### 3.1837 $$\int \frac{a d e+(c d^2+a e^2) x+c d e x^2}{(d+e x)^5} \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.0276398, 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} $-\frac{a-\frac{c d^2}{e^2}}{3 (d+e x)^3}-\frac{c d}{2 e^2 (d+e x)^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x)^5,x]

[Out]

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

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

Mathematica [A]  time = 0.0118837, size = 30, normalized size = 0.77 $-\frac{2 a e^2+c d (d+3 e x)}{6 e^2 (d+e x)^3}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x)^5,x]

[Out]

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

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Maple [A]  time = 0.044, size = 40, normalized size = 1. \begin{align*} -{\frac{cd}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}-{\frac{a{e}^{2}-c{d}^{2}}{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)^5,x)

[Out]

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

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Maxima [A]  time = 1.12517, size = 74, normalized size = 1.9 \begin{align*} -\frac{3 \, c d e x + c d^{2} + 2 \, a e^{2}}{6 \,{\left (e^{5} x^{3} + 3 \, d e^{4} x^{2} + 3 \, d^{2} e^{3} x + d^{3} 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)^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.56125, size = 113, normalized size = 2.9 \begin{align*} -\frac{3 \, c d e x + c d^{2} + 2 \, a e^{2}}{6 \,{\left (e^{5} x^{3} + 3 \, d e^{4} x^{2} + 3 \, d^{2} e^{3} x + d^{3} 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)^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.85635, size = 58, normalized size = 1.49 \begin{align*} - \frac{2 a e^{2} + c d^{2} + 3 c d e x}{6 d^{3} e^{2} + 18 d^{2} e^{3} x + 18 d e^{4} x^{2} + 6 e^{5} 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)/(e*x+d)**5,x)

[Out]

-(2*a*e**2 + c*d**2 + 3*c*d*e*x)/(6*d**3*e**2 + 18*d**2*e**3*x + 18*d*e**4*x**2 + 6*e**5*x**3)

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Giac [A]  time = 1.27447, size = 57, normalized size = 1.46 \begin{align*} -\frac{c d e^{\left (-2\right )}}{2 \,{\left (x e + d\right )}^{2}} + \frac{c d^{2} e^{\left (-2\right )}}{3 \,{\left (x e + d\right )}^{3}} - \frac{a}{3 \,{\left (x e + d\right )}^{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)/(e*x+d)^5,x, algorithm="giac")

[Out]

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