3.2290 \(\int \frac{a+b \sqrt [3]{x}}{x} \, dx\)

Optimal. Leaf size=13 \[ a \log (x)+3 b \sqrt [3]{x} \]

[Out]

3*b*x^(1/3) + a*Log[x]

________________________________________________________________________________________

Rubi [A]  time = 0.0049706, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {14} \[ a \log (x)+3 b \sqrt [3]{x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))/x,x]

[Out]

3*b*x^(1/3) + a*Log[x]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \frac{a+b \sqrt [3]{x}}{x} \, dx &=\int \left (\frac{a}{x}+\frac{b}{x^{2/3}}\right ) \, dx\\ &=3 b \sqrt [3]{x}+a \log (x)\\ \end{align*}

Mathematica [A]  time = 0.005113, size = 13, normalized size = 1. \[ a \log (x)+3 b \sqrt [3]{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))/x,x]

[Out]

3*b*x^(1/3) + a*Log[x]

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 12, normalized size = 0.9 \begin{align*} 3\,b\sqrt [3]{x}+a\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*b*x^(1/3)+a*ln(x)

________________________________________________________________________________________

Maxima [A]  time = 0.971428, size = 15, normalized size = 1.15 \begin{align*} a \log \left (x\right ) + 3 \, b x^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))/x,x, algorithm="maxima")

[Out]

a*log(x) + 3*b*x^(1/3)

________________________________________________________________________________________

Fricas [A]  time = 1.48211, size = 43, normalized size = 3.31 \begin{align*} 3 \, a \log \left (x^{\frac{1}{3}}\right ) + 3 \, b x^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))/x,x, algorithm="fricas")

[Out]

3*a*log(x^(1/3)) + 3*b*x^(1/3)

________________________________________________________________________________________

Sympy [A]  time = 0.315951, size = 12, normalized size = 0.92 \begin{align*} a \log{\left (x \right )} + 3 b \sqrt [3]{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/3))/x,x)

[Out]

a*log(x) + 3*b*x**(1/3)

________________________________________________________________________________________

Giac [A]  time = 1.09804, size = 16, normalized size = 1.23 \begin{align*} a \log \left ({\left | x \right |}\right ) + 3 \, b x^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))/x,x, algorithm="giac")

[Out]

a*log(abs(x)) + 3*b*x^(1/3)