\(\int \sqrt {c+d x} (c^2-d^2 x^2)^p (A+B x+C x^2+D x^3) \, dx\) [243]

Optimal result

Integrand size = 39, antiderivative size = 373 \[ \int \sqrt {c+d x} \left (c^2-d^2 x^2\right )^p \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {2 \left (4 c C d \left (9+13 p+4 p^2\right )-c^2 D \left (59+60 p+16 p^2\right )-B d^2 \left (63+64 p+16 p^2\right )\right ) \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{1+p}}{d^4 (5+4 p) (7+4 p) (9+4 p)}-\frac {2 (C d (9+4 p)-c D (13+8 p)) (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{1+p}}{d^4 (7+4 p) (9+4 p)}-\frac {2 D (c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{1+p}}{d^4 (9+4 p)}-\frac {2^{\frac {1}{2}+p} \left (3 c^3 D (13+8 p)+B c d^2 \left (63+64 p+16 p^2\right )+c^2 C d \left (99+116 p+32 p^2\right )+A d^3 \left (315+572 p+336 p^2+64 p^3\right )\right ) \sqrt {c+d x} \left (1+\frac {d x}{c}\right )^{-\frac {3}{2}-p} \left (c^2-d^2 x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2}-p,1+p,2+p,\frac {c-d x}{2 c}\right )}{c d^4 (1+p) (5+4 p) (7+4 p) (9+4 p)} \] Output:

2*(4*c*C*d*(4*p^2+13*p+9)-c^2*D*(16*p^2+60*p+59)-B*d^2*(16*p^2+64*p+63))*( 
d*x+c)^(1/2)*(-d^2*x^2+c^2)^(p+1)/d^4/(5+4*p)/(7+4*p)/(9+4*p)-2*(C*d*(9+4* 
p)-c*D*(13+8*p))*(d*x+c)^(3/2)*(-d^2*x^2+c^2)^(p+1)/d^4/(7+4*p)/(9+4*p)-2* 
D*(d*x+c)^(5/2)*(-d^2*x^2+c^2)^(p+1)/d^4/(9+4*p)-2^(1/2+p)*(3*c^3*D*(13+8* 
p)+B*c*d^2*(16*p^2+64*p+63)+c^2*C*d*(32*p^2+116*p+99)+A*d^3*(64*p^3+336*p^ 
2+572*p+315))*(d*x+c)^(1/2)*(1+d*x/c)^(-3/2-p)*(-d^2*x^2+c^2)^(p+1)*hyperg 
eom([p+1, -1/2-p],[2+p],1/2*(-d*x+c)/c)/c/d^4/(p+1)/(5+4*p)/(7+4*p)/(9+4*p 
)
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 2.30 (sec) , antiderivative size = 306, normalized size of antiderivative = 0.82 \[ \int \sqrt {c+d x} \left (c^2-d^2 x^2\right )^p \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {\sqrt {c+d x} \left (1-\frac {d x}{c}\right )^{-p} \left (1+\frac {d x}{c}\right )^{-\frac {1}{2}-2 p} \left (6 B d (1+p) x^2 (c-d x)^p (c+d x)^p \left (1+\frac {d x}{c}\right )^p \operatorname {AppellF1}\left (2,-p,-\frac {1}{2}-p,3,\frac {d x}{c},-\frac {d x}{c}\right )+4 C d (1+p) x^3 (c-d x)^p (c+d x)^p \left (1+\frac {d x}{c}\right )^p \operatorname {AppellF1}\left (3,-p,-\frac {1}{2}-p,4,\frac {d x}{c},-\frac {d x}{c}\right )+3 d D (1+p) x^4 (c-d x)^p (c+d x)^p \left (1+\frac {d x}{c}\right )^p \operatorname {AppellF1}\left (4,-p,-\frac {1}{2}-p,5,\frac {d x}{c},-\frac {d x}{c}\right )-3\ 2^{\frac {5}{2}+p} A (c-d x) \left (c^2-d^2 x^2\right )^p \left (1-\frac {d^2 x^2}{c^2}\right )^p \operatorname {Hypergeometric2F1}\left (-\frac {1}{2}-p,1+p,2+p,\frac {c-d x}{2 c}\right )\right )}{12 d (1+p)} \] Input:

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

Output:

