*x)^(7/2)*(c^2 - d^2*x^2)^(5/2))/(17*d^4) + ((-2*d*(17*C*d - 2 
7*c*D)*(c + d*x)^(5/2)*(c^2 - d^2*x^2)^(5/2))/15 + (d^2*((2*(34*c*C*d - 51 
*B*d^2 - 45*c^2*D)*(c + d*x)^(3/2)*(c^2 - d^2*x^2)^(5/2))/(13*d) + ((119*c 
^2*C*d + 153*B*c*d^2 + 663*A*d^3 + 57*c^3*D)*((-2*Sqrt[c + d*x]*(c^2 - d^2 
*x^2)^(5/2))/(11*d) + (12*c*((-2*(c^2 - d^2*x^2)^(5/2))/(9*d*Sqrt[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)/(17*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 = 251, normalized size of antiderivative = 0.68

method	result	size
gosper	\(-\frac {2 \left (-d x +c \right ) \left (45045 D d^{6} x^{6}+51051 C \,d^{6} x^{5}+189189 D c \,d^{5} x^{5}+58905 B \,d^{6} x^{4}+215985 C c \,d^{5} x^{4}+322245 D c^{2} d^{4} x^{4}+69615 A \,d^{6} x^{3}+251685 B c \,d^{5} x^{3}+365925 C \,c^{2} d^{4} x^{3}+303975 D c^{3} d^{3} x^{3}+301665 A c \,d^{5} x^{2}+423045 B \,c^{2} d^{4} x^{2}+329035 C \,c^{3} d^{3} x^{2}+202650 D c^{4} d^{2} x^{2}+500565 A \,c^{2} d^{4} x +351135 B \,c^{3} d^{3} x +188020 C \,c^{4} d^{2} x +115800 D c^{5} d x +353379 A \,c^{3} d^{3}+140454 B \,c^{4} d^{2}+75208 C \,c^{5} d +46320 D c^{6}\right ) \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{765765 d^{4} \left (d x +c \right )^{\frac {3}{2}}}\)	\(251\)
orering	\(-\frac {2 \left (-d x +c \right ) \left (45045 D d^{6} x^{6}+51051 C \,d^{6} x^{5}+189189 D c \,d^{5} x^{5}+58905 B \,d^{6} x^{4}+215985 C c \,d^{5} x^{4}+322245 D c^{2} d^{4} x^{4}+69615 A \,d^{6} x^{3}+251685 B c \,d^{5} x^{3}+365925 C \,c^{2} d^{4} x^{3}+303975 D c^{3} d^{3} x^{3}+301665 A c \,d^{5} x^{2}+423045 B \,c^{2} d^{4} x^{2}+329035 C \,c^{3} d^{3} x^{2}+202650 D c^{4} d^{2} x^{2}+500565 A \,c^{2} d^{4} x +351135 B \,c^{3} d^{3} x +188020 C \,c^{4} d^{2} x +115800 D c^{5} d x +353379 A \,c^{3} d^{3}+140454 B \,c^{4} d^{2}+75208 C \,c^{5} d +46320 D c^{6}\right ) \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{765765 d^{4} \left (d x +c \right )^{\frac {3}{2}}}\)	\(251\)
default	\(-\frac {2 \sqrt {-d^{2} x^{2}+c^{2}}\, \left (-d x +c \right )^{2} \left (45045 D d^{6} x^{6}+51051 C \,d^{6} x^{5}+189189 D c \,d^{5} x^{5}+58905 B \,d^{6} x^{4}+215985 C c \,d^{5} x^{4}+322245 D c^{2} d^{4} x^{4}+69615 A \,d^{6} x^{3}+251685 B c \,d^{5} x^{3}+365925 C \,c^{2} d^{4} x^{3}+303975 D c^{3} d^{3} x^{3}+301665 A c \,d^{5} x^{2}+423045 B \,c^{2} d^{4} x^{2}+329035 C \,c^{3} d^{3} x^{2}+202650 D c^{4} d^{2} x^{2}+500565 A \,c^{2} d^{4} x +351135 B \,c^{3} d^{3} x +188020 C \,c^{4} d^{2} x +115800 D c^{5} d x +353379 A \,c^{3} d^{3}+140454 B \,c^{4} d^{2}+75208 C \,c^{5} d +46320 D c^{6}\right )}{765765 \sqrt {d x +c}\, d^{4}}\)	\(253\)

Input:

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

Output:

-2/765765*(-d*x+c)*(45045*D*d^6*x^6+51051*C*d^6*x^5+189189*D*c*d^5*x^5+589 
05*B*d^6*x^4+215985*C*c*d^5*x^4+322245*D*c^2*d^4*x^4+69615*A*d^6*x^3+25168 
5*B*c*d^5*x^3+365925*C*c^2*d^4*x^3+303975*D*c^3*d^3*x^3+301665*A*c*d^5*x^2 
+423045*B*c^2*d^4*x^2+329035*C*c^3*d^3*x^2+202650*D*c^4*d^2*x^2+500565*A*c 
^2*d^4*x+351135*B*c^3*d^3*x+188020*C*c^4*d^2*x+115800*D*c^5*d*x+353379*A*c 
^3*d^3+140454*B*c^4*d^2+75208*C*c^5*d+46320*D*c^6)*(-d^2*x^2+c^2)^(3/2)/d^ 
4/(d*x+c)^(3/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 322, normalized size of antiderivative = 0.88 \[ \int (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=-\frac {2 \, {\left (45045 \, D d^{8} x^{8} + 46320 \, D c^{8} + 75208 \, C c^{7} d + 140454 \, B c^{6} d^{2} + 353379 \, A c^{5} d^{3} + 3003 \, {\left (33 \, D c d^{7} + 17 \, C d^{8}\right )} x^{7} - 231 \, {\left (48 \, D c^{2} d^{6} - 493 \, C c d^{7} - 255 \, B d^{8}\right )} x^{6} - 63 \, {\left (2402 \, D c^{3} d^{5} + 238 \, C c^{2} d^{6} - 2125 \, B c d^{7} - 1105 \, A d^{8}\right )} x^{5} - 35 \, {\left (2373 \, D c^{4} d^{4} + 5338 \, C c^{3} d^{5} + 612 \, B c^{2} d^{6} - 4641 \, A c d^{7}\right )} x^{4} + 5 \, {\left (2895 \, D c^{5} d^{3} - 20825 \, C c^{4} d^{4} - 48654 \, B c^{3} d^{5} - 6630 \, A c^{2} d^{6}\right )} x^{3} + 3 \, {\left (5790 \, D c^{6} d^{2} + 9401 \, C c^{5} d^{3} - 46257 \, B c^{4} d^{4} - 115362 \, A c^{3} d^{5}\right )} x^{2} + {\left (23160 \, D c^{7} d + 37604 \, C c^{6} d^{2} + 70227 \, B c^{5} d^{3} - 206193 \, A c^{4} d^{4}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}} \sqrt {d x + c}}{765765 \, {\left (d^{5} x + c d^{4}\right )}} \] Input:

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

Output:

-2/765765*(45045*D*d^8*x^8 + 46320*D*c^8 + 75208*C*c^7*d + 140454*B*c^6*d^ 
2 + 353379*A*c^5*d^3 + 3003*(33*D*c*d^7 + 17*C*d^8)*x^7 - 231*(48*D*c^2*d^ 
6 - 493*C*c*d^7 - 255*B*d^8)*x^6 - 63*(2402*D*c^3*d^5 + 238*C*c^2*d^6 - 21 
25*B*c*d^7 - 1105*A*d^8)*x^5 - 35*(2373*D*c^4*d^4 + 5338*C*c^3*d^5 + 612*B 
*c^2*d^6 - 4641*A*c*d^7)*x^4 + 5*(2895*D*c^5*d^3 - 20825*C*c^4*d^4 - 48654 
*B*c^3*d^5 - 6630*A*c^2*d^6)*x^3 + 3*(5790*D*c^6*d^2 + 9401*C*c^5*d^3 - 46 
257*B*c^4*d^4 - 115362*A*c^3*d^5)*x^2 + (23160*D*c^7*d + 37604*C*c^6*d^2 + 
 70227*B*c^5*d^3 - 206193*A*c^4*d^4)*x)*sqrt(-d^2*x^2 + c^2)*sqrt(d*x + c) 
/(d^5*x + c*d^4)
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Maxima [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.06 \[ \int (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=-\frac {2 \, {\left (105 \, d^{5} x^{5} + 245 \, c d^{4} x^{4} - 50 \, c^{2} d^{3} x^{3} - 522 \, c^{3} d^{2} x^{2} - 311 \, c^{4} d x + 533 \, c^{5}\right )} {\left (d x + c\right )} \sqrt {-d x + c} A}{1155 \, {\left (d^{2} x + c d\right )}} - \frac {2 \, {\left (385 \, d^{6} x^{6} + 875 \, c d^{5} x^{5} - 140 \, c^{2} d^{4} x^{4} - 1590 \, c^{3} d^{3} x^{3} - 907 \, c^{4} d^{2} x^{2} + 459 \, c^{5} d x + 918 \, c^{6}\right )} {\left (d x + c\right )} \sqrt {-d x + c} B}{5005 \, {\left (d^{3} x + c d^{2}\right )}} - \frac {2 \, {\left (429 \, d^{7} x^{7} + 957 \, c d^{6} x^{6} - 126 \, c^{2} d^{5} x^{5} - 1570 \, c^{3} d^{4} x^{4} - 875 \, c^{4} d^{3} x^{3} + 237 \, c^{5} d^{2} x^{2} + 316 \, c^{6} d x + 632 \, c^{7}\right )} {\left (d x + c\right )} \sqrt {-d x + c} C}{6435 \, {\left (d^{4} x + c d^{3}\right )}} - \frac {2 \, {\left (15015 \, d^{8} x^{8} + 33033 \, c d^{7} x^{7} - 3696 \, c^{2} d^{6} x^{6} - 50442 \, c^{3} d^{5} x^{5} - 27685 \, c^{4} d^{4} x^{4} + 4825 \, c^{5} d^{3} x^{3} + 5790 \, c^{6} d^{2} x^{2} + 7720 \, c^{7} d x + 15440 \, c^{8}\right )} {\left (d x + c\right )} \sqrt {-d x + c} D}{255255 \, {\left (d^{5} x + c d^{4}\right )}} \] Input:

