### 3.1856 $$\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^3}{(d+e x)^3} \, dx$$

Optimal. Leaf size=20 $\frac{(a e+c d x)^4}{4 c d}$

[Out]

(a*e + c*d*x)^4/(4*c*d)

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Rubi [A]  time = 0.0133856, 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} $\frac{(a e+c d x)^4}{4 c d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0025022, size = 20, normalized size = 1. $\frac{(a e+c d x)^4}{4 c d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(a*e + c*d*x)^4/(4*c*d)

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Maple [A]  time = 0.04, size = 19, normalized size = 1. \begin{align*}{\frac{ \left ( cdx+ae \right ) ^{4}}{4\,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)^3/(e*x+d)^3,x)

[Out]

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

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Maxima [B]  time = 1.08492, size = 61, normalized size = 3.05 \begin{align*} \frac{1}{4} \, c^{3} d^{3} x^{4} + a c^{2} d^{2} e x^{3} + \frac{3}{2} \, a^{2} c d e^{2} x^{2} + a^{3} e^{3} 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)^3/(e*x+d)^3,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.52203, size = 93, normalized size = 4.65 \begin{align*} \frac{1}{4} \, c^{3} d^{3} x^{4} + a c^{2} d^{2} e x^{3} + \frac{3}{2} \, a^{2} c d e^{2} x^{2} + a^{3} e^{3} 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)^3/(e*x+d)^3,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.137744, size = 49, normalized size = 2.45 \begin{align*} a^{3} e^{3} x + \frac{3 a^{2} c d e^{2} x^{2}}{2} + a c^{2} d^{2} e x^{3} + \frac{c^{3} d^{3} x^{4}}{4} \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)**3/(e*x+d)**3,x)

[Out]

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

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Giac [B]  time = 1.23946, size = 69, normalized size = 3.45 \begin{align*} \frac{1}{4} \,{\left (c^{3} d^{3} x^{4} e^{12} + 4 \, a c^{2} d^{2} x^{3} e^{13} + 6 \, a^{2} c d x^{2} e^{14} + 4 \, a^{3} x e^{15}\right )} e^{\left (-12\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)^3/(e*x+d)^3,x, algorithm="giac")

[Out]

1/4*(c^3*d^3*x^4*e^12 + 4*a*c^2*d^2*x^3*e^13 + 6*a^2*c*d*x^2*e^14 + 4*a^3*x*e^15)*e^(-12)