3.1651 \(\int \frac{a+\frac{b}{x}}{x^{3/2}} \, dx\)

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

[Out]

(-2*b)/(3*x^(3/2)) - (2*a)/Sqrt[x]

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Rubi [A]  time = 0.0040228, antiderivative size = 19, normalized size of antiderivative = 1., 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} \[ -\frac{2 a}{\sqrt{x}}-\frac{2 b}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)/x^(3/2),x]

[Out]

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

Mathematica [A]  time = 0.0049352, size = 15, normalized size = 0.79 \[ -\frac{2 (3 a x+b)}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)/x^(3/2),x]

[Out]

(-2*(b + 3*a*x))/(3*x^(3/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.953224, size = 18, normalized size = 0.95 \begin{align*} -\frac{2 \, a}{\sqrt{x}} - \frac{2 \, b}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a/sqrt(x) - 2/3*b/x^(3/2)

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Fricas [A]  time = 1.93764, size = 35, normalized size = 1.84 \begin{align*} -\frac{2 \,{\left (3 \, a x + b\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(3*a*x + b)/x^(3/2)

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Sympy [A]  time = 0.47052, size = 19, normalized size = 1. \begin{align*} - \frac{2 a}{\sqrt{x}} - \frac{2 b}{3 x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a/sqrt(x) - 2*b/(3*x**(3/2))

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Giac [A]  time = 1.10062, size = 15, normalized size = 0.79 \begin{align*} -\frac{2 \,{\left (3 \, a x + b\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(3*a*x + b)/x^(3/2)