3.1200 \(\int \frac{1-2 x}{(2+3 x) (3+5 x)} \, dx\)

Optimal. Leaf size=21 \[ \frac{11}{5} \log (5 x+3)-\frac{7}{3} \log (3 x+2) \]

[Out]

(-7*Log[2 + 3*x])/3 + (11*Log[3 + 5*x])/5

________________________________________________________________________________________

Rubi [A]  time = 0.0084839, 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} \[ \frac{11}{5} \log (5 x+3)-\frac{7}{3} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*Log[2 + 3*x])/3 + (11*Log[3 + 5*x])/5

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

Mathematica [A]  time = 0.0046402, size = 21, normalized size = 1. \[ \frac{11}{5} \log (5 x+3)-\frac{7}{3} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*Log[2 + 3*x])/3 + (11*Log[3 + 5*x])/5

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 18, normalized size = 0.9 \begin{align*} -{\frac{7\,\ln \left ( 2+3\,x \right ) }{3}}+{\frac{11\,\ln \left ( 3+5\,x \right ) }{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-7/3*ln(2+3*x)+11/5*ln(3+5*x)

________________________________________________________________________________________

Maxima [A]  time = 1.28072, size = 23, normalized size = 1.1 \begin{align*} \frac{11}{5} \, \log \left (5 \, x + 3\right ) - \frac{7}{3} \, \log \left (3 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

11/5*log(5*x + 3) - 7/3*log(3*x + 2)

________________________________________________________________________________________

Fricas [A]  time = 1.52817, size = 51, normalized size = 2.43 \begin{align*} \frac{11}{5} \, \log \left (5 \, x + 3\right ) - \frac{7}{3} \, \log \left (3 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

11/5*log(5*x + 3) - 7/3*log(3*x + 2)

________________________________________________________________________________________

Sympy [A]  time = 0.102219, size = 19, normalized size = 0.9 \begin{align*} \frac{11 \log{\left (x + \frac{3}{5} \right )}}{5} - \frac{7 \log{\left (x + \frac{2}{3} \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

11*log(x + 3/5)/5 - 7*log(x + 2/3)/3

________________________________________________________________________________________

Giac [A]  time = 1.52308, size = 26, normalized size = 1.24 \begin{align*} \frac{11}{5} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac{7}{3} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

11/5*log(abs(5*x + 3)) - 7/3*log(abs(3*x + 2))