\(\int \sqrt {a+b x} (c+d x)^n (A+B x+C x^2+D x^3) \, dx\) [183]

Optimal result

Integrand size = 32, antiderivative size = 484 \[ \int \sqrt {a+b x} (c+d x)^n \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {2 \left (a^2 d^2 D \left (89+66 n+12 n^2\right )+b^2 \left (35 c^2 D-5 c C d (9+2 n)+B d^2 \left (63+32 n+4 n^2\right )\right )+a b d \left (5 c D (13+6 n)-C d \left (81+54 n+8 n^2\right )\right )\right ) (a+b x)^{3/2} (c+d x)^{1+n}}{b^3 d^3 (5+2 n) (7+2 n) (9+2 n)}-\frac {2 (7 b c D-b C d (9+2 n)+2 a d D (10+3 n)) (a+b x)^{5/2} (c+d x)^{1+n}}{b^3 d^2 (7+2 n) (9+2 n)}+\frac {2 D (a+b x)^{7/2} (c+d x)^{1+n}}{b^3 d (9+2 n)}-\frac {2 \left (8 a^3 d^3 D \left (6+11 n+6 n^2+n^3\right )+4 a^2 b d^2 \left (2+3 n+n^2\right ) (9 c D-C d (9+2 n))+2 a b^2 d (1+n) \left (45 c^2 D-6 c C d (9+2 n)+B d^2 \left (63+32 n+4 n^2\right )\right )+b^3 \left (105 c^3 D-15 c^2 C d (9+2 n)+3 B c d^2 \left (63+32 n+4 n^2\right )-A d^3 \left (315+286 n+84 n^2+8 n^3\right )\right )\right ) (a+b x)^{3/2} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-n,\frac {5}{2},-\frac {d (a+b x)}{b c-a d}\right )}{3 b^4 d^3 (5+2 n) (7+2 n) (9+2 n)} \] Output:

2*(a^2*d^2*D*(12*n^2+66*n+89)+b^2*(35*D*c^2-5*c*C*d*(9+2*n)+B*d^2*(4*n^2+3 
2*n+63))+a*b*d*(5*c*D*(13+6*n)-C*d*(8*n^2+54*n+81)))*(b*x+a)^(3/2)*(d*x+c) 
^(1+n)/b^3/d^3/(5+2*n)/(7+2*n)/(9+2*n)-2*(7*D*b*c-b*C*d*(9+2*n)+2*a*d*D*(1 
0+3*n))*(b*x+a)^(5/2)*(d*x+c)^(1+n)/b^3/d^2/(7+2*n)/(9+2*n)+2*D*(b*x+a)^(7 
/2)*(d*x+c)^(1+n)/b^3/d/(9+2*n)-2/3*(8*a^3*d^3*D*(n^3+6*n^2+11*n+6)+4*a^2* 
b*d^2*(n^2+3*n+2)*(9*D*c-C*d*(9+2*n))+2*a*b^2*d*(1+n)*(45*D*c^2-6*c*C*d*(9 
+2*n)+B*d^2*(4*n^2+32*n+63))+b^3*(105*D*c^3-15*c^2*C*d*(9+2*n)+3*B*c*d^2*( 
4*n^2+32*n+63)-A*d^3*(8*n^3+84*n^2+286*n+315)))*(b*x+a)^(3/2)*(d*x+c)^n*hy 
pergeom([3/2, -n],[5/2],-d*(b*x+a)/(-a*d+b*c))/b^4/d^3/(5+2*n)/(7+2*n)/(9+ 
2*n)/((b*(d*x+c)/(-a*d+b*c))^n)
 

Mathematica [A] (verified)

Time = 0.80 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.48 \[ \int \sqrt {a+b x} (c+d x)^n \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {2 (a+b x)^{3/2} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (105 \left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-n,\frac {5}{2},\frac {d (a+b x)}{-b c+a d}\right )+(a+b x) \left (63 \left (b^2 B-2 a b C+3 a^2 D\right ) \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-n,\frac {7}{2},\frac {d (a+b x)}{-b c+a d}\right )-5 (a+b x) \left (-9 (b C-3 a D) \operatorname {Hypergeometric2F1}\left (\frac {7}{2},-n,\frac {9}{2},\frac {d (a+b x)}{-b c+a d}\right )-7 D (a+b x) \operatorname {Hypergeometric2F1}\left (\frac {9}{2},-n,\frac {11}{2},\frac {d (a+b x)}{-b c+a d}\right )\right )\right )\right )}{315 b^4} \] Input:

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

Output:

(2*(a + b*x)^(3/2)*(c + d*x)^n*(105*(A*b^3 - a*(b^2*B - a*b*C + a^2*D))*Hy 
pergeometric2F1[3/2, -n, 5/2, (d*(a + b*x))/(-(b*c) + a*d)] + (a + b*x)*(6 
3*(b^2*B - 2*a*b*C + 3*a^2*D)*Hypergeometric2F1[5/2, -n, 7/2, (d*(a + b*x) 
)/(-(b*c) + a*d)] - 5*(a + b*x)*(-9*(b*C - 3*a*D)*Hypergeometric2F1[7/2, - 
n, 9/2, (d*(a + b*x))/(-(b*c) + a*d)] - 7*D*(a + b*x)*Hypergeometric2F1[9/ 
2, -n, 11/2, (d*(a + b*x))/(-(b*c) + a*d)]))))/(315*b^4*((b*(c + d*x))/(b* 
c - a*d))^n)
 

Rubi [A] (verified)

Time = 0.91 (sec) , antiderivative size = 470, normalized size of antiderivative = 0.97, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {2125, 27, 1194, 27, 90, 80, 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.

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

\(\Big \downarrow \) 2125

\(\displaystyle \frac {2 \int \frac {1}{2} \sqrt {a+b x} (c+d x)^n \left (A d (2 n+9) b^3-(7 b c D+2 a d (3 n+10) D-b C d (2 n+9)) x^2 b^2-\left (d D (6 n+13) a^2+14 b c D a-b^2 B d (2 n+9)\right ) x b-a^2 D (7 b c+2 a d (n+1))\right )dx}{b^3 d (2 n+9)}+\frac {2 D (a+b x)^{7/2} (c+d x)^{n+1}}{b^3 d (2 n+9)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \sqrt {a+b x} (c+d x)^n \left (A d (2 n+9) b^3-(7 b c D+2 a d (3 n+10) D-b C d (2 n+9)) x^2 b^2-\left (d D (6 n+13) a^2+14 b c D a-b^2 B d (2 n+9)\right ) x b-a^2 D (7 b c+2 a d (n+1))\right )dx}{b^3 d (2 n+9)}+\frac {2 D (a+b x)^{7/2} (c+d x)^{n+1}}{b^3 d (2 n+9)}\)

\(\Big \downarrow \) 1194

\(\displaystyle \frac {\frac {2 \int \frac {1}{2} b^2 \sqrt {a+b x} (c+d x)^n \left (2 d^2 D \left (4 n^2+17 n+13\right ) a^3+b d \left (5 c D (6 n+13)-2 C d \left (2 n^2+11 n+9\right )\right ) a^2+5 b^2 c (7 c D-C d (2 n+9)) a+A b^3 d^2 \left (4 n^2+32 n+63\right )+b \left (\left (35 D c^2-5 C d (2 n+9) c+B d^2 \left (4 n^2+32 n+63\right )\right ) b^2+a d \left (5 c D (6 n+13)-C d \left (8 n^2+54 n+81\right )\right ) b+a^2 d^2 D \left (12 n^2+66 n+89\right )\right ) x\right )dx}{b^2 d (2 n+7)}-\frac {2 (a+b x)^{5/2} (c+d x)^{n+1} (2 a d D (3 n+10)+7 b c D-b C d (2 n+9))}{d (2 n+7)}}{b^3 d (2 n+9)}+\frac {2 D (a+b x)^{7/2} (c+d x)^{n+1}}{b^3 d (2 n+9)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \sqrt {a+b x} (c+d x)^n \left (2 d^2 D \left (4 n^2+17 n+13\right ) a^3+b d \left (5 c D (6 n+13)-2 C d \left (2 n^2+11 n+9\right )\right ) a^2+5 b^2 c (7 c D-C d (2 n+9)) a+A b^3 d^2 \left (4 n^2+32 n+63\right )+b \left (\left (35 D c^2-5 C d (2 n+9) c+B d^2 \left (4 n^2+32 n+63\right )\right ) b^2+a d \left (5 c D (6 n+13)-C d \left (8 n^2+54 n+81\right )\right ) b+a^2 d^2 D \left (12 n^2+66 n+89\right )\right ) x\right )dx}{d (2 n+7)}-\frac {2 (a+b x)^{5/2} (c+d x)^{n+1} (2 a d D (3 n+10)+7 b c D-b C d (2 n+9))}{d (2 n+7)}}{b^3 d (2 n+9)}+\frac {2 D (a+b x)^{7/2} (c+d x)^{n+1}}{b^3 d (2 n+9)}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {\frac {\frac {2 (a+b x)^{3/2} (c+d x)^{n+1} \left (a^2 d^2 D \left (12 n^2+66 n+89\right )+a b d \left (5 c D (6 n+13)-C d \left (8 n^2+54 n+81\right )\right )+b^2 \left (B d^2 \left (4 n^2+32 n+63\right )+35 c^2 D-5 c C d (2 n+9)\right )\right )}{d (2 n+5)}-\frac {\left (8 a^3 d^3 D \left (n^3+6 n^2+11 n+6\right )+4 a^2 b d^2 \left (n^2+3 n+2\right ) (9 c D-C d (2 n+9))+2 a b^2 d (n+1) \left (B d^2 \left (4 n^2+32 n+63\right )+45 c^2 D-6 c C d (2 n+9)\right )+b^3 \left (-A d^3 \left (8 n^3+84 n^2+286 n+315\right )+3 B c d^2 \left (4 n^2+32 n+63\right )+105 c^3 D-15 c^2 C d (2 n+9)\right )\right ) \int \sqrt {a+b x} (c+d x)^ndx}{d (2 n+5)}}{d (2 n+7)}-\frac {2 (a+b x)^{5/2} (c+d x)^{n+1} (2 a d D (3 n+10)+7 b c D-b C d (2 n+9))}{d (2 n+7)}}{b^3 d (2 n+9)}+\frac {2 D (a+b x)^{7/2} (c+d x)^{n+1}}{b^3 d (2 n+9)}\)

\(\Big \downarrow \) 80

\(\displaystyle \frac {\frac {\frac {2 (a+b x)^{3/2} (c+d x)^{n+1} \left (a^2 d^2 D \left (12 n^2+66 n+89\right )+a b d \left (5 c D (6 n+13)-C d \left (8 n^2+54 n+81\right )\right )+b^2 \left (B d^2 \left (4 n^2+32 n+63\right )+35 c^2 D-5 c C d (2 n+9)\right )\right )}{d (2 n+5)}-\frac {(c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (8 a^3 d^3 D \left (n^3+6 n^2+11 n+6\right )+4 a^2 b d^2 \left (n^2+3 n+2\right ) (9 c D-C d (2 n+9))+2 a b^2 d (n+1) \left (B d^2 \left (4 n^2+32 n+63\right )+45 c^2 D-6 c C d (2 n+9)\right )+b^3 \left (-A d^3 \left (8 n^3+84 n^2+286 n+315\right )+3 B c d^2 \left (4 n^2+32 n+63\right )+105 c^3 D-15 c^2 C d (2 n+9)\right )\right ) \int \sqrt {a+b x} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^ndx}{d (2 n+5)}}{d (2 n+7)}-\frac {2 (a+b x)^{5/2} (c+d x)^{n+1} (2 a d D (3 n+10)+7 b c D-b C d (2 n+9))}{d (2 n+7)}}{b^3 d (2 n+9)}+\frac {2 D (a+b x)^{7/2} (c+d x)^{n+1}}{b^3 d (2 n+9)}\)

\(\Big \downarrow \) 79

\(\displaystyle \frac {\frac {\frac {2 (a+b x)^{3/2} (c+d x)^{n+1} \left (a^2 d^2 D \left (12 n^2+66 n+89\right )+a b d \left (5 c D (6 n+13)-C d \left (8 n^2+54 n+81\right )\right )+b^2 \left (B d^2 \left (4 n^2+32 n+63\right )+35 c^2 D-5 c C d (2 n+9)\right )\right )}{d (2 n+5)}-\frac {2 (a+b x)^{3/2} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-n,\frac {5}{2},-\frac {d (a+b x)}{b c-a d}\right ) \left (8 a^3 d^3 D \left (n^3+6 n^2+11 n+6\right )+4 a^2 b d^2 \left (n^2+3 n+2\right ) (9 c D-C d (2 n+9))+2 a b^2 d (n+1) \left (B d^2 \left (4 n^2+32 n+63\right )+45 c^2 D-6 c C d (2 n+9)\right )+b^3 \left (-A d^3 \left (8 n^3+84 n^2+286 n+315\right )+3 B c d^2 \left (4 n^2+32 n+63\right )+105 c^3 D-15 c^2 C d (2 n+9)\right )\right )}{3 b d (2 n+5)}}{d (2 n+7)}-\frac {2 (a+b x)^{5/2} (c+d x)^{n+1} (2 a d D (3 n+10)+7 b c D-b C d (2 n+9))}{d (2 n+7)}}{b^3 d (2 n+9)}+\frac {2 D (a+b x)^{7/2} (c+d x)^{n+1}}{b^3 d (2 n+9)}\)

Input:

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

Output:

(2*D*(a + b*x)^(7/2)*(c + d*x)^(1 + n))/(b^3*d*(9 + 2*n)) + ((-2*(7*b*c*D 
- b*C*d*(9 + 2*n) + 2*a*d*D*(10 + 3*n))*(a + b*x)^(5/2)*(c + d*x)^(1 + n)) 
/(d*(7 + 2*n)) + ((2*(a^2*d^2*D*(89 + 66*n + 12*n^2) + b^2*(35*c^2*D - 5*c 
*C*d*(9 + 2*n) + B*d^2*(63 + 32*n + 4*n^2)) + a*b*d*(5*c*D*(13 + 6*n) - C* 
d*(81 + 54*n + 8*n^2)))*(a + b*x)^(3/2)*(c + d*x)^(1 + n))/(d*(5 + 2*n)) - 
 (2*(8*a^3*d^3*D*(6 + 11*n + 6*n^2 + n^3) + 4*a^2*b*d^2*(2 + 3*n + n^2)*(9 
*c*D - C*d*(9 + 2*n)) + 2*a*b^2*d*(1 + n)*(45*c^2*D - 6*c*C*d*(9 + 2*n) + 
B*d^2*(63 + 32*n + 4*n^2)) + b^3*(105*c^3*D - 15*c^2*C*d*(9 + 2*n) + 3*B*c 
*d^2*(63 + 32*n + 4*n^2) - A*d^3*(315 + 286*n + 84*n^2 + 8*n^3)))*(a + b*x 
)^(3/2)*(c + d*x)^n*Hypergeometric2F1[3/2, -n, 5/2, -((d*(a + b*x))/(b*c - 
 a*d))])/(3*b*d*(5 + 2*n)*((b*(c + d*x))/(b*c - a*d))^n))/(d*(7 + 2*n)))/( 
b^3*d*(9 + 2*n))
 

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 80

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

rule 90

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

rule 1194

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) 
 + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[c^p*(d + e*x)^(m + 2*p)*((f + g*x 
)^(n + 1)/(g*e^(2*p)*(m + n + 2*p + 1))), x] + Simp[1/(g*e^(2*p)*(m + n + 2 
*p + 1))   Int[(d + e*x)^m*(f + g*x)^n*ExpandToSum[g*(m + n + 2*p + 1)*(e^( 
2*p)*(a + b*x + c*x^2)^p - c^p*(d + e*x)^(2*p)) - c^p*(e*f - d*g)*(m + 2*p) 
*(d + e*x)^(2*p - 1), x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ 
[p, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && NeQ[m + n + 2*p + 1, 0]
 

rule 2125

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] 
:> With[{q = Expon[Px, x], k = Coeff[Px, x, Expon[Px, x]]}, Simp[k*(a + b*x 
)^(m + q)*((c + d*x)^(n + 1)/(d*b^q*(m + n + q + 1))), x] + Simp[1/(d*b^q*( 
m + n + q + 1))   Int[(a + b*x)^m*(c + d*x)^n*ExpandToSum[d*b^q*(m + n + q 
+ 1)*Px - d*k*(m + n + q + 1)*(a + b*x)^q - k*(b*c - a*d)*(m + q)*(a + b*x) 
^(q - 1), x], x], x] /; NeQ[m + n + q + 1, 0]] /; FreeQ[{a, b, c, d, m, n}, 
 x] && PolyQ[Px, x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((D*x^3 + C*x^2 + B*x + A)*sqrt(b*x + a)*(d*x + c)^n, x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

integrate((D*x^3 + C*x^2 + B*x + A)*sqrt(b*x + a)*(d*x + c)^n, x)
 

Giac [F]

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

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

Output:

integrate((D*x^3 + C*x^2 + B*x + A)*sqrt(b*x + a)*(d*x + c)^n, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*( - 8*(c + d*x)**n*sqrt(a + b*x)*a**4*c*d**3*n**2 - 40*(c + d*x)**n*sqr 
t(a + b*x)*a**4*c*d**3*n - 48*(c + d*x)**n*sqrt(a + b*x)*a**4*c*d**3 + 8*( 
c + d*x)**n*sqrt(a + b*x)*a**4*d**4*n**3*x + 40*(c + d*x)**n*sqrt(a + b*x) 
*a**4*d**4*n**2*x + 48*(c + d*x)**n*sqrt(a + b*x)*a**4*d**4*n*x + 24*(c + 
d*x)**n*sqrt(a + b*x)*a**3*b*c**2*d**2*n**2 + 88*(c + d*x)**n*sqrt(a + b*x 
)*a**3*b*c**2*d**2*n + 72*(c + d*x)**n*sqrt(a + b*x)*a**3*b*c**2*d**2 - 24 
*(c + d*x)**n*sqrt(a + b*x)*a**3*b*c*d**3*n**3*x - 84*(c + d*x)**n*sqrt(a 
+ b*x)*a**3*b*c*d**3*n**2*x - 52*(c + d*x)**n*sqrt(a + b*x)*a**3*b*c*d**3* 
n*x + 24*(c + d*x)**n*sqrt(a + b*x)*a**3*b*c*d**3*x - 8*(c + d*x)**n*sqrt( 
a + b*x)*a**3*b*d**4*n**3*x**2 - 36*(c + d*x)**n*sqrt(a + b*x)*a**3*b*d**4 
*n**2*x**2 - 36*(c + d*x)**n*sqrt(a + b*x)*a**3*b*d**4*n*x**2 + 16*(c + d* 
x)**n*sqrt(a + b*x)*a**2*b**3*c*d**2*n**4 + 176*(c + d*x)**n*sqrt(a + b*x) 
*a**2*b**3*c*d**2*n**3 + 648*(c + d*x)**n*sqrt(a + b*x)*a**2*b**3*c*d**2*n 
**2 + 852*(c + d*x)**n*sqrt(a + b*x)*a**2*b**3*c*d**2*n + 189*(c + d*x)**n 
*sqrt(a + b*x)*a**2*b**3*c*d**2 + 16*(c + d*x)**n*sqrt(a + b*x)*a**2*b**3* 
d**3*n**4*x + 176*(c + d*x)**n*sqrt(a + b*x)*a**2*b**3*d**3*n**3*x + 636*( 
c + d*x)**n*sqrt(a + b*x)*a**2*b**3*d**3*n**2*x + 756*(c + d*x)**n*sqrt(a 
+ b*x)*a**2*b**3*d**3*n*x - 24*(c + d*x)**n*sqrt(a + b*x)*a**2*b**2*c**3*d 
*n**2 - 24*(c + d*x)**n*sqrt(a + b*x)*a**2*b**2*c**3*d*n + 24*(c + d*x)**n 
*sqrt(a + b*x)*a**2*b**2*c**2*d**2*n**3*x + 12*(c + d*x)**n*sqrt(a + b*...