\(\int (c+d x)^n (A+B x+C x^2) (c^2-d^2 x^2)^p \, dx\) [251]

Optimal result

Integrand size = 32, antiderivative size = 253 \[ \int (c+d x)^n \left (A+B x+C x^2\right ) \left (c^2-d^2 x^2\right )^p \, dx=\frac {(2 c C (1+p)-B d (3+n+2 p)) (c+d x)^n \left (c^2-d^2 x^2\right )^{1+p}}{d^3 (2+n+2 p) (3+n+2 p)}-\frac {C (c+d x)^{1+n} \left (c^2-d^2 x^2\right )^{1+p}}{d^3 (3+n+2 p)}-\frac {2^{n+p} \left (B c d n (3+n+2 p)+c^2 C \left (2+n+n^2+2 p\right )+A d^2 \left (6+n^2+10 p+4 p^2+n (5+4 p)\right )\right ) (c+d x)^n \left (1+\frac {d x}{c}\right )^{-1-n-p} \left (c^2-d^2 x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (-n-p,1+p,2+p,\frac {c-d x}{2 c}\right )}{c d^3 (1+p) (2+n+2 p) (3+n+2 p)} \] Output:

(2*c*C*(p+1)-B*d*(3+n+2*p))*(d*x+c)^n*(-d^2*x^2+c^2)^(p+1)/d^3/(2+n+2*p)/( 
3+n+2*p)-C*(d*x+c)^(1+n)*(-d^2*x^2+c^2)^(p+1)/d^3/(3+n+2*p)-2^(n+p)*(B*c*d 
*n*(3+n+2*p)+c^2*C*(n^2+n+2*p+2)+A*d^2*(6+n^2+10*p+4*p^2+n*(5+4*p)))*(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^3/(p+1)/(2+n+2*p)/(3+n+2*p)
 

Mathematica [A] (verified)

Time = 0.77 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.76 \[ \int (c+d x)^n \left (A+B x+C x^2\right ) \left (c^2-d^2 x^2\right )^p \, dx=\frac {(c-d x) (c+d x)^n \left (c^2-d^2 x^2\right )^p \left (-C (c+d x)^2+\frac {(1+p) (2 c C (1+p)-B d (3+n+2 p)) (c+d x)-2^{n+p} \left (B c d n (3+n+2 p)+c^2 C \left (2+n+n^2+2 p\right )+A d^2 \left (6+n^2+10 p+4 p^2+n (5+4 p)\right )\right ) \left (1+\frac {d x}{c}\right )^{-n-p} \operatorname {Hypergeometric2F1}\left (-n-p,1+p,2+p,\frac {c-d x}{2 c}\right )}{(1+p) (2+n+2 p)}\right )}{d^3 (3+n+2 p)} \] Input:

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

Output:

((c - d*x)*(c + d*x)^n*(c^2 - d^2*x^2)^p*(-(C*(c + d*x)^2) + ((1 + p)*(2*c 
*C*(1 + p) - B*d*(3 + n + 2*p))*(c + d*x) - 2^(n + p)*(B*c*d*n*(3 + n + 2* 
p) + c^2*C*(2 + n + n^2 + 2*p) + A*d^2*(6 + n^2 + 10*p + 4*p^2 + n*(5 + 4* 
p)))*(1 + (d*x)/c)^(-n - p)*Hypergeometric2F1[-n - p, 1 + p, 2 + p, (c - d 
*x)/(2*c)])/((1 + p)*(2 + n + 2*p))))/(d^3*(3 + n + 2*p))
 

Rubi [A] (verified)

Time = 0.88 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {2170, 25, 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.

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

\(\Big \downarrow \) 2170

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 672

\(\displaystyle \frac {\frac {\left (A d^2 \left (n^2+n (4 p+5)+4 p^2+10 p+6\right )+B c d n (n+2 p+3)+c^2 C \left (n^2+n+2 p+2\right )\right ) \int (c+d x)^n \left (c^2-d^2 x^2\right )^pdx}{n+2 p+2}+\frac {(c+d x)^n \left (c^2-d^2 x^2\right )^{p+1} (2 c C (p+1)-B d (n+2 p+3))}{d (n+2 p+2)}}{d^2 (n+2 p+3)}-\frac {C (c+d x)^{n+1} \left (c^2-d^2 x^2\right )^{p+1}}{d^3 (n+2 p+3)}\)

\(\Big \downarrow \) 474

\(\displaystyle \frac {\frac {(c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \left (A d^2 \left (n^2+n (4 p+5)+4 p^2+10 p+6\right )+B c d n (n+2 p+3)+c^2 C \left (n^2+n+2 p+2\right )\right ) \int \left (\frac {d x}{c}+1\right )^n \left (c^2-d^2 x^2\right )^pdx}{n+2 p+2}+\frac {(c+d x)^n \left (c^2-d^2 x^2\right )^{p+1} (2 c C (p+1)-B d (n+2 p+3))}{d (n+2 p+2)}}{d^2 (n+2 p+3)}-\frac {C (c+d x)^{n+1} \left (c^2-d^2 x^2\right )^{p+1}}{d^3 (n+2 p+3)}\)

