Optimal. Leaf size=65 \[ a^3 x+\frac{3}{4} a^2 b x \left (c x^n\right )^{3/n}+\frac{3}{7} a b^2 x \left (c x^n\right )^{6/n}+\frac{1}{10} b^3 x \left (c x^n\right )^{9/n} \]
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Rubi [A] time = 0.0517521, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ a^3 x+\frac{3}{4} a^2 b x \left (c x^n\right )^{3/n}+\frac{3}{7} a b^2 x \left (c x^n\right )^{6/n}+\frac{1}{10} b^3 x \left (c x^n\right )^{9/n} \]
Antiderivative was successfully verified.
[In] Int[(a + b*(c*x^n)^(3/n))^3,x]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{3 a^{2} b x \left (c x^{n}\right )^{\frac{3}{n}}}{4} + \frac{3 a b^{2} x \left (c x^{n}\right )^{\frac{6}{n}}}{7} + \frac{b^{3} x \left (c x^{n}\right )^{\frac{9}{n}}}{10} + x \left (c x^{n}\right )^{- \frac{1}{n}} \int ^{\left (c x^{n}\right )^{\frac{1}{n}}} a^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b*(c*x**n)**(3/n))**3,x)
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Mathematica [A] time = 0.210039, size = 65, normalized size = 1. \[ a^3 x+\frac{3}{4} a^2 b x \left (c x^n\right )^{3/n}+\frac{3}{7} a b^2 x \left (c x^n\right )^{6/n}+\frac{1}{10} b^3 x \left (c x^n\right )^{9/n} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*(c*x^n)^(3/n))^3,x]
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Maple [F] time = 0.043, size = 0, normalized size = 0. \[ \int \left ( a+b \left ( c{x}^{n} \right ) ^{3\,{n}^{-1}} \right ) ^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b*(c*x^n)^(3/n))^3,x)
[Out]
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Maxima [A] time = 1.50232, size = 72, normalized size = 1.11 \[ \frac{1}{10} \, b^{3} c^{\frac{9}{n}} x^{10} + \frac{3}{7} \, a b^{2} c^{\frac{6}{n}} x^{7} + \frac{3}{4} \, a^{2} b c^{\frac{3}{n}} x^{4} + a^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((c*x^n)^(3/n)*b + a)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.249848, size = 72, normalized size = 1.11 \[ \frac{1}{10} \, b^{3} c^{\frac{9}{n}} x^{10} + \frac{3}{7} \, a b^{2} c^{\frac{6}{n}} x^{7} + \frac{3}{4} \, a^{2} b c^{\frac{3}{n}} x^{4} + a^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((c*x^n)^(3/n)*b + a)^3,x, algorithm="fricas")
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Sympy [A] time = 2.34324, size = 66, normalized size = 1.02 \[ a^{3} x + \frac{3 a^{2} b c^{\frac{3}{n}} x \left (x^{n}\right )^{\frac{3}{n}}}{4} + \frac{3 a b^{2} c^{\frac{6}{n}} x \left (x^{n}\right )^{\frac{6}{n}}}{7} + \frac{b^{3} c^{\frac{9}{n}} x \left (x^{n}\right )^{\frac{9}{n}}}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b*(c*x**n)**(3/n))**3,x)
[Out]
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GIAC/XCAS [A] time = 0.224421, size = 76, normalized size = 1.17 \[ \frac{1}{10} \, b^{3} x^{10} e^{\left (\frac{9 \,{\rm ln}\left (c\right )}{n}\right )} + \frac{3}{7} \, a b^{2} x^{7} e^{\left (\frac{6 \,{\rm ln}\left (c\right )}{n}\right )} + \frac{3}{4} \, a^{2} b x^{4} e^{\left (\frac{3 \,{\rm ln}\left (c\right )}{n}\right )} + a^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((c*x^n)^(3/n)*b + a)^3,x, algorithm="giac")
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