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]

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

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Rubi [A]  time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 31, normalized size = 1.00 \[ -\frac {2}{77} \log (1-2 x)-\frac {3}{7} \log (3 x+2)+\frac {5}{11} \log (5 x+3) \]

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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fricas [A]  time = 0.79, size = 25, normalized size = 0.81 \[ \frac {5}{11} \, \log \left (5 \, x + 3\right ) - \frac {3}{7} \, \log \left (3 \, x + 2\right ) - \frac {2}{77} \, \log \left (2 \, x - 1\right ) \]

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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giac [A]  time = 0.90, size = 28, normalized size = 0.90 \[ \frac {5}{11} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {3}{7} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {2}{77} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]

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))

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maple [A]  time = 0.01, size = 26, normalized size = 0.84 \[ -\frac {2 \ln \left (2 x -1\right )}{77}-\frac {3 \ln \left (3 x +2\right )}{7}+\frac {5 \ln \left (5 x +3\right )}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.52, size = 25, normalized size = 0.81 \[ \frac {5}{11} \, \log \left (5 \, x + 3\right ) - \frac {3}{7} \, \log \left (3 \, x + 2\right ) - \frac {2}{77} \, \log \left (2 \, x - 1\right ) \]

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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mupad [B]  time = 1.19, size = 19, normalized size = 0.61 \[ \frac {5\,\ln \left (x+\frac {3}{5}\right )}{11}-\frac {3\,\ln \left (x+\frac {2}{3}\right )}{7}-\frac {2\,\ln \left (x-\frac {1}{2}\right )}{77} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.15, size = 29, normalized size = 0.94 \[ - \frac {2 \log {\left (x - \frac {1}{2} \right )}}{77} + \frac {5 \log {\left (x + \frac {3}{5} \right )}}{11} - \frac {3 \log {\left (x + \frac {2}{3} \right )}}{7} \]

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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