3.49.70 \(\int -\frac {40}{40 x+11 x^2} \, dx\)

Optimal. Leaf size=17 \[ \log \left (\frac {4 \left (4+\frac {11 x}{10}\right )}{x \log (4)}\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 11, normalized size of antiderivative = 0.65, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {12, 615} \begin {gather*} \log (11 x+40)-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-40/(40*x + 11*x^2),x]

[Out]

-Log[x] + Log[40 + 11*x]

Rule 12

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

Rule 615

Int[((b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[Log[x]/b, x] - Simp[Log[RemoveContent[b + c*x, x]]/b,
x] /; FreeQ[{b, c}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\left (40 \int \frac {1}{40 x+11 x^2} \, dx\right )\\ &=-\log (x)+\log (40+11 x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 19, normalized size = 1.12 \begin {gather*} -40 \left (\frac {\log (x)}{40}-\frac {1}{40} \log (40+11 x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-40/(40*x + 11*x^2),x]

[Out]

-40*(Log[x]/40 - Log[40 + 11*x]/40)

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fricas [A]  time = 0.71, size = 11, normalized size = 0.65 \begin {gather*} \log \left (11 \, x + 40\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-40/(11*x^2+40*x),x, algorithm="fricas")

[Out]

log(11*x + 40) - log(x)

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giac [A]  time = 0.13, size = 13, normalized size = 0.76 \begin {gather*} \log \left ({\left | 11 \, x + 40 \right |}\right ) - \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-40/(11*x^2+40*x),x, algorithm="giac")

[Out]

log(abs(11*x + 40)) - log(abs(x))

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maple [A]  time = 0.11, size = 12, normalized size = 0.71

method	result	size
default	\(\ln \left (11 x +40\right )-\ln \relax (x )\)	\(12\)
norman	\(\ln \left (11 x +40\right )-\ln \relax (x )\)	\(12\)
risch	\(\ln \left (11 x +40\right )-\ln \relax (x )\)	\(12\)
meijerg	\(\ln \left (1+\frac {11 x}{40}\right )-\ln \relax (x )-\ln \left (11\right )+3 \ln \relax (2)+\ln \relax (5)\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-40/(11*x^2+40*x),x,method=_RETURNVERBOSE)

[Out]

ln(11*x+40)-ln(x)

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maxima [A]  time = 0.35, size = 11, normalized size = 0.65 \begin {gather*} \log \left (11 \, x + 40\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-40/(11*x^2+40*x),x, algorithm="maxima")

[Out]

log(11*x + 40) - log(x)

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mupad [B]  time = 0.10, size = 8, normalized size = 0.47 \begin {gather*} 2\,\mathrm {atanh}\left (\frac {11\,x}{20}+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-40/(40*x + 11*x^2),x)

[Out]

2*atanh((11*x)/20 + 1)

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sympy [A]  time = 0.10, size = 8, normalized size = 0.47 \begin {gather*} - \log {\relax (x )} + \log {\left (x + \frac {40}{11} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-40/(11*x**2+40*x),x)

[Out]

-log(x) + log(x + 40/11)

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