### 3.220 $$\int \frac{a+2 b x}{a x+b x^2} \, dx$$

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

[Out]

Log[a*x + b*x^2]

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Rubi [A]  time = 0.0039895, 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} $\log \left (a x+b x^2\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + 2*b*x)/(a*x + b*x^2),x]

[Out]

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

Mathematica [A]  time = 0.0039173, size = 9, normalized size = 0.9 $\log (a+b x)+\log (x)$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

Log[x] + Log[a + b*x]

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Maple [A]  time = 0.043, size = 9, normalized size = 0.9 \begin{align*} \ln \left ( x \left ( bx+a \right ) \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(x*(b*x+a))

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Maxima [A]  time = 1.11328, size = 14, normalized size = 1.4 \begin{align*} \log \left (b x^{2} + a x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(b*x^2 + a*x)

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Fricas [A]  time = 1.55282, size = 24, normalized size = 2.4 \begin{align*} \log \left (b x^{2} + a x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(b*x^2 + a*x)

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Sympy [A]  time = 0.981668, size = 8, normalized size = 0.8 \begin{align*} \log{\left (a x + b x^{2} \right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(a*x + b*x**2)

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Giac [A]  time = 1.33478, size = 15, normalized size = 1.5 \begin{align*} \log \left ({\left | b x + a \right |}\right ) + \log \left ({\left | x \right |}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(b*x + a)) + log(abs(x))