3.49.70 $\int -\frac{40}{40x+11x^2}dx$

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

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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} $$\begin{array}{c}\log (11x+40)-\log (x)\end{array}$$

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =-(40\int \frac{1}{40x+11x^2}dx)\\ & =-\log (x)+\log (40+11x)\end{array}\end{array}$$

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Mathematica [A]  time = 0.00, size = 19, normalized size = 1.12 $$\begin{array}{c}-40(\frac{\log (x)}{40}-\frac{1}{40}\log (40+11x))\end{array}$$

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 $$\begin{array}{c}\log (11x+40)-\log (x)\end{array}$$

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 $$\begin{array}{c}\log \left(|11x+40|\right)-\log \left(\left|x\right|\right)\end{array}$$

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	$\ln (11x+40)-\ln (x)$	$12$
norman	$\ln (11x+40)-\ln (x)$	$12$
risch	$\ln (11x+40)-\ln (x)$	$12$
meijerg	$\ln (1+\frac{11x}{40})-\ln (x)-\ln \left(11\right)+3\ln (2)+\ln (5)$	$22$

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 $$\begin{array}{c}\log (11x+40)-\log (x)\end{array}$$

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 $$\begin{array}{c}2\mathrm{atanh}(\frac{11x}{20}+1)\end{array}$$

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 $$\begin{array}{c}-\log (x)+\log (x+\frac{40}{11})\end{array}$$

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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