\(\int (a+a x)^m (c-c x)^m \, dx\) [1233]

Optimal result

Integrand size = 16, antiderivative size = 41 \[ \int (a+a x)^m (c-c x)^m \, dx=x (a+a x)^m (c-c x)^m \left (1-x^2\right )^{-m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m,\frac {3}{2},x^2\right ) \]

[Out]

x*(a*x+a)^m*(-c*x+c)^m*hypergeom([1/2, -m],[3/2],x^2)/((-x^2+1)^m)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {42, 252, 251} \[ \int (a+a x)^m (c-c x)^m \, dx=x \left (1-x^2\right )^{-m} (a x+a)^m (c-c x)^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m,\frac {3}{2},x^2\right ) \]

[In]

Int[(a + a*x)^m*(c - c*x)^m,x]

[Out]

(x*(a + a*x)^m*(c - c*x)^m*Hypergeometric2F1[1/2, -m, 3/2, x^2])/(1 - x^2)^m

Rule 42

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Dist[(a + b*x)^FracPart[m]*((c + d*x)^Frac
Part[m]/(a*c + b*d*x^2)^FracPart[m]), Int[(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]

Rule 251

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)],
x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p
] || GtQ[a, 0])

Rule 252

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^Fra
cPart[p]), Int[(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILt
Q[Simplify[1/n + p], 0] &&  !(IntegerQ[p] || GtQ[a, 0])

Rubi steps \begin{align*} \text {integral}& = \left ((a+a x)^m (c-c x)^m \left (a c-a c x^2\right )^{-m}\right ) \int \left (a c-a c x^2\right )^m \, dx \\ & = \left ((a+a x)^m (c-c x)^m \left (1-x^2\right )^{-m}\right ) \int \left (1-x^2\right )^m \, dx \\ & = x (a+a x)^m (c-c x)^m \left (1-x^2\right )^{-m} \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};x^2\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.29 \[ \int (a+a x)^m (c-c x)^m \, dx=\frac {2^m (-1+x) (1+x)^{-m} (a (1+x))^m (c-c x)^m \operatorname {Hypergeometric2F1}\left (-m,1+m,2+m,\frac {1}{2}-\frac {x}{2}\right )}{1+m} \]

[In]

Integrate[(a + a*x)^m*(c - c*x)^m,x]

[Out]

(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)

Maple [F]

\[\int \left (a x +a \right )^{m} \left (-c x +c \right )^{m}d x\]

[In]

int((a*x+a)^m*(-c*x+c)^m,x)

[Out]

int((a*x+a)^m*(-c*x+c)^m,x)

Fricas [F]

\[ \int (a+a x)^m (c-c x)^m \, dx=\int { {\left (a x + a\right )}^{m} {\left (-c x + c\right )}^{m} \,d x } \]

[In]

integrate((a*x+a)^m*(-c*x+c)^m,x, algorithm="fricas")

[Out]

integral((a*x + a)^m*(-c*x + c)^m, x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 4.73 (sec) , antiderivative size = 124, normalized size of antiderivative = 3.02 \[ \int (a+a x)^m (c-c x)^m \, dx=\frac {a^{m} c^{m} {G_{6, 6}^{5, 3}\left (\begin {matrix} - \frac {m}{2}, \frac {1}{2} - \frac {m}{2}, 1 & \frac {1}{2}, - m, \frac {1}{2} - m \\- m - \frac {1}{2}, - m, - \frac {m}{2}, \frac {1}{2} - m, \frac {1}{2} - \frac {m}{2} & 0 \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{x^{2}}} \right )} e^{- i \pi m}}{4 \pi \Gamma \left (- m\right )} - \frac {a^{m} c^{m} {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, 0, \frac {1}{2}, - \frac {m}{2} - \frac {1}{2}, - \frac {m}{2}, 1 & \\- \frac {m}{2} - \frac {1}{2}, - \frac {m}{2} & - \frac {1}{2}, 0, - m - \frac {1}{2}, 0 \end {matrix} \middle | {\frac {1}{x^{2}}} \right )}}{4 \pi \Gamma \left (- m\right )} \]

[In]

integrate((a*x+a)**m*(-c*x+c)**m,x)

[Out]

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))

Maxima [F]

\[ \int (a+a x)^m (c-c x)^m \, dx=\int { {\left (a x + a\right )}^{m} {\left (-c x + c\right )}^{m} \,d x } \]

[In]

integrate((a*x+a)^m*(-c*x+c)^m,x, algorithm="maxima")

[Out]

integrate((a*x + a)^m*(-c*x + c)^m, x)

Giac [F]

\[ \int (a+a x)^m (c-c x)^m \, dx=\int { {\left (a x + a\right )}^{m} {\left (-c x + c\right )}^{m} \,d x } \]

[In]

integrate((a*x+a)^m*(-c*x+c)^m,x, algorithm="giac")

[Out]

integrate((a*x + a)^m*(-c*x + c)^m, x)

Mupad [F(-1)]

Timed out. \[ \int (a+a x)^m (c-c x)^m \, dx=\int {\left (a+a\,x\right )}^m\,{\left (c-c\,x\right )}^m \,d x \]

[In]

int((a + a*x)^m*(c - c*x)^m,x)

[Out]

int((a + a*x)^m*(c - c*x)^m, x)