### 3.1843 $$\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^2}{(d+e x)^2} \, dx$$

Optimal. Leaf size=20 $\frac{(a e+c d x)^3}{3 c d}$

[Out]

(a*e + c*d*x)^3/(3*c*d)

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Rubi [A]  time = 0.012909, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 35, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.057, Rules used = {626, 32} $\frac{(a e+c d x)^3}{3 c d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(a*e + c*d*x)^3/(3*c*d)

Rule 626

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c*x)/e)^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&
IntegerQ[p]

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

Mathematica [A]  time = 0.0019549, size = 20, normalized size = 1. $\frac{(a e+c d x)^3}{3 c d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(a*e + c*d*x)^3/(3*c*d)

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Maple [A]  time = 0.039, size = 19, normalized size = 1. \begin{align*}{\frac{ \left ( cdx+ae \right ) ^{3}}{3\,cd}} \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)^2/(e*x+d)^2,x)

[Out]

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

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Maxima [A]  time = 0.988521, size = 38, normalized size = 1.9 \begin{align*} \frac{1}{3} \, c^{2} d^{2} x^{3} + a c d e x^{2} + a^{2} e^{2} 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)^2/(e*x+d)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.51214, size = 58, normalized size = 2.9 \begin{align*} \frac{1}{3} \, c^{2} d^{2} x^{3} + a c d e x^{2} + a^{2} e^{2} 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)^2/(e*x+d)^2,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.14883, size = 29, normalized size = 1.45 \begin{align*} a^{2} e^{2} x + a c d e x^{2} + \frac{c^{2} d^{2} x^{3}}{3} \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)**2/(e*x+d)**2,x)

[Out]

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

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Giac [B]  time = 1.19405, size = 116, normalized size = 5.8 \begin{align*} \frac{1}{3} \,{\left (c^{2} d^{2} - \frac{3 \,{\left (c^{2} d^{3} e - a c d e^{3}\right )} e^{\left (-1\right )}}{x e + d} + \frac{3 \,{\left (c^{2} d^{4} e^{2} - 2 \, a c d^{2} e^{4} + a^{2} e^{6}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}}\right )}{\left (x e + d\right )}^{3} e^{\left (-3\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)^2/(e*x+d)^2,x, algorithm="giac")

[Out]

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