Integrand size = 13, antiderivative size = 33 \[ \int (2-3 x)^p x^{m+p} \, dx=\frac {2^p x^{1+m+p} \operatorname {Hypergeometric2F1}\left (-p,1+m+p,2+m+p,\frac {3 x}{2}\right )}{1+m+p} \] Output:
2^p*x^(1+m+p)*hypergeom([-p, 1+m+p],[2+m+p],3/2*x)/(1+m+p)
Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int (2-3 x)^p x^{m+p} \, dx=\frac {2^p x^{1+m+p} \operatorname {Hypergeometric2F1}\left (-p,1+m+p,2+m+p,\frac {3 x}{2}\right )}{1+m+p} \] Input:
Integrate[(2 - 3*x)^p*x^(m + p),x]
Output:
(2^p*x^(1 + m + p)*Hypergeometric2F1[-p, 1 + m + p, 2 + m + p, (3*x)/2])/( 1 + m + p)
Time = 0.24 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {74}
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.
\(\displaystyle \int (2-3 x)^p x^{m+p} \, dx\) |
\(\Big \downarrow \) 74 |
\(\displaystyle \frac {2^p x^{m+p+1} \operatorname {Hypergeometric2F1}\left (-p,m+p+1,m+p+2,\frac {3 x}{2}\right )}{m+p+1}\) |
Input:
Int[(2 - 3*x)^p*x^(m + p),x]
Output:
(2^p*x^(1 + m + p)*Hypergeometric2F1[-p, 1 + m + p, 2 + m + p, (3*x)/2])/( 1 + m + p)
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x )^(m + 1)/(b*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[m] && (IntegerQ[n] || (GtQ[c, 0] && !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))
Time = 0.39 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.03
method | result | size |
meijerg | \(\frac {2^{p} x^{1+m +p} \operatorname {hypergeom}\left (\left [-p , 1+m +p \right ], \left [2+m +p \right ], \frac {3 x}{2}\right )}{1+m +p}\) | \(34\) |
Input:
int((2-3*x)^p*x^(m+p),x,method=_RETURNVERBOSE)
Output:
2^p*x^(1+m+p)*hypergeom([-p,1+m+p],[2+m+p],3/2*x)/(1+m+p)
\[ \int (2-3 x)^p x^{m+p} \, dx=\int { x^{m + p} {\left (-3 \, x + 2\right )}^{p} \,d x } \] Input:
integrate((2-3*x)^p*x^(m+p),x, algorithm="fricas")
Output:
integral(x^(m + p)*(-3*x + 2)^p, x)
Result contains complex when optimal does not.
Time = 1.23 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.33 \[ \int (2-3 x)^p x^{m+p} \, dx=\frac {2^{p} x^{m + p + 1} \Gamma \left (m + p + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, m + p + 1 \\ m + p + 2 \end {matrix}\middle | {\frac {3 x e^{2 i \pi }}{2}} \right )}}{\Gamma \left (m + p + 2\right )} \] Input:
integrate((2-3*x)**p*x**(m+p),x)
Output:
2**p*x**(m + p + 1)*gamma(m + p + 1)*hyper((-p, m + p + 1), (m + p + 2,), 3*x*exp_polar(2*I*pi)/2)/gamma(m + p + 2)
\[ \int (2-3 x)^p x^{m+p} \, dx=\int { x^{m + p} {\left (-3 \, x + 2\right )}^{p} \,d x } \] Input:
integrate((2-3*x)^p*x^(m+p),x, algorithm="maxima")
Output:
integrate(x^(m + p)*(-3*x + 2)^p, x)
\[ \int (2-3 x)^p x^{m+p} \, dx=\int { x^{m + p} {\left (-3 \, x + 2\right )}^{p} \,d x } \] Input:
integrate((2-3*x)^p*x^(m+p),x, algorithm="giac")
Output:
integrate(x^(m + p)*(-3*x + 2)^p, x)
Timed out. \[ \int (2-3 x)^p x^{m+p} \, dx=\int x^{m+p}\,{\left (2-3\,x\right )}^p \,d x \] Input:
int(x^(m + p)*(2 - 3*x)^p,x)
Output:
