3.1510 \(\int \frac{1}{(1-2 x) (2+3 x) (3+5 x)^2} \, dx\)

Optimal. Leaf size=42 \[ -\frac{5}{11 (5 x+3)}-\frac{4}{847} \log (1-2 x)+\frac{9}{7} \log (3 x+2)-\frac{155}{121} \log (5 x+3) \]

[Out]

-5/(11*(3 + 5*x)) - (4*Log[1 - 2*x])/847 + (9*Log[2 + 3*x])/7 - (155*Log[3 + 5*x])/121

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Rubi [A]  time = 0.0202411, antiderivative size = 42, 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} \[ -\frac{5}{11 (5 x+3)}-\frac{4}{847} \log (1-2 x)+\frac{9}{7} \log (3 x+2)-\frac{155}{121} \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-5/(11*(3 + 5*x)) - (4*Log[1 - 2*x])/847 + (9*Log[2 + 3*x])/7 - (155*Log[3 + 5*x])/121

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)^2} \, dx &=\int \left (-\frac{8}{847 (-1+2 x)}+\frac{27}{7 (2+3 x)}+\frac{25}{11 (3+5 x)^2}-\frac{775}{121 (3+5 x)}\right ) \, dx\\ &=-\frac{5}{11 (3+5 x)}-\frac{4}{847} \log (1-2 x)+\frac{9}{7} \log (2+3 x)-\frac{155}{121} \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0208269, size = 38, normalized size = 0.9 \[ \frac{1}{847} \left (-\frac{385}{5 x+3}-4 \log (1-2 x)+1089 \log (6 x+4)-1085 \log (10 x+6)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-385/(3 + 5*x) - 4*Log[1 - 2*x] + 1089*Log[4 + 6*x] - 1085*Log[6 + 10*x])/847

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Maple [A]  time = 0.008, size = 35, normalized size = 0.8 \begin{align*} -{\frac{4\,\ln \left ( 2\,x-1 \right ) }{847}}+{\frac{9\,\ln \left ( 2+3\,x \right ) }{7}}-{\frac{5}{33+55\,x}}-{\frac{155\,\ln \left ( 3+5\,x \right ) }{121}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/847*ln(2*x-1)+9/7*ln(2+3*x)-5/11/(3+5*x)-155/121*ln(3+5*x)

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Maxima [A]  time = 1.08209, size = 46, normalized size = 1.1 \begin{align*} -\frac{5}{11 \,{\left (5 \, x + 3\right )}} - \frac{155}{121} \, \log \left (5 \, x + 3\right ) + \frac{9}{7} \, \log \left (3 \, x + 2\right ) - \frac{4}{847} \, \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)^2,x, algorithm="maxima")

[Out]

-5/11/(5*x + 3) - 155/121*log(5*x + 3) + 9/7*log(3*x + 2) - 4/847*log(2*x - 1)

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Fricas [A]  time = 1.57755, size = 150, normalized size = 3.57 \begin{align*} -\frac{1085 \,{\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) - 1089 \,{\left (5 \, x + 3\right )} \log \left (3 \, x + 2\right ) + 4 \,{\left (5 \, x + 3\right )} \log \left (2 \, x - 1\right ) + 385}{847 \,{\left (5 \, x + 3\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)^2,x, algorithm="fricas")

[Out]

-1/847*(1085*(5*x + 3)*log(5*x + 3) - 1089*(5*x + 3)*log(3*x + 2) + 4*(5*x + 3)*log(2*x - 1) + 385)/(5*x + 3)

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Sympy [A]  time = 0.162598, size = 36, normalized size = 0.86 \begin{align*} - \frac{4 \log{\left (x - \frac{1}{2} \right )}}{847} - \frac{155 \log{\left (x + \frac{3}{5} \right )}}{121} + \frac{9 \log{\left (x + \frac{2}{3} \right )}}{7} - \frac{5}{55 x + 33} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*log(x - 1/2)/847 - 155*log(x + 3/5)/121 + 9*log(x + 2/3)/7 - 5/(55*x + 33)

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Giac [A]  time = 2.45049, size = 54, normalized size = 1.29 \begin{align*} -\frac{5}{11 \,{\left (5 \, x + 3\right )}} + \frac{9}{7} \, \log \left ({\left | -\frac{1}{5 \, x + 3} - 3 \right |}\right ) - \frac{4}{847} \, \log \left ({\left | -\frac{11}{5 \, x + 3} + 2 \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)^2,x, algorithm="giac")

[Out]

-5/11/(5*x + 3) + 9/7*log(abs(-1/(5*x + 3) - 3)) - 4/847*log(abs(-11/(5*x + 3) + 2))