3.1493 \(\int \frac {1}{(1-2 x) (3+5 x)} \, dx\)

Optimal. Leaf size=21 \[ \frac {1}{11} \log (5 x+3)-\frac {1}{11} \log (1-2 x) \]

[Out]

-1/11*ln(1-2*x)+1/11*ln(3+5*x)

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Rubi [A]  time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {36, 31} \[ \frac {1}{11} \log (5 x+3)-\frac {1}{11} \log (1-2 x) \]

Antiderivative was successfully verified.

[In]

Int[1/((1 - 2*x)*(3 + 5*x)),x]

[Out]

-Log[1 - 2*x]/11 + Log[3 + 5*x]/11

Rule 31

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

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rubi steps

\begin {align*} \int \frac {1}{(1-2 x) (3+5 x)} \, dx &=\frac {2}{11} \int \frac {1}{1-2 x} \, dx+\frac {5}{11} \int \frac {1}{3+5 x} \, dx\\ &=-\frac {1}{11} \log (1-2 x)+\frac {1}{11} \log (3+5 x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 21, normalized size = 1.00 \[ \frac {1}{11} \log (5 x+3)-\frac {1}{11} \log (1-2 x) \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - 2*x)*(3 + 5*x)),x]

[Out]

-1/11*Log[1 - 2*x] + Log[3 + 5*x]/11

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fricas [A]  time = 0.70, size = 17, normalized size = 0.81 \[ \frac {1}{11} \, \log \left (5 \, x + 3\right ) - \frac {1}{11} \, \log \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)/(3+5*x),x, algorithm="fricas")

[Out]

1/11*log(5*x + 3) - 1/11*log(2*x - 1)

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giac [A]  time = 0.85, size = 19, normalized size = 0.90 \[ \frac {1}{11} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {1}{11} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)/(3+5*x),x, algorithm="giac")

[Out]

1/11*log(abs(5*x + 3)) - 1/11*log(abs(2*x - 1))

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maple [A]  time = 0.00, size = 18, normalized size = 0.86 \[ -\frac {\ln \left (2 x -1\right )}{11}+\frac {\ln \left (5 x +3\right )}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-2*x)/(5*x+3),x)

[Out]

1/11*ln(5*x+3)-1/11*ln(2*x-1)

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maxima [A]  time = 0.79, size = 17, normalized size = 0.81 \[ \frac {1}{11} \, \log \left (5 \, x + 3\right ) - \frac {1}{11} \, \log \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)/(3+5*x),x, algorithm="maxima")

[Out]

1/11*log(5*x + 3) - 1/11*log(2*x - 1)

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mupad [B]  time = 0.13, size = 16, normalized size = 0.76 \[ \frac {\ln \left (\frac {5\,x+3}{2\,x-1}\right )}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/((2*x - 1)*(5*x + 3)),x)

[Out]

log((5*x + 3)/(2*x - 1))/11

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sympy [A]  time = 0.11, size = 15, normalized size = 0.71 \[ - \frac {\log {\left (x - \frac {1}{2} \right )}}{11} + \frac {\log {\left (x + \frac {3}{5} \right )}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)/(3+5*x),x)

[Out]

-log(x - 1/2)/11 + log(x + 3/5)/11

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