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

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 = 0.0280078, 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 \[ -\frac{2^{2 p+1} (-2 x-2)^{-p-1} \left (x^2+4 x+3\right )^{p+1} \, _2F_1\left (-p,p+1;p+2;\frac{x+3}{2}\right )}{p+1} \]

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))

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Rubi in Sympy [A]  time = 1.93508, size = 44, normalized size = 0.81 \[ - \frac{\left (- \frac{x}{2} - \frac{1}{2}\right )^{- p - 1} \left (x^{2} + 4 x + 3\right )^{p + 1}{{}_{2}F_{1}\left (\begin{matrix} - p, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{x}{2} + \frac{3}{2}} \right )}}{2 \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**2+4*x+3)**p,x)

[Out]

-(-x/2 - 1/2)**(-p - 1)*(x**2 + 4*x + 3)**(p + 1)*hyper((-p, p + 1), (p + 2,), x
/2 + 3/2)/(2*(p + 1))

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Mathematica [A]  time = 0.0292909, 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 = 0.068, size = 0, normalized size = 0. \[ \int \left ({x}^{2}+4\,x+3 \right ) ^{p}\, dx \]

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. \[ \int{\left (x^{2} + 4 \, x + 3\right )}^{p}\,{d x} \]

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. \[{\rm integral}\left ({\left (x^{2} + 4 \, x + 3\right )}^{p}, x\right ) \]

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. \[ \int \left (x^{2} + 4 x + 3\right )^{p}\, dx \]

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/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (x^{2} + 4 \, x + 3\right )}^{p}\,{d x} \]

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)