3.1234 \(\int (a+b x)^m (a c-b c x)^m \, dx\)

Optimal. Leaf size=57 \[ x (a+b x)^m \left (1-\frac{b^2 x^2}{a^2}\right )^{-m} (a c-b c x)^m \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};\frac{b^2 x^2}{a^2}\right ) \]

[Out]

(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

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Rubi [A]  time = 0.0179656, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {42, 246, 245} \[ x (a+b x)^m \left (1-\frac{b^2 x^2}{a^2}\right )^{-m} (a c-b c x)^m \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};\frac{b^2 x^2}{a^2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^m*(a*c - b*c*x)^m,x]

[Out]

(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

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

Mathematica [A]  time = 0.0239115, size = 72, normalized size = 1.26 \[ -\frac{2^m (a-b x) (a+b x)^m \left (\frac{a+b x}{a}\right )^{-m} (c (a-b x))^m \, _2F_1\left (-m,m+1;m+2;\frac{a-b x}{2 a}\right )}{b (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((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))

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Maple [F]  time = 0.084, size = 0, normalized size = 0. \begin{align*} \int \left ( bx+a \right ) ^{m} \left ( -bcx+ac \right ) ^{m}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((-b*c*x + a*c)^m*(b*x + a)^m, x)

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Sympy [C]  time = 5.12157, size = 146, normalized size = 2.56 \begin{align*} \frac{a a^{2 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{a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )} e^{- i \pi m}}{4 \pi b \Gamma \left (- m\right )} - \frac{a a^{2 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{a^{2}}{b^{2} x^{2}}} \right )}}{4 \pi b \Gamma \left (- m\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*a**(2*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,)), a**2*exp_polar(-2*I*pi)/(b**2*x**2))*exp(-I*pi*m)/(4*pi*b*gamma(-m)) - a*a**(2*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)), a**2/(b**2*x**2))/(4*pi*b*ga
mma(-m))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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