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

Optimal. Leaf size=21 \[ (x+1) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-2 (x+1)^2\right ) \]

[Out]

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

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Rubi [A]  time = 0.0227508, antiderivative size = 21, 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 \[ (x+1) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-2 (x+1)^2\right ) \]

Antiderivative was successfully verified.

[In]  Int[(3 + 4*x + 2*x^2)^p,x]

[Out]

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

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Rubi in Sympy [A]  time = 1.72968, size = 17, normalized size = 0.81 \[ \left (x + 1\right ){{}_{2}F_{1}\left (\begin{matrix} - p, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{- 2 \left (x + 1\right )^{2}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.100943, size = 92, normalized size = 4.38 \[ \frac{2^{\frac{3 p}{2}-1} \left (2 x-i \sqrt{2}+2\right ) \left (-2 i x+\sqrt{2}-2 i\right )^{-p} \left (2 x^2+4 x+3\right )^p \, _2F_1\left (-p,p+1;p+2;\frac{2 i x+\sqrt{2}+2 i}{2 \sqrt{2}}\right )}{p+1} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.228, size = 0, normalized size = 0. \[ \int \left ( 2\,{x}^{2}+4\,x+3 \right ) ^{p}\, dx \]

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)

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

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)

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

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)

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

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)

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

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)