### 3.1001 $$\int \frac{(d+e x)^2}{c d^2+2 c d e x+c e^2 x^2} \, dx$$

Optimal. Leaf size=5 $\frac{x}{c}$

[Out]

x/c

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Rubi [A]  time = 0.0023776, 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} $\frac{x}{c}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

x/c

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

Mathematica [A]  time = 0.0002426, size = 5, normalized size = 1. $\frac{x}{c}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

x/c

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Maple [A]  time = 0.037, size = 6, normalized size = 1.2 \begin{align*}{\frac{x}{c}} \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),x)

[Out]

x/c

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Maxima [A]  time = 1.25048, size = 7, normalized size = 1.4 \begin{align*} \frac{x}{c} \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),x, algorithm="maxima")

[Out]

x/c

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Fricas [A]  time = 2.0046, size = 7, normalized size = 1.4 \begin{align*} \frac{x}{c} \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),x, algorithm="fricas")

[Out]

x/c

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Sympy [A]  time = 0.096697, size = 2, normalized size = 0.4 \begin{align*} \frac{x}{c} \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),x)

[Out]

x/c

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

[Out]

Exception raised: NotImplementedError