3.3030 \(\int (a+b (c x^n)^{2/n}) \, dx\)

Optimal. Leaf size=21 \[ a x+\frac{1}{3} b x \left (c x^n\right )^{2/n} \]

[Out]

a*x + (b*x*(c*x^n)^(2/n))/3

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Rubi [A]  time = 0.0056403, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {15, 30} \[ a x+\frac{1}{3} b x \left (c x^n\right )^{2/n} \]

Antiderivative was successfully verified.

[In]

Int[a + b*(c*x^n)^(2/n),x]

[Out]

a*x + (b*x*(c*x^n)^(2/n))/3

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \left (a+b \left (c x^n\right )^{2/n}\right ) \, dx &=a x+b \int \left (c x^n\right )^{2/n} \, dx\\ &=a x+\frac{\left (b \left (c x^n\right )^{2/n}\right ) \int x^2 \, dx}{x^2}\\ &=a x+\frac{1}{3} b x \left (c x^n\right )^{2/n}\\ \end{align*}

Mathematica [A]  time = 0.0020764, size = 21, normalized size = 1. \[ a x+\frac{1}{3} b x \left (c x^n\right )^{2/n} \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*(c*x^n)^(2/n),x]

[Out]

a*x + (b*x*(c*x^n)^(2/n))/3

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Maple [A]  time = 0.018, size = 23, normalized size = 1.1 \begin{align*} ax+{\frac{bx}{3}{{\rm e}^{2\,{\frac{\ln \left ( c{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{n}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} b c^{\frac{2}{n}} \int{\left (x^{n}\right )}^{\frac{2}{n}}\,{d x} + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.3099, size = 34, normalized size = 1.62 \begin{align*} \frac{1}{3} \, b c^{\frac{2}{n}} x^{3} + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b*c^(2/n)*x^3 + a*x

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Sympy [A]  time = 0.230187, size = 19, normalized size = 0.9 \begin{align*} a x + \frac{b c^{\frac{2}{n}} x \left (x^{n}\right )^{\frac{2}{n}}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x + b*c**(2/n)*x*(x**n)**(2/n)/3

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Giac [A]  time = 1.0945, size = 23, normalized size = 1.1 \begin{align*} \frac{1}{3} \, b c^{\frac{2}{n}} x^{3} + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b*c^(2/n)*x^3 + a*x