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 = 0.0469697, 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 \[ x (a+b x)^m \left (1-\frac{b^2 x^2}{a^2}\right )^{-m} (a c-b c x)^m \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};\frac{b^2 x^2}{a^2}\right ) \]

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

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Rubi in Sympy [A]  time = 22.6077, size = 48, normalized size = 0.84 \[ x \left (1 - \frac{b^{2} x^{2}}{a^{2}}\right )^{- m} \left (a + b x\right )^{m} \left (a c - b c x\right )^{m}{{}_{2}F_{1}\left (\begin{matrix} - m, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b^{2} x^{2}}{a^{2}}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0456993, size = 56, normalized size = 0.98 \[ x (a+b x)^m \left (1-\frac{b^2 x^2}{a^2}\right )^{-m} (c (a-b x))^m \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};\frac{b^2 x^2}{a^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x*(c*(a - b*x))^m*(a + b*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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Maple [F]  time = 0.142, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{m} \left ( -bcx+ac \right ) ^{m}\, dx \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*c*x + a*c)^m*(b*x + a)^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. \[{\rm integral}\left ({\left (-b c x + a c\right )}^{m}{\left (b x + a\right )}^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 17.9697, size = 146, normalized size = 2.56 \[ \frac{a a^{2 m} c^{m}{G_{6, 6}^{5, 3}\left (\begin{matrix} - \frac{m}{2}, - \frac{m}{2} + \frac{1}{2}, 1 & \frac{1}{2}, - m, - m + \frac{1}{2} \\- m - \frac{1}{2}, - m, - m + \frac{1}{2}, - \frac{m}{2}, - \frac{m}{2} + \frac{1}{2} & 0 \end{matrix} \middle |{\frac{a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )} e^{- i \pi m}}{4 \pi b \Gamma \left (- m\right )} - \frac{a a^{2 m} c^{m}{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, 0, \frac{1}{2}, - \frac{m}{2} - \frac{1}{2}, - \frac{m}{2}, 1 & \\- \frac{m}{2} - \frac{1}{2}, - \frac{m}{2} & - \frac{1}{2}, 0, - m - \frac{1}{2}, 0 \end{matrix} \middle |{\frac{a^{2}}{b^{2} x^{2}}} \right )}}{4 \pi b \Gamma \left (- m\right )} \]

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, -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,)), a**2*exp_polar(-2*I*pi)/(b**2*x**2))*e
xp(-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*gamma(-m))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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