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

Optimal. Leaf size=31 \[ -5^p (1-x) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{2}{5} (1-x)^2\right ) \]

[Out]

-(5^p*(1 - x)*Hypergeometric2F1[1/2, -p, 3/2, (2*(1 - x)^2)/5])

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Rubi [A]  time = 0.0264776, 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 \[ -5^p (1-x) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{2}{5} (1-x)^2\right ) \]

Antiderivative was successfully verified.

[In]  Int[(3 + 4*x - 2*x^2)^p,x]

[Out]

-(5^p*(1 - x)*Hypergeometric2F1[1/2, -p, 3/2, (2*(1 - x)^2)/5])

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Rubi in Sympy [A]  time = 2.03983, size = 26, normalized size = 0.84 \[ - \frac{5^{p} \left (- 4 x + 4\right ){{}_{2}F_{1}\left (\begin{matrix} - p, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{\left (- 4 x + 4\right )^{2}}{40}} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5**p*(-4*x + 4)*hyper((-p, 1/2), (3/2,), (-4*x + 4)**2/40)/4

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Mathematica [B]  time = 0.172804, size = 86, normalized size = 2.77 \[ -\frac{2^{\frac{3 p}{2}-1} 5^{p/2} \left (-2 x+\sqrt{10}+2\right ) \left (2 x+\sqrt{10}-2\right )^{-p} \left (-2 x^2+4 x+3\right )^p \, _2F_1\left (-p,p+1;p+2;-\frac{x}{\sqrt{10}}+\frac{1}{\sqrt{10}}+\frac{1}{2}\right )}{p+1} \]

Warning: Unable to verify antiderivative.

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

[Out]

-((2^(-1 + (3*p)/2)*5^(p/2)*(2 + Sqrt[10] - 2*x)*(3 + 4*x - 2*x^2)^p*Hypergeomet
ric2F1[-p, 1 + p, 2 + p, 1/2 + 1/Sqrt[10] - x/Sqrt[10]])/((1 + p)*(-2 + Sqrt[10]
+ 2*x)^p))

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Maple [F]  time = 0.154, 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)