3.1836 \(\int \frac{a d e+(c d^2+a e^2) x+c d e x^2}{(d+e x)^4} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0137279, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used = {24, 37} \[ \frac{(a e+c d x)^2}{2 (d+e x)^2 \left (c d^2-a e^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x)^4,x]

[Out]

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

Mathematica [A]  time = 0.0109906, size = 29, normalized size = 0.83 \[ -\frac{a e^2+c d (d+2 e x)}{2 e^2 (d+e x)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x)^4,x]

[Out]

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

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Maple [A]  time = 0.043, size = 40, normalized size = 1.1 \begin{align*} -{\frac{a{e}^{2}-c{d}^{2}}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}-{\frac{cd}{{e}^{2} \left ( ex+d \right ) }} \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)^4,x)

[Out]

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

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Maxima [A]  time = 1.05566, size = 58, normalized size = 1.66 \begin{align*} -\frac{2 \, c d e x + c d^{2} + a e^{2}}{2 \,{\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} 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)^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.59144, size = 89, normalized size = 2.54 \begin{align*} -\frac{2 \, c d e x + c d^{2} + a e^{2}}{2 \,{\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} 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)^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.830981, size = 44, normalized size = 1.26 \begin{align*} - \frac{a e^{2} + c d^{2} + 2 c d e x}{2 d^{2} e^{2} + 4 d e^{3} x + 2 e^{4} x^{2}} \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)**4,x)

[Out]

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

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Giac [A]  time = 1.17246, size = 62, normalized size = 1.77 \begin{align*} -\frac{{\left (2 \, c d x^{2} e^{2} + 3 \, c d^{2} x e + c d^{3} + a x e^{3} + a d e^{2}\right )} e^{\left (-2\right )}}{2 \,{\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)^4,x, algorithm="giac")

[Out]

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