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

Optimal. Leaf size=31 \[ -7^p (2-x) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{1}{7} (2-x)^2\right ) \]

[Out]

-(7^p*(2 - x)*Hypergeometric2F1[1/2, -p, 3/2, (2 - x)^2/7])

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Rubi [A]  time = 0.0279102, 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 \[ -7^p (2-x) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{1}{7} (2-x)^2\right ) \]

Antiderivative was successfully verified.

[In]  Int[(3 + 4*x - x^2)^p,x]

[Out]

-(7^p*(2 - x)*Hypergeometric2F1[1/2, -p, 3/2, (2 - x)^2/7])

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Rubi in Sympy [A]  time = 1.95643, size = 26, normalized size = 0.84 \[ - \frac{7^{p} \left (- 2 x + 4\right ){{}_{2}F_{1}\left (\begin{matrix} - p, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{\left (- 2 x + 4\right )^{2}}{28}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7**p*(-2*x + 4)*hyper((-p, 1/2), (3/2,), (-2*x + 4)**2/28)/2

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Mathematica [B]  time = 0.0801263, size = 83, normalized size = 2.68 \[ -\frac{\left (-x+\sqrt{7}+2\right ) \left (-x^2+4 x+3\right )^p \left (\frac{x-\sqrt{7}-2}{2 \sqrt{7}}+1\right )^{-p} \, _2F_1\left (-p,p+1;p+2;-\frac{x-\sqrt{7}-2}{2 \sqrt{7}}\right )}{p+1} \]

Antiderivative was successfully verified.

[In]  Integrate[(3 + 4*x - x^2)^p,x]

[Out]

-(((2 + Sqrt[7] - x)*(3 + 4*x - x^2)^p*Hypergeometric2F1[-p, 1 + p, 2 + p, -(-2
- Sqrt[7] + x)/(2*Sqrt[7])])/((1 + p)*(1 + (-2 - Sqrt[7] + x)/(2*Sqrt[7]))^p))

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Maple [F]  time = 0.151, 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)