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

Optimal. Leaf size=11 \[ b \log (x)-\frac{a}{x} \]

[Out]

-(a/x) + b*Log[x]

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Rubi [A]  time = 0.0043358, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {43} \[ b \log (x)-\frac{a}{x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/x^2,x]

[Out]

-(a/x) + b*Log[x]

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

Mathematica [A]  time = 0.0019118, size = 11, normalized size = 1. \[ b \log (x)-\frac{a}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/x^2,x]

[Out]

-(a/x) + b*Log[x]

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Maple [A]  time = 0.004, size = 12, normalized size = 1.1 \begin{align*} -{\frac{a}{x}}+b\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a/x+b*ln(x)

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Maxima [A]  time = 1.01865, size = 15, normalized size = 1.36 \begin{align*} b \log \left (x\right ) - \frac{a}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*log(x) - a/x

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Fricas [A]  time = 1.53353, size = 27, normalized size = 2.45 \begin{align*} \frac{b x \log \left (x\right ) - a}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x*log(x) - a)/x

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Sympy [A]  time = 0.379399, size = 7, normalized size = 0.64 \begin{align*} - \frac{a}{x} + b \log{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a/x + b*log(x)

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Giac [A]  time = 1.15994, size = 16, normalized size = 1.45 \begin{align*} b \log \left ({\left | x \right |}\right ) - \frac{a}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*log(abs(x)) - a/x