### 3.1847 $$\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^2}{(d+e x)^6} \, dx$$

Optimal. Leaf size=35 $\frac{(a e+c d x)^3}{3 (d+e x)^3 \left (c d^2-a e^2\right )}$

[Out]

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

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Rubi [A]  time = 0.0141029, antiderivative size = 35, 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, 37} $\frac{(a e+c d x)^3}{3 (d+e x)^3 \left (c d^2-a e^2\right )}$

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)^6,x]

[Out]

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

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 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[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)^6} \, dx &=\int \frac{(a e+c d x)^2}{(d+e x)^4} \, dx\\ &=\frac{(a e+c d x)^3}{3 \left (c d^2-a e^2\right ) (d+e x)^3}\\ \end{align*}

Mathematica [A]  time = 0.0277713, size = 59, normalized size = 1.69 $-\frac{a^2 e^4+a c d e^2 (d+3 e x)+c^2 d^2 \left (d^2+3 d e x+3 e^2 x^2\right )}{3 e^3 (d+e x)^3}$

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)^6,x]

[Out]

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

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Maple [B]  time = 0.043, size = 83, normalized size = 2.4 \begin{align*} -{\frac{cd \left ( a{e}^{2}-c{d}^{2} \right ) }{{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{{c}^{2}{d}^{2}}{{e}^{3} \left ( ex+d \right ) }}-{\frac{{a}^{2}{e}^{4}-2\,ac{d}^{2}{e}^{2}+{c}^{2}{d}^{4}}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}} \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)^6,x)

[Out]

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

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Maxima [B]  time = 1.0719, size = 127, normalized size = 3.63 \begin{align*} -\frac{3 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + a c d^{2} e^{2} + a^{2} e^{4} + 3 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x}{3 \,{\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{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)^6,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.66687, size = 186, normalized size = 5.31 \begin{align*} -\frac{3 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + a c d^{2} e^{2} + a^{2} e^{4} + 3 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x}{3 \,{\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{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)^6,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 1.54047, size = 99, normalized size = 2.83 \begin{align*} - \frac{a^{2} e^{4} + a c d^{2} e^{2} + c^{2} d^{4} + 3 c^{2} d^{2} e^{2} x^{2} + x \left (3 a c d e^{3} + 3 c^{2} d^{3} e\right )}{3 d^{3} e^{3} + 9 d^{2} e^{4} x + 9 d e^{5} x^{2} + 3 e^{6} x^{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)**6,x)

[Out]

-(a**2*e**4 + a*c*d**2*e**2 + c**2*d**4 + 3*c**2*d**2*e**2*x**2 + x*(3*a*c*d*e**3 + 3*c**2*d**3*e))/(3*d**3*e*
*3 + 9*d**2*e**4*x + 9*d*e**5*x**2 + 3*e**6*x**3)

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Giac [B]  time = 1.22456, size = 185, normalized size = 5.29 \begin{align*} -\frac{{\left (3 \, c^{2} d^{2} x^{4} e^{4} + 9 \, c^{2} d^{3} x^{3} e^{3} + 10 \, c^{2} d^{4} x^{2} e^{2} + 5 \, c^{2} d^{5} x e + c^{2} d^{6} + 3 \, a c d x^{3} e^{5} + 7 \, a c d^{2} x^{2} e^{4} + 5 \, a c d^{3} x e^{3} + a c d^{4} e^{2} + a^{2} x^{2} e^{6} + 2 \, a^{2} d x e^{5} + a^{2} d^{2} e^{4}\right )} e^{\left (-3\right )}}{3 \,{\left (x e + d\right )}^{5}} \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)^6,x, algorithm="giac")

[Out]

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