Integrand size = 19, antiderivative size = 67 \[ \int (a+b x)^m (a c-b c x)^m \, dx=\frac {2^m (a+b x)^{1+m} \left (1-\frac {b x}{a}\right )^{-m} (a c-b c x)^m \operatorname {Hypergeometric2F1}\left (-m,1+m,2+m,\frac {a+b x}{2 a}\right )}{b (1+m)} \] Output:
2^m*(b*x+a)^(1+m)*(-b*c*x+a*c)^m*hypergeom([-m, 1+m],[2+m],1/2*(b*x+a)/a)/ b/(1+m)/((1-b*x/a)^m)
Time = 0.04 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.07 \[ \int (a+b x)^m (a c-b c x)^m \, dx=-\frac {2^m (a-b x) (c (a-b x))^m (a+b x)^m \left (\frac {a+b x}{a}\right )^{-m} \operatorname {Hypergeometric2F1}\left (-m,1+m,2+m,\frac {a-b x}{2 a}\right )}{b (1+m)} \] Input:
Integrate[(a + b*x)^m*(a*c - b*c*x)^m,x]
Output:
-((2^m*(a - b*x)*(c*(a - b*x))^m*(a + b*x)^m*Hypergeometric2F1[-m, 1 + m, 2 + m, (a - b*x)/(2*a)])/(b*(1 + m)*((a + b*x)/a)^m))
Time = 0.16 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.85, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, 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.
\(\displaystyle \int (a+b x)^m (a c-b c x)^m \, dx\) |
\(\Big \downarrow \) 46 |
\(\displaystyle (a+b x)^m (a c-b c x)^m \left (a^2 c-b^2 c x^2\right )^{-m} \int \left (a^2 c-b^2 c x^2\right )^mdx\) |
\(\Big \downarrow \) 238 |
\(\displaystyle (a+b x)^m \left (1-\frac {b^2 x^2}{a^2}\right )^{-m} (a c-b c x)^m \int \left (1-\frac {b^2 x^2}{a^2}\right )^mdx\) |
\(\Big \downarrow \) 237 |
\(\displaystyle x (a+b x)^m \left (1-\frac {b^2 x^2}{a^2}\right )^{-m} (a c-b c x)^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m,\frac {3}{2},\frac {b^2 x^2}{a^2}\right )\) |
Input:
Int[(a + b*x)^m*(a*c - b*c*x)^m,x]
Output:
(x*(a + b*x)^m*(a*c - b*c*x)^m*Hypergeometric2F1[1/2, -m, 3/2, (b^2*x^2)/a ^2])/(1 - (b^2*x^2)/a^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]
\[\int \left (b x +a \right )^{m} \left (-b c x +a c \right )^{m}d x\]
Input:
int((b*x+a)^m*(-b*c*x+a*c)^m,x)
Output:
int((b*x+a)^m*(-b*c*x+a*c)^m,x)
\[ \int (a+b x)^m (a c-b c x)^m \, dx=\int { {\left (-b c x + a c\right )}^{m} {\left (b x + a\right )}^{m} \,d x } \] Input:
integrate((b*x+a)^m*(-b*c*x+a*c)^m,x, algorithm="fricas")
Output:
integral((-b*c*x + a*c)^m*(b*x + a)^m, x)
Result contains complex when optimal does not.
Time = 2.51 (sec) , antiderivative size = 146, normalized size of antiderivative = 2.18 \[ \int (a+b x)^m (a c-b c x)^m \, dx=\frac {a^{2 m + 1} c^{m} {G_{6, 6}^{5, 3}\left (\begin {matrix} - \frac {m}{2}, \frac {1}{2} - \frac {m}{2}, 1 & \frac {1}{2}, - m, \frac {1}{2} - m \\- m - \frac {1}{2}, - m, - \frac {m}{2}, \frac {1}{2} - m, \frac {1}{2} - \frac {m}{2} & 0 \end {matrix} \middle | {\frac {a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )} e^{- i \pi m}}{4 \pi b \Gamma \left (- m\right )} - \frac {a^{2 m + 1} c^{m} {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, 0, \frac {1}{2}, - \frac {m}{2} - \frac {1}{2}, - \frac {m}{2}, 1 & \\- \frac {m}{2} - \frac {1}{2}, - \frac {m}{2} & - \frac {1}{2}, 0, - m - \frac {1}{2}, 0 \end {matrix} \middle | {\frac {a^{2}}{b^{2} x^{2}}} \right )}}{4 \pi b \Gamma \left (- m\right )} \] Input:
integrate((b*x+a)**m*(-b*c*x+a*c)**m,x)
Output:
a**(2*m + 1)*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,)), a**2*exp_polar(-2*I*pi)/(b** 2*x**2))*exp(-I*pi*m)/(4*pi*b*gamma(-m)) - a**(2*m + 1)*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)), a**2/(b**2*x**2))/(4*pi*b*gamma(-m))
\[ \int (a+b x)^m (a c-b c x)^m \, dx=\int { {\left (-b c x + a c\right )}^{m} {\left (b x + a\right )}^{m} \,d x } \] Input:
integrate((b*x+a)^m*(-b*c*x+a*c)^m,x, algorithm="maxima")
Output:
integrate((-b*c*x + a*c)^m*(b*x + a)^m, x)
\[ \int (a+b x)^m (a c-b c x)^m \, dx=\int { {\left (-b c x + a c\right )}^{m} {\left (b x + a\right )}^{m} \,d x } \] Input:
integrate((b*x+a)^m*(-b*c*x+a*c)^m,x, algorithm="giac")
Output:
integrate((-b*c*x + a*c)^m*(b*x + a)^m, x)
Timed out. \[ \int (a+b x)^m (a c-b c x)^m \, dx=\int {\left (a\,c-b\,c\,x\right )}^m\,{\left (a+b\,x\right )}^m \,d x \] Input:
int((a*c - b*c*x)^m*(a + b*x)^m,x)
Output:
int((a*c - b*c*x)^m*(a + b*x)^m, x)
\[ \int (a+b x)^m (a c-b c x)^m \, dx=\frac {\left (b x +a \right )^{m} \left (-b c x +a c \right )^{m} x +4 \left (\int \frac {\left (b x +a \right )^{m} \left (-b c x +a c \right )^{m}}{-2 b^{2} m \,x^{2}-b^{2} x^{2}+2 a^{2} m +a^{2}}d x \right ) a^{2} m^{2}+2 \left (\int \frac {\left (b x +a \right )^{m} \left (-b c x +a c \right )^{m}}{-2 b^{2} m \,x^{2}-b^{2} x^{2}+2 a^{2} m +a^{2}}d x \right ) a^{2} m}{2 m +1} \] Input:
int((b*x+a)^m*(-b*c*x+a*c)^m,x)
Output:
((a + b*x)**m*(a*c - b*c*x)**m*x + 4*int(((a + b*x)**m*(a*c - b*c*x)**m)/( 2*a**2*m + a**2 - 2*b**2*m*x**2 - b**2*x**2),x)*a**2*m**2 + 2*int(((a + b* x)**m*(a*c - b*c*x)**m)/(2*a**2*m + a**2 - 2*b**2*m*x**2 - b**2*x**2),x)*a **2*m)/(2*m + 1)