Integrand size = 16, antiderivative size = 55 \[ \int (a+a x)^m (c-c x)^m \, dx=\frac {2^m (1-x)^{-m} (a+a x)^{1+m} (c-c x)^m \operatorname {Hypergeometric2F1}\left (-m,1+m,2+m,\frac {1+x}{2}\right )}{a (1+m)} \] Output:
2^m*(a*x+a)^(1+m)*(-c*x+c)^m*hypergeom([-m, 1+m],[2+m],1/2+1/2*x)/a/(1+m)/ ((1-x)^m)
Time = 0.07 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.96 \[ \int (a+a x)^m (c-c x)^m \, dx=\frac {2^m (-1+x) (1+x)^{-m} (a (1+x))^m (c-c x)^m \operatorname {Hypergeometric2F1}\left (-m,1+m,2+m,\frac {1}{2}-\frac {x}{2}\right )}{1+m} \] Input:
Integrate[(a + a*x)^m*(c - c*x)^m,x]
Output:
(2^m*(-1 + x)*(a*(1 + x))^m*(c - c*x)^m*Hypergeometric2F1[-m, 1 + m, 2 + m , 1/2 - x/2])/((1 + m)*(1 + x)^m)
Time = 0.15 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.75, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {46, 238, 237}
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.
\(\displaystyle \int (a x+a)^m (c-c x)^m \, dx\) |
\(\Big \downarrow \) 46 |
\(\displaystyle (a x+a)^m (c-c x)^m \left (a c-a c x^2\right )^{-m} \int \left (a c-a c x^2\right )^mdx\) |
\(\Big \downarrow \) 238 |
\(\displaystyle \left (1-x^2\right )^{-m} (a x+a)^m (c-c x)^m \int \left (1-x^2\right )^mdx\) |
\(\Big \downarrow \) 237 |
\(\displaystyle x \left (1-x^2\right )^{-m} (a x+a)^m (c-c x)^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m,\frac {3}{2},x^2\right )\) |
Input:
Int[(a + a*x)^m*(c - c*x)^m,x]
Output:
(x*(a + a*x)^m*(c - c*x)^m*Hypergeometric2F1[1/2, -m, 3/2, x^2])/(1 - x^2) ^m
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^FracPart[m]*((c + d*x)^FracPart[m]/(a*c + b*d*x^2)^FracPart[m]) I nt[(a*c + b*d*x^2)^m, x], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b*c + a*d, 0] && !IntegerQ[2*m]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[- p, 1/2, 1/2 + 1, (-b)*(x^2/a)], x] /; FreeQ[{a, b, p}, x] && !IntegerQ[2*p ] && GtQ[a, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^2) ^FracPart[p]/(1 + b*(x^2/a))^FracPart[p]) Int[(1 + b*(x^2/a))^p, x], x] / ; FreeQ[{a, b, p}, x] && !IntegerQ[2*p] && !GtQ[a, 0]
\[\int \left (a x +a \right )^{m} \left (-c x +c \right )^{m}d x\]
Input:
int((a*x+a)^m*(-c*x+c)^m,x)
Output:
int((a*x+a)^m*(-c*x+c)^m,x)
\[ \int (a+a x)^m (c-c x)^m \, dx=\int { {\left (a x + a\right )}^{m} {\left (-c x + c\right )}^{m} \,d x } \] Input:
integrate((a*x+a)^m*(-c*x+c)^m,x, algorithm="fricas")
Output:
integral((a*x + a)^m*(-c*x + c)^m, x)
Result contains complex when optimal does not.
Time = 1.96 (sec) , antiderivative size = 124, normalized size of antiderivative = 2.25 \[ \int (a+a x)^m (c-c x)^m \, dx=\frac {a^{m} c^{m} {G_{6, 6}^{5, 3}\left (\begin {matrix} - \frac {m}{2}, \frac {1}{2} - \frac {m}{2}, 1 & \frac {1}{2}, - m, \frac {1}{2} - m \\- m - \frac {1}{2}, - m, - \frac {m}{2}, \frac {1}{2} - m, \frac {1}{2} - \frac {m}{2} & 0 \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{x^{2}}} \right )} e^{- i \pi m}}{4 \pi \Gamma \left (- m\right )} - \frac {a^{m} c^{m} {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, 0, \frac {1}{2}, - \frac {m}{2} - \frac {1}{2}, - \frac {m}{2}, 1 & \\- \frac {m}{2} - \frac {1}{2}, - \frac {m}{2} & - \frac {1}{2}, 0, - m - \frac {1}{2}, 0 \end {matrix} \middle | {\frac {1}{x^{2}}} \right )}}{4 \pi \Gamma \left (- m\right )} \] Input:
integrate((a*x+a)**m*(-c*x+c)**m,x)
Output:
a**m*c**m*meijerg(((-m/2, 1/2 - m/2, 1), (1/2, -m, 1/2 - m)), ((-m - 1/2, -m, -m/2, 1/2 - m, 1/2 - m/2), (0,)), exp_polar(-2*I*pi)/x**2)*exp(-I*pi*m )/(4*pi*gamma(-m)) - a**m*c**m*meijerg(((-1/2, 0, 1/2, -m/2 - 1/2, -m/2, 1 ), ()), ((-m/2 - 1/2, -m/2), (-1/2, 0, -m - 1/2, 0)), x**(-2))/(4*pi*gamma (-m))
\[ \int (a+a x)^m (c-c x)^m \, dx=\int { {\left (a x + a\right )}^{m} {\left (-c x + c\right )}^{m} \,d x } \] Input:
integrate((a*x+a)^m*(-c*x+c)^m,x, algorithm="maxima")
Output:
integrate((a*x + a)^m*(-c*x + c)^m, x)
\[ \int (a+a x)^m (c-c x)^m \, dx=\int { {\left (a x + a\right )}^{m} {\left (-c x + c\right )}^{m} \,d x } \] Input:
integrate((a*x+a)^m*(-c*x+c)^m,x, algorithm="giac")
Output:
integrate((a*x + a)^m*(-c*x + c)^m, x)
Timed out. \[ \int (a+a x)^m (c-c x)^m \, dx=\int {\left (a+a\,x\right )}^m\,{\left (c-c\,x\right )}^m \,d x \] Input:
int((a + a*x)^m*(c - c*x)^m,x)
Output:
int((a + a*x)^m*(c - c*x)^m, x)
\[ \int (a+a x)^m (c-c x)^m \, dx=\frac {\left (a x +a \right )^{m} \left (-c x +c \right )^{m} x -4 \left (\int \frac {\left (a x +a \right )^{m} \left (-c x +c \right )^{m}}{2 m \,x^{2}+x^{2}-2 m -1}d x \right ) m^{2}-2 \left (\int \frac {\left (a x +a \right )^{m} \left (-c x +c \right )^{m}}{2 m \,x^{2}+x^{2}-2 m -1}d x \right ) m}{2 m +1} \] Input:
int((a*x+a)^m*(-c*x+c)^m,x)
Output:
((a*x + a)**m*( - c*x + c)**m*x - 4*int(((a*x + a)**m*( - c*x + c)**m)/(2* m*x**2 - 2*m + x**2 - 1),x)*m**2 - 2*int(((a*x + a)**m*( - c*x + c)**m)/(2 *m*x**2 - 2*m + x**2 - 1),x)*m)/(2*m + 1)