### 3.993 $$\int \frac{(c d^2+2 c d e x+c e^2 x^2)^2}{(d+e x)^4} \, dx$$

Optimal. Leaf size=5 $c^2 x$

[Out]

c^2*x

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Rubi [A]  time = 0.0022651, antiderivative size = 5, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.067, Rules used = {27, 8} $c^2 x$

Antiderivative was successfully veriﬁed.

[In]

Int[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2/(d + e*x)^4,x]

[Out]

c^2*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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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)^4} \, dx &=\int c^2 \, dx\\ &=c^2 x\\ \end{align*}

Mathematica [A]  time = 0.0002489, size = 5, normalized size = 1. $c^2 x$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2/(d + e*x)^4,x]

[Out]

c^2*x

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Maple [A]  time = 0.04, size = 6, normalized size = 1.2 \begin{align*}{c}^{2}x \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)^4,x)

[Out]

c^2*x

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Maxima [A]  time = 1.14836, size = 7, normalized size = 1.4 \begin{align*} c^{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)^4,x, algorithm="maxima")

[Out]

c^2*x

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Fricas [A]  time = 2.01003, size = 9, normalized size = 1.8 \begin{align*} c^{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)^4,x, algorithm="fricas")

[Out]

c^2*x

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Sympy [A]  time = 0.095872, size = 3, normalized size = 0.6 \begin{align*} c^{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)**4,x)

[Out]

c**2*x

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Giac [A]  time = 1.16299, size = 7, normalized size = 1.4 \begin{align*} c^{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)^4,x, algorithm="giac")

[Out]

c^2*x