3.3021 \(\int \left (a+b \left (c x^n\right )^{2/n}\right )^3 \, dx\)

Optimal. Leaf size=62 \[ a^3 x+a^2 b x \left (c x^n\right )^{2/n}+\frac{3}{5} a b^2 x \left (c x^n\right )^{4/n}+\frac{1}{7} b^3 x \left (c x^n\right )^{6/n} \]

[Out]

a^3*x + a^2*b*x*(c*x^n)^(2/n) + (3*a*b^2*x*(c*x^n)^(4/n))/5 + (b^3*x*(c*x^n)^(6/
n))/7

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Rubi [A]  time = 0.0507829, antiderivative size = 62, 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 \[ a^3 x+a^2 b x \left (c x^n\right )^{2/n}+\frac{3}{5} a b^2 x \left (c x^n\right )^{4/n}+\frac{1}{7} b^3 x \left (c x^n\right )^{6/n} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*(c*x^n)^(2/n))^3,x]

[Out]

a^3*x + a^2*b*x*(c*x^n)^(2/n) + (3*a*b^2*x*(c*x^n)^(4/n))/5 + (b^3*x*(c*x^n)^(6/
n))/7

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ a^{2} b x \left (c x^{n}\right )^{\frac{2}{n}} + \frac{3 a b^{2} x \left (c x^{n}\right )^{\frac{4}{n}}}{5} + \frac{b^{3} x \left (c x^{n}\right )^{\frac{6}{n}}}{7} + x \left (c x^{n}\right )^{- \frac{1}{n}} \int ^{\left (c x^{n}\right )^{\frac{1}{n}}} a^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*(c*x**n)**(2/n))**3,x)

[Out]

a**2*b*x*(c*x**n)**(2/n) + 3*a*b**2*x*(c*x**n)**(4/n)/5 + b**3*x*(c*x**n)**(6/n)
/7 + x*(c*x**n)**(-1/n)*Integral(a**3, (x, (c*x**n)**(1/n)))

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Mathematica [A]  time = 0.213451, size = 62, normalized size = 1. \[ a^3 x+a^2 b x \left (c x^n\right )^{2/n}+\frac{3}{5} a b^2 x \left (c x^n\right )^{4/n}+\frac{1}{7} b^3 x \left (c x^n\right )^{6/n} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*(c*x^n)^(2/n))^3,x]

[Out]

a^3*x + a^2*b*x*(c*x^n)^(2/n) + (3*a*b^2*x*(c*x^n)^(4/n))/5 + (b^3*x*(c*x^n)^(6/
n))/7

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Maple [F]  time = 0.042, size = 0, normalized size = 0. \[ \int \left ( a+b \left ( c{x}^{n} \right ) ^{2\,{n}^{-1}} \right ) ^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*(c*x^n)^(2/n))^3,x)

[Out]

int((a+b*(c*x^n)^(2/n))^3,x)

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Maxima [A]  time = 1.4868, size = 70, normalized size = 1.13 \[ \frac{1}{7} \, b^{3} c^{\frac{6}{n}} x^{7} + \frac{3}{5} \, a b^{2} c^{\frac{4}{n}} x^{5} + a^{2} b c^{\frac{2}{n}} x^{3} + a^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((c*x^n)^(2/n)*b + a)^3,x, algorithm="maxima")

[Out]

1/7*b^3*c^(6/n)*x^7 + 3/5*a*b^2*c^(4/n)*x^5 + a^2*b*c^(2/n)*x^3 + a^3*x

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Fricas [A]  time = 0.234977, size = 70, normalized size = 1.13 \[ \frac{1}{7} \, b^{3} c^{\frac{6}{n}} x^{7} + \frac{3}{5} \, a b^{2} c^{\frac{4}{n}} x^{5} + a^{2} b c^{\frac{2}{n}} x^{3} + a^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((c*x^n)^(2/n)*b + a)^3,x, algorithm="fricas")

[Out]

1/7*b^3*c^(6/n)*x^7 + 3/5*a*b^2*c^(4/n)*x^5 + a^2*b*c^(2/n)*x^3 + a^3*x

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Sympy [A]  time = 2.32689, size = 63, normalized size = 1.02 \[ a^{3} x + a^{2} b c^{\frac{2}{n}} x \left (x^{n}\right )^{\frac{2}{n}} + \frac{3 a b^{2} c^{\frac{4}{n}} x \left (x^{n}\right )^{\frac{4}{n}}}{5} + \frac{b^{3} c^{\frac{6}{n}} x \left (x^{n}\right )^{\frac{6}{n}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*(c*x**n)**(2/n))**3,x)

[Out]

a**3*x + a**2*b*c**(2/n)*x*(x**n)**(2/n) + 3*a*b**2*c**(4/n)*x*(x**n)**(4/n)/5 +
b**3*c**(6/n)*x*(x**n)**(6/n)/7

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GIAC/XCAS [A]  time = 0.236137, size = 74, normalized size = 1.19 \[ \frac{1}{7} \, b^{3} x^{7} e^{\left (\frac{6 \,{\rm ln}\left (c\right )}{n}\right )} + \frac{3}{5} \, a b^{2} x^{5} e^{\left (\frac{4 \,{\rm ln}\left (c\right )}{n}\right )} + a^{2} b x^{3} e^{\left (\frac{2 \,{\rm ln}\left (c\right )}{n}\right )} + a^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((c*x^n)^(2/n)*b + a)^3,x, algorithm="giac")

[Out]

1/7*b^3*x^7*e^(6*ln(c)/n) + 3/5*a*b^2*x^5*e^(4*ln(c)/n) + a^2*b*x^3*e^(2*ln(c)/n
) + a^3*x