\(\int \frac {(1-x)^m ((2-x) x)^p}{x} \, dx\) [617]

Optimal result

Integrand size = 20, antiderivative size = 87 \[ \int \frac {(1-x)^m ((2-x) x)^p}{x} \, dx=-\frac {(1-x)^{1+m} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},1-p,\frac {3+m}{2},(1-x)^2\right )}{1+m}-\frac {(1-x)^{2+m} \operatorname {Hypergeometric2F1}\left (\frac {2+m}{2},1-p,\frac {4+m}{2},(1-x)^2\right )}{2+m} \] Output:

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

Mathematica [C] (verified)

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

Time = 0.12 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.47 \[ \int \frac {(1-x)^m ((2-x) x)^p}{x} \, dx=\frac {2^p (2-x)^{-p} (-((-2+x) x))^p \operatorname {AppellF1}\left (p,-m,-p,1+p,x,\frac {x}{2}\right )}{p} \] Input:

Integrate[((1 - x)^m*((2 - x)*x)^p)/x,x]
 

Output:

(2^p*(-((-2 + x)*x))^p*AppellF1[p, -m, -p, 1 + p, x, x/2])/(p*(2 - x)^p)
 

Rubi [C] (verified)

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

Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.51, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2048, 1261, 150}

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.

\(\displaystyle \int \frac {(1-x)^m ((2-x) x)^p}{x} \, dx\)

\(\Big \downarrow \) 2048

\(\displaystyle \int \frac {(1-x)^m \left (2 x-x^2\right )^p}{x}dx\)

\(\Big \downarrow \) 1261

\(\displaystyle (2-x)^{-p} x^{-p} \left (2 x-x^2\right )^p \int (1-x)^m (2-x)^p x^{p-1}dx\)

\(\Big \downarrow \) 150

\(\displaystyle \frac {2^p (2-x)^{-p} \left (2 x-x^2\right )^p \operatorname {AppellF1}\left (p,-m,-p,p+1,x,\frac {x}{2}\right )}{p}\)

Input:

Int[((1 - x)^m*((2 - x)*x)^p)/x,x]
 

Output:

(2^p*(2*x - x^2)^p*AppellF1[p, -m, -p, 1 + p, x, x/2])/(p*(2 - x)^p)
 

Defintions of rubi rules used

rule 150

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 
, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] &&  !In 
tegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
 

rule 1261

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

rule 2048

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))*((c_) + (d_.)*(x_)^(n_.)))^(p_) 
, x_Symbol] :> Int[u*(a*c*e + (b*c + a*d)*e*x^n + b*d*e*x^(2*n))^p, x] /; F 
reeQ[{a, b, c, d, e, n, p}, x]
 

Maple [F]

\[\int \frac {\left (1-x \right )^{m} \left (\left (-x +2\right ) x \right )^{p}}{x}d x\]

Input:

int((1-x)^m*((-x+2)*x)^p/x,x)
 

Output:

int((1-x)^m*((-x+2)*x)^p/x,x)
 

Fricas [F]

\[ \int \frac {(1-x)^m ((2-x) x)^p}{x} \, dx=\int { \frac {\left (-{\left (x - 2\right )} x\right )^{p} {\left (-x + 1\right )}^{m}}{x} \,d x } \] Input:

integrate((1-x)^m*((2-x)*x)^p/x,x, algorithm="fricas")
 

Output:

integral((-x^2 + 2*x)^p*(-x + 1)^m/x, x)
 

Sympy [F]

\[ \int \frac {(1-x)^m ((2-x) x)^p}{x} \, dx=\int \frac {\left (- x \left (x - 2\right )\right )^{p} \left (1 - x\right )^{m}}{x}\, dx \] Input:

integrate((1-x)**m*((2-x)*x)**p/x,x)
 

Output:

Integral((-x*(x - 2))**p*(1 - x)**m/x, x)
 

Maxima [F]

\[ \int \frac {(1-x)^m ((2-x) x)^p}{x} \, dx=\int { \frac {\left (-{\left (x - 2\right )} x\right )^{p} {\left (-x + 1\right )}^{m}}{x} \,d x } \] Input:

integrate((1-x)^m*((2-x)*x)^p/x,x, algorithm="maxima")
 

