Integrand size = 20, antiderivative size = 127 \[ \int (1-x)^m x^2 ((2-x) x)^p \, dx=\frac {(1-x)^{1+m} (2-x)^{1+p} x^{1+p}}{3+m+2 p}-\frac {2 (2+m+p) (1-x)^{1+m} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},-p,\frac {3+m}{2},(1-x)^2\right )}{(1+m) (3+m+2 p)}+\frac {2 (1-x)^{2+m} \operatorname {Hypergeometric2F1}\left (\frac {2+m}{2},-p,\frac {4+m}{2},(1-x)^2\right )}{2+m} \] Output:
(1-x)^(1+m)*(2-x)^(p+1)*x^(p+1)/(3+m+2*p)-2*(2+m+p)*(1-x)^(1+m)*hypergeom( [-p, 1/2+1/2*m],[3/2+1/2*m],(1-x)^2)/(1+m)/(3+m+2*p)+2*(1-x)^(2+m)*hyperge om([-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 = 0.38 \[ \int (1-x)^m x^2 ((2-x) x)^p \, dx=\frac {2^p (2-x)^{-p} x^3 (-((-2+x) x))^p \operatorname {AppellF1}\left (3+p,-m,-p,4+p,x,\frac {x}{2}\right )}{3+p} \] Input:
Integrate[(1 - x)^m*x^2*((2 - x)*x)^p,x]
Output:
(2^p*x^3*(-((-2 + x)*x))^p*AppellF1[3 + p, -m, -p, 4 + p, x, x/2])/((3 + p )*(2 - x)^p)
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 0.27 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.40, 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.
\(\displaystyle \int x^2 (1-x)^m ((2-x) x)^p \, dx\) |
\(\Big \downarrow \) 2048 |
\(\displaystyle \int x^2 (1-x)^m \left (2 x-x^2\right )^pdx\) |
\(\Big \downarrow \) 1261 |
\(\displaystyle (2-x)^{-p} x^{-p} \left (2 x-x^2\right )^p \int (1-x)^m (2-x)^p x^{p+2}dx\) |
\(\Big \downarrow \) 150 |
\(\displaystyle \frac {2^p x^3 (2-x)^{-p} \left (2 x-x^2\right )^p \operatorname {AppellF1}\left (p+3,-m,-p,p+4,x,\frac {x}{2}\right )}{p+3}\) |
Input:
Int[(1 - x)^m*x^2*((2 - x)*x)^p,x]
Output:
(2^p*x^3*(2*x - x^2)^p*AppellF1[3 + p, -m, -p, 4 + p, x, x/2])/((3 + p)*(2 - x)^p)
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])
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]
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]
\[\int \left (1-x \right )^{m} x^{2} \left (\left (-x +2\right ) x \right )^{p}d x\]
Input:
int((1-x)^m*x^2*((-x+2)*x)^p,x)
Output:
int((1-x)^m*x^2*((-x+2)*x)^p,x)
\[ \int (1-x)^m x^2 ((2-x) x)^p \, dx=\int { \left (-{\left (x - 2\right )} x\right )^{p} x^{2} {\left (-x + 1\right )}^{m} \,d x } \] Input:
integrate((1-x)^m*x^2*((2-x)*x)^p,x, algorithm="fricas")
Output:
integral((-x^2 + 2*x)^p*x^2*(-x + 1)^m, x)
\[ \int (1-x)^m x^2 ((2-x) x)^p \, dx=\int x^{2} \left (- x \left (x - 2\right )\right )^{p} \left (1 - x\right )^{m}\, dx \] Input:
integrate((1-x)**m*x**2*((2-x)*x)**p,x)
