3.655 \(\int \frac{a+b x}{x^{2/3}} \, dx\)

Optimal. Leaf size=19 \[ 3 a \sqrt [3]{x}+\frac{3}{4} b x^{4/3} \]

[Out]

3*a*x^(1/3) + (3*b*x^(4/3))/4

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Rubi [A]  time = 0.0034785, antiderivative size = 19, 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} \[ 3 a \sqrt [3]{x}+\frac{3}{4} b x^{4/3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/x^(2/3),x]

[Out]

3*a*x^(1/3) + (3*b*x^(4/3))/4

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}{x^{2/3}} \, dx &=\int \left (\frac{a}{x^{2/3}}+b \sqrt [3]{x}\right ) \, dx\\ &=3 a \sqrt [3]{x}+\frac{3}{4} b x^{4/3}\\ \end{align*}

Mathematica [A]  time = 0.0045585, size = 16, normalized size = 0.84 \[ \frac{3}{4} \sqrt [3]{x} (4 a+b x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/x^(2/3),x]

[Out]

(3*x^(1/3)*(4*a + b*x))/4

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Maple [A]  time = 0.002, size = 13, normalized size = 0.7 \begin{align*}{\frac{3\,bx+12\,a}{4}\sqrt [3]{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)/x^(2/3),x)

[Out]

3/4*x^(1/3)*(b*x+4*a)

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Maxima [A]  time = 1.01697, size = 18, normalized size = 0.95 \begin{align*} \frac{3}{4} \, b x^{\frac{4}{3}} + 3 \, a x^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/x^(2/3),x, algorithm="maxima")

[Out]

3/4*b*x^(4/3) + 3*a*x^(1/3)

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Fricas [A]  time = 1.53769, size = 34, normalized size = 1.79 \begin{align*} \frac{3}{4} \,{\left (b x + 4 \, a\right )} x^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/x^(2/3),x, algorithm="fricas")

[Out]

3/4*(b*x + 4*a)*x^(1/3)

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Sympy [A]  time = 1.14289, size = 17, normalized size = 0.89 \begin{align*} 3 a \sqrt [3]{x} + \frac{3 b x^{\frac{4}{3}}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/x**(2/3),x)

[Out]

3*a*x**(1/3) + 3*b*x**(4/3)/4

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Giac [A]  time = 1.05519, size = 18, normalized size = 0.95 \begin{align*} \frac{3}{4} \, b x^{\frac{4}{3}} + 3 \, a x^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/x^(2/3),x, algorithm="giac")

[Out]

3/4*b*x^(4/3) + 3*a*x^(1/3)