\(\int x (c-d x)^p (c+d x)^{-3+p} \, dx\) [347]

Optimal result

Integrand size = 19, antiderivative size = 106 \[ \int x (c-d x)^p (c+d x)^{-3+p} \, dx=\frac {(c-d x)^{1+p} (c+d x)^{-2+p}}{d^2 (1-2 p)}-\frac {3\ 2^p c (c-d x)^p (c+d x)^{-2+p} \left (1-\frac {d x}{c}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-2+p,-p,-1+p,\frac {c+d x}{2 c}\right )}{d^2 (1-2 p) (2-p)} \] Output:

(-d*x+c)^(p+1)*(d*x+c)^(-2+p)/d^2/(1-2*p)-3*2^p*c*(-d*x+c)^p*(d*x+c)^(-2+p 
)*hypergeom([-p, -2+p],[-1+p],1/2*(d*x+c)/c)/d^2/(1-2*p)/(2-p)/((1-d*x/c)^ 
p)
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.01 \[ \int x (c-d x)^p (c+d x)^{-3+p} \, dx=\frac {(c-d x)^{1+p} (c+d x)^{-2+p} \left (1+\frac {d x}{c}\right )^{-p} \left (-4 c^2 (1+p) \left (1+\frac {d x}{c}\right )^p+3\ 2^p (c+d x)^2 \operatorname {Hypergeometric2F1}\left (2-p,1+p,2+p,\frac {c-d x}{2 c}\right )\right )}{8 c^2 d^2 (-2+p) (1+p)} \] Input:

Integrate[x*(c - d*x)^p*(c + d*x)^(-3 + p),x]
 

Output:

((c - d*x)^(1 + p)*(c + d*x)^(-2 + p)*(-4*c^2*(1 + p)*(1 + (d*x)/c)^p + 3* 
2^p*(c + d*x)^2*Hypergeometric2F1[2 - p, 1 + p, 2 + p, (c - d*x)/(2*c)]))/ 
(8*c^2*d^2*(-2 + p)*(1 + p)*(1 + (d*x)/c)^p)
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.08, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {88, 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.

\(\displaystyle \int x (c-d x)^p (c+d x)^{p-3} \, dx\)

\(\Big \downarrow \) 88

\(\displaystyle \frac {3 \int (c-d x)^p (c+d x)^{p-2}dx}{2 d (2-p)}+\frac {(c-d x)^{p+1} (c+d x)^{p-2}}{2 d^2 (2-p)}\)

\(\Big \downarrow \) 80

\(\displaystyle \frac {3\ 2^{p-3} \left (\frac {c+d x}{c}\right )^{-p} (c+d x)^p \int (c-d x)^p \left (\frac {d x}{2 c}+\frac {1}{2}\right )^{p-2}dx}{c^2 d (2-p)}+\frac {(c-d x)^{p+1} (c+d x)^{p-2}}{2 d^2 (2-p)}\)

\(\Big \downarrow \) 79

\(\displaystyle \frac {(c-d x)^{p+1} (c+d x)^{p-2}}{2 d^2 (2-p)}-\frac {3\ 2^{p-3} (c-d x)^{p+1} (c+d x)^p \left (\frac {c+d x}{c}\right )^{-p} \operatorname {Hypergeometric2F1}\left (2-p,p+1,p+2,\frac {c-d x}{2 c}\right )}{c^2 d^2 (2-p) (p+1)}\)

Input:

Int[x*(c - d*x)^p*(c + d*x)^(-3 + p),x]
 

Output:

((c - d*x)^(1 + p)*(c + d*x)^(-2 + p))/(2*d^2*(2 - p)) - (3*2^(-3 + p)*(c 
- d*x)^(1 + p)*(c + d*x)^p*Hypergeometric2F1[2 - p, 1 + p, 2 + p, (c - d*x 
)/(2*c)])/(c^2*d^2*(2 - p)*(1 + p)*((c + d*x)/c)^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 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 88

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

Maple [F]

\[\int x \left (-x d +c \right )^{p} \left (x d +c \right )^{-3+p}d x\]

Input:

int(x*(-d*x+c)^p*(d*x+c)^(-3+p),x)
 

Output:

int(x*(-d*x+c)^p*(d*x+c)^(-3+p),x)
 

Fricas [F]

\[ \int x (c-d x)^p (c+d x)^{-3+p} \, dx=\int { {\left (d x + c\right )}^{p - 3} {\left (-d x + c\right )}^{p} x \,d x } \] Input:

integrate(x*(-d*x+c)^p*(d*x+c)^(-3+p),x, algorithm="fricas")
 

Output:

integral((d*x + c)^(p - 3)*(-d*x + c)^p*x, x)
 

Sympy [F]

\[ \int x (c-d x)^p (c+d x)^{-3+p} \, dx=\int x \left (c - d x\right )^{p} \left (c + d x\right )^{p - 3}\, dx \] Input:

integrate(x*(-d*x+c)**p*(d*x+c)**(-3+p),x)
 

Output:

Integral(x*(c - d*x)**p*(c + d*x)**(p - 3), x)
 

Maxima [F]

\[ \int x (c-d x)^p (c+d x)^{-3+p} \, dx=\int { {\left (d x + c\right )}^{p - 3} {\left (-d x + c\right )}^{p} x \,d x } \] Input:

integrate(x*(-d*x+c)^p*(d*x+c)^(-3+p),x, algorithm="maxima")
 

Output:

integrate((d*x + c)^(p - 3)*(-d*x + c)^p*x, x)
 

Giac [F]

\[ \int x (c-d x)^p (c+d x)^{-3+p} \, dx=\int { {\left (d x + c\right )}^{p - 3} {\left (-d x + c\right )}^{p} x \,d x } \] Input:

integrate(x*(-d*x+c)^p*(d*x+c)^(-3+p),x, algorithm="giac")
 

Output:

integrate((d*x + c)^(p - 3)*(-d*x + c)^p*x, x)
 

Mupad [F(-1)]

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

int(x*(c + d*x)^(p - 3)*(c - d*x)^p,x)
 

Output:

int(x*(c + d*x)^(p - 3)*(c - d*x)^p, x)
 

Reduce [F]

\[ \int x (c-d x)^p (c+d x)^{-3+p} \, dx=\frac {\left (d x +c \right )^{p} \left (-d x +c \right )^{p} c +2 \left (d x +c \right )^{p} \left (-d x +c \right )^{p} d x +12 \left (\int \frac {\left (d x +c \right )^{p} \left (-d x +c \right )^{p} x}{-2 d^{4} p \,x^{4}-4 c \,d^{3} p \,x^{3}+d^{4} x^{4}+2 c \,d^{3} x^{3}+4 c^{3} d p x +2 c^{4} p -2 c^{3} d x -c^{4}}d x \right ) c^{3} d^{2} p^{2}-6 \left (\int \frac {\left (d x +c \right )^{p} \left (-d x +c \right )^{p} x}{-2 d^{4} p \,x^{4}-4 c \,d^{3} p \,x^{3}+d^{4} x^{4}+2 c \,d^{3} x^{3}+4 c^{3} d p x +2 c^{4} p -2 c^{3} d x -c^{4}}d x \right ) c^{3} d^{2} p +24 \left (\int \frac {\left (d x +c \right )^{p} \left (-d x +c \right )^{p} x}{-2 d^{4} p \,x^{4}-4 c \,d^{3} p \,x^{3}+d^{4} x^{4}+2 c \,d^{3} x^{3}+4 c^{3} d p x +2 c^{4} p -2 c^{3} d x -c^{4}}d x \right ) c^{2} d^{3} p^{2} x -12 \left (\int \frac {\left (d x +c \right )^{p} \left (-d x +c \right )^{p} x}{-2 d^{4} p \,x^{4}-4 c \,d^{3} p \,x^{3}+d^{4} x^{4}+2 c \,d^{3} x^{3}+4 c^{3} d p x +2 c^{4} p -2 c^{3} d x -c^{4}}d x \right ) c^{2} d^{3} p x +12 \left (\int \frac {\left (d x +c \right )^{p} \left (-d x +c \right )^{p} x}{-2 d^{4} p \,x^{4}-4 c \,d^{3} p \,x^{3}+d^{4} x^{4}+2 c \,d^{3} x^{3}+4 c^{3} d p x +2 c^{4} p -2 c^{3} d x -c^{4}}d x \right ) c \,d^{4} p^{2} x^{2}-6 \left (\int \frac {\left (d x +c \right )^{p} \left (-d x +c \right )^{p} x}{-2 d^{4} p \,x^{4}-4 c \,d^{3} p \,x^{3}+d^{4} x^{4}+2 c \,d^{3} x^{3}+4 c^{3} d p x +2 c^{4} p -2 c^{3} d x -c^{4}}d x \right ) c \,d^{4} p \,x^{2}}{2 d^{2} \left (2 d^{2} p \,x^{2}+4 c d p x -d^{2} x^{2}+2 c^{2} p -2 c d x -c^{2}\right )} \] Input:

