### 3.1838 $$\int \frac{a d e+(c d^2+a e^2) x+c d e x^2}{(d+e x)^6} \, dx$$

Optimal. Leaf size=39 $-\frac{a-\frac{c d^2}{e^2}}{4 (d+e x)^4}-\frac{c d}{3 e^2 (d+e x)^3}$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0278199, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.061, Rules used = {24, 43} $-\frac{a-\frac{c d^2}{e^2}}{4 (d+e x)^4}-\frac{c d}{3 e^2 (d+e x)^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x)^6,x]

[Out]

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

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]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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)^6} \, dx &=\frac{\int \frac{a e^3+c d e^2 x}{(d+e x)^5} \, dx}{e^2}\\ &=\frac{\int \left (\frac{-c d^2 e+a e^3}{(d+e x)^5}+\frac{c d e}{(d+e x)^4}\right ) \, dx}{e^2}\\ &=-\frac{a-\frac{c d^2}{e^2}}{4 (d+e x)^4}-\frac{c d}{3 e^2 (d+e x)^3}\\ \end{align*}

Mathematica [A]  time = 0.012282, size = 30, normalized size = 0.77 $-\frac{3 a e^2+c d (d+4 e x)}{12 e^2 (d+e x)^4}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x)^6,x]

[Out]

-(3*a*e^2 + c*d*(d + 4*e*x))/(12*e^2*(d + e*x)^4)

________________________________________________________________________________________

Maple [A]  time = 0.043, size = 40, normalized size = 1. \begin{align*} -{\frac{a{e}^{2}-c{d}^{2}}{4\,{e}^{2} \left ( ex+d \right ) ^{4}}}-{\frac{cd}{3\,{e}^{2} \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)/(e*x+d)^6,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.17402, size = 89, normalized size = 2.28 \begin{align*} -\frac{4 \, c d e x + c d^{2} + 3 \, a e^{2}}{12 \,{\left (e^{6} x^{4} + 4 \, d e^{5} x^{3} + 6 \, d^{2} e^{4} x^{2} + 4 \, d^{3} e^{3} x + d^{4} e^{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)^6,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.44876, size = 136, normalized size = 3.49 \begin{align*} -\frac{4 \, c d e x + c d^{2} + 3 \, a e^{2}}{12 \,{\left (e^{6} x^{4} + 4 \, d e^{5} x^{3} + 6 \, d^{2} e^{4} x^{2} + 4 \, d^{3} e^{3} x + d^{4} e^{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)^6,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [B]  time = 0.842324, size = 70, normalized size = 1.79 \begin{align*} - \frac{3 a e^{2} + c d^{2} + 4 c d e x}{12 d^{4} e^{2} + 48 d^{3} e^{3} x + 72 d^{2} e^{4} x^{2} + 48 d e^{5} x^{3} + 12 e^{6} x^{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)/(e*x+d)**6,x)

[Out]

-(3*a*e**2 + c*d**2 + 4*c*d*e*x)/(12*d**4*e**2 + 48*d**3*e**3*x + 72*d**2*e**4*x**2 + 48*d*e**5*x**3 + 12*e**6
*x**4)

________________________________________________________________________________________

Giac [A]  time = 1.22475, size = 65, normalized size = 1.67 \begin{align*} -\frac{{\left (4 \, c d x^{2} e^{2} + 5 \, c d^{2} x e + c d^{3} + 3 \, a x e^{3} + 3 \, a d e^{2}\right )} e^{\left (-2\right )}}{12 \,{\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)/(e*x+d)^6,x, algorithm="giac")

[Out]

-1/12*(4*c*d*x^2*e^2 + 5*c*d^2*x*e + c*d^3 + 3*a*x*e^3 + 3*a*d*e^2)*e^(-2)/(x*e + d)^5