3.1492 \(\int \frac{2+3 x}{(1-2 x) (3+5 x)} \, dx\)

Optimal. Leaf size=21 \[ \frac{1}{55} \log (5 x+3)-\frac{7}{22} \log (1-2 x) \]

[Out]

(-7*Log[1 - 2*x])/22 + Log[3 + 5*x]/55

________________________________________________________________________________________

Rubi [A]  time = 0.0104349, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {72} \[ \frac{1}{55} \log (5 x+3)-\frac{7}{22} \log (1-2 x) \]

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)/((1 - 2*x)*(3 + 5*x)),x]

[Out]

(-7*Log[1 - 2*x])/22 + Log[3 + 5*x]/55

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

\begin{align*} \int \frac{2+3 x}{(1-2 x) (3+5 x)} \, dx &=\int \left (-\frac{7}{11 (-1+2 x)}+\frac{1}{11 (3+5 x)}\right ) \, dx\\ &=-\frac{7}{22} \log (1-2 x)+\frac{1}{55} \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.004192, size = 21, normalized size = 1. \[ \frac{1}{55} \log (5 x+3)-\frac{7}{22} \log (1-2 x) \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)/((1 - 2*x)*(3 + 5*x)),x]

[Out]

(-7*Log[1 - 2*x])/22 + Log[3 + 5*x]/55

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 18, normalized size = 0.9 \begin{align*} -{\frac{7\,\ln \left ( 2\,x-1 \right ) }{22}}+{\frac{\ln \left ( 3+5\,x \right ) }{55}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-7/22*ln(2*x-1)+1/55*ln(3+5*x)

________________________________________________________________________________________

Maxima [A]  time = 2.14937, size = 23, normalized size = 1.1 \begin{align*} \frac{1}{55} \, \log \left (5 \, x + 3\right ) - \frac{7}{22} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/55*log(5*x + 3) - 7/22*log(2*x - 1)

________________________________________________________________________________________

Fricas [A]  time = 1.23127, size = 53, normalized size = 2.52 \begin{align*} \frac{1}{55} \, \log \left (5 \, x + 3\right ) - \frac{7}{22} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/55*log(5*x + 3) - 7/22*log(2*x - 1)

________________________________________________________________________________________

Sympy [A]  time = 0.10671, size = 17, normalized size = 0.81 \begin{align*} - \frac{7 \log{\left (x - \frac{1}{2} \right )}}{22} + \frac{\log{\left (x + \frac{3}{5} \right )}}{55} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-7*log(x - 1/2)/22 + log(x + 3/5)/55

________________________________________________________________________________________

Giac [A]  time = 2.21171, size = 26, normalized size = 1.24 \begin{align*} \frac{1}{55} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac{7}{22} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/55*log(abs(5*x + 3)) - 7/22*log(abs(2*x - 1))