### 3.982 $$\int \frac{c d^2+2 c d e x+c e^2 x^2}{(d+e x)^2} \, dx$$

Optimal. Leaf size=3 $c x$

[Out]

c*x

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Rubi [A]  time = 0.0037241, antiderivative size = 3, 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, 8} $c x$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

c*x

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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

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

Mathematica [A]  time = 0.0003464, size = 3, normalized size = 1. $c x$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

c*x

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Maple [A]  time = 0.038, size = 4, normalized size = 1.3 \begin{align*} cx \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)^2,x)

[Out]

c*x

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Maxima [A]  time = 1.12504, size = 4, normalized size = 1.33 \begin{align*} c x \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)^2,x, algorithm="maxima")

[Out]

c*x

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Fricas [A]  time = 1.9908, size = 7, normalized size = 2.33 \begin{align*} c x \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)^2,x, algorithm="fricas")

[Out]

c*x

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Sympy [A]  time = 0.081061, size = 2, normalized size = 0.67 \begin{align*} c x \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)**2,x)

[Out]

c*x

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Giac [C]  time = 1.16001, size = 149, normalized size = 49.67 \begin{align*} -2 \,{\left (e^{\left (-1\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) - \frac{d e^{\left (-1\right )}}{x e + d}\right )} c d +{\left (2 \, d e^{\left (-3\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) +{\left (x e + d\right )} e^{\left (-3\right )} - \frac{d^{2} e^{\left (-3\right )}}{x e + d}\right )} c e^{2} - \frac{c d^{2} e^{\left (-1\right )}}{x e + d} \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)^2,x, algorithm="giac")

[Out]

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