Integrand size = 18, antiderivative size = 41 \[ \int (1-x)^m (2-x)^p x^p \, dx=-\frac {(1-x)^{1+m} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},-p,\frac {3+m}{2},(1-x)^2\right )}{1+m} \] Output:
-(1-x)^(1+m)*hypergeom([-p, 1/2+1/2*m],[3/2+1/2*m],(1-x)^2)/(1+m)
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 0.10 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.12 \[ \int (1-x)^m (2-x)^p x^p \, dx=\frac {2^p (2-x)^{-p} x (-((-2+x) x))^p \operatorname {AppellF1}\left (1+p,-m,-p,2+p,x,\frac {x}{2}\right )}{1+p} \] Input:
Integrate[(1 - x)^m*(2 - x)^p*x^p,x]
Output:
(2^p*x*(-((-2 + x)*x))^p*AppellF1[1 + p, -m, -p, 2 + p, x, x/2])/((1 + p)* (2 - x)^p)
Time = 0.21 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.56, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {146, 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.
\(\displaystyle \int (1-x)^m (2-x)^p x^p \, dx\) |
\(\Big \downarrow \) 146 |
\(\displaystyle -(2-x)^p x^p \left (2 x-x^2\right )^{-p} \int \left (1-(1-x)^2\right )^p (1-x)^md(1-x)\) |
\(\Big \downarrow \) 278 |
\(\displaystyle -\frac {(1-x)^{m+1} (2-x)^p x^p \left (2 x-x^2\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+1}{2},-p,\frac {m+3}{2},(1-x)^2\right )}{m+1}\) |
Input:
Int[(1 - x)^m*(2 - x)^p*x^p,x]
Output:
-(((1 - x)^(1 + m)*(2 - x)^p*x^p*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/ 2, (1 - x)^2])/((1 + m)*(2*x - x^2)^p))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(c + d*x)^n*((e + f*x)^p/(b*(c*e + (d*e + c*f)*x + d*f* x^2)^n)) Subst[Int[x^m*(c*e - (d*e + c*f)^2/(4*d*f) + d*f*(x^2/b^2))^n, x ], x, a + b*x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[p, n] && EqQ[b*d*e + b*c*f - 2*a*d*f, 0]
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])
\[\int \left (1-x \right )^{m} \left (-x +2\right )^{p} x^{p}d x\]
Input:
int((1-x)^m*(-x+2)^p*x^p,x)
Output:
int((1-x)^m*(-x+2)^p*x^p,x)
\[ \int (1-x)^m (2-x)^p x^p \, dx=\int { x^{p} {\left (-x + 2\right )}^{p} {\left (-x + 1\right )}^{m} \,d x } \] Input:
integrate((1-x)^m*(2-x)^p*x^p,x, algorithm="fricas")
Output:
integral(x^p*(-x + 2)^p*(-x + 1)^m, x)
Result contains complex when optimal does not.
Time = 19.37 (sec) , antiderivative size = 189, normalized size of antiderivative = 4.61 \[ \int (1-x)^m (2-x)^p x^p \, dx=\frac {{G_{6, 6}^{5, 3}\left (\begin {matrix} - \frac {m}{2} - \frac {p}{2}, - \frac {m}{2} - \frac {p}{2} + \frac {1}{2}, 1 & \frac {1}{2} - \frac {m}{2}, - \frac {m}{2} - p, - \frac {m}{2} - p + \frac {1}{2} \\- \frac {m}{2} - p - \frac {1}{2}, - \frac {m}{2} - p, - \frac {m}{2} - \frac {p}{2}, - \frac {m}{2} - p + \frac {1}{2}, - \frac {m}{2} - \frac {p}{2} + \frac {1}{2} & 0 \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{\left (x - 1\right )^{2}}} \right )} e^{- i \pi p}}{4 \pi \Gamma \left (- p\right )} - \frac {{G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {m}{2} - \frac {1}{2}, - \frac {m}{2}, \frac {1}{2} - \frac {m}{2}, - \frac {m}{2} - \frac {p}{2} - \frac {1}{2}, - \frac {m}{2} - \frac {p}{2}, 1 & \\- \frac {m}{2} - \frac {p}{2} - \frac {1}{2}, - \frac {m}{2} - \frac {p}{2} & - \frac {m}{2} - \frac {1}{2}, - \frac {m}{2}, - \frac {m}{2} - p - \frac {1}{2}, 0 \end {matrix} \middle | {\frac {1}{\left (x - 1\right )^{2}}} \right )} e^{i \pi m}}{4 \pi \Gamma \left (- p\right )} \] Input:
integrate((1-x)**m*(2-x)**p*x**p,x)
Output:
meijerg(((-m/2 - p/2, -m/2 - p/2 + 1/2, 1), (1/2 - m/2, -m/2 - p, -m/2 - p + 1/2)), ((-m/2 - p - 1/2, -m/2 - p, -m/2 - p/2, -m/2 - p + 1/2, -m/2 - p /2 + 1/2), (0,)), exp_polar(-2*I*pi)/(x - 1)**2)*exp(-I*pi*p)/(4*pi*gamma( -p)) - meijerg(((-m/2 - 1/2, -m/2, 1/2 - m/2, -m/2 - p/2 - 1/2, -m/2 - p/2 , 1), ()), ((-m/2 - p/2 - 1/2, -m/2 - p/2), (-m/2 - 1/2, -m/2, -m/2 - p - 1/2, 0)), (x - 1)**(-2))*exp(I*pi*m)/(4*pi*gamma(-p))
\[ \int (1-x)^m (2-x)^p x^p \, dx=\int { x^{p} {\left (-x + 2\right )}^{p} {\left (-x + 1\right )}^{m} \,d x } \] Input:
integrate((1-x)^m*(2-x)^p*x^p,x, algorithm="maxima")
Output:
integrate(x^p*(-x + 2)^p*(-x + 1)^m, x)
\[ \int (1-x)^m (2-x)^p x^p \, dx=\int { x^{p} {\left (-x + 2\right )}^{p} {\left (-x + 1\right )}^{m} \,d x } \] Input:
integrate((1-x)^m*(2-x)^p*x^p,x, algorithm="giac")
Output:
integrate(x^p*(-x + 2)^p*(-x + 1)^m, x)
Timed out. \[ \int (1-x)^m (2-x)^p x^p \, dx=\int x^p\,{\left (1-x\right )}^m\,{\left (2-x\right )}^p \,d x \] Input:
int(x^p*(1 - x)^m*(2 - x)^p,x)
Output:
int(x^p*(1 - x)^m*(2 - x)^p, x)
\[ \int (1-x)^m (2-x)^p x^p \, dx=\text {too large to display} \] Input:
int((1-x)^m*(2-x)^p*x^p,x)
Output:
(x**p*( - x + 1)**m*( - x + 2)**p*m*x - x**p*( - x + 1)**m*( - x + 2)**p*m + 2*x**p*( - x + 1)**m*( - x + 2)**p*p*x - x**p*( - x + 1)**m*( - x + 2)* *p*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 + m*x**2 - 3*m*x + 2*m + 4*p**2*x**2 - 12*p**2*x + 8*p**2 + 2*p*x**2 - 6*p*x + 4*p),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 - 12*m*p*x + 8*m*p + m*x**2 - 3*m*x + 2*m + 4*p**2*x**2 - 12*p**2*x + 8*p**2 + 2*p*x**2 - 6*p*x + 4*p),x)*m**2*p**2 - 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 + m*x**2 - 3*m*x + 2*m + 4*p**2*x**2 - 12*p**2*x + 8*p**2 + 2*p*x**2 - 6*p*x + 4*p),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 + m*x**2 - 3*m*x + 2* m + 4*p**2*x**2 - 12*p**2*x + 8*p**2 + 2*p*x**2 - 6*p*x + 4*p),x)*m*p**3 - 4*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 + m*x**2 - 3*m*x + 2*m + 4*p**2*x**2 - 12 *p**2*x + 8*p**2 + 2*p*x**2 - 6*p*x + 4*p),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**2 + 4*m*p*x**2 - 12*m *p*x + 8*m*p + m*x**2 - 3*m*x + 2*m + 4*p**2*x**2 - 12*p**2*x + 8*p**2 + 2 *p*x**2 - 6*p*x + 4*p),x)*p**4 - 4*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 + m*x*...