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]

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

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Rubi [A]  time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 18, normalized size = 1.00 \[ \frac {\log (x)}{b}-\frac {\log (b+c x)}{b} \]

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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fricas [A]  time = 1.03, size = 16, normalized size = 0.89 \[ -\frac {\log \left (c x + b\right ) - \log \relax (x)}{b} \]

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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giac [A]  time = 0.15, size = 20, normalized size = 1.11 \[ -\frac {\log \left ({\left | c x + b \right |}\right )}{b} + \frac {\log \left ({\left | x \right |}\right )}{b} \]

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

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maple [A]  time = 0.05, size = 19, normalized size = 1.06 \[ \frac {\ln \relax (x )}{b}-\frac {\ln \left (c x +b \right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.31, size = 18, normalized size = 1.00 \[ -\frac {\log \left (c x + b\right )}{b} + \frac {\log \relax (x)}{b} \]

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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mupad [B]  time = 0.16, size = 15, normalized size = 0.83 \[ -\frac {2\,\mathrm {atanh}\left (\frac {2\,c\,x}{b}+1\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x + c*x^2),x)

[Out]

-(2*atanh((2*c*x)/b + 1))/b

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sympy [A]  time = 0.15, size = 10, normalized size = 0.56 \[ \frac {\log {\relax (x )} - \log {\left (\frac {b}{c} + x \right )}}{b} \]

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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