Optimal. Leaf size=126 \[ \frac{a^2 x^3 \left (c x^n\right )^{-3/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+1}}{b^3 (p+1)}-\frac{2 a x^3 \left (c x^n\right )^{-3/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+2}}{b^3 (p+2)}+\frac{x^3 \left (c x^n\right )^{-3/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+3}}{b^3 (p+3)} \]
[Out]
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Rubi [A] time = 0.104915, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{a^2 x^3 \left (c x^n\right )^{-3/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+1}}{b^3 (p+1)}-\frac{2 a x^3 \left (c x^n\right )^{-3/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+2}}{b^3 (p+2)}+\frac{x^3 \left (c x^n\right )^{-3/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+3}}{b^3 (p+3)} \]
Antiderivative was successfully verified.
[In] Int[x^2*(a + b*(c*x^n)^n^(-1))^p,x]
[Out]
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Rubi in Sympy [A] time = 18.5395, size = 107, normalized size = 0.85 \[ \frac{a^{2} x^{3} \left (c x^{n}\right )^{- \frac{3}{n}} \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{p + 1}}{b^{3} \left (p + 1\right )} - \frac{2 a x^{3} \left (c x^{n}\right )^{- \frac{3}{n}} \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{p + 2}}{b^{3} \left (p + 2\right )} + \frac{x^{3} \left (c x^{n}\right )^{- \frac{3}{n}} \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{p + 3}}{b^{3} \left (p + 3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(a+b*(c*x**n)**(1/n))**p,x)
[Out]
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Mathematica [A] time = 0.355206, size = 207, normalized size = 1.64 \[ \frac{x^3 \left (c x^n\right )^{-3/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^p \left (\frac{b \left (c x^n\right )^{\frac{1}{n}}}{a}+1\right )^{-p} \left (2 a^3 \left (\left (\frac{b \left (c x^n\right )^{\frac{1}{n}}}{a}+1\right )^p-1\right )-2 a^2 b p \left (c x^n\right )^{\frac{1}{n}} \left (\frac{b \left (c x^n\right )^{\frac{1}{n}}}{a}+1\right )^p+b^3 \left (p^2+3 p+2\right ) \left (c x^n\right )^{3/n} \left (\frac{b \left (c x^n\right )^{\frac{1}{n}}}{a}+1\right )^p+a b^2 p (p+1) \left (c x^n\right )^{2/n} \left (\frac{b \left (c x^n\right )^{\frac{1}{n}}}{a}+1\right )^p\right )}{b^3 (p+1) (p+2) (p+3)} \]
Antiderivative was successfully verified.
[In] Integrate[x^2*(a + b*(c*x^n)^n^(-1))^p,x]
[Out]
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Maple [C] time = 140.953, size = 2258, normalized size = 17.9 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(a+b*(c*x^n)^(1/n))^p,x)
[Out]
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Maxima [A] time = 7.56487, size = 134, normalized size = 1.06 \[ \frac{{\left ({\left (p^{2} + 3 \, p + 2\right )} b^{3} c^{\frac{3}{n}} x^{3} +{\left (p^{2} + p\right )} a b^{2} c^{\frac{2}{n}} x^{2} - 2 \, a^{2} b c^{\left (\frac{1}{n}\right )} p x + 2 \, a^{3}\right )}{\left (b c^{\left (\frac{1}{n}\right )} x + a\right )}^{p} c^{-\frac{3}{n}}}{{\left (p^{3} + 6 \, p^{2} + 11 \, p + 6\right )} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((c*x^n)^(1/n)*b + a)^p*x^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.239885, size = 174, normalized size = 1.38 \[ -\frac{{\left (2 \, a^{2} b c^{\left (\frac{1}{n}\right )} p x -{\left (b^{3} p^{2} + 3 \, b^{3} p + 2 \, b^{3}\right )} c^{\frac{3}{n}} x^{3} -{\left (a b^{2} p^{2} + a b^{2} p\right )} c^{\frac{2}{n}} x^{2} - 2 \, a^{3}\right )}{\left (b c^{\left (\frac{1}{n}\right )} x + a\right )}^{p}}{{\left (b^{3} p^{3} + 6 \, b^{3} p^{2} + 11 \, b^{3} p + 6 \, b^{3}\right )} c^{\frac{3}{n}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((c*x^n)^(1/n)*b + a)^p*x^2,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int x^{2} \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*(a+b*(c*x**n)**(1/n))**p,x)
[Out]
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GIAC/XCAS [A] time = 23.5303, size = 381, normalized size = 3.02 \[ \frac{b^{3} p^{2} x^{3} e^{\left (p{\rm ln}\left (b x e^{\left (\frac{{\rm ln}\left (c\right )}{n}\right )} + a\right ) + \frac{3 \,{\rm ln}\left (c\right )}{n}\right )} + 3 \, b^{3} p x^{3} e^{\left (p{\rm ln}\left (b x e^{\left (\frac{{\rm ln}\left (c\right )}{n}\right )} + a\right ) + \frac{3 \,{\rm ln}\left (c\right )}{n}\right )} + a b^{2} p^{2} x^{2} e^{\left (p{\rm ln}\left (b x e^{\left (\frac{{\rm ln}\left (c\right )}{n}\right )} + a\right ) + \frac{2 \,{\rm ln}\left (c\right )}{n}\right )} + 2 \, b^{3} x^{3} e^{\left (p{\rm ln}\left (b x e^{\left (\frac{{\rm ln}\left (c\right )}{n}\right )} + a\right ) + \frac{3 \,{\rm ln}\left (c\right )}{n}\right )} + a b^{2} p x^{2} e^{\left (p{\rm ln}\left (b x e^{\left (\frac{{\rm ln}\left (c\right )}{n}\right )} + a\right ) + \frac{2 \,{\rm ln}\left (c\right )}{n}\right )} - 2 \, a^{2} b p x e^{\left (p{\rm ln}\left (b x e^{\left (\frac{{\rm ln}\left (c\right )}{n}\right )} + a\right ) + \frac{{\rm ln}\left (c\right )}{n}\right )} + 2 \, a^{3} e^{\left (p{\rm ln}\left (b x e^{\left (\frac{{\rm ln}\left (c\right )}{n}\right )} + a\right )\right )}}{b^{3} p^{3} e^{\left (\frac{3 \,{\rm ln}\left (c\right )}{n}\right )} + 6 \, b^{3} p^{2} e^{\left (\frac{3 \,{\rm ln}\left (c\right )}{n}\right )} + 11 \, b^{3} p e^{\left (\frac{3 \,{\rm ln}\left (c\right )}{n}\right )} + 6 \, b^{3} e^{\left (\frac{3 \,{\rm ln}\left (c\right )}{n}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((c*x^n)^(1/n)*b + a)^p*x^2,x, algorithm="giac")
[Out]