3.1650 \(\int \frac {a+\frac {b}{x}}{\sqrt {x}} \, dx\)

Optimal. Leaf size=17 \[ 2 a \sqrt {x}-\frac {2 b}{\sqrt {x}} \]

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, 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} \[ 2 a \sqrt {x}-\frac {2 b}{\sqrt {x}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)/Sqrt[x],x]

[Out]

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

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Mathematica [A]  time = 0.01, size = 14, normalized size = 0.82 \[ \frac {2 (a x-b)}{\sqrt {x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)/Sqrt[x],x]

[Out]

(2*(-b + a*x))/Sqrt[x]

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fricas [A]  time = 0.65, size = 12, normalized size = 0.71 \[ \frac {2 \, {\left (a x - b\right )}}{\sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(a*x - b)/sqrt(x)

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giac [A]  time = 0.15, size = 13, normalized size = 0.76 \[ 2 \, a \sqrt {x} - \frac {2 \, b}{\sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*sqrt(x) - 2*b/sqrt(x)

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maple [A]  time = 0.00, size = 13, normalized size = 0.76 \[ \frac {2 a x -2 b}{\sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.07, size = 13, normalized size = 0.76 \[ 2 \, a \sqrt {x} - \frac {2 \, b}{\sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*sqrt(x) - 2*b/sqrt(x)

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mupad [B]  time = 0.03, size = 11, normalized size = 0.65 \[ -\frac {2\,\left (b-a\,x\right )}{\sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x)/x^(1/2),x)

[Out]

-(2*(b - a*x))/x^(1/2)

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sympy [A]  time = 0.36, size = 15, normalized size = 0.88 \[ 2 a \sqrt {x} - \frac {2 b}{\sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)/x**(1/2),x)

[Out]

2*a*sqrt(x) - 2*b/sqrt(x)

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