Output:
Integral(x**2*(-x*(x - 2))**p*(1 - x)**m, x)
\[ \int (1-x)^m x^2 ((2-x) x)^p \, dx=\int { \left (-{\left (x - 2\right )} x\right )^{p} x^{2} {\left (-x + 1\right )}^{m} \,d x } \] Input:
integrate((1-x)^m*x^2*((2-x)*x)^p,x, algorithm="maxima")
Output:
integrate((-(x - 2)*x)^p*x^2*(-x + 1)^m, x)
\[ \int (1-x)^m x^2 ((2-x) x)^p \, dx=\int { \left (-{\left (x - 2\right )} x\right )^{p} x^{2} {\left (-x + 1\right )}^{m} \,d x } \] Input:
integrate((1-x)^m*x^2*((2-x)*x)^p,x, algorithm="giac")
Output:
integrate((-(x - 2)*x)^p*x^2*(-x + 1)^m, x)
Timed out. \[ \int (1-x)^m x^2 ((2-x) x)^p \, dx=\int x^2\,{\left (-x\,\left (x-2\right )\right )}^p\,{\left (1-x\right )}^m \,d x \] Input:
int(x^2*(-x*(x - 2))^p*(1 - x)^m,x)
Output:
int(x^2*(-x*(x - 2))^p*(1 - x)^m, x)
\[ \int (1-x)^m x^2 ((2-x) x)^p \, dx=\text {too large to display} \] Input:
int((1-x)^m*x^2*((2-x)*x)^p,x)
Output:
(x**p*( - x + 1)**m*( - x + 2)**p*m**3*x**3 - x**p*( - x + 1)**m*( - x + 2 )**p*m**3*x**2 + 6*x**p*( - x + 1)**m*( - x + 2)**p*m**2*p*x**3 - 6*x**p*( - x + 1)**m*( - x + 2)**p*m**2*p*x**2 - 2*x**p*( - x + 1)**m*( - x + 2)** p*m**2*p*x + 3*x**p*( - x + 1)**m*( - x + 2)**p*m**2*x**3 - x**p*( - x + 1 )**m*( - x + 2)**p*m**2*x**2 - 2*x**p*( - x + 1)**m*( - x + 2)**p*m**2*x + 12*x**p*( - x + 1)**m*( - x + 2)**p*m*p**2*x**3 - 12*x**p*( - x + 1)**m*( - x + 2)**p*m*p**2*x**2 - 8*x**p*( - x + 1)**m*( - x + 2)**p*m*p**2*x - 2 *x**p*( - x + 1)**m*( - x + 2)**p*m*p**2 + 12*x**p*( - x + 1)**m*( - x + 2 )**p*m*p*x**3 - 4*x**p*( - x + 1)**m*( - x + 2)**p*m*p*x**2 - 12*x**p*( - x + 1)**m*( - x + 2)**p*m*p*x - 4*x**p*( - x + 1)**m*( - x + 2)**p*m*p + 2 *x**p*( - x + 1)**m*( - x + 2)**p*m*x**3 - 2*x**p*( - x + 1)**m*( - x + 2) **p*m + 8*x**p*( - x + 1)**m*( - x + 2)**p*p**3*x**3 - 8*x**p*( - x + 1)** m*( - x + 2)**p*p**3*x**2 - 8*x**p*( - x + 1)**m*( - x + 2)**p*p**3*x - 4* x**p*( - x + 1)**m*( - x + 2)**p*p**3 + 12*x**p*( - x + 1)**m*( - x + 2)** p*p**2*x**3 - 4*x**p*( - x + 1)**m*( - x + 2)**p*p**2*x**2 - 16*x**p*( - x + 1)**m*( - x + 2)**p*p**2*x - 12*x**p*( - x + 1)**m*( - x + 2)**p*p**2 + 4*x**p*( - x + 1)**m*( - x + 2)**p*p*x**3 - 8*x**p*( - x + 1)**m*( - x + 2)**p*p - 2*int((x**p*( - x + 1)**m*( - x + 2)**p*x)/(m**4*x**2 - 3*m**4*x + 2*m**4 + 8*m**3*p*x**2 - 24*m**3*p*x + 16*m**3*p + 6*m**3*x**2 - 18*m** 3*x + 12*m**3 + 24*m**2*p**2*x**2 - 72*m**2*p**2*x + 48*m**2*p**2 + 36*...