### 3.113 $$\int \frac{b+2 c x}{-a+b x+c x^2} \, dx$$

Optimal. Leaf size=13 $\log \left (a-b x-c x^2\right )$

[Out]

Log[a - b*x - c*x^2]

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Rubi [A]  time = 0.0047574, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.048, Rules used = {628} $\log \left (a-b x-c x^2\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[(b + 2*c*x)/(-a + b*x + c*x^2),x]

[Out]

Log[a - 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}{-a+b x+c x^2} \, dx &=\log \left (a-b x-c x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0045095, size = 12, normalized size = 0.92 $\log (x (b+c x)-a)$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

Log[-a + x*(b + c*x)]

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Maple [A]  time = 0., size = 14, normalized size = 1.1 \begin{align*} \ln \left ( c{x}^{2}+bx-a \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.18616, size = 18, normalized size = 1.38 \begin{align*} \log \left (c x^{2} + b x - a\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(c*x^2 + b*x - a)

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Fricas [A]  time = 0.950366, size = 30, normalized size = 2.31 \begin{align*} \log \left (c x^{2} + b x - a\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(c*x^2 + b*x - a)

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Sympy [A]  time = 0.393052, size = 10, normalized size = 0.77 \begin{align*} \log{\left (- a + 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-a),x)

[Out]

log(-a + b*x + c*x**2)

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Giac [A]  time = 1.12131, size = 18, normalized size = 1.38 \begin{align*} \log \left (c x^{2} + b x - a\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(c*x^2 + b*x - a)