3.11.2 \(\int \frac {d+e x}{c d^2+2 c d e x+c e^2 x^2} \, dx\) [1002]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {\log (d+e x)}{c e} \end {gather*}

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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 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*}

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Mathematica [A]
time = 0.00, size = 16, normalized size = 1.23 \begin {gather*} \frac {\log (c d+c e x)}{c e} \end {gather*}

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 = 0.53, size = 14, normalized size = 1.08

method	result	size
default	\(\frac {\ln \left (e x +d \right )}{c e}\)	\(14\)
norman	\(\frac {\ln \left (e x +d \right )}{c e}\)	\(14\)
risch	\(\frac {\ln \left (e x +d \right )}{c e}\)	\(14\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 28 vs. \(2 (13) = 26\).
time = 0.28, size = 28, normalized size = 2.15 \begin {gather*} \frac {e^{\left (-1\right )} \log \left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}{2 \, c} \end {gather*}

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

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Fricas [A]
time = 2.61, size = 13, normalized size = 1.00 \begin {gather*} \frac {e^{\left (-1\right )} \log \left (x e + d\right )}{c} \end {gather*}

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]

e^(-1)*log(x*e + d)/c

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Sympy [A]
time = 0.02, size = 12, normalized size = 0.92 \begin {gather*} \frac {\log {\left (c d + c e x \right )}}{c e} \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 30 vs. \(2 (13) = 26\).
time = 1.80, size = 30, normalized size = 2.31 \begin {gather*} \frac {e^{\left (-1\right )} \log \left ({\left | c d^{2} + {\left (x^{2} e + 2 \, d x\right )} c e \right |}\right )}{2 \, c} \end {gather*}

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]

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

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Mupad [B]
time = 0.03, size = 13, normalized size = 1.00 \begin {gather*} \frac {\ln \left (d+e\,x\right )}{c\,e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(d + e*x)/(c*e)

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