Integrand size = 18, antiderivative size = 82 \[ \int (1-x)^m x ((2-x) x)^p \, dx=-\frac {(1-x)^{1+m} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},-p,\frac {3+m}{2},(1-x)^2\right )}{1+m}+\frac {(1-x)^{2+m} \operatorname {Hypergeometric2F1}\left (\frac {2+m}{2},-p,\frac {4+m}{2},(1-x)^2\right )}{2+m} \] Output:
-(1-x)^(1+m)*hypergeom([-p, 1/2+1/2*m],[3/2+1/2*m],(1-x)^2)/(1+m)+(1-x)^(2 +m)*hypergeom([-p, 1+1/2*m],[2+1/2*m],(1-x)^2)/(2+m)
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 0.12 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.59 \[ \int (1-x)^m x ((2-x) x)^p \, dx=\frac {2^p (2-x)^{-p} x^2 (-((-2+x) x))^p \operatorname {AppellF1}\left (2+p,-m,-p,3+p,x,\frac {x}{2}\right )}{2+p} \] Input:
Integrate[(1 - x)^m*x*((2 - x)*x)^p,x]
Output:
(2^p*x^2*(-((-2 + x)*x))^p*AppellF1[2 + p, -m, -p, 3 + p, x, x/2])/((2 + p )*(2 - x)^p)
Time = 0.29 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2048, 1269, 1118, 278}
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.
| \(\displaystyle \int x (1-x)^m ((2-x) x)^p \, dx\) |
| \(\Big \downarrow \) 2048 |
| \(\displaystyle \int x (1-x)^m \left (2 x-x^2\right )^pdx\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \int (1-x)^m \left (2 x-x^2\right )^pdx-\int (1-x)^{m+1} \left (2 x-x^2\right )^pdx\) |
| \(\Big \downarrow \) 1118 |
| \(\displaystyle \int \left (1-(1-x)^2\right )^p (1-x)^{m+1}d(1-x)-\int \left (1-(1-x)^2\right )^p (1-x)^md(1-x)\) |
| \(\Big \downarrow \) 278 |
| \(\displaystyle \frac {(1-x)^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {m+2}{2},-p,\frac {m+4}{2},(1-x)^2\right )}{m+2}-\frac {(1-x)^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {m+1}{2},-p,\frac {m+3}{2},(1-x)^2\right )}{m+1}\) |
Input:
Int[(1 - x)^m*x*((2 - x)*x)^p,x]
Output:
-(((1 - x)^(1 + m)*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, (1 - x)^2]) /(1 + m)) + ((1 - x)^(2 + m)*Hypergeometric2F1[(2 + m)/2, -p, (4 + m)/2, ( 1 - x)^2])/(2 + m)
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( -b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && (ILtQ[p, 0 ] || GtQ[a, 0])
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S ymbol] :> Simp[1/e Subst[Int[x^m*(a - b^2/(4*c) + (c*x^2)/e^2)^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
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]
\[\int \left (1-x \right )^{m} x \left (\left (-x +2\right ) x \right )^{p}d x\]
Input:
int((1-x)^m*x*((-x+2)*x)^p,x)
Output:
int((1-x)^m*x*((-x+2)*x)^p,x)
\[ \int (1-x)^m x ((2-x) x)^p \, dx=\int { \left (-{\left (x - 2\right )} x\right )^{p} x {\left (-x + 1\right )}^{m} \,d x } \] Input:
integrate((1-x)^m*x*((2-x)*x)^p,x, algorithm="fricas")
Output:
integral((-x^2 + 2*x)^p*x*(-x + 1)^m, x)
\[ \int (1-x)^m x ((2-x) x)^p \, dx=\int x \left (- x \left (x - 2\right )\right )^{p} \left (1 - x\right )^{m}\, dx \] Input:
integrate((1-x)**m*x*((2-x)*x)**p,x)
Output:
Integral(x*(-x*(x - 2))**p*(1 - x)**m, x)
\[ \int (1-x)^m x ((2-x) x)^p \, dx=\int { \left (-{\left (x - 2\right )} x\right )^{p} x {\left (-x + 1\right )}^{m} \,d x } \] Input:
integrate((1-x)^m*x*((2-x)*x)^p,x, algorithm="maxima")
Output:
integrate((-(x - 2)*x)^p*x*(-x + 1)^m, x)
\[ \int (1-x)^m x ((2-x) x)^p \, dx=\int { \left (-{\left (x - 2\right )} x\right )^{p} x {\left (-x + 1\right )}^{m} \,d x } \] Input:
integrate((1-x)^m*x*((2-x)*x)^p,x, algorithm="giac")
Output:
integrate((-(x - 2)*x)^p*x*(-x + 1)^m, x)
Timed out. \[ \int (1-x)^m x ((2-x) x)^p \, dx=\int x\,{\left (-x\,\left (x-2\right )\right )}^p\,{\left (1-x\right )}^m \,d x \] Input:
int(x*(-x*(x - 2))^p*(1 - x)^m,x)
Output:
int(x*(-x*(x - 2))^p*(1 - x)^m, x)
\[ \int (1-x)^m x ((2-x) x)^p \, dx=\text {too large to display} \] Input:
int((1-x)^m*x*((2-x)*x)^p,x)
Output:
(x**p*( - x + 1)**m*( - x + 2)**p*m*x**2 - x**p*( - x + 1)**m*( - x + 2)** p*m*x + 2*x**p*( - x + 1)**m*( - x + 2)**p*p*x**2 - 2*x**p*( - x + 1)**m*( - x + 2)**p*p*x - x**p*( - x + 1)**m*( - x + 2)**p*p + x**p*( - x + 1)**m *( - x + 2)**p*x**2 - x**p*( - x + 1)**m*( - x + 2)**p - int((x**p*( - x + 1)**m*( - x + 2)**p*x)/(m**2*x**2 - 3*m**2*x + 2*m**2 + 4*m*p*x**2 - 12*m *p*x + 8*m*p + 3*m*x**2 - 9*m*x + 6*m + 4*p**2*x**2 - 12*p**2*x + 8*p**2 + 6*p*x**2 - 18*p*x + 12*p + 2*x**2 - 6*x + 4),x)*m**3*p - 6*int((x**p*( - x + 1)**m*( - x + 2)**p*x)/(m**2*x**2 - 3*m**2*x + 2*m**2 + 4*m*p*x**2 - 1 2*m*p*x + 8*m*p + 3*m*x**2 - 9*m*x + 6*m + 4*p**2*x**2 - 12*p**2*x + 8*p** 2 + 6*p*x**2 - 18*p*x + 12*p + 2*x**2 - 6*x + 4),x)*m**2*p**2 - 5*int((x** p*( - x + 1)**m*( - x + 2)**p*x)/(m**2*x**2 - 3*m**2*x + 2*m**2 + 4*m*p*x* *2 - 12*m*p*x + 8*m*p + 3*m*x**2 - 9*m*x + 6*m + 4*p**2*x**2 - 12*p**2*x + 8*p**2 + 6*p*x**2 - 18*p*x + 12*p + 2*x**2 - 6*x + 4),x)*m**2*p - 12*int( (x**p*( - x + 1)**m*( - x + 2)**p*x)/(m**2*x**2 - 3*m**2*x + 2*m**2 + 4*m* p*x**2 - 12*m*p*x + 8*m*p + 3*m*x**2 - 9*m*x + 6*m + 4*p**2*x**2 - 12*p**2 *x + 8*p**2 + 6*p*x**2 - 18*p*x + 12*p + 2*x**2 - 6*x + 4),x)*m*p**3 - 20* int((x**p*( - x + 1)**m*( - x + 2)**p*x)/(m**2*x**2 - 3*m**2*x + 2*m**2 + 4*m*p*x**2 - 12*m*p*x + 8*m*p + 3*m*x**2 - 9*m*x + 6*m + 4*p**2*x**2 - 12* p**2*x + 8*p**2 + 6*p*x**2 - 18*p*x + 12*p + 2*x**2 - 6*x + 4),x)*m*p**2 - 8*int((x**p*( - x + 1)**m*( - x + 2)**p*x)/(m**2*x**2 - 3*m**2*x + 2*m...