### 3.1012 $$\int \frac{(d+e x)^2}{(c d^2+2 c d e x+c e^2 x^2)^2} \, dx$$

Optimal. Leaf size=15 $-\frac{1}{c^2 e (d+e x)}$

[Out]

-(1/(c^2*e*(d + e*x)))

________________________________________________________________________________________

Rubi [A]  time = 0.0048716, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 30, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.1, Rules used = {27, 12, 32} $-\frac{1}{c^2 e (d+e x)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(1/(c^2*e*(d + e*x)))

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

Mathematica [A]  time = 0.0021272, size = 15, normalized size = 1. $-\frac{1}{c^2 e (d+e x)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(1/(c^2*e*(d + e*x)))

________________________________________________________________________________________

Maple [A]  time = 0.039, size = 16, normalized size = 1.1 \begin{align*} -{\frac{1}{{c}^{2}e \left ( ex+d \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.08718, size = 26, normalized size = 1.73 \begin{align*} -\frac{1}{c^{2} e^{2} x + c^{2} d e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(c^2*e^2*x + c^2*d*e)

________________________________________________________________________________________

Fricas [A]  time = 1.92588, size = 35, normalized size = 2.33 \begin{align*} -\frac{1}{c^{2} e^{2} x + c^{2} d e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(c^2*e^2*x + c^2*d*e)

________________________________________________________________________________________

Sympy [A]  time = 0.52727, size = 17, normalized size = 1.13 \begin{align*} - \frac{1}{c^{2} d e + c^{2} e^{2} x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(c**2*d*e + c**2*e**2*x)

________________________________________________________________________________________

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

[Out]

Exception raised: NotImplementedError