\(\int (c+d x)^n (c^2-d^2 x^2)^p \, dx\) [249]

Optimal result

Integrand size = 22, antiderivative size = 85 \[ \int (c+d x)^n \left (c^2-d^2 x^2\right )^p \, dx=-\frac {2^{n+p} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-1-n-p} \left (c^2-d^2 x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (-n-p,1+p,2+p,\frac {c-d x}{2 c}\right )}{c d (1+p)} \] Output:

-2^(n+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/(p+1)
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00 \[ \int (c+d x)^n \left (c^2-d^2 x^2\right )^p \, dx=\frac {2^{n+p} (-c+d x) (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n-p} \left (c^2-d^2 x^2\right )^p \operatorname {Hypergeometric2F1}\left (-n-p,1+p,2+p,\frac {c-d x}{2 c}\right )}{d (1+p)} \] Input:

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

Output:

(2^(n + p)*(-c + d*x)*(c + d*x)^n*(1 + (d*x)/c)^(-n - p)*(c^2 - d^2*x^2)^p 
*Hypergeometric2F1[-n - p, 1 + p, 2 + p, (c - d*x)/(2*c)])/(d*(1 + p))
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {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.

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

\(\Big \downarrow \) 474

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

\(\Big \downarrow \) 473

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

\(\Big \downarrow \) 79

\(\displaystyle -\frac {2^{n+p} (c+d x)^n \left (c^2-d^2 x^2\right )^{p+1} \left (\frac {d x}{c}+1\right )^{-n-p-1} \operatorname {Hypergeometric2F1}\left (-n-p,p+1,p+2,\frac {c-d x}{2 c}\right )}{c d (p+1)}\)

Input:

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

Output:

-((2^(n + 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)))
 

Defintions of rubi rules used

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])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral((-(-c + d*x)*(c + d*x))**p*(c + d*x)**n, x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (c+d x)^n \left (c^2-d^2 x^2\right )^p \, dx=\frac {\left (d x +c \right )^{n} \left (-d^{2} x^{2}+c^{2}\right )^{p} c n +2 \left (d x +c \right )^{n} \left (-d^{2} x^{2}+c^{2}\right )^{p} c p +\left (d x +c \right )^{n} \left (-d^{2} x^{2}+c^{2}\right )^{p} d n x +4 \left (\int \frac {\left (d x +c \right )^{n} \left (-d^{2} x^{2}+c^{2}\right )^{p} x}{-d^{2} n \,x^{2}-2 d^{2} p \,x^{2}-d^{2} x^{2}+c^{2} n +2 c^{2} p +c^{2}}d x \right ) c \,d^{2} n^{2} p +12 \left (\int \frac {\left (d x +c \right )^{n} \left (-d^{2} x^{2}+c^{2}\right )^{p} x}{-d^{2} n \,x^{2}-2 d^{2} p \,x^{2}-d^{2} x^{2}+c^{2} n +2 c^{2} p +c^{2}}d x \right ) c \,d^{2} n \,p^{2}+4 \left (\int \frac {\left (d x +c \right )^{n} \left (-d^{2} x^{2}+c^{2}\right )^{p} x}{-d^{2} n \,x^{2}-2 d^{2} p \,x^{2}-d^{2} x^{2}+c^{2} n +2 c^{2} p +c^{2}}d x \right ) c \,d^{2} n p +8 \left (\int \frac {\left (d x +c \right )^{n} \left (-d^{2} x^{2}+c^{2}\right )^{p} x}{-d^{2} n \,x^{2}-2 d^{2} p \,x^{2}-d^{2} x^{2}+c^{2} n +2 c^{2} p +c^{2}}d x \right ) c \,d^{2} p^{3}+4 \left (\int \frac {\left (d x +c \right )^{n} \left (-d^{2} x^{2}+c^{2}\right )^{p} x}{-d^{2} n \,x^{2}-2 d^{2} p \,x^{2}-d^{2} x^{2}+c^{2} n +2 c^{2} p +c^{2}}d x \right ) c \,d^{2} p^{2}}{d n \left (n +2 p +1\right )} \] Input:

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

Output:

((c + d*x)**n*(c**2 - d**2*x**2)**p*c*n + 2*(c + d*x)**n*(c**2 - d**2*x**2 
)**p*c*p + (c + d*x)**n*(c**2 - d**2*x**2)**p*d*n*x + 4*int(((c + d*x)**n* 
(c**2 - d**2*x**2)**p*x)/(c**2*n + 2*c**2*p + c**2 - d**2*n*x**2 - 2*d**2* 
p*x**2 - d**2*x**2),x)*c*d**2*n**2*p + 12*int(((c + d*x)**n*(c**2 - d**2*x 
**2)**p*x)/(c**2*n + 2*c**2*p + c**2 - d**2*n*x**2 - 2*d**2*p*x**2 - d**2* 
x**2),x)*c*d**2*n*p**2 + 4*int(((c + d*x)**n*(c**2 - d**2*x**2)**p*x)/(c** 
2*n + 2*c**2*p + c**2 - d**2*n*x**2 - 2*d**2*p*x**2 - d**2*x**2),x)*c*d**2 
*n*p + 8*int(((c + d*x)**n*(c**2 - d**2*x**2)**p*x)/(c**2*n + 2*c**2*p + c 
**2 - d**2*n*x**2 - 2*d**2*p*x**2 - d**2*x**2),x)*c*d**2*p**3 + 4*int(((c 
+ d*x)**n*(c**2 - d**2*x**2)**p*x)/(c**2*n + 2*c**2*p + c**2 - d**2*n*x**2 
 - 2*d**2*p*x**2 - d**2*x**2),x)*c*d**2*p**2)/(d*n*(n + 2*p + 1))