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

Optimal. Leaf size=10 \[ \log \left (b x+c x^2\right ) \]

[Out]

ln(c*x^2+b*x)

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Rubi [A]  time = 0.00, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {628} \[ \log \left (b x+c x^2\right ) \]

Antiderivative was successfully verified.

[In]

Int[(b + 2*c*x)/(b*x + c*x^2),x]

[Out]

Log[b*x + c*x^2]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 9, normalized size = 0.90 \[ \log (b+c x)+\log (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[x] + Log[b + c*x]

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fricas [A]  time = 0.78, size = 10, normalized size = 1.00 \[ \log \left (c x^{2} + b x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(c*x^2 + b*x)

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giac [A]  time = 0.23, size = 11, normalized size = 1.10 \[ \log \left ({\left | c x^{2} + b x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(c*x^2 + b*x))

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maple [A]  time = 0.00, size = 9, normalized size = 0.90 \[ \ln \left (\left (c x +b \right ) x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.55, size = 10, normalized size = 1.00 \[ \log \left (c x^{2} + b x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(c*x^2 + b*x)

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mupad [B]  time = 0.05, size = 8, normalized size = 0.80 \[ \ln \left (x\,\left (b+c\,x\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b + 2*c*x)/(b*x + c*x^2),x)

[Out]

log(x*(b + c*x))

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sympy [A]  time = 0.13, size = 8, normalized size = 0.80 \[ \log {\left (b x + c x^{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(b*x + c*x**2)

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