3.656 \(\int \frac{a+b x}{x^{4/3}} \, dx\)

Optimal. Leaf size=19 \[ \frac{3}{2} b x^{2/3}-\frac{3 a}{\sqrt [3]{x}} \]

[Out]

(-3*a)/x^(1/3) + (3*b*x^(2/3))/2

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Rubi [A]  time = 0.0036323, 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} \[ \frac{3}{2} b x^{2/3}-\frac{3 a}{\sqrt [3]{x}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/x^(4/3),x]

[Out]

(-3*a)/x^(1/3) + (3*b*x^(2/3))/2

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

Mathematica [A]  time = 0.0051168, size = 16, normalized size = 0.84 \[ \frac{3 (b x-2 a)}{2 \sqrt [3]{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/x^(4/3),x]

[Out]

(3*(-2*a + b*x))/(2*x^(1/3))

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Maple [A]  time = 0.003, size = 14, normalized size = 0.7 \begin{align*} -{\frac{-3\,bx+6\,a}{2}{\frac{1}{\sqrt [3]{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)/x^(4/3),x)

[Out]

-3/2*(-b*x+2*a)/x^(1/3)

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Maxima [A]  time = 1.07173, size = 18, normalized size = 0.95 \begin{align*} \frac{3}{2} \, b x^{\frac{2}{3}} - \frac{3 \, a}{x^{\frac{1}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/x^(4/3),x, algorithm="maxima")

[Out]

3/2*b*x^(2/3) - 3*a/x^(1/3)

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Fricas [A]  time = 1.53729, size = 34, normalized size = 1.79 \begin{align*} \frac{3 \,{\left (b x - 2 \, a\right )}}{2 \, x^{\frac{1}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/x^(4/3),x, algorithm="fricas")

[Out]

3/2*(b*x - 2*a)/x^(1/3)

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Sympy [A]  time = 0.437249, size = 17, normalized size = 0.89 \begin{align*} - \frac{3 a}{\sqrt [3]{x}} + \frac{3 b x^{\frac{2}{3}}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/x**(4/3),x)

[Out]

-3*a/x**(1/3) + 3*b*x**(2/3)/2

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Giac [A]  time = 1.06447, size = 18, normalized size = 0.95 \begin{align*} \frac{3}{2} \, b x^{\frac{2}{3}} - \frac{3 \, a}{x^{\frac{1}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/x^(4/3),x, algorithm="giac")

[Out]

3/2*b*x^(2/3) - 3*a/x^(1/3)