3.1056 \(\int \frac{b+2 c x}{x (b+c x)} \, dx\)

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

[Out]

Log[x*(b + c*x)]

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Rubi [A]  time = 0.0056778, antiderivative size = 9, normalized size of antiderivative = 1.12, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {72} \[ \log (b+c x)+\log (x) \]

Antiderivative was successfully verified.

[In]

Int[(b + 2*c*x)/(x*(b + c*x)),x]

[Out]

Log[x] + Log[b + c*x]

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

\begin{align*} \int \frac{b+2 c x}{x (b+c x)} \, dx &=\int \left (\frac{1}{x}+\frac{c}{b+c x}\right ) \, dx\\ &=\log (x)+\log (b+c x)\\ \end{align*}

Mathematica [A]  time = 0.0047065, size = 9, normalized size = 1.12 \[ \log (b+c x)+\log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(b + 2*c*x)/(x*(b + c*x)),x]

[Out]

Log[x] + Log[b + c*x]

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Maple [A]  time = 0.003, size = 9, normalized size = 1.1 \begin{align*} \ln \left ( x \left ( cx+b \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)/x/(c*x+b),x)

[Out]

ln(x*(c*x+b))

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Maxima [A]  time = 0.924054, size = 12, normalized size = 1.5 \begin{align*} \log \left (c x + b\right ) + \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(c*x + b) + log(x)

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Fricas [A]  time = 0.940105, size = 24, normalized size = 3. \begin{align*} \log \left (c x^{2} + b x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(c*x^2 + b*x)

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Sympy [A]  time = 0.27032, size = 8, normalized size = 1. \begin{align*} \log{\left (b x + c x^{2} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(b*x + c*x**2)

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Giac [A]  time = 1.06982, size = 15, normalized size = 1.88 \begin{align*} \log \left ({\left | c x + b \right |}\right ) + \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(c*x + b)) + log(abs(x))