### 3.999 $$\int \frac{(d+e x)^4}{c d^2+2 c d e x+c e^2 x^2} \, dx$$

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

[Out]

(d + e*x)^3/(3*c*e)

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Rubi [A]  time = 0.0047593, 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} $\frac{(d+e x)^3}{3 c e}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^4/(c*d^2 + 2*c*d*e*x + c*e^2*x^2),x]

[Out]

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

Mathematica [A]  time = 0.0008047, size = 17, normalized size = 1. $\frac{(d+e x)^3}{3 c e}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^4/(c*d^2 + 2*c*d*e*x + c*e^2*x^2),x]

[Out]

(d + e*x)^3/(3*c*e)

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Maple [A]  time = 0.039, size = 16, normalized size = 0.9 \begin{align*}{\frac{ \left ( ex+d \right ) ^{3}}{3\,ce}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.10039, size = 35, normalized size = 2.06 \begin{align*} \frac{e^{2} x^{3} + 3 \, d e x^{2} + 3 \, d^{2} x}{3 \, c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4/(c*e^2*x^2+2*c*d*e*x+c*d^2),x, algorithm="maxima")

[Out]

1/3*(e^2*x^3 + 3*d*e*x^2 + 3*d^2*x)/c

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Fricas [A]  time = 1.92547, size = 53, normalized size = 3.12 \begin{align*} \frac{e^{2} x^{3} + 3 \, d e x^{2} + 3 \, d^{2} x}{3 \, c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4/(c*e^2*x^2+2*c*d*e*x+c*d^2),x, algorithm="fricas")

[Out]

1/3*(e^2*x^3 + 3*d*e*x^2 + 3*d^2*x)/c

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Sympy [B]  time = 0.117196, size = 24, normalized size = 1.41 \begin{align*} \frac{d^{2} x}{c} + \frac{d e x^{2}}{c} + \frac{e^{2} x^{3}}{3 c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**4/(c*e**2*x**2+2*c*d*e*x+c*d**2),x)

[Out]

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

[Out]

Exception raised: NotImplementedError