### 3.983 $$\int \frac{c d^2+2 c d e x+c e^2 x^2}{(d+e x)^3} \, dx$$

Optimal. Leaf size=11 $\frac{c \log (d+e x)}{e}$

[Out]

(c*Log[d + e*x])/e

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Rubi [A]  time = 0.0061493, antiderivative size = 11, 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, 31} $\frac{c \log (d+e x)}{e}$

Antiderivative was successfully veriﬁed.

[In]

Int[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)/(d + e*x)^3,x]

[Out]

(c*Log[d + e*x])/e

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 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.0013314, size = 11, normalized size = 1. $\frac{c \log (d+e x)}{e}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)/(d + e*x)^3,x]

[Out]

(c*Log[d + e*x])/e

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Maple [A]  time = 0.038, size = 12, normalized size = 1.1 \begin{align*}{\frac{c\ln \left ( ex+d \right ) }{e}} \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)^3,x)

[Out]

c*ln(e*x+d)/e

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Maxima [A]  time = 1.15981, size = 15, normalized size = 1.36 \begin{align*} \frac{c \log \left (e x + d\right )}{e} \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)^3,x, algorithm="maxima")

[Out]

c*log(e*x + d)/e

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Fricas [A]  time = 2.05157, size = 24, normalized size = 2.18 \begin{align*} \frac{c \log \left (e x + d\right )}{e} \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)^3,x, algorithm="fricas")

[Out]

c*log(e*x + d)/e

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Sympy [A]  time = 0.083459, size = 8, normalized size = 0.73 \begin{align*} \frac{c \log{\left (d + e x \right )}}{e} \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)**3,x)

[Out]

c*log(d + e*x)/e

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Giac [A]  time = 1.20382, size = 16, normalized size = 1.45 \begin{align*} c e^{\left (-1\right )} \log \left ({\left | x e + d \right |}\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)^3,x, algorithm="giac")

[Out]

c*e^(-1)*log(abs(x*e + d))