### 3.1025 $$\int \frac{d+e x}{(c d^2+2 c d e x+c e^2 x^2)^3} \, dx$$

Optimal. Leaf size=17 $-\frac{1}{4 c^3 e (d+e x)^4}$

[Out]

-1/(4*c^3*e*(d + e*x)^4)

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Rubi [A]  time = 0.0044241, 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} $-\frac{1}{4 c^3 e (d+e x)^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-1/(4*c^3*e*(d + e*x)^4)

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 )^3} \, dx &=\int \frac{1}{c^3 (d+e x)^5} \, dx\\ &=\frac{\int \frac{1}{(d+e x)^5} \, dx}{c^3}\\ &=-\frac{1}{4 c^3 e (d+e x)^4}\\ \end{align*}

Mathematica [A]  time = 0.0026144, size = 17, normalized size = 1. $-\frac{1}{4 c^3 e (d+e x)^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-1/(4*c^3*e*(d + e*x)^4)

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Maple [A]  time = 0.042, size = 16, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,{c}^{3}e \left ( ex+d \right ) ^{4}}} \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)^3,x)

[Out]

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

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Maxima [B]  time = 0.97872, size = 41, normalized size = 2.41 \begin{align*} -\frac{1}{4 \,{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{2} 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)^3,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.02453, size = 119, normalized size = 7. \begin{align*} -\frac{1}{4 \,{\left (c^{3} e^{5} x^{4} + 4 \, c^{3} d e^{4} x^{3} + 6 \, c^{3} d^{2} e^{3} x^{2} + 4 \, c^{3} d^{3} e^{2} x + c^{3} d^{4} 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)^3,x, algorithm="fricas")

[Out]

-1/4/(c^3*e^5*x^4 + 4*c^3*d*e^4*x^3 + 6*c^3*d^2*e^3*x^2 + 4*c^3*d^3*e^2*x + c^3*d^4*e)

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Sympy [B]  time = 0.772306, size = 66, normalized size = 3.88 \begin{align*} - \frac{1}{4 c^{3} d^{4} e + 16 c^{3} d^{3} e^{2} x + 24 c^{3} d^{2} e^{3} x^{2} + 16 c^{3} d e^{4} x^{3} + 4 c^{3} e^{5} x^{4}} \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)**3,x)

[Out]

-1/(4*c**3*d**4*e + 16*c**3*d**3*e**2*x + 24*c**3*d**2*e**3*x**2 + 16*c**3*d*e**4*x**3 + 4*c**3*e**5*x**4)

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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)^3,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError