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]
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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]
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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]
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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]
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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)
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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]
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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]
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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]
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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]