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

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)

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Rubi [A]  time = 0.0128489, 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, Rules used = {125, 364} \[ \frac{18^n x^{m+1} \, _2F_1\left (\frac{m+1}{2},-n;\frac{m+3}{2};\frac{4 a^2 x^2}{9}\right )}{m+1} \]

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)

Rule 125

Int[((f_.)*(x_))^(p_.)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[(a*c + b*d*x^2)
^m*(f*x)^p, x] /; FreeQ[{a, b, c, d, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[m - n, 0] && GtQ[a, 0] && GtQ
[c, 0]

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int x^m (3-2 a x)^n (6+4 a x)^n \, dx &=\int x^m \left (18-8 a^2 x^2\right )^n \, dx\\ &=\frac{18^n x^{1+m} \, _2F_1\left (\frac{1+m}{2},-n;\frac{3+m}{2};\frac{4 a^2 x^2}{9}\right )}{1+m}\\ \end{align*}

Mathematica [A]  time = 0.0145827, size = 69, normalized size = 1.64 \[ \frac{x^{m+1} (54-36 a x)^n (4 a x+6)^n \left (18-8 a^2 x^2\right )^{-n} \, _2F_1\left (\frac{m+1}{2},-n;\frac{m+3}{2};\frac{4 a^2 x^2}{9}\right )}{m+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^(1 + m)*(54 - 36*a*x)^n*(6 + 4*a*x)^n*Hypergeometric2F1[(1 + m)/2, -n, (3 + m)/2, (4*a^2*x^2)/9])/((1 + m)*
(18 - 8*a^2*x^2)^n)

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Maple [F]  time = 0.141, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( -2\,ax+3 \right ) ^{n} \left ( 4\,ax+6 \right ) ^{n}\, dx \end{align*}

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)

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

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, algorithm="maxima")

[Out]

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

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

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, algorithm="fricas")

[Out]

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

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Sympy [C]  time = 154.564, size = 238, normalized size = 5.67 \begin{align*} - \frac{3 \cdot 18^{n} 4^{- \frac{m}{2}} \cdot 9^{\frac{m}{2}} a^{- m}{G_{6, 6}^{5, 3}\left (\begin{matrix} - \frac{m}{2} - \frac{n}{2}, - \frac{m}{2} - \frac{n}{2} + \frac{1}{2}, 1 & \frac{1}{2} - \frac{m}{2}, - \frac{m}{2} - n, - \frac{m}{2} - n + \frac{1}{2} \\- \frac{m}{2} - n - \frac{1}{2}, - \frac{m}{2} - n, - \frac{m}{2} - \frac{n}{2}, - \frac{m}{2} - n + \frac{1}{2}, - \frac{m}{2} - \frac{n}{2} + \frac{1}{2} & 0 \end{matrix} \middle |{\frac{9}{4 a^{2} x^{2}}} \right )} e^{i \pi n}}{8 \pi a \Gamma \left (- n\right )} + \frac{3 \cdot 18^{n} 4^{- \frac{m}{2}} \cdot 9^{\frac{m}{2}} a^{- m}{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{m}{2} - \frac{1}{2}, - \frac{m}{2}, \frac{1}{2} - \frac{m}{2}, - \frac{m}{2} - \frac{n}{2} - \frac{1}{2}, - \frac{m}{2} - \frac{n}{2}, 1 & \\- \frac{m}{2} - \frac{n}{2} - \frac{1}{2}, - \frac{m}{2} - \frac{n}{2} & - \frac{m}{2} - \frac{1}{2}, - \frac{m}{2}, - \frac{m}{2} - n - \frac{1}{2}, 0 \end{matrix} \middle |{\frac{9 e^{- 2 i \pi }}{4 a^{2} x^{2}}} \right )} e^{- i \pi m}}{8 \pi a \Gamma \left (- n\right )} \end{align*}

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]

-3*18**n*4**(-m/2)*9**(m/2)*a**(-m)*meijerg(((-m/2 - n/2, -m/2 - n/2 + 1/2, 1), (1/2 - m/2, -m/2 - n, -m/2 - n
+ 1/2)), ((-m/2 - n - 1/2, -m/2 - n, -m/2 - n/2, -m/2 - n + 1/2, -m/2 - n/2 + 1/2), (0,)), 9/(4*a**2*x**2))*e
xp(I*pi*n)/(8*pi*a*gamma(-n)) + 3*18**n*4**(-m/2)*9**(m/2)*a**(-m)*meijerg(((-m/2 - 1/2, -m/2, 1/2 - m/2, -m/2
- n/2 - 1/2, -m/2 - n/2, 1), ()), ((-m/2 - n/2 - 1/2, -m/2 - n/2), (-m/2 - 1/2, -m/2, -m/2 - n - 1/2, 0)), 9*
exp_polar(-2*I*pi)/(4*a**2*x**2))*exp(-I*pi*m)/(8*pi*a*gamma(-n))

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

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, algorithm="giac")

[Out]

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