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

Optimal. Leaf size=21 \[ \frac{3}{2} a x^{2/3}+\frac{3}{5} b x^{5/3} \]

[Out]

(3*a*x^(2/3))/2 + (3*b*x^(5/3))/5

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Rubi [A]  time = 0.0038604, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ \frac{3}{2} a x^{2/3}+\frac{3}{5} b x^{5/3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/x^(1/3),x]

[Out]

(3*a*x^(2/3))/2 + (3*b*x^(5/3))/5

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{a+b x}{\sqrt [3]{x}} \, dx &=\int \left (\frac{a}{\sqrt [3]{x}}+b x^{2/3}\right ) \, dx\\ &=\frac{3}{2} a x^{2/3}+\frac{3}{5} b x^{5/3}\\ \end{align*}

Mathematica [A]  time = 0.0045825, size = 17, normalized size = 0.81 \[ \frac{3}{10} x^{2/3} (5 a+2 b x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/x^(1/3),x]

[Out]

(3*x^(2/3)*(5*a + 2*b*x))/10

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Maple [A]  time = 0.002, size = 14, normalized size = 0.7 \begin{align*}{\frac{6\,bx+15\,a}{10}{x}^{{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/10*x^(2/3)*(2*b*x+5*a)

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Maxima [A]  time = 1.06304, size = 18, normalized size = 0.86 \begin{align*} \frac{3}{5} \, b x^{\frac{5}{3}} + \frac{3}{2} \, a x^{\frac{2}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/5*b*x^(5/3) + 3/2*a*x^(2/3)

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Fricas [A]  time = 1.53034, size = 38, normalized size = 1.81 \begin{align*} \frac{3}{10} \,{\left (2 \, b x + 5 \, a\right )} x^{\frac{2}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/10*(2*b*x + 5*a)*x^(2/3)

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Sympy [A]  time = 1.32003, size = 19, normalized size = 0.9 \begin{align*} \frac{3 a x^{\frac{2}{3}}}{2} + \frac{3 b x^{\frac{5}{3}}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*a*x**(2/3)/2 + 3*b*x**(5/3)/5

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Giac [A]  time = 1.07289, size = 18, normalized size = 0.86 \begin{align*} \frac{3}{5} \, b x^{\frac{5}{3}} + \frac{3}{2} \, a x^{\frac{2}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/5*b*x^(5/3) + 3/2*a*x^(2/3)