### 3.121 $$\int \frac{b+2 c x}{b x+c x^2} \, dx$$

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

[Out]

Log[b*x + c*x^2]

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Rubi [A]  time = 0.0042626, antiderivative size = 10, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0038842, size = 9, normalized size = 0.9 $\log (b+c x)+\log (x)$

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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Maple [A]  time = 0.002, size = 9, normalized size = 0.9 \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)/(c*x^2+b*x),x)

[Out]

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

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Maxima [A]  time = 1.02648, size = 14, normalized size = 1.4 \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)/(c*x^2+b*x),x, algorithm="maxima")

[Out]

log(c*x^2 + b*x)

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Fricas [A]  time = 1.09618, size = 24, normalized size = 2.4 \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)/(c*x^2+b*x),x, algorithm="fricas")

[Out]

log(c*x^2 + b*x)

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Sympy [A]  time = 0.352951, size = 8, normalized size = 0.8 \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)/(c*x**2+b*x),x)

[Out]

log(b*x + c*x**2)

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Giac [A]  time = 1.09606, size = 15, normalized size = 1.5 \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)/(c*x^2+b*x),x, algorithm="giac")

[Out]

log(abs(c*x + b)) + log(abs(x))