### 3.1020 $$\int \frac{(d+e x)^6}{(c d^2+2 c d e x+c e^2 x^2)^3} \, dx$$

Optimal. Leaf size=5 $\frac{x}{c^3}$

[Out]

x/c^3

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Rubi [A]  time = 0.002476, 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} $\frac{x}{c^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^6/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^3,x]

[Out]

x/c^3

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)^6}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx &=\int \frac{1}{c^3} \, dx\\ &=\frac{x}{c^3}\\ \end{align*}

Mathematica [A]  time = 0.0003703, size = 5, normalized size = 1. $\frac{x}{c^3}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^6/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^3,x]

[Out]

x/c^3

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Maple [A]  time = 0.038, size = 6, normalized size = 1.2 \begin{align*}{\frac{x}{{c}^{3}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^6/(c*e^2*x^2+2*c*d*e*x+c*d^2)^3,x)

[Out]

x/c^3

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Maxima [A]  time = 0.987178, size = 7, normalized size = 1.4 \begin{align*} \frac{x}{c^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^6/(c*e^2*x^2+2*c*d*e*x+c*d^2)^3,x, algorithm="maxima")

[Out]

x/c^3

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Fricas [A]  time = 2.01358, size = 9, normalized size = 1.8 \begin{align*} \frac{x}{c^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^6/(c*e^2*x^2+2*c*d*e*x+c*d^2)^3,x, algorithm="fricas")

[Out]

x/c^3

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Sympy [A]  time = 0.134455, size = 3, normalized size = 0.6 \begin{align*} \frac{x}{c^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**6/(c*e**2*x**2+2*c*d*e*x+c*d**2)**3,x)

[Out]

x/c**3

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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)^6/(c*e^2*x^2+2*c*d*e*x+c*d^2)^3,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError