3.2118 \(\int \frac{a+b \sqrt{x}}{x^3} \, dx\)

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

[Out]

-a/(2*x^2) - (2*b)/(3*x^(3/2))

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Rubi [A]  time = 0.0051484, 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} \[ -\frac{a}{2 x^2}-\frac{2 b}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])/x^3,x]

[Out]

-a/(2*x^2) - (2*b)/(3*x^(3/2))

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

Mathematica [A]  time = 0.0064661, size = 19, normalized size = 1. \[ -\frac{a}{2 x^2}-\frac{2 b}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])/x^3,x]

[Out]

-a/(2*x^2) - (2*b)/(3*x^(3/2))

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Maple [A]  time = 0.001, size = 14, normalized size = 0.7 \begin{align*} -{\frac{a}{2\,{x}^{2}}}-{\frac{2\,b}{3}{x}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.942523, size = 20, normalized size = 1.05 \begin{align*} -\frac{4 \, b \sqrt{x} + 3 \, a}{6 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))/x^3,x, algorithm="maxima")

[Out]

-1/6*(4*b*sqrt(x) + 3*a)/x^2

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Fricas [A]  time = 1.43819, size = 41, normalized size = 2.16 \begin{align*} -\frac{4 \, b \sqrt{x} + 3 \, a}{6 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))/x^3,x, algorithm="fricas")

[Out]

-1/6*(4*b*sqrt(x) + 3*a)/x^2

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Sympy [A]  time = 0.590651, size = 17, normalized size = 0.89 \begin{align*} - \frac{a}{2 x^{2}} - \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**(1/2))/x**3,x)

[Out]

-a/(2*x**2) - 2*b/(3*x**(3/2))

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Giac [A]  time = 1.11444, size = 20, normalized size = 1.05 \begin{align*} -\frac{4 \, b \sqrt{x} + 3 \, a}{6 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))/x^3,x, algorithm="giac")

[Out]

-1/6*(4*b*sqrt(x) + 3*a)/x^2