int(x*(-d*x+c)^p*(d*x+c)^(-3+p),x)
 

Output:

((c + d*x)**p*(c - d*x)**p*c + 2*(c + d*x)**p*(c - d*x)**p*d*x + 12*int((( 
c + d*x)**p*(c - d*x)**p*x)/(2*c**4*p - c**4 + 4*c**3*d*p*x - 2*c**3*d*x - 
 4*c*d**3*p*x**3 + 2*c*d**3*x**3 - 2*d**4*p*x**4 + d**4*x**4),x)*c**3*d**2 
*p**2 - 6*int(((c + d*x)**p*(c - d*x)**p*x)/(2*c**4*p - c**4 + 4*c**3*d*p* 
x - 2*c**3*d*x - 4*c*d**3*p*x**3 + 2*c*d**3*x**3 - 2*d**4*p*x**4 + d**4*x* 
*4),x)*c**3*d**2*p + 24*int(((c + d*x)**p*(c - d*x)**p*x)/(2*c**4*p - c**4 
 + 4*c**3*d*p*x - 2*c**3*d*x - 4*c*d**3*p*x**3 + 2*c*d**3*x**3 - 2*d**4*p* 
x**4 + d**4*x**4),x)*c**2*d**3*p**2*x - 12*int(((c + d*x)**p*(c - d*x)**p* 
x)/(2*c**4*p - c**4 + 4*c**3*d*p*x - 2*c**3*d*x - 4*c*d**3*p*x**3 + 2*c*d* 
*3*x**3 - 2*d**4*p*x**4 + d**4*x**4),x)*c**2*d**3*p*x + 12*int(((c + d*x)* 
*p*(c - d*x)**p*x)/(2*c**4*p - c**4 + 4*c**3*d*p*x - 2*c**3*d*x - 4*c*d**3 
*p*x**3 + 2*c*d**3*x**3 - 2*d**4*p*x**4 + d**4*x**4),x)*c*d**4*p**2*x**2 - 
 6*int(((c + d*x)**p*(c - d*x)**p*x)/(2*c**4*p - c**4 + 4*c**3*d*p*x - 2*c 
**3*d*x - 4*c*d**3*p*x**3 + 2*c*d**3*x**3 - 2*d**4*p*x**4 + d**4*x**4),x)* 
c*d**4*p*x**2)/(2*d**2*(2*c**2*p - c**2 + 4*c*d*p*x - 2*c*d*x + 2*d**2*p*x 
**2 - d**2*x**2))