3.1233 \(\int (a+a x)^m (c-c x)^m \, dx\)

Optimal. Leaf size=41 \[ x \left (1-x^2\right )^{-m} (a x+a)^m (c-c x)^m \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};x^2\right ) \]

[Out]

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

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Rubi [A]  time = 0.0318149, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ x \left (1-x^2\right )^{-m} (a x+a)^m (c-c x)^m \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};x^2\right ) \]

Antiderivative was successfully verified.

[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

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Rubi in Sympy [A]  time = 5.25611, size = 31, normalized size = 0.76 \[ x \left (- x^{2} + 1\right )^{- m} \left (a x + a\right )^{m} \left (- c x + c\right )^{m}{{}_{2}F_{1}\left (\begin{matrix} - m, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{x^{2}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*x+a)**m*(-c*x+c)**m,x)

[Out]

x*(-x**2 + 1)**(-m)*(a*x + a)**m*(-c*x + c)**m*hyper((-m, 1/2), (3/2,), x**2)

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Mathematica [A]  time = 0.0285089, size = 41, normalized size = 1. \[ x \left (1-x^2\right )^{-m} (a (x+1))^m (c-c x)^m \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};x^2\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + a*x)^m*(c - c*x)^m,x]

[Out]

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

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Maple [F]  time = 0.148, size = 0, normalized size = 0. \[ \int \left ( ax+a \right ) ^{m} \left ( -cx+c \right ) ^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*x+a)^m*(-c*x+c)^m,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (a x + a\right )}^{m}{\left (-c x + c\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (a x + a\right )}^{m}{\left (-c x + c\right )}^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((a*x + a)^m*(-c*x + c)^m, x)

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Sympy [A]  time = 15.206, size = 124, normalized size = 3.02 \[ \frac{a^{m} c^{m}{G_{6, 6}^{5, 3}\left (\begin{matrix} - \frac{m}{2}, - \frac{m}{2} + \frac{1}{2}, 1 & \frac{1}{2}, - m, - m + \frac{1}{2} \\- m - \frac{1}{2}, - m, - m + \frac{1}{2}, - \frac{m}{2}, - \frac{m}{2} + \frac{1}{2} & 0 \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{x^{2}}} \right )} e^{- i \pi m}}{4 \pi \Gamma \left (- m\right )} - \frac{a^{m} c^{m}{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, 0, \frac{1}{2}, - \frac{m}{2} - \frac{1}{2}, - \frac{m}{2}, 1 & \\- \frac{m}{2} - \frac{1}{2}, - \frac{m}{2} & - \frac{1}{2}, 0, - m - \frac{1}{2}, 0 \end{matrix} \middle |{\frac{1}{x^{2}}} \right )}}{4 \pi \Gamma \left (- m\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x+a)**m*(-c*x+c)**m,x)

[Out]

a**m*c**m*meijerg(((-m/2, -m/2 + 1/2, 1), (1/2, -m, -m + 1/2)), ((-m - 1/2, -m,
-m + 1/2, -m/2, -m/2 + 1/2), (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))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (a x + a\right )}^{m}{\left (-c x + c\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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