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

Optimal. Leaf size=54 \[ -\frac{2^{2 p+1} (-2 x-2)^{-p-1} \left (x^2+4 x+3\right )^{p+1} \, _2F_1\left (-p,p+1;p+2;\frac{x+3}{2}\right )}{p+1} \]

[Out]

-((2^(1 + 2*p)*(-2 - 2*x)^(-1 - p)*(3 + 4*x + x^2)^(1 + p)*Hypergeometric2F1[-p, 1 + p, 2 + p, (3 + x)/2])/(1
+ p))

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Rubi [A]  time = 0.0102118, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {624} \[ -\frac{2^{2 p+1} (-2 x-2)^{-p-1} \left (x^2+4 x+3\right )^{p+1} \, _2F_1\left (-p,p+1;p+2;\frac{x+3}{2}\right )}{p+1} \]

Antiderivative was successfully verified.

[In]

Int[(3 + 4*x + x^2)^p,x]

[Out]

-((2^(1 + 2*p)*(-2 - 2*x)^(-1 - p)*(3 + 4*x + x^2)^(1 + p)*Hypergeometric2F1[-p, 1 + p, 2 + p, (3 + x)/2])/(1
+ p))

Rule 624

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, -Simp[((a + b*x + c*
x^2)^(p + 1)*Hypergeometric2F1[-p, p + 1, p + 2, (b + q + 2*c*x)/(2*q)])/(q*(p + 1)*((q - b - 2*c*x)/(2*q))^(p
+ 1)), x]] /; FreeQ[{a, b, c, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !IntegerQ[4*p]

Rubi steps

\begin{align*} \int \left (3+4 x+x^2\right )^p \, dx &=-\frac{2^{1+2 p} (-2-2 x)^{-1-p} \left (3+4 x+x^2\right )^{1+p} \, _2F_1\left (-p,1+p;2+p;\frac{3+x}{2}\right )}{1+p}\\ \end{align*}

Mathematica [A]  time = 0.0203232, size = 48, normalized size = 0.89 \[ \frac{2^p (x+1) (x+3)^{-p} \left (x^2+4 x+3\right )^p \, _2F_1\left (-p,p+1;p+2;\frac{1}{2} (-x-1)\right )}{p+1} \]

Antiderivative was successfully verified.

[In]

Integrate[(3 + 4*x + x^2)^p,x]

[Out]

(2^p*(1 + x)*(3 + 4*x + x^2)^p*Hypergeometric2F1[-p, 1 + p, 2 + p, (-1 - x)/2])/((1 + p)*(3 + x)^p)

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Maple [F]  time = 0.841, size = 0, normalized size = 0. \begin{align*} \int \left ({x}^{2}+4\,x+3 \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x^2 + 4*x + 3)^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((x^2 + 4*x + 3)^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((x**2 + 4*x + 3)**p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x^2 + 4*x + 3)^p, x)