### 3.1161 $$\int \frac{b d+2 c d x}{a+b x+c x^2} \, dx$$

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

[Out]

d*Log[a + b*x + c*x^2]

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Rubi [A]  time = 0.0048115, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.045, Rules used = {628} $d \log \left (a+b x+c x^2\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[(b*d + 2*c*d*x)/(a + b*x + c*x^2),x]

[Out]

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

Mathematica [A]  time = 0.002052, size = 12, normalized size = 0.92 $d \log (a+x (b+c x))$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

d*Log[a + x*(b + c*x)]

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Maple [A]  time = 0.04, size = 14, normalized size = 1.1 \begin{align*} d\ln \left ( c{x}^{2}+bx+a \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.29871, size = 18, normalized size = 1.38 \begin{align*} d \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*d*x+b*d)/(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

d*log(c*x^2 + b*x + a)

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Fricas [A]  time = 1.7289, size = 32, normalized size = 2.46 \begin{align*} d \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*d*x+b*d)/(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

d*log(c*x^2 + b*x + a)

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Sympy [A]  time = 0.333279, size = 12, normalized size = 0.92 \begin{align*} d \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*d*x+b*d)/(c*x**2+b*x+a),x)

[Out]

d*log(a + b*x + c*x**2)

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Giac [A]  time = 1.22793, size = 18, normalized size = 1.38 \begin{align*} d \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*d*x+b*d)/(c*x^2+b*x+a),x, algorithm="giac")

[Out]

d*log(c*x^2 + b*x + a)