3.657 \(\int \frac{a+b x}{x^{5/3}} \, dx\)

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

[Out]

(-3*a)/(2*x^(2/3)) + 3*b*x^(1/3)

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Rubi [A]  time = 0.0035201, 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} \[ 3 b \sqrt [3]{x}-\frac{3 a}{2 x^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/x^(5/3),x]

[Out]

(-3*a)/(2*x^(2/3)) + 3*b*x^(1/3)

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

Mathematica [A]  time = 0.0052261, size = 19, normalized size = 1. \[ 3 b \sqrt [3]{x}-\frac{3 a}{2 x^{2/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/x^(5/3),x]

[Out]

(-3*a)/(2*x^(2/3)) + 3*b*x^(1/3)

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Maple [A]  time = 0.002, size = 12, normalized size = 0.6 \begin{align*} -{\frac{-6\,bx+3\,a}{2}{x}^{-{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)/x^(5/3),x)

[Out]

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

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Maxima [A]  time = 1.00163, size = 18, normalized size = 0.95 \begin{align*} 3 \, b x^{\frac{1}{3}} - \frac{3 \, a}{2 \, x^{\frac{2}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/x^(5/3),x, algorithm="maxima")

[Out]

3*b*x^(1/3) - 3/2*a/x^(2/3)

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Fricas [A]  time = 1.4177, size = 34, normalized size = 1.79 \begin{align*} \frac{3 \,{\left (2 \, b x - a\right )}}{2 \, x^{\frac{2}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/x^(5/3),x, algorithm="fricas")

[Out]

3/2*(2*b*x - a)/x^(2/3)

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Sympy [A]  time = 0.486049, size = 17, normalized size = 0.89 \begin{align*} - \frac{3 a}{2 x^{\frac{2}{3}}} + 3 b \sqrt [3]{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/x**(5/3),x)

[Out]

-3*a/(2*x**(2/3)) + 3*b*x**(1/3)

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Giac [A]  time = 1.05946, size = 18, normalized size = 0.95 \begin{align*} 3 \, b x^{\frac{1}{3}} - \frac{3 \, a}{2 \, x^{\frac{2}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/x^(5/3),x, algorithm="giac")

[Out]

3*b*x^(1/3) - 3/2*a/x^(2/3)