3.140 \(\int (3+4 x-2 x^2)^p \, dx\)

Optimal. Leaf size=31 \[ -5^p (1-x) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{2}{5} (1-x)^2\right ) \]

[Out]

-(5^p*(1 - x)*Hypergeometric2F1[1/2, -p, 3/2, (2*(1 - x)^2)/5])

________________________________________________________________________________________

Rubi [A]  time = 0.0107076, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {619, 245} \[ -5^p (1-x) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{2}{5} (1-x)^2\right ) \]

Antiderivative was successfully verified.

[In]

Int[(3 + 4*x - 2*x^2)^p,x]

[Out]

-(5^p*(1 - x)*Hypergeometric2F1[1/2, -p, 3/2, (2*(1 - x)^2)/5])

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 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 \left (3+4 x-2 x^2\right )^p \, dx &=-\left (\frac{1}{4} 5^p \operatorname{Subst}\left (\int \left (1-\frac{x^2}{40}\right )^p \, dx,x,4-4 x\right )\right )\\ &=-5^p (1-x) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{2}{5} (1-x)^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0066819, size = 26, normalized size = 0.84 \[ 5^p (x-1) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{2}{5} (x-1)^2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(3 + 4*x - 2*x^2)^p,x]

[Out]

5^p*(-1 + x)*Hypergeometric2F1[1/2, -p, 3/2, (2*(-1 + x)^2)/5]

________________________________________________________________________________________

Maple [F]  time = 0.933, size = 0, normalized size = 0. \begin{align*} \int \left ( -2\,{x}^{2}+4\,x+3 \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((-2*x^2 + 4*x + 3)^p, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((-2*x^2 + 4*x + 3)^p, x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (- 2 x^{2} + 4 x + 3\right )^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x**2+4*x+3)**p,x)

[Out]

Integral((-2*x**2 + 4*x + 3)**p, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((-2*x^2 + 4*x + 3)^p, x)