### 3.1002 $$\int \frac{d+e x}{c d^2+2 c d e x+c e^2 x^2} \, dx$$

Optimal. Leaf size=13 $\frac{\log (d+e x)}{c e}$

[Out]

Log[d + e*x]/(c*e)

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Rubi [A]  time = 0.0040553, antiderivative size = 13, 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 = {27, 12, 31} $\frac{\log (d+e x)}{c e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

Log[d + e*x]/(c*e)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[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{d+e x}{c d^2+2 c d e x+c e^2 x^2} \, dx &=\int \frac{1}{c (d+e x)} \, dx\\ &=\frac{\int \frac{1}{d+e x} \, dx}{c}\\ &=\frac{\log (d+e x)}{c e}\\ \end{align*}

Mathematica [A]  time = 0.0017076, size = 16, normalized size = 1.23 $\frac{\log (c d+c e x)}{c e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

Log[c*d + c*e*x]/(c*e)

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Maple [A]  time = 0.039, size = 14, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( ex+d \right ) }{ce}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.17815, size = 39, normalized size = 3. \begin{align*} \frac{\log \left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}{2 \, c e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.91153, size = 27, normalized size = 2.08 \begin{align*} \frac{\log \left (e x + d\right )}{c e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(e*x + d)/(c*e)

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Sympy [A]  time = 0.137631, size = 12, normalized size = 0.92 \begin{align*} \frac{\log{\left (c d + c e x \right )}}{c e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(c*d + c*e*x)/(c*e)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError