3.2998 \(\int \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^3 \, dx\)

Optimal. Leaf size=34 \[ \frac{x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^4}{4 b} \]

[Out]

(x*(a + b*(c*x^n)^n^(-1))^4)/(4*b*(c*x^n)^n^(-1))

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Rubi [A]  time = 0.0216235, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^4}{4 b} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*(c*x^n)^n^(-1))^3,x]

[Out]

(x*(a + b*(c*x^n)^n^(-1))^4)/(4*b*(c*x^n)^n^(-1))

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Rubi in Sympy [A]  time = 2.29812, size = 26, normalized size = 0.76 \[ \frac{x \left (c x^{n}\right )^{- \frac{1}{n}} \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{4}}{4 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*(c*x**n)**(1/n))**3,x)

[Out]

x*(c*x**n)**(-1/n)*(a + b*(c*x**n)**(1/n))**4/(4*b)

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Mathematica [A]  time = 0.0171997, size = 34, normalized size = 1. \[ \frac{x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^4}{4 b} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*(c*x^n)^n^(-1))^3,x]

[Out]

(x*(a + b*(c*x^n)^n^(-1))^4)/(4*b*(c*x^n)^n^(-1))

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Maple [F]  time = 0.038, size = 0, normalized size = 0. \[ \int \left ( a+b\sqrt [n]{c{x}^{n}} \right ) ^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*(c*x^n)^(1/n))^3,x)

[Out]

int((a+b*(c*x^n)^(1/n))^3,x)

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Maxima [A]  time = 1.45539, size = 68, normalized size = 2. \[ \frac{1}{4} \, b^{3} c^{\frac{3}{n}} x^{4} + a b^{2} c^{\frac{2}{n}} x^{3} + \frac{3}{2} \, a^{2} b c^{\left (\frac{1}{n}\right )} x^{2} + a^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((c*x^n)^(1/n)*b + a)^3,x, algorithm="maxima")

[Out]

1/4*b^3*c^(3/n)*x^4 + a*b^2*c^(2/n)*x^3 + 3/2*a^2*b*c^(1/n)*x^2 + a^3*x

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Fricas [A]  time = 0.252329, size = 68, normalized size = 2. \[ \frac{1}{4} \, b^{3} c^{\frac{3}{n}} x^{4} + a b^{2} c^{\frac{2}{n}} x^{3} + \frac{3}{2} \, a^{2} b c^{\left (\frac{1}{n}\right )} x^{2} + a^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((c*x^n)^(1/n)*b + a)^3,x, algorithm="fricas")

[Out]

1/4*b^3*c^(3/n)*x^4 + a*b^2*c^(2/n)*x^3 + 3/2*a^2*b*c^(1/n)*x^2 + a^3*x

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Sympy [A]  time = 2.28096, size = 63, normalized size = 1.85 \[ a^{3} x + \frac{3 a^{2} b c^{\frac{1}{n}} x \left (x^{n}\right )^{\frac{1}{n}}}{2} + a b^{2} c^{\frac{2}{n}} x \left (x^{n}\right )^{\frac{2}{n}} + \frac{b^{3} c^{\frac{3}{n}} x \left (x^{n}\right )^{\frac{3}{n}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*(c*x**n)**(1/n))**3,x)

[Out]

a**3*x + 3*a**2*b*c**(1/n)*x*(x**n)**(1/n)/2 + a*b**2*c**(2/n)*x*(x**n)**(2/n) +
b**3*c**(3/n)*x*(x**n)**(3/n)/4

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GIAC/XCAS [A]  time = 0.230625, size = 73, normalized size = 2.15 \[ \frac{1}{4} \, b^{3} x^{4} e^{\left (\frac{3 \,{\rm ln}\left (c\right )}{n}\right )} + a b^{2} x^{3} e^{\left (\frac{2 \,{\rm ln}\left (c\right )}{n}\right )} + \frac{3}{2} \, a^{2} b x^{2} e^{\left (\frac{{\rm ln}\left (c\right )}{n}\right )} + a^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((c*x^n)^(1/n)*b + a)^3,x, algorithm="giac")

[Out]

1/4*b^3*x^4*e^(3*ln(c)/n) + a*b^2*x^3*e^(2*ln(c)/n) + 3/2*a^2*b*x^2*e^(ln(c)/n)
+ a^3*x