Output:

integrate((-(x - 2)*x)^p*(-x + 1)^m/x, x)
 

Giac [F]

\[ \int \frac {(1-x)^m ((2-x) x)^p}{x} \, dx=\int { \frac {\left (-{\left (x - 2\right )} x\right )^{p} {\left (-x + 1\right )}^{m}}{x} \,d x } \] Input:

integrate((1-x)^m*((2-x)*x)^p/x,x, algorithm="giac")
 

Output:

integrate((-(x - 2)*x)^p*(-x + 1)^m/x, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(1-x)^m ((2-x) x)^p}{x} \, dx=\int \frac {{\left (-x\,\left (x-2\right )\right )}^p\,{\left (1-x\right )}^m}{x} \,d x \] Input:

int(((-x*(x - 2))^p*(1 - x)^m)/x,x)
 

Output:

int(((-x*(x - 2))^p*(1 - x)^m)/x, x)
 

Reduce [F]

\[ \int \frac {(1-x)^m ((2-x) x)^p}{x} \, dx=\frac {3 x^{p} \left (1-x \right )^{m} \left (-x +2\right )^{p}-\left (\int \frac {x^{p} \left (1-x \right )^{m} \left (-x +2\right )^{p} x}{m \,x^{2}+2 p \,x^{2}-3 m x -6 p x +2 m +4 p}d x \right ) m^{2}-4 \left (\int \frac {x^{p} \left (1-x \right )^{m} \left (-x +2\right )^{p} x}{m \,x^{2}+2 p \,x^{2}-3 m x -6 p x +2 m +4 p}d x \right ) m p -4 \left (\int \frac {x^{p} \left (1-x \right )^{m} \left (-x +2\right )^{p} x}{m \,x^{2}+2 p \,x^{2}-3 m x -6 p x +2 m +4 p}d x \right ) p^{2}+4 \left (\int \frac {x^{p} \left (1-x \right )^{m} \left (-x +2\right )^{p}}{m \,x^{3}+2 p \,x^{3}-3 m \,x^{2}-6 p \,x^{2}+2 m x +4 p x}d x \right ) m^{2}+10 \left (\int \frac {x^{p} \left (1-x \right )^{m} \left (-x +2\right )^{p}}{m \,x^{3}+2 p \,x^{3}-3 m \,x^{2}-6 p \,x^{2}+2 m x +4 p x}d x \right ) m p +4 \left (\int \frac {x^{p} \left (1-x \right )^{m} \left (-x +2\right )^{p}}{m \,x^{3}+2 p \,x^{3}-3 m \,x^{2}-6 p \,x^{2}+2 m x +4 p x}d x \right ) p^{2}}{2 m +4 p} \] Input:

int((1-x)^m*((2-x)*x)^p/x,x)
 

Output:

(3*x**p*( - x + 1)**m*( - x + 2)**p - int((x**p*( - x + 1)**m*( - x + 2)** 
p*x)/(m*x**2 - 3*m*x + 2*m + 2*p*x**2 - 6*p*x + 4*p),x)*m**2 - 4*int((x**p 
*( - x + 1)**m*( - x + 2)**p*x)/(m*x**2 - 3*m*x + 2*m + 2*p*x**2 - 6*p*x + 
 4*p),x)*m*p - 4*int((x**p*( - x + 1)**m*( - x + 2)**p*x)/(m*x**2 - 3*m*x 
+ 2*m + 2*p*x**2 - 6*p*x + 4*p),x)*p**2 + 4*int((x**p*( - x + 1)**m*( - x 
+ 2)**p)/(m*x**3 - 3*m*x**2 + 2*m*x + 2*p*x**3 - 6*p*x**2 + 4*p*x),x)*m**2 
 + 10*int((x**p*( - x + 1)**m*( - x + 2)**p)/(m*x**3 - 3*m*x**2 + 2*m*x + 
2*p*x**3 - 6*p*x**2 + 4*p*x),x)*m*p + 4*int((x**p*( - x + 1)**m*( - x + 2) 
**p)/(m*x**3 - 3*m*x**2 + 2*m*x + 2*p*x**3 - 6*p*x**2 + 4*p*x),x)*p**2)/(2 
*(m + 2*p))