### 3.1004 $$\int \frac{1}{(d+e x) (c d^2+2 c d e x+c e^2 x^2)} \, dx$$

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

[Out]

-1/(2*c*e*(d + e*x)^2)

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Rubi [A]  time = 0.0047097, antiderivative size = 17, 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} $-\frac{1}{2 c e (d+e x)^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/((d + e*x)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)),x]

[Out]

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

Mathematica [A]  time = 0.0020022, size = 17, normalized size = 1. $-\frac{1}{2 c e (d+e x)^2}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/((d + e*x)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)),x]

[Out]

-1/(2*c*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\,ce \left ( ex+d \right ) ^{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.08266, size = 36, normalized size = 2.12 \begin{align*} -\frac{1}{2 \,{\left (c e^{3} x^{2} + 2 \, c d e^{2} x + c d^{2} e\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(c*e^2*x^2+2*c*d*e*x+c*d^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.09091, size = 57, normalized size = 3.35 \begin{align*} -\frac{1}{2 \,{\left (c e^{3} x^{2} + 2 \, c d e^{2} x + c d^{2} e\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(c*e^2*x^2+2*c*d*e*x+c*d^2),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.506623, size = 31, normalized size = 1.82 \begin{align*} - \frac{1}{2 c d^{2} e + 4 c d e^{2} x + 2 c e^{3} x^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(c*e**2*x**2+2*c*d*e*x+c*d**2),x)

[Out]

-1/(2*c*d**2*e + 4*c*d*e**2*x + 2*c*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(1/(e*x+d)/(c*e^2*x^2+2*c*d*e*x+c*d^2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError