3.980 \(\int x^m (3-2 a x)^n (6+4 a x)^n \, dx\)

Optimal. Leaf size=42 \[ \frac{18^n x^{m+1} \, _2F_1\left (\frac{m+1}{2},-n;\frac{m+3}{2};\frac{4 a^2 x^2}{9}\right )}{m+1} \]

[Out]

(18^n*x^(1 + m)*Hypergeometric2F1[(1 + m)/2, -n, (3 + m)/2, (4*a^2*x^2)/9])/(1 +
m)

_______________________________________________________________________________________

Rubi [A]  time = 0.0448511, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{18^n x^{m+1} \, _2F_1\left (\frac{m+1}{2},-n;\frac{m+3}{2};\frac{4 a^2 x^2}{9}\right )}{m+1} \]

Antiderivative was successfully verified.

[In]  Int[x^m*(3 - 2*a*x)^n*(6 + 4*a*x)^n,x]

[Out]

(18^n*x^(1 + m)*Hypergeometric2F1[(1 + m)/2, -n, (3 + m)/2, (4*a^2*x^2)/9])/(1 +
m)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 7.86994, size = 34, normalized size = 0.81 \[ \frac{18^{n} x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} - n, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{\frac{4 a^{2} x^{2}}{9}} \right )}}{m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*(-2*a*x+3)**n*(4*a*x+6)**n,x)

[Out]

18**n*x**(m + 1)*hyper((-n, m/2 + 1/2), (m/2 + 3/2,), 4*a**2*x**2/9)/(m + 1)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0629963, size = 73, normalized size = 1.74 \[ \frac{x^{m+1} (3-2 a x)^n (4 a x+6)^n \left (1-\frac{4 a^2 x^2}{9}\right )^{-n} \, _2F_1\left (\frac{m+1}{2},-n;\frac{m+1}{2}+1;\frac{4 a^2 x^2}{9}\right )}{m+1} \]

Antiderivative was successfully verified.

[In]  Integrate[x^m*(3 - 2*a*x)^n*(6 + 4*a*x)^n,x]

[Out]

(x^(1 + m)*(3 - 2*a*x)^n*(6 + 4*a*x)^n*Hypergeometric2F1[(1 + m)/2, -n, 1 + (1 +
m)/2, (4*a^2*x^2)/9])/((1 + m)*(1 - (4*a^2*x^2)/9)^n)

_______________________________________________________________________________________

Maple [F]  time = 0.221, size = 0, normalized size = 0. \[ \int{x}^{m} \left ( -2\,ax+3 \right ) ^{n} \left ( 4\,ax+6 \right ) ^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(-2*a*x+3)^n*(4*a*x+6)^n,x)

[Out]

int(x^m*(-2*a*x+3)^n*(4*a*x+6)^n,x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (4 \, a x + 6\right )}^{n}{\left (-2 \, a x + 3\right )}^{n} x^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*a*x + 6)^n*(-2*a*x + 3)^n*x^m,x, algorithm="maxima")

[Out]

integrate((4*a*x + 6)^n*(-2*a*x + 3)^n*x^m, x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (4 \, a x + 6\right )}^{n}{\left (-2 \, a x + 3\right )}^{n} x^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*a*x + 6)^n*(-2*a*x + 3)^n*x^m,x, algorithm="fricas")

[Out]

integral((4*a*x + 6)^n*(-2*a*x + 3)^n*x^m, x)

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**m*(-2*a*x+3)**n*(4*a*x+6)**n,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (4 \, a x + 6\right )}^{n}{\left (-2 \, a x + 3\right )}^{n} x^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*a*x + 6)^n*(-2*a*x + 3)^n*x^m,x, algorithm="giac")

[Out]

integrate((4*a*x + 6)^n*(-2*a*x + 3)^n*x^m, x)