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

Optimal. Leaf size=38 \[ -3^{-p-1} 13^p (2-3 x) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{1}{13} (2-3 x)^2\right ) \]

[Out]

-(3^(-1 - p)*13^p*(2 - 3*x)*Hypergeometric2F1[1/2, -p, 3/2, (2 - 3*x)^2/13])

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Rubi [A]  time = 0.0317743, antiderivative size = 38, 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 \[ -3^{-p-1} 13^p (2-3 x) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{1}{13} (2-3 x)^2\right ) \]

Antiderivative was successfully verified.

[In]  Int[(3 + 4*x - 3*x^2)^p,x]

[Out]

-(3^(-1 - p)*13^p*(2 - 3*x)*Hypergeometric2F1[1/2, -p, 3/2, (2 - 3*x)^2/13])

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Rubi in Sympy [A]  time = 2.06834, size = 27, normalized size = 0.71 \[ - \frac{\left (\frac{13}{3}\right )^{p} \left (- 6 x + 4\right ){{}_{2}F_{1}\left (\begin{matrix} - p, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{\left (- 6 x + 4\right )^{2}}{52}} \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(13/3)**p*(-6*x + 4)*hyper((-p, 1/2), (3/2,), (-6*x + 4)**2/52)/6

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Mathematica [B]  time = 0.112104, size = 81, normalized size = 2.13 \[ -\frac{13^{p/2} \left (-3 x+\sqrt{13}+2\right ) \left (3 x+\sqrt{13}-2\right )^{-p} \left (-6 x^2+8 x+6\right )^p \, _2F_1\left (-p,p+1;p+2;\frac{-3 x+\sqrt{13}+2}{2 \sqrt{13}}\right )}{3 (p+1)} \]

Warning: Unable to verify antiderivative.

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

[Out]

-(13^(p/2)*(2 + Sqrt[13] - 3*x)*(6 + 8*x - 6*x^2)^p*Hypergeometric2F1[-p, 1 + p,
2 + p, (2 + Sqrt[13] - 3*x)/(2*Sqrt[13])])/(3*(1 + p)*(-2 + Sqrt[13] + 3*x)^p)

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Maple [F]  time = 0.157, size = 0, normalized size = 0. \[ \int \left ( -3\,{x}^{2}+4\,x+3 \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-3*x^2+4*x+3)^p,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((-3*x^2 + 4*x + 3)^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((-3*x^2 + 4*x + 3)^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-3*x**2+4*x+3)**p,x)

[Out]

Integral((-3*x**2 + 4*x + 3)**p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((-3*x^2 + 4*x + 3)^p, x)