3.1494 \(\int \frac{1}{(1-2 x) (2+3 x) (3+5 x)} \, dx\)

Optimal. Leaf size=31 \[ -\frac{2}{77} \log (1-2 x)-\frac{3}{7} \log (3 x+2)+\frac{5}{11} \log (5 x+3) \]

[Out]

(-2*Log[1 - 2*x])/77 - (3*Log[2 + 3*x])/7 + (5*Log[3 + 5*x])/11

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Rubi [A]  time = 0.0143793, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {72} \[ -\frac{2}{77} \log (1-2 x)-\frac{3}{7} \log (3 x+2)+\frac{5}{11} \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Log[1 - 2*x])/77 - (3*Log[2 + 3*x])/7 + (5*Log[3 + 5*x])/11

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}{(1-2 x) (2+3 x) (3+5 x)} \, dx &=\int \left (-\frac{4}{77 (-1+2 x)}-\frac{9}{7 (2+3 x)}+\frac{25}{11 (3+5 x)}\right ) \, dx\\ &=-\frac{2}{77} \log (1-2 x)-\frac{3}{7} \log (2+3 x)+\frac{5}{11} \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0068216, size = 31, normalized size = 1. \[ -\frac{2}{77} \log (1-2 x)-\frac{3}{7} \log (3 x+2)+\frac{5}{11} \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Log[1 - 2*x])/77 - (3*Log[2 + 3*x])/7 + (5*Log[3 + 5*x])/11

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Maple [A]  time = 0.006, size = 26, normalized size = 0.8 \begin{align*} -{\frac{2\,\ln \left ( 2\,x-1 \right ) }{77}}-{\frac{3\,\ln \left ( 2+3\,x \right ) }{7}}+{\frac{5\,\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)/(2+3*x)/(3+5*x),x)

[Out]

-2/77*ln(2*x-1)-3/7*ln(2+3*x)+5/11*ln(3+5*x)

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Maxima [A]  time = 1.02492, size = 34, normalized size = 1.1 \begin{align*} \frac{5}{11} \, \log \left (5 \, x + 3\right ) - \frac{3}{7} \, \log \left (3 \, x + 2\right ) - \frac{2}{77} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/11*log(5*x + 3) - 3/7*log(3*x + 2) - 2/77*log(2*x - 1)

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Fricas [A]  time = 1.32895, size = 78, normalized size = 2.52 \begin{align*} \frac{5}{11} \, \log \left (5 \, x + 3\right ) - \frac{3}{7} \, \log \left (3 \, x + 2\right ) - \frac{2}{77} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/11*log(5*x + 3) - 3/7*log(3*x + 2) - 2/77*log(2*x - 1)

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Sympy [A]  time = 0.133795, size = 29, normalized size = 0.94 \begin{align*} - \frac{2 \log{\left (x - \frac{1}{2} \right )}}{77} + \frac{5 \log{\left (x + \frac{3}{5} \right )}}{11} - \frac{3 \log{\left (x + \frac{2}{3} \right )}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*log(x - 1/2)/77 + 5*log(x + 3/5)/11 - 3*log(x + 2/3)/7

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Giac [A]  time = 1.20787, size = 38, normalized size = 1.23 \begin{align*} \frac{5}{11} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac{3}{7} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac{2}{77} \, \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)/(2+3*x)/(3+5*x),x, algorithm="giac")

[Out]

5/11*log(abs(5*x + 3)) - 3/7*log(abs(3*x + 2)) - 2/77*log(abs(2*x - 1))