\(\Big \downarrow \) 473

\(\displaystyle \frac {\frac {(c+d x)^n \left (c^2-c d x\right )^{-p-1} \left (c^2-d^2 x^2\right )^{p+1} \left (\frac {d x}{c}+1\right )^{-n-p-1} \left (A d^2 \left (n^2+n (4 p+5)+4 p^2+10 p+6\right )+B c d n (n+2 p+3)+c^2 C \left (n^2+n+2 p+2\right )\right ) \int \left (\frac {d x}{c}+1\right )^{n+p} \left (c^2-c d x\right )^pdx}{n+2 p+2}+\frac {(c+d x)^n \left (c^2-d^2 x^2\right )^{p+1} (2 c C (p+1)-B d (n+2 p+3))}{d (n+2 p+2)}}{d^2 (n+2 p+3)}-\frac {C (c+d x)^{n+1} \left (c^2-d^2 x^2\right )^{p+1}}{d^3 (n+2 p+3)}\)

\(\Big \downarrow \) 79

\(\displaystyle \frac {\frac {(c+d x)^n \left (c^2-d^2 x^2\right )^{p+1} (2 c C (p+1)-B d (n+2 p+3))}{d (n+2 p+2)}-\frac {2^{n+p} (c+d x)^n \left (c^2-d^2 x^2\right )^{p+1} \left (\frac {d x}{c}+1\right )^{-n-p-1} \operatorname {Hypergeometric2F1}\left (-n-p,p+1,p+2,\frac {c-d x}{2 c}\right ) \left (A d^2 \left (n^2+n (4 p+5)+4 p^2+10 p+6\right )+B c d n (n+2 p+3)+c^2 C \left (n^2+n+2 p+2\right )\right )}{c d (p+1) (n+2 p+2)}}{d^2 (n+2 p+3)}-\frac {C (c+d x)^{n+1} \left (c^2-d^2 x^2\right )^{p+1}}{d^3 (n+2 p+3)}\)

Input:

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

Output:

-((C*(c + d*x)^(1 + n)*(c^2 - d^2*x^2)^(1 + p))/(d^3*(3 + n + 2*p))) + ((( 
2*c*C*(1 + p) - B*d*(3 + n + 2*p))*(c + d*x)^n*(c^2 - d^2*x^2)^(1 + p))/(d 
*(2 + n + 2*p)) - (2^(n + p)*(B*c*d*n*(3 + n + 2*p) + c^2*C*(2 + n + n^2 + 
 2*p) + A*d^2*(6 + n^2 + 10*p + 4*p^2 + n*(5 + 4*p)))*(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)))/(d^2*(3 + 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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral((-(-c + d*x)*(c + d*x))**p*(c + d*x)**n*(A + B*x + C*x**2), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

((c + d*x)**n*(c**2 - d**2*x**2)**p*a*c*d**2*n**3 + 6*(c + d*x)**n*(c**2 - 
 d**2*x**2)**p*a*c*d**2*n**2*p + 5*(c + d*x)**n*(c**2 - d**2*x**2)**p*a*c* 
d**2*n**2 + 12*(c + d*x)**n*(c**2 - d**2*x**2)**p*a*c*d**2*n*p**2 + 20*(c 
+ d*x)**n*(c**2 - d**2*x**2)**p*a*c*d**2*n*p + 6*(c + d*x)**n*(c**2 - d**2 
*x**2)**p*a*c*d**2*n + 8*(c + d*x)**n*(c**2 - d**2*x**2)**p*a*c*d**2*p**3 
+ 20*(c + d*x)**n*(c**2 - d**2*x**2)**p*a*c*d**2*p**2 + 12*(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**3*x + 4*(c + d*x)**n*(c**2 - d**2*x**2)**p*a*d**3*n**2*p*x + 5*(c + d* 
x)**n*(c**2 - d**2*x**2)**p*a*d**3*n**2*x + 4*(c + d*x)**n*(c**2 - d**2*x* 
*2)**p*a*d**3*n*p**2*x + 10*(c + d*x)**n*(c**2 - d**2*x**2)**p*a*d**3*n*p* 
x + 6*(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**2 - 2*(c + d*x)**n*(c**2 - d**2*x**2)**p*b*c**2 
*d*n*p - 3*(c + d*x)**n*(c**2 - d**2*x**2)**p*b*c**2*d*n + (c + d*x)**n*(c 
**2 - d**2*x**2)**p*b*c*d**2*n**3*x + 2*(c + d*x)**n*(c**2 - d**2*x**2)**p 
*b*c*d**2*n**2*p*x + 3*(c + d*x)**n*(c**2 - d**2*x**2)**p*b*c*d**2*n**2*x 
+ (c + d*x)**n*(c**2 - d**2*x**2)**p*b*d**3*n**3*x**2 + 4*(c + d*x)**n*(c* 
*2 - d**2*x**2)**p*b*d**3*n**2*p*x**2 + 4*(c + d*x)**n*(c**2 - d**2*x**2)* 
*p*b*d**3*n**2*x**2 + 4*(c + d*x)**n*(c**2 - d**2*x**2)**p*b*d**3*n*p**2*x 
**2 + 8*(c + d*x)**n*(c**2 - d**2*x**2)**p*b*d**3*n*p*x**2 + 3*(c + d*x)** 
n*(c**2 - d**2*x**2)**p*b*d**3*n*x**2 + 4*(c + d*x)**n*(c**2 - d**2*x**...