### 3.1891 $$\int \frac{(d+e x)^3}{(a d e+(c d^2+a e^2) x+c d e x^2)^3} \, dx$$

Optimal. Leaf size=20 $-\frac{1}{2 c d (a e+c d x)^2}$

[Out]

-1/(2*c*d*(a*e + c*d*x)^2)

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Rubi [A]  time = 0.0102619, 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} $-\frac{1}{2 c d (a e+c d x)^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-1/(2*c*d*(a*e + c*d*x)^2)

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

Mathematica [A]  time = 0.0035411, size = 20, normalized size = 1. $-\frac{1}{2 c d (a e+c d x)^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-1/(2*c*d*(a*e + c*d*x)^2)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x)

[Out]

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

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Maxima [A]  time = 1.0685, size = 47, normalized size = 2.35 \begin{align*} -\frac{1}{2 \,{\left (c^{3} d^{3} x^{2} + 2 \, a c^{2} d^{2} e x + a^{2} c d e^{2}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.94773, size = 70, normalized size = 3.5 \begin{align*} -\frac{1}{2 \,{\left (c^{3} d^{3} x^{2} + 2 \, a c^{2} d^{2} e x + a^{2} c d e^{2}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.450763, size = 39, normalized size = 1.95 \begin{align*} - \frac{1}{2 a^{2} c d e^{2} + 4 a c^{2} d^{2} e x + 2 c^{3} d^{3} x^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)

[Out]

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

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Giac [B]  time = 1.37128, size = 350, normalized size = 17.5 \begin{align*} -\frac{c^{4} d^{8} x^{2} e^{2} + 2 \, c^{4} d^{9} x e + c^{4} d^{10} - 4 \, a c^{3} d^{6} x^{2} e^{4} - 8 \, a c^{3} d^{7} x e^{3} - 4 \, a c^{3} d^{8} e^{2} + 6 \, a^{2} c^{2} d^{4} x^{2} e^{6} + 12 \, a^{2} c^{2} d^{5} x e^{5} + 6 \, a^{2} c^{2} d^{6} e^{4} - 4 \, a^{3} c d^{2} x^{2} e^{8} - 8 \, a^{3} c d^{3} x e^{7} - 4 \, a^{3} c d^{4} e^{6} + a^{4} x^{2} e^{10} + 2 \, a^{4} d x e^{9} + a^{4} d^{2} e^{8}}{2 \,{\left (c^{5} d^{9} - 4 \, a c^{4} d^{7} e^{2} + 6 \, a^{2} c^{3} d^{5} e^{4} - 4 \, a^{3} c^{2} d^{3} e^{6} + a^{4} c d e^{8}\right )}{\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(c^4*d^8*x^2*e^2 + 2*c^4*d^9*x*e + c^4*d^10 - 4*a*c^3*d^6*x^2*e^4 - 8*a*c^3*d^7*x*e^3 - 4*a*c^3*d^8*e^2 +
6*a^2*c^2*d^4*x^2*e^6 + 12*a^2*c^2*d^5*x*e^5 + 6*a^2*c^2*d^6*e^4 - 4*a^3*c*d^2*x^2*e^8 - 8*a^3*c*d^3*x*e^7 -
4*a^3*c*d^4*e^6 + a^4*x^2*e^10 + 2*a^4*d*x*e^9 + a^4*d^2*e^8)/((c^5*d^9 - 4*a*c^4*d^7*e^2 + 6*a^2*c^3*d^5*e^4
- 4*a^3*c^2*d^3*e^6 + a^4*c*d*e^8)*(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)^2)