### 3.1186 $$\int \frac{b d+2 c d x}{(a+b x+c x^2)^3} \, dx$$

Optimal. Leaf size=17 $-\frac{d}{2 \left (a+b x+c x^2\right )^2}$

[Out]

-d/(2*(a + b*x + c*x^2)^2)

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Rubi [A]  time = 0.0048227, antiderivative size = 17, 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} $-\frac{d}{2 \left (a+b x+c x^2\right )^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(b*d + 2*c*d*x)/(a + b*x + c*x^2)^3,x]

[Out]

-d/(2*(a + b*x + c*x^2)^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 )^3} \, dx &=-\frac{d}{2 \left (a+b x+c x^2\right )^2}\\ \end{align*}

Mathematica [A]  time = 0.0064625, size = 16, normalized size = 0.94 $-\frac{d}{2 (a+x (b+c x))^2}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(b*d + 2*c*d*x)/(a + b*x + c*x^2)^3,x]

[Out]

-d/(2*(a + x*(b + c*x))^2)

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Maple [A]  time = 0.038, size = 16, normalized size = 0.9 \begin{align*} -{\frac{d}{2\, \left ( c{x}^{2}+bx+a \right ) ^{2}}} \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)^3,x)

[Out]

-1/2*d/(c*x^2+b*x+a)^2

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Maxima [A]  time = 1.33547, size = 20, normalized size = 1.18 \begin{align*} -\frac{d}{2 \,{\left (c x^{2} + b x + a\right )}^{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)^3,x, algorithm="maxima")

[Out]

-1/2*d/(c*x^2 + b*x + a)^2

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Fricas [B]  time = 2.02136, size = 89, normalized size = 5.24 \begin{align*} -\frac{d}{2 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{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)^3,x, algorithm="fricas")

[Out]

-1/2*d/(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)

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Sympy [B]  time = 1.81857, size = 44, normalized size = 2.59 \begin{align*} - \frac{d}{2 a^{2} + 4 a b x + 4 b c x^{3} + 2 c^{2} x^{4} + x^{2} \left (4 a c + 2 b^{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)**3,x)

[Out]

-d/(2*a**2 + 4*a*b*x + 4*b*c*x**3 + 2*c**2*x**4 + x**2*(4*a*c + 2*b**2))

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Giac [A]  time = 1.22106, size = 20, normalized size = 1.18 \begin{align*} -\frac{d}{2 \,{\left (c x^{2} + b x + a\right )}^{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)^3,x, algorithm="giac")

[Out]

-1/2*d/(c*x^2 + b*x + a)^2