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

Optimal. Leaf size=8 \[ a \log (x)+b x \]

[Out]

b*x + a*Log[x]

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Rubi [A]  time = 0.0030192, antiderivative size = 8, 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} \[ a \log (x)+b x \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/x,x]

[Out]

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

Mathematica [A]  time = 0.0008298, size = 8, normalized size = 1. \[ a \log (x)+b x \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/x,x]

[Out]

b*x + a*Log[x]

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Maple [A]  time = 0.002, size = 9, normalized size = 1.1 \begin{align*} bx+a\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x+a*ln(x)

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Maxima [A]  time = 1.0344, size = 11, normalized size = 1.38 \begin{align*} b x + a \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x + a*log(x)

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Fricas [A]  time = 1.55437, size = 22, normalized size = 2.75 \begin{align*} b x + a \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x + a*log(x)

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Sympy [A]  time = 0.116574, size = 7, normalized size = 0.88 \begin{align*} a \log{\left (x \right )} + b x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*log(x) + b*x

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Giac [A]  time = 1.15128, size = 12, normalized size = 1.5 \begin{align*} b x + a \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x + a*log(abs(x))