3.985 \(\int \frac{c d^2+2 c d e x+c e^2 x^2}{(d+e x)^5} \, dx\)

Optimal. Leaf size=15 \[ -\frac{c}{2 e (d+e x)^2} \]

[Out]

-c/(2*e*(d + e*x)^2)

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Rubi [A]  time = 0.0063687, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {24, 21, 32} \[ -\frac{c}{2 e (d+e x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-c/(2*e*(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 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

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{c d^2+2 c d e x+c e^2 x^2}{(d+e x)^5} \, dx &=\frac{\int \frac{c d e^2+c e^3 x}{(d+e x)^4} \, dx}{e^2}\\ &=c \int \frac{1}{(d+e x)^3} \, dx\\ &=-\frac{c}{2 e (d+e x)^2}\\ \end{align*}

Mathematica [A]  time = 0.0033624, size = 15, normalized size = 1. \[ -\frac{c}{2 e (d+e x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-c/(2*e*(d + e*x)^2)

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Maple [A]  time = 0.037, size = 14, normalized size = 0.9 \begin{align*} -{\frac{c}{2\,e \left ( ex+d \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*e^2*x^2+2*c*d*e*x+c*d^2)/(e*x+d)^5,x)

[Out]

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

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Maxima [A]  time = 1.18599, size = 34, normalized size = 2.27 \begin{align*} -\frac{c}{2 \,{\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)/(e*x+d)^5,x, algorithm="maxima")

[Out]

-1/2*c/(e^3*x^2 + 2*d*e^2*x + d^2*e)

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Fricas [A]  time = 1.93154, size = 51, normalized size = 3.4 \begin{align*} -\frac{c}{2 \,{\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)/(e*x+d)^5,x, algorithm="fricas")

[Out]

-1/2*c/(e^3*x^2 + 2*d*e^2*x + d^2*e)

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Sympy [B]  time = 0.41448, size = 26, normalized size = 1.73 \begin{align*} - \frac{c}{2 d^{2} e + 4 d e^{2} x + 2 e^{3} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*e**2*x**2+2*c*d*e*x+c*d**2)/(e*x+d)**5,x)

[Out]

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

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Giac [A]  time = 1.16165, size = 18, normalized size = 1.2 \begin{align*} -\frac{c e^{\left (-1\right )}}{2 \,{\left (x e + d\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)/(e*x+d)^5,x, algorithm="giac")

[Out]

-1/2*c*e^(-1)/(x*e + d)^2