### 3.1833 $$\int \frac{a d e+(c d^2+a e^2) x+c d e x^2}{d+e x} \, dx$$

Optimal. Leaf size=14 $a e x+\frac{1}{2} c d x^2$

[Out]

a*e*x + (c*d*x^2)/2

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Rubi [A]  time = 0.0079364, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 33, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.03, Rules used = {24} $a e x+\frac{1}{2} c d x^2$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

a*e*x + (c*d*x^2)/2

Rule 24

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((A_.) + (B_.)*(v_) + (C_.)*(v_)^2), x_Symbol] :> Dist[1/b^2, Int[u*(a + b*
v)^(m + 1)*Simp[b*B - a*C + b*C*v, x], x], x] /; FreeQ[{a, b, A, B, C}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0] &&
LeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0008907, size = 14, normalized size = 1. $a e x+\frac{1}{2} c d x^2$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

a*e*x + (c*d*x^2)/2

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Maple [A]  time = 0.038, size = 13, normalized size = 0.9 \begin{align*} aex+{\frac{cd{x}^{2}}{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*e*x+1/2*c*d*x^2

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Maxima [A]  time = 1.03206, size = 16, normalized size = 1.14 \begin{align*} \frac{1}{2} \, c d x^{2} + a e x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*c*d*x^2 + a*e*x

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Fricas [A]  time = 1.59621, size = 28, normalized size = 2. \begin{align*} \frac{1}{2} \, c d x^{2} + a e x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*c*d*x^2 + a*e*x

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Sympy [A]  time = 0.128321, size = 12, normalized size = 0.86 \begin{align*} a e x + \frac{c d x^{2}}{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*e*x + c*d*x**2/2

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Giac [A]  time = 1.15146, size = 26, normalized size = 1.86 \begin{align*} \frac{1}{2} \,{\left (c d x^{2} e^{2} + 2 \, a x e^{3}\right )} e^{\left (-2\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(c*d*x^2*e^2 + 2*a*x*e^3)*e^(-2)