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]

-Log[1 - 2*x]/11 + Log[3 + 5*x]/11

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Rubi [A]  time = 0.0038877, antiderivative size = 21, normalized size of antiderivative = 1., 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 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]

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{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*}

Mathematica [A]  time = 0.0033789, size = 21, normalized size = 1. \[ \frac{1}{11} \log (5 x+3)-\frac{1}{11} \log (1-2 x) \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - 2*x)*(3 + 5*x)),x]

[Out]

-Log[1 - 2*x]/11 + Log[3 + 5*x]/11

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Maple [A]  time = 0.005, size = 18, normalized size = 0.9 \begin{align*} -{\frac{\ln \left ( 2\,x-1 \right ) }{11}}+{\frac{\ln \left ( 3+5\,x \right ) }{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.05572, size = 23, normalized size = 1.1 \begin{align*} \frac{1}{11} \, \log \left (5 \, x + 3\right ) - \frac{1}{11} \, \log \left (2 \, x - 1\right ) \end{align*}

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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Fricas [A]  time = 1.24473, size = 53, normalized size = 2.52 \begin{align*} \frac{1}{11} \, \log \left (5 \, x + 3\right ) - \frac{1}{11} \, \log \left (2 \, x - 1\right ) \end{align*}

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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Sympy [A]  time = 0.097193, size = 15, normalized size = 0.71 \begin{align*} - \frac{\log{\left (x - \frac{1}{2} \right )}}{11} + \frac{\log{\left (x + \frac{3}{5} \right )}}{11} \end{align*}

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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Giac [A]  time = 2.63052, size = 26, normalized size = 1.24 \begin{align*} \frac{1}{11} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac{1}{11} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end{align*}

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))