### 3.1013 $$\int \frac{d+e x}{(c d^2+2 c d e x+c e^2 x^2)^2} \, dx$$

Optimal. Leaf size=17 $-\frac{1}{2 c^2 e (d+e x)^2}$

[Out]

-1/(2*c^2*e*(d + e*x)^2)

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Rubi [A]  time = 0.0043401, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.107, Rules used = {27, 12, 32} $-\frac{1}{2 c^2 e (d+e x)^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-1/(2*c^2*e*(d + e*x)^2)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int \frac{d+e x}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx &=\int \frac{1}{c^2 (d+e x)^3} \, dx\\ &=\frac{\int \frac{1}{(d+e x)^3} \, dx}{c^2}\\ &=-\frac{1}{2 c^2 e (d+e x)^2}\\ \end{align*}

Mathematica [A]  time = 0.0020644, size = 17, normalized size = 1. $-\frac{1}{2 c^2 e (d+e x)^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-1/(2*c^2*e*(d + e*x)^2)

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Maple [A]  time = 0.041, size = 16, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,{c}^{2}e \left ( ex+d \right ) ^{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.08769, size = 41, normalized size = 2.41 \begin{align*} -\frac{1}{2 \,{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )} c e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(c*e^2*x^2+2*c*d*e*x+c*d^2)^2,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.03139, size = 65, normalized size = 3.82 \begin{align*} -\frac{1}{2 \,{\left (c^{2} e^{3} x^{2} + 2 \, c^{2} d e^{2} x + c^{2} d^{2} e\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(c*e^2*x^2+2*c*d*e*x+c*d^2)^2,x, algorithm="fricas")

[Out]

-1/2/(c^2*e^3*x^2 + 2*c^2*d*e^2*x + c^2*d^2*e)

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Sympy [B]  time = 0.621539, size = 36, normalized size = 2.12 \begin{align*} - \frac{1}{2 c^{2} d^{2} e + 4 c^{2} d e^{2} x + 2 c^{2} e^{3} x^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(c*e**2*x**2+2*c*d*e*x+c*d**2)**2,x)

[Out]

-1/(2*c**2*d**2*e + 4*c**2*d*e**2*x + 2*c**2*e**3*x**2)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(c*e^2*x^2+2*c*d*e*x+c*d^2)^2,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError