### 3.1173 $$\int \frac{b d+2 c d x}{(a+b x+c x^2)^2} \, dx$$

Optimal. Leaf size=15 $-\frac{d}{a+b x+c x^2}$

[Out]

-(d/(a + b*x + c*x^2))

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Rubi [A]  time = 0.0051263, antiderivative size = 15, 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 = {629} $-\frac{d}{a+b x+c x^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(d/(a + b*x + c*x^2))

Rule 629

Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d*(a + b*x + c*x^2)^(p +
1))/(b*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps

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

Mathematica [A]  time = 0.0039089, size = 14, normalized size = 0.93 $-\frac{d}{a+x (b+c x)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(d/(a + x*(b + c*x)))

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Maple [A]  time = 0.037, size = 16, normalized size = 1.1 \begin{align*} -{\frac{d}{c{x}^{2}+bx+a}} \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)^2,x)

[Out]

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

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Maxima [A]  time = 1.05552, size = 20, normalized size = 1.33 \begin{align*} -\frac{d}{c x^{2} + b x + a} \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)^2,x, algorithm="maxima")

[Out]

-d/(c*x^2 + b*x + a)

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Fricas [A]  time = 1.93995, size = 30, normalized size = 2. \begin{align*} -\frac{d}{c x^{2} + b x + a} \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)^2,x, algorithm="fricas")

[Out]

-d/(c*x^2 + b*x + a)

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Sympy [A]  time = 0.594069, size = 12, normalized size = 0.8 \begin{align*} - \frac{d}{a + b x + c x^{2}} \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)**2,x)

[Out]

-d/(a + b*x + c*x**2)

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Giac [A]  time = 1.18969, size = 20, normalized size = 1.33 \begin{align*} -\frac{d}{c x^{2} + b x + a} \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)^2,x, algorithm="giac")

[Out]

-d/(c*x^2 + b*x + a)