int(x^(m + p)*(2 - 3*x)^p, x)
\[ \int (2-3 x)^p x^{m+p} \, dx=\frac {3 x^{m +p} \left (-3 x +2\right )^{p} m x +6 x^{m +p} \left (-3 x +2\right )^{p} p x -2 x^{m +p} \left (-3 x +2\right )^{p} p -4 \left (\int \frac {x^{m +p} \left (-3 x +2\right )^{p}}{3 m^{2} x^{2}+12 m p \,x^{2}+12 p^{2} x^{2}-2 m^{2} x -8 m p x +3 m \,x^{2}-8 p^{2} x +6 p \,x^{2}-2 m x -4 p x}d x \right ) m^{3} p -20 \left (\int \frac {x^{m +p} \left (-3 x +2\right )^{p}}{3 m^{2} x^{2}+12 m p \,x^{2}+12 p^{2} x^{2}-2 m^{2} x -8 m p x +3 m \,x^{2}-8 p^{2} x +6 p \,x^{2}-2 m x -4 p x}d x \right ) m^{2} p^{2}-4 \left (\int \frac {x^{m +p} \left (-3 x +2\right )^{p}}{3 m^{2} x^{2}+12 m p \,x^{2}+12 p^{2} x^{2}-2 m^{2} x -8 m p x +3 m \,x^{2}-8 p^{2} x +6 p \,x^{2}-2 m x -4 p x}d x \right ) m^{2} p -32 \left (\int \frac {x^{m +p} \left (-3 x +2\right )^{p}}{3 m^{2} x^{2}+12 m p \,x^{2}+12 p^{2} x^{2}-2 m^{2} x -8 m p x +3 m \,x^{2}-8 p^{2} x +6 p \,x^{2}-2 m x -4 p x}d x \right ) m \,p^{3}-12 \left (\int \frac {x^{m +p} \left (-3 x +2\right )^{p}}{3 m^{2} x^{2}+12 m p \,x^{2}+12 p^{2} x^{2}-2 m^{2} x -8 m p x +3 m \,x^{2}-8 p^{2} x +6 p \,x^{2}-2 m x -4 p x}d x \right ) m \,p^{2}-16 \left (\int \frac {x^{m +p} \left (-3 x +2\right )^{p}}{3 m^{2} x^{2}+12 m p \,x^{2}+12 p^{2} x^{2}-2 m^{2} x -8 m p x +3 m \,x^{2}-8 p^{2} x +6 p \,x^{2}-2 m x -4 p x}d x \right ) p^{4}-8 \left (\int \frac {x^{m +p} \left (-3 x +2\right )^{p}}{3 m^{2} x^{2}+12 m p \,x^{2}+12 p^{2} x^{2}-2 m^{2} x -8 m p x +3 m \,x^{2}-8 p^{2} x +6 p \,x^{2}-2 m x -4 p x}d x \right ) p^{3}}{3 m^{2}+12 m p +12 p^{2}+3 m +6 p} \] Input:
int((2-3*x)^p*x^(m+p),x)
Output:
(3*x**(m + p)*( - 3*x + 2)**p*m*x + 6*x**(m + p)*( - 3*x + 2)**p*p*x - 2*x **(m + p)*( - 3*x + 2)**p*p - 4*int((x**(m + p)*( - 3*x + 2)**p)/(3*m**2*x **2 - 2*m**2*x + 12*m*p*x**2 - 8*m*p*x + 3*m*x**2 - 2*m*x + 12*p**2*x**2 - 8*p**2*x + 6*p*x**2 - 4*p*x),x)*m**3*p - 20*int((x**(m + p)*( - 3*x + 2)* *p)/(3*m**2*x**2 - 2*m**2*x + 12*m*p*x**2 - 8*m*p*x + 3*m*x**2 - 2*m*x + 1 2*p**2*x**2 - 8*p**2*x + 6*p*x**2 - 4*p*x),x)*m**2*p**2 - 4*int((x**(m + p )*( - 3*x + 2)**p)/(3*m**2*x**2 - 2*m**2*x + 12*m*p*x**2 - 8*m*p*x + 3*m*x **2 - 2*m*x + 12*p**2*x**2 - 8*p**2*x + 6*p*x**2 - 4*p*x),x)*m**2*p - 32*i nt((x**(m + p)*( - 3*x + 2)**p)/(3*m**2*x**2 - 2*m**2*x + 12*m*p*x**2 - 8* m*p*x + 3*m*x**2 - 2*m*x + 12*p**2*x**2 - 8*p**2*x + 6*p*x**2 - 4*p*x),x)* m*p**3 - 12*int((x**(m + p)*( - 3*x + 2)**p)/(3*m**2*x**2 - 2*m**2*x + 12* m*p*x**2 - 8*m*p*x + 3*m*x**2 - 2*m*x + 12*p**2*x**2 - 8*p**2*x + 6*p*x**2 - 4*p*x),x)*m*p**2 - 16*int((x**(m + p)*( - 3*x + 2)**p)/(3*m**2*x**2 - 2 *m**2*x + 12*m*p*x**2 - 8*m*p*x + 3*m*x**2 - 2*m*x + 12*p**2*x**2 - 8*p**2 *x + 6*p*x**2 - 4*p*x),x)*p**4 - 8*int((x**(m + p)*( - 3*x + 2)**p)/(3*m** 2*x**2 - 2*m**2*x + 12*m*p*x**2 - 8*m*p*x + 3*m*x**2 - 2*m*x + 12*p**2*x** 2 - 8*p**2*x + 6*p*x**2 - 4*p*x),x)*p**3)/(3*(m**2 + 4*m*p + m + 4*p**2 + 2*p))