Integrand size = 21, antiderivative size = 42 \[ \int x^m \left (1-\frac {a x}{2}\right )^n (2+a x)^n \, dx=\frac {2^n x^{1+m} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},-n,\frac {3+m}{2},\frac {a^2 x^2}{4}\right )}{1+m} \] Output:
2^n*x^(1+m)*hypergeom([-n, 1/2+1/2*m],[3/2+1/2*m],1/4*a^2*x^2)/(1+m)
Time = 0.08 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00 \[ \int x^m \left (1-\frac {a x}{2}\right )^n (2+a x)^n \, dx=\frac {2^n x^{1+m} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},-n,\frac {3+m}{2},\frac {a^2 x^2}{4}\right )}{1+m} \] Input:
Integrate[x^m*(1 - (a*x)/2)^n*(2 + a*x)^n,x]
Output:
(2^n*x^(1 + m)*Hypergeometric2F1[(1 + m)/2, -n, (3 + m)/2, (a^2*x^2)/4])/( 1 + m)
Time = 0.15 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {135, 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.
\(\displaystyle \int x^m \left (1-\frac {a x}{2}\right )^n (a x+2)^n \, dx\) |
\(\Big \downarrow \) 135 |
\(\displaystyle \int x^m \left (2-\frac {a^2 x^2}{2}\right )^ndx\) |
\(\Big \downarrow \) 278 |
\(\displaystyle \frac {2^n x^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {m+1}{2},-n,\frac {m+3}{2},\frac {a^2 x^2}{4}\right )}{m+1}\) |
Input:
Int[x^m*(1 - (a*x)/2)^n*(2 + a*x)^n,x]
Output:
(2^n*x^(1 + m)*Hypergeometric2F1[(1 + m)/2, -n, (3 + m)/2, (a^2*x^2)/4])/( 1 + m)
Int[((f_.)*(x_))^(p_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_] :> Int[(a*c + b*d*x^2)^m*(f*x)^p, x] /; FreeQ[{a, b, c, d, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && GtQ[a, 0] && GtQ[c, 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])
\[\int x^{m} \left (1-\frac {a x}{2}\right )^{n} \left (a x +2\right )^{n}d x\]
Input:
int(x^m*(1-1/2*a*x)^n*(a*x+2)^n,x)
Output:
int(x^m*(1-1/2*a*x)^n*(a*x+2)^n,x)
\[ \int x^m \left (1-\frac {a x}{2}\right )^n (2+a x)^n \, dx=\int { {\left (a x + 2\right )}^{n} {\left (-\frac {1}{2} \, a x + 1\right )}^{n} x^{m} \,d x } \] Input:
integrate(x^m*(1-1/2*a*x)^n*(a*x+2)^n,x, algorithm="fricas")
Output:
integral((a*x + 2)^n*(-1/2*a*x + 1)^n*x^m, x)
Timed out. \[ \int x^m \left (1-\frac {a x}{2}\right )^n (2+a x)^n \, dx=\text {Timed out} \] Input:
integrate(x**m*(1-1/2*a*x)**n*(a*x+2)**n,x)
Output:
Timed out
\[ \int x^m \left (1-\frac {a x}{2}\right )^n (2+a x)^n \, dx=\int { {\left (a x + 2\right )}^{n} {\left (-\frac {1}{2} \, a x + 1\right )}^{n} x^{m} \,d x } \] Input:
integrate(x^m*(1-1/2*a*x)^n*(a*x+2)^n,x, algorithm="maxima")
Output:
integrate((a*x + 2)^n*(-1/2*a*x + 1)^n*x^m, x)
\[ \int x^m \left (1-\frac {a x}{2}\right )^n (2+a x)^n \, dx=\int { {\left (a x + 2\right )}^{n} {\left (-\frac {1}{2} \, a x + 1\right )}^{n} x^{m} \,d x } \] Input:
integrate(x^m*(1-1/2*a*x)^n*(a*x+2)^n,x, algorithm="giac")
Output:
integrate((a*x + 2)^n*(-1/2*a*x + 1)^n*x^m, x)
Timed out. \[ \int x^m \left (1-\frac {a x}{2}\right )^n (2+a x)^n \, dx=\int x^m\,{\left (a\,x+2\right )}^n\,{\left (1-\frac {a\,x}{2}\right )}^n \,d x \] Input:
int(x^m*(a*x + 2)^n*(1 - (a*x)/2)^n,x)
Output:
int(x^m*(a*x + 2)^n*(1 - (a*x)/2)^n, x)
\[ \int x^m \left (1-\frac {a x}{2}\right )^n (2+a x)^n \, dx=\frac {x^{m} \left (a x +2\right )^{n} \left (-a x +2\right )^{n} x -8 \left (\int \frac {x^{m} \left (a x +2\right )^{n} \left (-a x +2\right )^{n}}{a^{2} m \,x^{2}+2 a^{2} n \,x^{2}+a^{2} x^{2}-4 m -8 n -4}d x \right ) m n -16 \left (\int \frac {x^{m} \left (a x +2\right )^{n} \left (-a x +2\right )^{n}}{a^{2} m \,x^{2}+2 a^{2} n \,x^{2}+a^{2} x^{2}-4 m -8 n -4}d x \right ) n^{2}-8 \left (\int \frac {x^{m} \left (a x +2\right )^{n} \left (-a x +2\right )^{n}}{a^{2} m \,x^{2}+2 a^{2} n \,x^{2}+a^{2} x^{2}-4 m -8 n -4}d x \right ) n}{2^{n} \left (m +2 n +1\right )} \] Input:
int(x^m*(1-1/2*a*x)^n*(a*x+2)^n,x)
Output:
(x**m*(a*x + 2)**n*( - a*x + 2)**n*x - 8*int((x**m*(a*x + 2)**n*( - a*x + 2)**n)/(a**2*m*x**2 + 2*a**2*n*x**2 + a**2*x**2 - 4*m - 8*n - 4),x)*m*n - 16*int((x**m*(a*x + 2)**n*( - a*x + 2)**n)/(a**2*m*x**2 + 2*a**2*n*x**2 + a**2*x**2 - 4*m - 8*n - 4),x)*n**2 - 8*int((x**m*(a*x + 2)**n*( - a*x + 2) **n)/(a**2*m*x**2 + 2*a**2*n*x**2 + a**2*x**2 - 4*m - 8*n - 4),x)*n)/(2**n *(m + 2*n + 1))