3.2117 \(\int \frac {a+b \sqrt {x}}{x^2} \, dx\)

Optimal. Leaf size=15 \[ -\frac {a}{x}-\frac {2 b}{\sqrt {x}} \]

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 15, 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} \[ -\frac {a}{x}-\frac {2 b}{\sqrt {x}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])/x^2,x]

[Out]

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

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Mathematica [A]  time = 0.01, size = 15, normalized size = 1.00 \[ -\frac {a}{x}-\frac {2 b}{\sqrt {x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])/x^2,x]

[Out]

-(a/x) - (2*b)/Sqrt[x]

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fricas [A]  time = 1.01, size = 13, normalized size = 0.87 \[ -\frac {2 \, b \sqrt {x} + a}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*b*sqrt(x) + a)/x

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giac [A]  time = 0.15, size = 13, normalized size = 0.87 \[ -\frac {2 \, b \sqrt {x} + a}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*b*sqrt(x) + a)/x

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maple [A]  time = 0.00, size = 14, normalized size = 0.93 \[ -\frac {2 b}{\sqrt {x}}-\frac {a}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.79, size = 13, normalized size = 0.87 \[ -\frac {2 \, b \sqrt {x} + a}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*b*sqrt(x) + a)/x

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mupad [B]  time = 0.04, size = 13, normalized size = 0.87 \[ -\frac {a}{x}-\frac {2\,b}{\sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^(1/2))/x^2,x)

[Out]

- a/x - (2*b)/x^(1/2)

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sympy [A]  time = 0.44, size = 12, normalized size = 0.80 \[ - \frac {a}{x} - \frac {2 b}{\sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a/x - 2*b/sqrt(x)

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