### 3.995 $$\int \frac{(c d^2+2 c d e x+c e^2 x^2)^2}{(d+e x)^6} \, dx$$

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

[Out]

-(c^2/(e*(d + e*x)))

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Rubi [A]  time = 0.0046743, 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} $-\frac{c^2}{e (d+e x)}$

Antiderivative was successfully veriﬁed.

[In]

Int[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2/(d + e*x)^6,x]

[Out]

-(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{\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{(d+e x)^6} \, dx &=\int \frac{c^2}{(d+e x)^2} \, dx\\ &=c^2 \int \frac{1}{(d+e x)^2} \, dx\\ &=-\frac{c^2}{e (d+e x)}\\ \end{align*}

Mathematica [A]  time = 0.0017567, size = 15, normalized size = 1. $-\frac{c^2}{e (d+e x)}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2/(d + e*x)^6,x]

[Out]

-(c^2/(e*(d + e*x)))

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*e^2*x^2+2*c*d*e*x+c*d^2)^2/(e*x+d)^6,x)

[Out]

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

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Maxima [A]  time = 1.17471, size = 22, normalized size = 1.47 \begin{align*} -\frac{c^{2}}{e^{2} x + d e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)^2/(e*x+d)^6,x, algorithm="maxima")

[Out]

-c^2/(e^2*x + d*e)

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Fricas [A]  time = 2.03494, size = 27, normalized size = 1.8 \begin{align*} -\frac{c^{2}}{e^{2} x + d e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)^2/(e*x+d)^6,x, algorithm="fricas")

[Out]

-c^2/(e^2*x + d*e)

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Sympy [A]  time = 0.314408, size = 12, normalized size = 0.8 \begin{align*} - \frac{c^{2}}{d e + e^{2} x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*e**2*x**2+2*c*d*e*x+c*d**2)**2/(e*x+d)**6,x)

[Out]

-c**2/(d*e + e**2*x)

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Giac [B]  time = 1.20877, size = 89, normalized size = 5.93 \begin{align*} -\frac{{\left (c^{2} x^{4} e^{8} + 4 \, c^{2} d x^{3} e^{7} + 6 \, c^{2} d^{2} x^{2} e^{6} + 4 \, c^{2} d^{3} x e^{5} + c^{2} d^{4} e^{4}\right )} e^{\left (-5\right )}}{{\left (x e + d\right )}^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)^2/(e*x+d)^6,x, algorithm="giac")

[Out]

-(c^2*x^4*e^8 + 4*c^2*d*x^3*e^7 + 6*c^2*d^2*x^2*e^6 + 4*c^2*d^3*x*e^5 + c^2*d^4*e^4)*e^(-5)/(x*e + d)^5