### 3.1009 $$\int \frac{(d+e x)^5}{(c d^2+2 c d e x+c e^2 x^2)^2} \, dx$$

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

[Out]

(d + e*x)^2/(2*c^2*e)

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Rubi [A]  time = 0.0035537, antiderivative size = 17, 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, 9} $\frac{(d+e x)^2}{2 c^2 e}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^5/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2,x]

[Out]

(d + e*x)^2/(2*c^2*e)

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 9

Int[(a_)*((b_) + (c_.)*(x_)), x_Symbol] :> Simp[(a*(b + c*x)^2)/(2*c), x] /; FreeQ[{a, b, c}, x]

Rubi steps

\begin{align*} \int \frac{(d+e x)^5}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx &=\int \frac{d+e x}{c^2} \, dx\\ &=\frac{(d+e x)^2}{2 c^2 e}\\ \end{align*}

Mathematica [A]  time = 0.0006573, size = 16, normalized size = 0.94 $\frac{d x+\frac{e x^2}{2}}{c^2}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^5/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2,x]

[Out]

(d*x + (e*x^2)/2)/c^2

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Maple [A]  time = 0.04, size = 15, normalized size = 0.9 \begin{align*}{\frac{1}{{c}^{2}} \left ({\frac{e{x}^{2}}{2}}+dx \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^5/(c*e^2*x^2+2*c*d*e*x+c*d^2)^2,x)

[Out]

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

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Maxima [A]  time = 1.11812, size = 20, normalized size = 1.18 \begin{align*} \frac{e x^{2} + 2 \, d x}{2 \, c^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(c*e^2*x^2+2*c*d*e*x+c*d^2)^2,x, algorithm="maxima")

[Out]

1/2*(e*x^2 + 2*d*x)/c^2

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Fricas [A]  time = 2.01251, size = 34, normalized size = 2. \begin{align*} \frac{e x^{2} + 2 \, d x}{2 \, c^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(c*e^2*x^2+2*c*d*e*x+c*d^2)^2,x, algorithm="fricas")

[Out]

1/2*(e*x^2 + 2*d*x)/c^2

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Sympy [A]  time = 0.138297, size = 15, normalized size = 0.88 \begin{align*} \frac{d x}{c^{2}} + \frac{e x^{2}}{2 c^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**5/(c*e**2*x**2+2*c*d*e*x+c*d**2)**2,x)

[Out]

d*x/c**2 + e*x**2/(2*c**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((e*x+d)^5/(c*e^2*x^2+2*c*d*e*x+c*d^2)^2,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError