3.259 \(\int x^2 \left (a+b x^3+c x^6\right )^p \, dx\)

Optimal. Leaf size=130 \[ -\frac{2^{p+1} \left (-\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}}\right )^{-p-1} \left (a+b x^3+c x^6\right )^{p+1} \, _2F_1\left (-p,p+1;p+2;\frac{2 c x^3+b+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{3 (p+1) \sqrt{b^2-4 a c}} \]

[Out]

-(2^(1 + p)*(-((b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/Sqrt[b^2 - 4*a*c]))^(-1 - p)*(a
+ b*x^3 + c*x^6)^(1 + p)*Hypergeometric2F1[-p, 1 + p, 2 + p, (b + Sqrt[b^2 - 4*
a*c] + 2*c*x^3)/(2*Sqrt[b^2 - 4*a*c])])/(3*Sqrt[b^2 - 4*a*c]*(1 + p))

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Rubi [A]  time = 0.152087, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{2^{p+1} \left (-\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}}\right )^{-p-1} \left (a+b x^3+c x^6\right )^{p+1} \, _2F_1\left (-p,p+1;p+2;\frac{2 c x^3+b+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{3 (p+1) \sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(a + b*x^3 + c*x^6)^p,x]

[Out]

-(2^(1 + p)*(-((b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/Sqrt[b^2 - 4*a*c]))^(-1 - p)*(a
+ b*x^3 + c*x^6)^(1 + p)*Hypergeometric2F1[-p, 1 + p, 2 + p, (b + Sqrt[b^2 - 4*
a*c] + 2*c*x^3)/(2*Sqrt[b^2 - 4*a*c])])/(3*Sqrt[b^2 - 4*a*c]*(1 + p))

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Rubi in Sympy [A]  time = 11.9058, size = 112, normalized size = 0.86 \[ - \frac{\left (\frac{- \frac{b}{2} - c x^{3} + \frac{\sqrt{- 4 a c + b^{2}}}{2}}{\sqrt{- 4 a c + b^{2}}}\right )^{- p - 1} \left (a + b x^{3} + c x^{6}\right )^{p + 1}{{}_{2}F_{1}\left (\begin{matrix} - p, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{\frac{b}{2} + c x^{3} + \frac{\sqrt{- 4 a c + b^{2}}}{2}}{\sqrt{- 4 a c + b^{2}}}} \right )}}{3 \left (p + 1\right ) \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(c*x**6+b*x**3+a)**p,x)

[Out]

-((-b/2 - c*x**3 + sqrt(-4*a*c + b**2)/2)/sqrt(-4*a*c + b**2))**(-p - 1)*(a + b*
x**3 + c*x**6)**(p + 1)*hyper((-p, p + 1), (p + 2,), (b/2 + c*x**3 + sqrt(-4*a*c
+ b**2)/2)/sqrt(-4*a*c + b**2))/(3*(p + 1)*sqrt(-4*a*c + b**2))

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Mathematica [A]  time = 0.175943, size = 138, normalized size = 1.06 \[ \frac{2^{p-1} \left (-\sqrt{b^2-4 a c}+b+2 c x^3\right ) \left (\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}}\right )^{-p} \left (a+b x^3+c x^6\right )^p \, _2F_1\left (-p,p+1;p+2;\frac{-2 c x^3-b+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{3 c (p+1)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*(a + b*x^3 + c*x^6)^p,x]

[Out]

(2^(-1 + p)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)*(a + b*x^3 + c*x^6)^p*Hypergeometr
ic2F1[-p, 1 + p, 2 + p, (-b + Sqrt[b^2 - 4*a*c] - 2*c*x^3)/(2*Sqrt[b^2 - 4*a*c])
])/(3*c*(1 + p)*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/Sqrt[b^2 - 4*a*c])^p)

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Maple [F]  time = 0.033, size = 0, normalized size = 0. \[ \int{x}^{2} \left ( c{x}^{6}+b{x}^{3}+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(c*x^6+b*x^3+a)^p,x)

[Out]

int(x^2*(c*x^6+b*x^3+a)^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{6} + b x^{3} + a\right )}^{p} x^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)^p*x^2,x, algorithm="maxima")

[Out]

integrate((c*x^6 + b*x^3 + a)^p*x^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c x^{6} + b x^{3} + a\right )}^{p} x^{2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)^p*x^2,x, algorithm="fricas")

[Out]

integral((c*x^6 + b*x^3 + a)^p*x^2, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(c*x**6+b*x**3+a)**p,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{6} + b x^{3} + a\right )}^{p} x^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)^p*x^2,x, algorithm="giac")

[Out]

integrate((c*x^6 + b*x^3 + a)^p*x^2, x)