3.3022 \(\int \left (a+b \left (c x^n\right )^{2/n}\right )^2 \, dx\)

Optimal. Leaf size=43 \[ a^2 x+\frac{2}{3} a b x \left (c x^n\right )^{2/n}+\frac{1}{5} b^2 x \left (c x^n\right )^{4/n} \]

[Out]

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

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Rubi [A]  time = 0.0352567, antiderivative size = 43, 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 \[ a^2 x+\frac{2}{3} a b x \left (c x^n\right )^{2/n}+\frac{1}{5} b^2 x \left (c x^n\right )^{4/n} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*(c*x^n)^(2/n))^2,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a*b*x*(c*x**n)**(2/n)/3 + b**2*x*(c*x**n)**(4/n)/5 + x*(c*x**n)**(-1/n)*Integr
al(a**2, (x, (c*x**n)**(1/n)))

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Mathematica [A]  time = 0.12235, size = 43, normalized size = 1. \[ a^2 x+\frac{2}{3} a b x \left (c x^n\right )^{2/n}+\frac{1}{5} b^2 x \left (c x^n\right )^{4/n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.04, size = 0, normalized size = 0. \[ \int \left ( a+b \left ( c{x}^{n} \right ) ^{2\,{n}^{-1}} \right ) ^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*(c*x^n)^(2/n))^2,x)

[Out]

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

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Maxima [A]  time = 1.4317, size = 47, normalized size = 1.09 \[ \frac{1}{5} \, b^{2} c^{\frac{4}{n}} x^{5} + \frac{2}{3} \, a b c^{\frac{2}{n}} x^{3} + a^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.226743, size = 47, normalized size = 1.09 \[ \frac{1}{5} \, b^{2} c^{\frac{4}{n}} x^{5} + \frac{2}{3} \, a b c^{\frac{2}{n}} x^{3} + a^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.24234, size = 42, normalized size = 0.98 \[ a^{2} x + \frac{2 a b c^{\frac{2}{n}} x \left (x^{n}\right )^{\frac{2}{n}}}{3} + \frac{b^{2} c^{\frac{4}{n}} x \left (x^{n}\right )^{\frac{4}{n}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.224011, size = 50, normalized size = 1.16 \[ \frac{1}{5} \, b^{2} x^{5} e^{\left (\frac{4 \,{\rm ln}\left (c\right )}{n}\right )} + \frac{2}{3} \, a b x^{3} e^{\left (\frac{2 \,{\rm ln}\left (c\right )}{n}\right )} + a^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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