(Sqrt[c + d*x]*(1 + (d*x)/c)^(-1/2 - 2*p)*(6*B*d*(1 + p)*x^2*(c - d*x)^p*( 
c + d*x)^p*(1 + (d*x)/c)^p*AppellF1[2, -p, -1/2 - p, 3, (d*x)/c, -((d*x)/c 
)] + 4*C*d*(1 + p)*x^3*(c - d*x)^p*(c + d*x)^p*(1 + (d*x)/c)^p*AppellF1[3, 
 -p, -1/2 - p, 4, (d*x)/c, -((d*x)/c)] + 3*d*D*(1 + p)*x^4*(c - d*x)^p*(c 
+ d*x)^p*(1 + (d*x)/c)^p*AppellF1[4, -p, -1/2 - p, 5, (d*x)/c, -((d*x)/c)] 
 - 3*2^(5/2 + p)*A*(c - d*x)*(c^2 - d^2*x^2)^p*(1 - (d^2*x^2)/c^2)^p*Hyper 
geometric2F1[-1/2 - p, 1 + p, 2 + p, (c - d*x)/(2*c)]))/(12*d*(1 + p)*(1 - 
 (d*x)/c)^p)
 

Rubi [A] (verified)

Time = 1.49 (sec) , antiderivative size = 360, normalized size of antiderivative = 0.97, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.205, Rules used = {2170, 27, 2170, 27, 672, 474, 473, 79}

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.

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

\(\Big \downarrow \) 2170

\(\displaystyle -\frac {2 \int -\frac {1}{2} \sqrt {c+d x} \left (c^2-d^2 x^2\right )^p \left ((C d (4 p+9)-c D (8 p+13)) x^2 d^4+\left (D (1-4 p) c^2+B d^2 (4 p+9)\right ) x d^3+\left (5 D c^3+A d^3 (4 p+9)\right ) d^2\right )dx}{d^5 (4 p+9)}-\frac {2 D (c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{p+1}}{d^4 (4 p+9)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \sqrt {c+d x} \left (c^2-d^2 x^2\right )^p \left ((C d (4 p+9)-c D (8 p+13)) x^2 d^4+\left (D (1-4 p) c^2+B d^2 (4 p+9)\right ) x d^3+\left (5 D c^3+A d^3 (4 p+9)\right ) d^2\right )dx}{d^5 (4 p+9)}-\frac {2 D (c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{p+1}}{d^4 (4 p+9)}\)

\(\Big \downarrow \) 2170

\(\displaystyle \frac {-\frac {2 \int \frac {1}{2} d^6 \sqrt {c+d x} \left (4 D (p+1) c^3-3 C d (4 p+9) c^2-A d^3 \left (16 p^2+64 p+63\right )+d \left (-D \left (16 p^2+60 p+59\right ) c^2+4 C d \left (4 p^2+13 p+9\right ) c-B d^2 \left (16 p^2+64 p+63\right )\right ) x\right ) \left (c^2-d^2 x^2\right )^pdx}{d^4 (4 p+7)}-\frac {2 d (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{p+1} (C d (4 p+9)-c D (8 p+13))}{4 p+7}}{d^5 (4 p+9)}-\frac {2 D (c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{p+1}}{d^4 (4 p+9)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {d^2 \int \sqrt {c+d x} \left (4 D (p+1) c^3-3 C d (4 p+9) c^2-A d^3 \left (16 p^2+64 p+63\right )+d \left (-D \left (16 p^2+60 p+59\right ) c^2+4 C d \left (4 p^2+13 p+9\right ) c-B d^2 \left (16 p^2+64 p+63\right )\right ) x\right ) \left (c^2-d^2 x^2\right )^pdx}{4 p+7}-\frac {2 d (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{p+1} (C d (4 p+9)-c D (8 p+13))}{4 p+7}}{d^5 (4 p+9)}-\frac {2 D (c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{p+1}}{d^4 (4 p+9)}\)

\(\Big \downarrow \) 672

\(\displaystyle \frac {-\frac {d^2 \left (-\frac {\left (A d^3 \left (64 p^3+336 p^2+572 p+315\right )+B c d^2 \left (16 p^2+64 p+63\right )+3 c^3 D (8 p+13)+c^2 C d \left (32 p^2+116 p+99\right )\right ) \int \sqrt {c+d x} \left (c^2-d^2 x^2\right )^pdx}{4 p+5}-\frac {2 \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{p+1} \left (-B d^2 \left (16 p^2+64 p+63\right )+c^2 (-D) \left (16 p^2+60 p+59\right )+4 c C d \left (4 p^2+13 p+9\right )\right )}{d (4 p+5)}\right )}{4 p+7}-\frac {2 d (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{p+1} (C d (4 p+9)-c D (8 p+13))}{4 p+7}}{d^5 (4 p+9)}-\frac {2 D (c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{p+1}}{d^4 (4 p+9)}\)

\(\Big \downarrow \) 474

\(\displaystyle \frac {-\frac {d^2 \left (-\frac {\sqrt {c+d x} \left (A d^3 \left (64 p^3+336 p^2+572 p+315\right )+B c d^2 \left (16 p^2+64 p+63\right )+3 c^3 D (8 p+13)+c^2 C d \left (32 p^2+116 p+99\right )\right ) \int \sqrt {\frac {d x}{c}+1} \left (c^2-d^2 x^2\right )^pdx}{(4 p+5) \sqrt {\frac {d x}{c}+1}}-\frac {2 \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{p+1} \left (-B d^2 \left (16 p^2+64 p+63\right )+c^2 (-D) \left (16 p^2+60 p+59\right )+4 c C d \left (4 p^2+13 p+9\right )\right )}{d (4 p+5)}\right )}{4 p+7}-\frac {2 d (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{p+1} (C d (4 p+9)-c D (8 p+13))}{4 p+7}}{d^5 (4 p+9)}-\frac {2 D (c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{p+1}}{d^4 (4 p+9)}\)

\(\Big \downarrow \) 473

\(\displaystyle \frac {-\frac {d^2 \left (-\frac {\sqrt {c+d x} \left (c^2-c d x\right )^{-p-1} \left (c^2-d^2 x^2\right )^{p+1} \left (\frac {d x}{c}+1\right )^{-p-\frac {3}{2}} \left (A d^3 \left (64 p^3+336 p^2+572 p+315\right )+B c d^2 \left (16 p^2+64 p+63\right )+3 c^3 D (8 p+13)+c^2 C d \left (32 p^2+116 p+99\right )\right ) \int \left (\frac {d x}{c}+1\right )^{p+\frac {1}{2}} \left (c^2-c d x\right )^pdx}{4 p+5}-\frac {2 \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{p+1} \left (-B d^2 \left (16 p^2+64 p+63\right )+c^2 (-D) \left (16 p^2+60 p+59\right )+4 c C d \left (4 p^2+13 p+9\right )\right )}{d (4 p+5)}\right )}{4 p+7}-\frac {2 d (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{p+1} (C d (4 p+9)-c D (8 p+13))}{4 p+7}}{d^5 (4 p+9)}-\frac {2 D (c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{p+1}}{d^4 (4 p+9)}\)

\(\Big \downarrow \) 79

\(\displaystyle \frac {-\frac {d^2 \left (\frac {2^{p+\frac {1}{2}} \sqrt {c+d x} \left (\frac {d x}{c}+1\right )^{-p-\frac {3}{2}} \left (c^2-d^2 x^2\right )^{p+1} \operatorname {Hypergeometric2F1}\left (-p-\frac {1}{2},p+1,p+2,\frac {c-d x}{2 c}\right ) \left (A d^3 \left (64 p^3+336 p^2+572 p+315\right )+B c d^2 \left (16 p^2+64 p+63\right )+3 c^3 D (8 p+13)+c^2 C d \left (32 p^2+116 p+99\right )\right )}{c d (p+1) (4 p+5)}-\frac {2 \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{p+1} \left (-B d^2 \left (16 p^2+64 p+63\right )+c^2 (-D) \left (16 p^2+60 p+59\right )+4 c C d \left (4 p^2+13 p+9\right )\right )}{d (4 p+5)}\right )}{4 p+7}-\frac {2 d (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{p+1} (C d (4 p+9)-c D (8 p+13))}{4 p+7}}{d^5 (4 p+9)}-\frac {2 D (c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{p+1}}{d^4 (4 p+9)}\)

Input:

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

Output:

(-2*D*(c + d*x)^(5/2)*(c^2 - d^2*x^2)^(1 + p))/(d^4*(9 + 4*p)) + ((-2*d*(C 
*d*(9 + 4*p) - c*D*(13 + 8*p))*(c + d*x)^(3/2)*(c^2 - d^2*x^2)^(1 + p))/(7 
 + 4*p) - (d^2*((-2*(4*c*C*d*(9 + 13*p + 4*p^2) - c^2*D*(59 + 60*p + 16*p^ 
2) - B*d^2*(63 + 64*p + 16*p^2))*Sqrt[c + d*x]*(c^2 - d^2*x^2)^(1 + p))/(d 
*(5 + 4*p)) + (2^(1/2 + p)*(3*c^3*D*(13 + 8*p) + B*c*d^2*(63 + 64*p + 16*p 
^2) + c^2*C*d*(99 + 116*p + 32*p^2) + A*d^3*(315 + 572*p + 336*p^2 + 64*p^ 
3))*Sqrt[c + d*x]*(1 + (d*x)/c)^(-3/2 - p)*(c^2 - d^2*x^2)^(1 + p)*Hyperge 
ometric2F1[-1/2 - p, 1 + p, 2 + p, (c - d*x)/(2*c)])/(c*d*(1 + p)*(5 + 4*p 
))))/(7 + 4*p))/(d^5*(9 + 4*p))
 

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 79

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( 
a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 
, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] 
&&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] 
 ||  !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
 

rule 473

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
c^(n - 1)*((a + b*x^2)^(p + 1)/((1 + d*(x/c))^(p + 1)*(a/c + (b*x)/d)^(p + 
1)))   Int[(1 + d*(x/c))^(n + p)*(a/c + (b/d)*x)^p, x], x] /; FreeQ[{a, b, 
c, d, n}, x] && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[n] || GtQ[c, 0]) &&  !Gt 
Q[a, 0] &&  !(IntegerQ[n] && (IntegerQ[3*p] || IntegerQ[4*p]))
 

rule 474

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*(256*sqrt(c + d*x)*(c**2 - d**2*x**2)**p*a*c*d**2*p**4 + 1408*sqrt(c + 
d*x)*(c**2 - d**2*x**2)**p*a*c*d**2*p**3 + 2624*sqrt(c + d*x)*(c**2 - d**2 
*x**2)**p*a*c*d**2*p**2 + 1832*sqrt(c + d*x)*(c**2 - d**2*x**2)**p*a*c*d** 
2*p + 315*sqrt(c + d*x)*(c**2 - d**2*x**2)**p*a*c*d**2 + 64*sqrt(c + d*x)* 
(c**2 - d**2*x**2)**p*a*d**3*p**3*x + 336*sqrt(c + d*x)*(c**2 - d**2*x**2) 
**p*a*d**3*p**2*x + 572*sqrt(c + d*x)*(c**2 - d**2*x**2)**p*a*d**3*p*x + 3 
15*sqrt(c + d*x)*(c**2 - d**2*x**2)**p*a*d**3*x - 32*sqrt(c + d*x)*(c**2 - 
 d**2*x**2)**p*b*c**2*d*p**2 - 128*sqrt(c + d*x)*(c**2 - d**2*x**2)**p*b*c 
**2*d*p - 126*sqrt(c + d*x)*(c**2 - d**2*x**2)**p*b*c**2*d + 16*sqrt(c + d 
*x)*(c**2 - d**2*x**2)**p*b*c*d**2*p**2*x + 64*sqrt(c + d*x)*(c**2 - d**2* 
x**2)**p*b*c*d**2*p*x + 63*sqrt(c + d*x)*(c**2 - d**2*x**2)**p*b*c*d**2*x 
+ 64*sqrt(c + d*x)*(c**2 - d**2*x**2)**p*b*d**3*p**3*x**2 + 304*sqrt(c + d 
*x)*(c**2 - d**2*x**2)**p*b*d**3*p**2*x**2 + 444*sqrt(c + d*x)*(c**2 - d** 
2*x**2)**p*b*d**3*p*x**2 + 189*sqrt(c + d*x)*(c**2 - d**2*x**2)**p*b*d**3* 
x**2 + 128*sqrt(c + d*x)*(c**2 - d**2*x**2)**p*c**4*p**3 + 512*sqrt(c + d* 
x)*(c**2 - d**2*x**2)**p*c**4*p**2 + 480*sqrt(c + d*x)*(c**2 - d**2*x**2)* 
*p*c**4*p + 24*sqrt(c + d*x)*(c**2 - d**2*x**2)**p*c**4 - 64*sqrt(c + d*x) 
*(c**2 - d**2*x**2)**p*c**3*d*p**3*x - 256*sqrt(c + d*x)*(c**2 - d**2*x**2 
)**p*c**3*d*p**2*x - 240*sqrt(c + d*x)*(c**2 - d**2*x**2)**p*c**3*d*p*x - 
12*sqrt(c + d*x)*(c**2 - d**2*x**2)**p*c**3*d*x - 64*sqrt(c + d*x)*(c**...