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

Output:

-2/1155*(105*d^5*x^5 + 245*c*d^4*x^4 - 50*c^2*d^3*x^3 - 522*c^3*d^2*x^2 - 
311*c^4*d*x + 533*c^5)*(d*x + c)*sqrt(-d*x + c)*A/(d^2*x + c*d) - 2/5005*( 
385*d^6*x^6 + 875*c*d^5*x^5 - 140*c^2*d^4*x^4 - 1590*c^3*d^3*x^3 - 907*c^4 
*d^2*x^2 + 459*c^5*d*x + 918*c^6)*(d*x + c)*sqrt(-d*x + c)*B/(d^3*x + c*d^ 
2) - 2/6435*(429*d^7*x^7 + 957*c*d^6*x^6 - 126*c^2*d^5*x^5 - 1570*c^3*d^4* 
x^4 - 875*c^4*d^3*x^3 + 237*c^5*d^2*x^2 + 316*c^6*d*x + 632*c^7)*(d*x + c) 
*sqrt(-d*x + c)*C/(d^4*x + c*d^3) - 2/255255*(15015*d^8*x^8 + 33033*c*d^7* 
x^7 - 3696*c^2*d^6*x^6 - 50442*c^3*d^5*x^5 - 27685*c^4*d^4*x^4 + 4825*c^5* 
d^3*x^3 + 5790*c^6*d^2*x^2 + 7720*c^7*d*x + 15440*c^8)*(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. 2288 vs. \(2 (326) = 652\).

Time = 0.14 (sec) , antiderivative size = 2288, normalized size of antiderivative = 6.22 \[ \int (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

-2/765765*(765765*sqrt(-d*x + c)*A*c^5*d^3 - 255255*((-d*x + c)^(3/2) - 3* 
sqrt(-d*x + c)*c)*B*c^5*d^2 - 255255*((-d*x + c)^(3/2) - 3*sqrt(-d*x + c)* 
c)*A*c^4*d^3 + 51051*(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^5*d + 51051*(3*(d*x - c)^2*sqrt(-d*x + c) - 
10*(-d*x + c)^(3/2)*c + 15*sqrt(-d*x + c)*c^2)*B*c^4*d^2 - 102102*(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 
^3*d^3 + 21879*(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^5 + 21879*(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)*C*c^4*d - 43758*(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^3*d^2 - 43758*(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)*A*c^2*d^3 + 2431*(35*(d*x - c)^4*sqrt(-d*x + c) + 180*(d*x - c)^3*sqr 
t(-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)*D*c^4 - 4862*(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^3*d - 4862*(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^...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.63 \[ \int (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {2 \sqrt {-d x +c}\, \left (-45045 d^{8} x^{8}-150150 c \,d^{7} x^{7}-58905 b \,d^{7} x^{6}-102795 c^{2} d^{6} x^{6}-69615 a \,d^{7} x^{5}-133875 b c \,d^{6} x^{5}+166320 c^{3} d^{5} x^{5}-162435 a c \,d^{6} x^{4}+21420 b \,c^{2} d^{5} x^{4}+269885 c^{4} d^{4} x^{4}+33150 a \,c^{2} d^{5} x^{3}+243270 b \,c^{3} d^{4} x^{3}+89650 c^{5} d^{3} x^{3}+346086 a \,c^{3} d^{4} x^{2}+138771 b \,c^{4} d^{3} x^{2}-45573 c^{6} d^{2} x^{2}+206193 a \,c^{4} d^{3} x -70227 b \,c^{5} d^{2} x -60764 c^{7} d x -353379 a \,c^{5} d^{2}-140454 b \,c^{6} d -121528 c^{8}\right )}{765765 d^{3}} \] Input:

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

Output:

(2*sqrt(c - d*x)*( - 353379*a*c**5*d**2 + 206193*a*c**4*d**3*x + 346086*a* 
c**3*d**4*x**2 + 33150*a*c**2*d**5*x**3 - 162435*a*c*d**6*x**4 - 69615*a*d 
**7*x**5 - 140454*b*c**6*d - 70227*b*c**5*d**2*x + 138771*b*c**4*d**3*x**2 
 + 243270*b*c**3*d**4*x**3 + 21420*b*c**2*d**5*x**4 - 133875*b*c*d**6*x**5 
 - 58905*b*d**7*x**6 - 121528*c**8 - 60764*c**7*d*x - 45573*c**6*d**2*x**2 
 + 89650*c**5*d**3*x**3 + 269885*c**4*d**4*x**4 + 166320*c**3*d**5*x**5 - 
102795*c**2*d**6*x**6 - 150150*c*d**7*x**7 - 45045*d**8*x**8))/(765765*d** 
3)