### 3.1018 $$\int \frac{(d+e x)^8}{(c d^2+2 c d e x+c e^2 x^2)^3} \, dx$$

Optimal. Leaf size=17 $\frac{(d+e x)^3}{3 c^3 e}$

[Out]

(d + e*x)^3/(3*c^3*e)

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Rubi [A]  time = 0.0048614, 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} $\frac{(d+e x)^3}{3 c^3 e}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^8/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^3,x]

[Out]

(d + e*x)^3/(3*c^3*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)^8}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx &=\int \frac{(d+e x)^2}{c^3} \, dx\\ &=\frac{\int (d+e x)^2 \, dx}{c^3}\\ &=\frac{(d+e x)^3}{3 c^3 e}\\ \end{align*}

Mathematica [A]  time = 0.0009025, size = 17, normalized size = 1. $\frac{(d+e x)^3}{3 c^3 e}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^8/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^3,x]

[Out]

(d + e*x)^3/(3*c^3*e)

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Maple [A]  time = 0.04, size = 16, normalized size = 0.9 \begin{align*}{\frac{ \left ( ex+d \right ) ^{3}}{3\,{c}^{3}e}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^8/(c*e^2*x^2+2*c*d*e*x+c*d^2)^3,x)

[Out]

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

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Maxima [A]  time = 1.07916, size = 35, normalized size = 2.06 \begin{align*} \frac{e^{2} x^{3} + 3 \, d e x^{2} + 3 \, d^{2} x}{3 \, c^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^8/(c*e^2*x^2+2*c*d*e*x+c*d^2)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.98393, size = 55, normalized size = 3.24 \begin{align*} \frac{e^{2} x^{3} + 3 \, d e x^{2} + 3 \, d^{2} x}{3 \, c^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^8/(c*e^2*x^2+2*c*d*e*x+c*d^2)^3,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.145524, size = 29, normalized size = 1.71 \begin{align*} \frac{d^{2} x}{c^{3}} + \frac{d e x^{2}}{c^{3}} + \frac{e^{2} x^{3}}{3 c^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**8/(c*e**2*x**2+2*c*d*e*x+c*d**2)**3,x)

[Out]

d**2*x/c**3 + d*e*x**2/c**3 + e**2*x**3/(3*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)^8/(c*e^2*x^2+2*c*d*e*x+c*d^2)^3,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError