\(\int (c+d x)^n (c^2-d^2 x^2)^p (A+B x+C x^2+D x^3) \, dx\) [252]

Optimal result

Integrand size = 37, antiderivative size = 462 \[ \int (c+d x)^n \left (c^2-d^2 x^2\right )^p \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {\left (2 c C d (1+p) (4+n+2 p)-c^2 D \left (12+5 n+n^2+14 p+2 n p+4 p^2\right )-B d^2 \left (12+7 n+n^2+14 p+4 n p+4 p^2\right )\right ) (c+d x)^n \left (c^2-d^2 x^2\right )^{1+p}}{d^4 (2+n+2 p) (3+n+2 p) (4+n+2 p)}-\frac {(C d (4+n+2 p)-c D (6+n+4 p)) (c+d x)^{1+n} \left (c^2-d^2 x^2\right )^{1+p}}{d^4 (3+n+2 p) (4+n+2 p)}-\frac {D (c+d x)^{2+n} \left (c^2-d^2 x^2\right )^{1+p}}{d^4 (4+n+2 p)}-\frac {2^{n+p} \left (c^3 D n \left (8+3 n+n^2+6 p\right )+B c d^2 n \left (12+7 n+n^2+14 p+4 n p+4 p^2\right )+c^2 C d \left (n^3+n^2 (5+2 p)+n (6+4 p)+4 \left (2+3 p+p^2\right )\right )+A d^3 \left (n^3+n^2 (9+6 p)+2 n \left (13+18 p+6 p^2\right )+4 \left (6+13 p+9 p^2+2 p^3\right )\right )\right ) (c+d x)^n \left (1+\frac {d x}{c}\right )^{-1-n-p} \left (c^2-d^2 x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (-n-p,1+p,2+p,\frac {c-d x}{2 c}\right )}{c d^4 (1+p) (2+n+2 p) (3+n+2 p) (4+n+2 p)} \] Output:

(2*c*C*d*(p+1)*(4+n+2*p)-c^2*D*(n^2+2*n*p+4*p^2+5*n+14*p+12)-B*d^2*(n^2+4* 
n*p+4*p^2+7*n+14*p+12))*(d*x+c)^n*(-d^2*x^2+c^2)^(p+1)/d^4/(2+n+2*p)/(3+n+ 
2*p)/(4+n+2*p)-(C*d*(4+n+2*p)-c*D*(6+n+4*p))*(d*x+c)^(1+n)*(-d^2*x^2+c^2)^ 
(p+1)/d^4/(3+n+2*p)/(4+n+2*p)-D*(d*x+c)^(2+n)*(-d^2*x^2+c^2)^(p+1)/d^4/(4+ 
n+2*p)-2^(n+p)*(c^3*D*n*(n^2+3*n+6*p+8)+B*c*d^2*n*(n^2+4*n*p+4*p^2+7*n+14* 
p+12)+c^2*C*d*(n^3+n^2*(5+2*p)+n*(6+4*p)+4*p^2+12*p+8)+A*d^3*(n^3+n^2*(9+6 
*p)+2*n*(6*p^2+18*p+13)+8*p^3+36*p^2+52*p+24))*(d*x+c)^n*(1+d*x/c)^(-1-n-p 
)*(-d^2*x^2+c^2)^(p+1)*hypergeom([p+1, -n-p],[2+p],1/2*(-d*x+c)/c)/c/d^4/( 
p+1)/(2+n+2*p)/(3+n+2*p)/(4+n+2*p)
 

Mathematica [C] (warning: unable to verify)

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

Time = 0.96 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.58 \[ \int (c+d x)^n \left (c^2-d^2 x^2\right )^p \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {(c+d x)^n \left (1-\frac {d x}{c}\right )^{-p} \left (1+\frac {d x}{c}\right )^{-n-p} \left (6 B d (1+p) x^2 (c-d x)^p (c+d x)^p \operatorname {AppellF1}\left (2,-p,-n-p,3,\frac {d x}{c},-\frac {d x}{c}\right )+4 C d (1+p) x^3 (c-d x)^p (c+d x)^p \operatorname {AppellF1}\left (3,-p,-n-p,4,\frac {d x}{c},-\frac {d x}{c}\right )+3 d D (1+p) x^4 (c-d x)^p (c+d x)^p \operatorname {AppellF1}\left (4,-p,-n-p,5,\frac {d x}{c},-\frac {d x}{c}\right )-3\ 2^{2+n+p} A (c-d x) \left (1-\frac {d x}{c}\right )^p \left (c^2-d^2 x^2\right )^p \operatorname {Hypergeometric2F1}\left (-n-p,1+p,2+p,\frac {c-d x}{2 c}\right )\right )}{12 d (1+p)} \] Input:

