3.263 \(\int \frac{1}{b x+c x^2} \, dx\)

Optimal. Leaf size=18 \[ \frac{\log (x)}{b}-\frac{\log (b+c x)}{b} \]

[Out]

Log[x]/b - Log[b + c*x]/b

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Rubi [A]  time = 0.0030771, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {615} \[ \frac{\log (x)}{b}-\frac{\log (b+c x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)^(-1),x]

[Out]

Log[x]/b - Log[b + c*x]/b

Rule 615

Int[((b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[Log[x]/b, x] - Simp[Log[RemoveContent[b + c*x, x]]/b,
x] /; FreeQ[{b, c}, x]

Rubi steps

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

Mathematica [A]  time = 0.0032633, size = 18, normalized size = 1. \[ \frac{\log (x)}{b}-\frac{\log (b+c x)}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x + c*x^2)^(-1),x]

[Out]

Log[x]/b - Log[b + c*x]/b

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Maple [A]  time = 0.045, size = 19, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( x \right ) }{b}}-{\frac{\ln \left ( cx+b \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.11035, size = 24, normalized size = 1.33 \begin{align*} -\frac{\log \left (c x + b\right )}{b} + \frac{\log \left (x\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(c*x + b)/b + log(x)/b

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Fricas [A]  time = 1.65764, size = 38, normalized size = 2.11 \begin{align*} -\frac{\log \left (c x + b\right ) - \log \left (x\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(log(c*x + b) - log(x))/b

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Sympy [A]  time = 0.589893, size = 10, normalized size = 0.56 \begin{align*} \frac{\log{\left (x \right )} - \log{\left (\frac{b}{c} + x \right )}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(x) - log(b/c + x))/b

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Giac [A]  time = 1.26521, size = 27, normalized size = 1.5 \begin{align*} -\frac{\log \left ({\left | c x + b \right |}\right )}{b} + \frac{\log \left ({\left | x \right |}\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(abs(c*x + b))/b + log(abs(x))/b