### 3.1007 $$\int \frac{(d+e x)^7}{(c d^2+2 c d e x+c e^2 x^2)^2} \, dx$$

Optimal. Leaf size=17 $\frac{(d+e x)^4}{4 c^2 e}$

[Out]

(d + e*x)^4/(4*c^2*e)

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Rubi [A]  time = 0.0049706, antiderivative size = 17, 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} $\frac{(d+e x)^4}{4 c^2 e}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^7/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2,x]

[Out]

(d + e*x)^4/(4*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 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{(d+e x)^7}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx &=\int \frac{(d+e x)^3}{c^2} \, dx\\ &=\frac{\int (d+e x)^3 \, dx}{c^2}\\ &=\frac{(d+e x)^4}{4 c^2 e}\\ \end{align*}

Mathematica [A]  time = 0.0010034, size = 17, normalized size = 1. $\frac{(d+e x)^4}{4 c^2 e}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^7/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2,x]

[Out]

(d + e*x)^4/(4*c^2*e)

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Maple [A]  time = 0.038, size = 16, normalized size = 0.9 \begin{align*}{\frac{ \left ( ex+d \right ) ^{4}}{4\,{c}^{2}e}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^7/(c*e^2*x^2+2*c*d*e*x+c*d^2)^2,x)

[Out]

1/4*(e*x+d)^4/c^2/e

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Maxima [B]  time = 1.02417, size = 50, normalized size = 2.94 \begin{align*} \frac{e^{3} x^{4} + 4 \, d e^{2} x^{3} + 6 \, d^{2} e x^{2} + 4 \, d^{3} x}{4 \, c^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^7/(c*e^2*x^2+2*c*d*e*x+c*d^2)^2,x, algorithm="maxima")

[Out]

1/4*(e^3*x^4 + 4*d*e^2*x^3 + 6*d^2*e*x^2 + 4*d^3*x)/c^2

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Fricas [B]  time = 2.06592, size = 77, normalized size = 4.53 \begin{align*} \frac{e^{3} x^{4} + 4 \, d e^{2} x^{3} + 6 \, d^{2} e x^{2} + 4 \, d^{3} x}{4 \, c^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^7/(c*e^2*x^2+2*c*d*e*x+c*d^2)^2,x, algorithm="fricas")

[Out]

1/4*(e^3*x^4 + 4*d*e^2*x^3 + 6*d^2*e*x^2 + 4*d^3*x)/c^2

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Sympy [B]  time = 0.162961, size = 46, normalized size = 2.71 \begin{align*} \frac{d^{3} x}{c^{2}} + \frac{3 d^{2} e x^{2}}{2 c^{2}} + \frac{d e^{2} x^{3}}{c^{2}} + \frac{e^{3} x^{4}}{4 c^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**7/(c*e**2*x**2+2*c*d*e*x+c*d**2)**2,x)

[Out]

d**3*x/c**2 + 3*d**2*e*x**2/(2*c**2) + d*e**2*x**3/c**2 + e**3*x**4/(4*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)^7/(c*e^2*x^2+2*c*d*e*x+c*d^2)^2,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError