### 3.990 $$\int \frac{(c d^2+2 c d e x+c e^2 x^2)^2}{d+e x} \, dx$$

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

[Out]

(c^2*(d + e*x)^4)/(4*e)

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Rubi [A]  time = 0.0047706, 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} $\frac{c^2 (d+e x)^4}{4 e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0014003, size = 17, normalized size = 1. $\frac{c^2 (d+e x)^4}{4 e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(c^2*(d + e*x)^4)/(4*e)

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Maple [B]  time = 0.039, size = 36, normalized size = 2.1 \begin{align*}{c}^{2} \left ({\frac{{e}^{3}{x}^{4}}{4}}+d{e}^{2}{x}^{3}+{\frac{3\,{d}^{2}e{x}^{2}}{2}}+{d}^{3}x \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.20283, size = 58, normalized size = 3.41 \begin{align*} \frac{1}{4} \, c^{2} e^{3} x^{4} + c^{2} d e^{2} x^{3} + \frac{3}{2} \, c^{2} d^{2} e x^{2} + c^{2} d^{3} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.9784, size = 88, normalized size = 5.18 \begin{align*} \frac{1}{4} \, c^{2} e^{3} x^{4} + c^{2} d e^{2} x^{3} + \frac{3}{2} \, c^{2} d^{2} e x^{2} + c^{2} d^{3} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.104091, size = 46, normalized size = 2.71 \begin{align*} c^{2} d^{3} x + \frac{3 c^{2} d^{2} e x^{2}}{2} + c^{2} d e^{2} x^{3} + \frac{c^{2} e^{3} x^{4}}{4} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**2*d**3*x + 3*c**2*d**2*e*x**2/2 + c**2*d*e**2*x**3 + c**2*e**3*x**4/4

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Giac [B]  time = 1.17281, size = 66, normalized size = 3.88 \begin{align*} \frac{1}{4} \,{\left (c^{2} x^{4} e^{7} + 4 \, c^{2} d x^{3} e^{6} + 6 \, c^{2} d^{2} x^{2} e^{5} + 4 \, c^{2} d^{3} x e^{4}\right )} e^{\left (-4\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(c^2*x^4*e^7 + 4*c^2*d*x^3*e^6 + 6*c^2*d^2*x^2*e^5 + 4*c^2*d^3*x*e^4)*e^(-4)