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

Output:

((c + d*x)^n*(1 + (d*x)/c)^(-n - p)*(6*B*d*(1 + p)*x^2*(c - d*x)^p*(c + d* 
x)^p*AppellF1[2, -p, -n - p, 3, (d*x)/c, -((d*x)/c)] + 4*C*d*(1 + p)*x^3*( 
c - d*x)^p*(c + d*x)^p*AppellF1[3, -p, -n - p, 4, (d*x)/c, -((d*x)/c)] + 3 
*d*D*(1 + p)*x^4*(c - d*x)^p*(c + d*x)^p*AppellF1[4, -p, -n - p, 5, (d*x)/ 
c, -((d*x)/c)] - 3*2^(2 + n + p)*A*(c - d*x)*(1 - (d*x)/c)^p*(c^2 - d^2*x^ 
2)^p*Hypergeometric2F1[-n - p, 1 + p, 2 + p, (c - d*x)/(2*c)]))/(12*d*(1 + 
 p)*(1 - (d*x)/c)^p)
 

Rubi [A] (verified)

Time = 1.71 (sec) , antiderivative size = 447, normalized size of antiderivative = 0.97, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.216, Rules used = {2170, 25, 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.

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

\(\Big \downarrow \) 2170

\(\displaystyle -\frac {\int -(c+d x)^n \left (c^2-d^2 x^2\right )^p \left ((C d (n+2 p+4)-c D (n+4 p+6)) x^2 d^4+\left (D (n-2 p) c^2+B d^2 (n+2 p+4)\right ) x d^3+\left (D (n+2) c^3+A d^3 (n+2 p+4)\right ) d^2\right )dx}{d^5 (n+2 p+4)}-\frac {D (c+d x)^{n+2} \left (c^2-d^2 x^2\right )^{p+1}}{d^4 (n+2 p+4)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int (c+d x)^n \left (c^2-d^2 x^2\right )^p \left ((C d (n+2 p+4)-c D (n+4 p+6)) x^2 d^4+\left (D (n-2 p) c^2+B d^2 (n+2 p+4)\right ) x d^3+\left (D (n+2) c^3+A d^3 (n+2 p+4)\right ) d^2\right )dx}{d^5 (n+2 p+4)}-\frac {D (c+d x)^{n+2} \left (c^2-d^2 x^2\right )^{p+1}}{d^4 (n+2 p+4)}\)

\(\Big \downarrow \) 2170

\(\displaystyle \frac {-\frac {\int d^6 (c+d x)^n \left (2 D n (p+1) c^3-C d (n+1) (n+2 p+4) c^2-A d^3 \left (n^2+4 p n+7 n+4 p^2+14 p+12\right )+d \left (-D \left (n^2+2 p n+5 n+4 p^2+14 p+12\right ) c^2+2 C d (p+1) (n+2 p+4) c-B d^2 \left (n^2+4 p n+7 n+4 p^2+14 p+12\right )\right ) x\right ) \left (c^2-d^2 x^2\right )^pdx}{d^4 (n+2 p+3)}-\frac {d (c+d x)^{n+1} \left (c^2-d^2 x^2\right )^{p+1} (C d (n+2 p+4)-c D (n+4 p+6))}{n+2 p+3}}{d^5 (n+2 p+4)}-\frac {D (c+d x)^{n+2} \left (c^2-d^2 x^2\right )^{p+1}}{d^4 (n+2 p+4)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {d^2 \int (c+d x)^n \left (2 D n (p+1) c^3-C d (n+1) (n+2 p+4) c^2-A d^3 \left (n^2+4 p n+7 n+4 p^2+14 p+12\right )+d \left (-D \left (n^2+2 p n+5 n+4 p^2+14 p+12\right ) c^2+2 C d (p+1) (n+2 p+4) c-B d^2 \left (n^2+4 p n+7 n+4 p^2+14 p+12\right )\right ) x\right ) \left (c^2-d^2 x^2\right )^pdx}{n+2 p+3}-\frac {d (c+d x)^{n+1} \left (c^2-d^2 x^2\right )^{p+1} (C d (n+2 p+4)-c D (n+4 p+6))}{n+2 p+3}}{d^5 (n+2 p+4)}-\frac {D (c+d x)^{n+2} \left (c^2-d^2 x^2\right )^{p+1}}{d^4 (n+2 p+4)}\)

\(\Big \downarrow \) 672

\(\displaystyle \frac {-\frac {d^2 \left (-\frac {\left (A d^3 \left (n^3+n^2 (6 p+9)+2 n \left (6 p^2+18 p+13\right )+4 \left (2 p^3+9 p^2+13 p+6\right )\right )+B c d^2 n \left (n^2+4 n p+7 n+4 p^2+14 p+12\right )+c^3 D n \left (n^2+3 n+6 p+8\right )+c^2 C d \left (n^3+n^2 (2 p+5)+n (4 p+6)+4 \left (p^2+3 p+2\right )\right )\right ) \int (c+d x)^n \left (c^2-d^2 x^2\right )^pdx}{n+2 p+2}-\frac {(c+d x)^n \left (c^2-d^2 x^2\right )^{p+1} \left (-B d^2 \left (n^2+4 n p+7 n+4 p^2+14 p+12\right )+c^2 (-D) \left (n^2+2 n p+5 n+4 p^2+14 p+12\right )+2 c C d (p+1) (n+2 p+4)\right )}{d (n+2 p+2)}\right )}{n+2 p+3}-\frac {d (c+d x)^{n+1} \left (c^2-d^2 x^2\right )^{p+1} (C d (n+2 p+4)-c D (n+4 p+6))}{n+2 p+3}}{d^5 (n+2 p+4)}-\frac {D (c+d x)^{n+2} \left (c^2-d^2 x^2\right )^{p+1}}{d^4 (n+2 p+4)}\)

\(\Big \downarrow \) 474

\(\displaystyle \frac {-\frac {d^2 \left (-\frac {(c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \left (A d^3 \left (n^3+n^2 (6 p+9)+2 n \left (6 p^2+18 p+13\right )+4 \left (2 p^3+9 p^2+13 p+6\right )\right )+B c d^2 n \left (n^2+4 n p+7 n+4 p^2+14 p+12\right )+c^3 D n \left (n^2+3 n+6 p+8\right )+c^2 C d \left (n^3+n^2 (2 p+5)+n (4 p+6)+4 \left (p^2+3 p+2\right )\right )\right ) \int \left (\frac {d x}{c}+1\right )^n \left (c^2-d^2 x^2\right )^pdx}{n+2 p+2}-\frac {(c+d x)^n \left (c^2-d^2 x^2\right )^{p+1} \left (-B d^2 \left (n^2+4 n p+7 n+4 p^2+14 p+12\right )+c^2 (-D) \left (n^2+2 n p+5 n+4 p^2+14 p+12\right )+2 c C d (p+1) (n+2 p+4)\right )}{d (n+2 p+2)}\right )}{n+2 p+3}-\frac {d (c+d x)^{n+1} \left (c^2-d^2 x^2\right )^{p+1} (C d (n+2 p+4)-c D (n+4 p+6))}{n+2 p+3}}{d^5 (n+2 p+4)}-\frac {D (c+d x)^{n+2} \left (c^2-d^2 x^2\right )^{p+1}}{d^4 (n+2 p+4)}\)

\(\Big \downarrow \) 473

\(\displaystyle \frac {-\frac {d^2 \left (-\frac {(c+d x)^n \left (c^2-c d x\right )^{-p-1} \left (c^2-d^2 x^2\right )^{p+1} \left (\frac {d x}{c}+1\right )^{-n-p-1} \left (A d^3 \left (n^3+n^2 (6 p+9)+2 n \left (6 p^2+18 p+13\right )+4 \left (2 p^3+9 p^2+13 p+6\right )\right )+B c d^2 n \left (n^2+4 n p+7 n+4 p^2+14 p+12\right )+c^3 D n \left (n^2+3 n+6 p+8\right )+c^2 C d \left (n^3+n^2 (2 p+5)+n (4 p+6)+4 \left (p^2+3 p+2\right )\right )\right ) \int \left (\frac {d x}{c}+1\right )^{n+p} \left (c^2-c d x\right )^pdx}{n+2 p+2}-\frac {(c+d x)^n \left (c^2-d^2 x^2\right )^{p+1} \left (-B d^2 \left (n^2+4 n p+7 n+4 p^2+14 p+12\right )+c^2 (-D) \left (n^2+2 n p+5 n+4 p^2+14 p+12\right )+2 c C d (p+1) (n+2 p+4)\right )}{d (n+2 p+2)}\right )}{n+2 p+3}-\frac {d (c+d x)^{n+1} \left (c^2-d^2 x^2\right )^{p+1} (C d (n+2 p+4)-c D (n+4 p+6))}{n+2 p+3}}{d^5 (n+2 p+4)}-\frac {D (c+d x)^{n+2} \left (c^2-d^2 x^2\right )^{p+1}}{d^4 (n+2 p+4)}\)

\(\Big \downarrow \) 79

\(\displaystyle \frac {-\frac {d^2 \left (\frac {2^{n+p} (c+d x)^n \left (c^2-d^2 x^2\right )^{p+1} \left (\frac {d x}{c}+1\right )^{-n-p-1} \operatorname {Hypergeometric2F1}\left (-n-p,p+1,p+2,\frac {c-d x}{2 c}\right ) \left (A d^3 \left (n^3+n^2 (6 p+9)+2 n \left (6 p^2+18 p+13\right )+4 \left (2 p^3+9 p^2+13 p+6\right )\right )+B c d^2 n \left (n^2+4 n p+7 n+4 p^2+14 p+12\right )+c^3 D n \left (n^2+3 n+6 p+8\right )+c^2 C d \left (n^3+n^2 (2 p+5)+n (4 p+6)+4 \left (p^2+3 p+2\right )\right )\right )}{c d (p+1) (n+2 p+2)}-\frac {(c+d x)^n \left (c^2-d^2 x^2\right )^{p+1} \left (-B d^2 \left (n^2+4 n p+7 n+4 p^2+14 p+12\right )+c^2 (-D) \left (n^2+2 n p+5 n+4 p^2+14 p+12\right )+2 c C d (p+1) (n+2 p+4)\right )}{d (n+2 p+2)}\right )}{n+2 p+3}-\frac {d (c+d x)^{n+1} \left (c^2-d^2 x^2\right )^{p+1} (C d (n+2 p+4)-c D (n+4 p+6))}{n+2 p+3}}{d^5 (n+2 p+4)}-\frac {D (c+d x)^{n+2} \left (c^2-d^2 x^2\right )^{p+1}}{d^4 (n+2 p+4)}\)

Input:

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

Output:

-((D*(c + d*x)^(2 + n)*(c^2 - d^2*x^2)^(1 + p))/(d^4*(4 + n + 2*p))) + (-( 
(d*(C*d*(4 + n + 2*p) - c*D*(6 + n + 4*p))*(c + d*x)^(1 + n)*(c^2 - d^2*x^ 
2)^(1 + p))/(3 + n + 2*p)) - (d^2*(-(((2*c*C*d*(1 + p)*(4 + n + 2*p) - c^2 
*D*(12 + 5*n + n^2 + 14*p + 2*n*p + 4*p^2) - B*d^2*(12 + 7*n + n^2 + 14*p 
+ 4*n*p + 4*p^2))*(c + d*x)^n*(c^2 - d^2*x^2)^(1 + p))/(d*(2 + n + 2*p))) 
+ (2^(n + p)*(c^3*D*n*(8 + 3*n + n^2 + 6*p) + B*c*d^2*n*(12 + 7*n + n^2 + 
14*p + 4*n*p + 4*p^2) + c^2*C*d*(n^3 + n^2*(5 + 2*p) + n*(6 + 4*p) + 4*(2 
+ 3*p + p^2)) + A*d^3*(n^3 + n^2*(9 + 6*p) + 2*n*(13 + 18*p + 6*p^2) + 4*( 
6 + 13*p + 9*p^2 + 2*p^3)))*(c + d*x)^n*(1 + (d*x)/c)^(-1 - n - p)*(c^2 - 
d^2*x^2)^(1 + p)*Hypergeometric2F1[-n - p, 1 + p, 2 + p, (c - d*x)/(2*c)]) 
/(c*d*(1 + p)*(2 + n + 2*p))))/(3 + n + 2*p))/(d^5*(4 + n + 2*p))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

integrate((d*x+c)^n*(-d^2*x^2+c^2)^p*(D*x^3+C*x^2+B*x+A),x, algorithm="fri 
cas")
 

Output:

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

Sympy [F]

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

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

Maxima [F]

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

integrate((d*x+c)^n*(-d^2*x^2+c^2)^p*(D*x^3+C*x^2+B*x+A),x, algorithm="max 
ima")
 

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

((c + d*x)**n*(c**2 - d**2*x**2)**p*a*c*d**2*n**4 + 8*(c + d*x)**n*(c**2 - 
 d**2*x**2)**p*a*c*d**2*n**3*p + 9*(c + d*x)**n*(c**2 - d**2*x**2)**p*a*c* 
d**2*n**3 + 24*(c + d*x)**n*(c**2 - d**2*x**2)**p*a*c*d**2*n**2*p**2 + 54* 
(c + d*x)**n*(c**2 - d**2*x**2)**p*a*c*d**2*n**2*p + 26*(c + d*x)**n*(c**2 
 - d**2*x**2)**p*a*c*d**2*n**2 + 32*(c + d*x)**n*(c**2 - d**2*x**2)**p*a*c 
*d**2*n*p**3 + 108*(c + d*x)**n*(c**2 - d**2*x**2)**p*a*c*d**2*n*p**2 + 10 
4*(c + d*x)**n*(c**2 - d**2*x**2)**p*a*c*d**2*n*p + 24*(c + d*x)**n*(c**2 
- d**2*x**2)**p*a*c*d**2*n + 16*(c + d*x)**n*(c**2 - d**2*x**2)**p*a*c*d** 
2*p**4 + 72*(c + d*x)**n*(c**2 - d**2*x**2)**p*a*c*d**2*p**3 + 104*(c + d* 
x)**n*(c**2 - d**2*x**2)**p*a*c*d**2*p**2 + 48*(c + d*x)**n*(c**2 - d**2*x 
**2)**p*a*c*d**2*p + (c + d*x)**n*(c**2 - d**2*x**2)**p*a*d**3*n**4*x + 6* 
(c + d*x)**n*(c**2 - d**2*x**2)**p*a*d**3*n**3*p*x + 9*(c + d*x)**n*(c**2 
- d**2*x**2)**p*a*d**3*n**3*x + 12*(c + d*x)**n*(c**2 - d**2*x**2)**p*a*d* 
*3*n**2*p**2*x + 36*(c + d*x)**n*(c**2 - d**2*x**2)**p*a*d**3*n**2*p*x + 2 
6*(c + d*x)**n*(c**2 - d**2*x**2)**p*a*d**3*n**2*x + 8*(c + d*x)**n*(c**2 
- d**2*x**2)**p*a*d**3*n*p**3*x + 36*(c + d*x)**n*(c**2 - d**2*x**2)**p*a* 
d**3*n*p**2*x + 52*(c + d*x)**n*(c**2 - d**2*x**2)**p*a*d**3*n*p*x + 24*(c 
 + d*x)**n*(c**2 - d**2*x**2)**p*a*d**3*n*x - (c + d*x)**n*(c**2 - d**2*x* 
*2)**p*b*c**2*d*n**3 - 4*(c + d*x)**n*(c**2 - d**2*x**2)**p*b*c**2*d*n**2* 
p - 7*(c + d*x)**n*(c**2 - d**2*x**2)**p*b*c**2*d*n**2 - 4*(c + d*x)**n...