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

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

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Rubi [A]  time = 0.01, 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} \begin {gather*} \log (b+c x)+\log (x) \end {gather*}

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*}

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Mathematica [A]  time = 0.02, size = 9, normalized size = 1.12 \begin {gather*} \log (b+c x)+\log (x) \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {b+2 c x}{x (b+c x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.40, size = 10, normalized size = 1.25 \begin {gather*} \log \left (c x^{2} + b x\right ) \end {gather*}

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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giac [A]  time = 0.15, size = 11, normalized size = 1.38 \begin {gather*} \log \left ({\left | c x + b \right |}\right ) + \log \left ({\left | x \right |}\right ) \end {gather*}

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))

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maple [A]  time = 0.04, size = 9, normalized size = 1.12 \begin {gather*} \ln \left (\left (c x +b \right ) x \right ) \end {gather*}

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 = 0.48, size = 9, normalized size = 1.12 \begin {gather*} \log \left (c x + b\right ) + \log \relax (x) \end {gather*}

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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mupad [B]  time = 4.66, size = 8, normalized size = 1.00 \begin {gather*} \ln \left (x\,\left (b+c\,x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b + 2*c*x)/(x*(b + c*x)),x)

[Out]

log(x*(b + c*x))

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sympy [A]  time = 0.12, size = 8, normalized size = 1.00 \begin {gather*} \log {\left (b x + c x^{2} \right )} \end {gather*}

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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