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 = 0.0116055, 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, Rules used = {42, 246, 245} \[ x \left (1-x^2\right )^{-m} (a x+a)^m (c-c x)^m \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};x^2\right ) \]

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

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 246

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^Fr
acPart[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])

Rule 245

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])

Rubi steps

\begin{align*} \int (a+a x)^m (c-c x)^m \, dx &=\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]  time = 0.0219677, size = 53, normalized size = 1.29 \[ \frac{2^m (x-1) (x+1)^{-m} (a (x+1))^m (c-c x)^m \, _2F_1\left (-m,m+1;m+2;\frac{1}{2}-\frac{x}{2}\right )}{m+1} \]

Antiderivative was successfully verified.

[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)

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Maple [F]  time = 0.1, size = 0, normalized size = 0. \begin{align*} \int \left ( ax+a \right ) ^{m} \left ( -cx+c \right ) ^{m}\, dx \end{align*}

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. \begin{align*} \int{\left (a x + a\right )}^{m}{\left (-c x + c\right )}^{m}\,{d x} \end{align*}

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. \begin{align*}{\rm integral}\left ({\left (a x + a\right )}^{m}{\left (-c x + c\right )}^{m}, x\right ) \end{align*}

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 [C]  time = 4.37577, size = 124, normalized size = 3.02 \begin{align*} \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 )} \end{align*}

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, 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))

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

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)