\(\int (3-6 x)^m (2+4 x)^m \, dx\) [1235]

Optimal result

Integrand size = 15, antiderivative size = 20 \[ \int (3-6 x)^m (2+4 x)^m \, dx=6^m x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m,\frac {3}{2},4 x^2\right ) \]

[Out]

6^m*x*hypergeom([1/2, -m],[3/2],4*x^2)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {41, 251} \[ \int (3-6 x)^m (2+4 x)^m \, dx=6^m x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m,\frac {3}{2},4 x^2\right ) \]

[In]

Int[(3 - 6*x)^m*(2 + 4*x)^m,x]

[Out]

6^m*x*Hypergeometric2F1[1/2, -m, 3/2, 4*x^2]

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

Rule 251

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*} \text {integral}& = \int \left (6-24 x^2\right )^m \, dx \\ & = 6^m x \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};4 x^2\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int (3-6 x)^m (2+4 x)^m \, dx=6^m x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m,\frac {3}{2},4 x^2\right ) \]

[In]

Integrate[(3 - 6*x)^m*(2 + 4*x)^m,x]

[Out]

6^m*x*Hypergeometric2F1[1/2, -m, 3/2, 4*x^2]

Maple [F]

\[\int \left (3-6 x \right )^{m} \left (2+4 x \right )^{m}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int (3-6 x)^m (2+4 x)^m \, dx=\int { {\left (4 \, x + 2\right )}^{m} {\left (-6 \, x + 3\right )}^{m} \,d x } \]

[In]

integrate((3-6*x)^m*(2+4*x)^m,x, algorithm="fricas")

[Out]

integral((4*x + 2)^m*(-6*x + 3)^m, x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.56 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.95 \[ \int (3-6 x)^m (2+4 x)^m \, dx=\frac {24^{m} \left (x + \frac {1}{2}\right )^{m + 1} \Gamma \left (m + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} - m, m + 1 \\ m + 2 \end {matrix}\middle | {\left (x + \frac {1}{2}\right ) e^{2 i \pi }} \right )}}{\Gamma \left (m + 2\right )} \]

[In]

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

[Out]

24**m*(x + 1/2)**(m + 1)*gamma(m + 1)*hyper((-m, m + 1), (m + 2,), (x + 1/2)*exp_polar(2*I*pi))/gamma(m + 2)

Maxima [F]

\[ \int (3-6 x)^m (2+4 x)^m \, dx=\int { {\left (4 \, x + 2\right )}^{m} {\left (-6 \, x + 3\right )}^{m} \,d x } \]

[In]

integrate((3-6*x)^m*(2+4*x)^m,x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int (3-6 x)^m (2+4 x)^m \, dx=\int { {\left (4 \, x + 2\right )}^{m} {\left (-6 \, x + 3\right )}^{m} \,d x } \]

[In]

integrate((3-6*x)^m*(2+4*x)^m,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int (3-6 x)^m (2+4 x)^m \, dx=\int {\left (4\,x+2\right )}^m\,{\left (3-6\,x\right )}^m \,d x \]

[In]

int((4*x + 2)^m*(3 - 6*x)^m,x)

[Out]

int((4*x + 2)^m*(3 - 6*x)^m, x)