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

Optimal. Leaf size=32 \[ 2^{p-1} (2 x+1) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{1}{2} (2 x+1)^2\right ) \]

[Out]

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

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Rubi [A]  time = 0.033132, antiderivative size = 32, 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 \[ 2^{p-1} (2 x+1) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{1}{2} (2 x+1)^2\right ) \]

Antiderivative was successfully verified.

[In]  Int[(3 + 4*x + 4*x^2)^p,x]

[Out]

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

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Rubi in Sympy [A]  time = 1.91681, size = 26, normalized size = 0.81 \[ \frac{2^{p} \left (8 x + 4\right ){{}_{2}F_{1}\left (\begin{matrix} - p, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{- \frac{\left (8 x + 4\right )^{2}}{32}} \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2**p*(8*x + 4)*hyper((-p, 1/2), (3/2,), -(8*x + 4)**2/32)/8

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Mathematica [C]  time = 0.145667, size = 94, normalized size = 2.94 \[ \frac{2^{\frac{3 p}{2}-1} \left (2 x-i \sqrt{2}+1\right ) \left (-2 i x+\sqrt{2}-i\right )^{-p} \left (4 x^2+4 x+3\right )^p \, _2F_1\left (-p,p+1;p+2;\frac{1}{4} \left (2 i \sqrt{2} x+i \sqrt{2}+2\right )\right )}{p+1} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.231, size = 0, normalized size = 0. \[ \int \left ( 4\,{x}^{2}+4\,x+3 \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((4*x^2+4*x+3)^p,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((4*x^2 + 4*x + 3)^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((4*x^2 + 4*x + 3)^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x**2+4*x+3)**p,x)

[Out]

Integral((4*x**2 + 4*x + 3)**p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((4*x^2 + 4*x + 3)^p, x)