3.3003 \(\int (a+b (c x^n)^{\frac{1}{n}}) \, dx\)

Optimal. Leaf size=19 \[ a x+\frac{1}{2} b x \left (c x^n\right )^{\frac{1}{n}} \]

[Out]

a*x + (b*x*(c*x^n)^n^(-1))/2

________________________________________________________________________________________

Rubi [A]  time = 0.0051805, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {15, 30} \[ a x+\frac{1}{2} b x \left (c x^n\right )^{\frac{1}{n}} \]

Antiderivative was successfully verified.

[In]

Int[a + b*(c*x^n)^n^(-1),x]

[Out]

a*x + (b*x*(c*x^n)^n^(-1))/2

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

Mathematica [A]  time = 0.0052905, size = 19, normalized size = 1. \[ a x+\frac{1}{2} b x \left (c x^n\right )^{\frac{1}{n}} \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*(c*x^n)^n^(-1),x]

[Out]

a*x + (b*x*(c*x^n)^n^(-1))/2

________________________________________________________________________________________

Maple [A]  time = 0.017, size = 22, normalized size = 1.2 \begin{align*} ax+{\frac{bx}{2}{{\rm e}^{{\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)^(1/n),x)

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} b c^{\left (\frac{1}{n}\right )} \int{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )}\,{d x} + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.27623, size = 34, normalized size = 1.79 \begin{align*} \frac{1}{2} \, b c^{\left (\frac{1}{n}\right )} x^{2} + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*c^(1/n)*x^2 + a*x

________________________________________________________________________________________

Sympy [A]  time = 0.237278, size = 19, normalized size = 1. \begin{align*} a x + \frac{b c^{\frac{1}{n}} x \left (x^{n}\right )^{\frac{1}{n}}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*(c*x**n)**(1/n),x)

[Out]

a*x + b*c**(1/n)*x*(x**n)**(1/n)/2

________________________________________________________________________________________

Giac [A]  time = 1.26899, size = 20, normalized size = 1.05 \begin{align*} \frac{1}{2} \, b c^{\left (\frac{1}{n}\right )} x^{2} + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*c^(1/n)*x^2 + a*x