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*ln(x)

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Rubi [A]  time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 13, normalized size = 1.00 \[ a \log (x)+3 b \sqrt [3]{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))/x,x]

[Out]

3*b*x^(1/3) + a*Log[x]

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fricas [A]  time = 0.68, size = 14, normalized size = 1.08 \[ 3 \, a \log \left (x^{\frac {1}{3}}\right ) + 3 \, b x^{\frac {1}{3}} \]

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)

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giac [A]  time = 0.15, size = 12, normalized size = 0.92 \[ a \log \left ({\left | x \right |}\right ) + 3 \, b x^{\frac {1}{3}} \]

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)

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maple [A]  time = 0.00, size = 12, normalized size = 0.92 \[ a \ln \relax (x )+3 b \,x^{\frac {1}{3}} \]

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)

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maxima [A]  time = 0.90, size = 11, normalized size = 0.85 \[ a \log \relax (x) + 3 \, b x^{\frac {1}{3}} \]

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)

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mupad [B]  time = 1.10, size = 11, normalized size = 0.85 \[ 3\,b\,x^{1/3}+a\,\ln \relax (x) \]

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*log(x)

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sympy [A]  time = 0.27, size = 12, normalized size = 0.92 \[ a \log {\relax (x )} + 3 b \sqrt [3]{x} \]

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)

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