Optimal. Leaf size=58 \[ \frac{x^2 \left (a+b x^3\right ) \left (a^2+2 a b x^3+b^2 x^6\right )^p \, _2F_1\left (1,2 p+\frac{5}{3};\frac{5}{3};-\frac{b x^3}{a}\right )}{2 a} \]
[Out]
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Rubi [A] time = 0.0433727, antiderivative size = 60, normalized size of antiderivative = 1.03, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{1}{2} x^2 \left (\frac{b x^3}{a}+1\right )^{-2 p} \left (a^2+2 a b x^3+b^2 x^6\right )^p \, _2F_1\left (\frac{2}{3},-2 p;\frac{5}{3};-\frac{b x^3}{a}\right ) \]
Antiderivative was successfully verified.
[In] Int[x*(a^2 + 2*a*b*x^3 + b^2*x^6)^p,x]
[Out]
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Rubi in Sympy [A] time = 15.6771, size = 53, normalized size = 0.91 \[ \frac{x^{2} \left (1 + \frac{b x^{3}}{a}\right )^{- 2 p} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - 2 p, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(b**2*x**6+2*a*b*x**3+a**2)**p,x)
[Out]
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Mathematica [A] time = 0.0182643, size = 51, normalized size = 0.88 \[ \frac{1}{2} x^2 \left (\left (a+b x^3\right )^2\right )^p \left (\frac{b x^3}{a}+1\right )^{-2 p} \, _2F_1\left (\frac{2}{3},-2 p;\frac{5}{3};-\frac{b x^3}{a}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x*(a^2 + 2*a*b*x^3 + b^2*x^6)^p,x]
[Out]
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Maple [F] time = 0.041, size = 0, normalized size = 0. \[ \int x \left ({b}^{2}{x}^{6}+2\,ab{x}^{3}+{a}^{2} \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(b^2*x^6+2*a*b*x^3+a^2)^p,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^p*x,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p} x, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^p*x,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int x \left (\left (a + b x^{3}\right )^{2}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(b**2*x**6+2*a*b*x**3+a**2)**p,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^p*x,x, algorithm="giac")
[Out]