3.1509 \(\int \frac{1}{(1-2 x) (3+5 x)^2} \, dx\)

Optimal. Leaf size=32 \[ -\frac{1}{11 (5 x+3)}-\frac{2}{121} \log (1-2 x)+\frac{2}{121} \log (5 x+3) \]

[Out]

-1/(11*(3 + 5*x)) - (2*Log[1 - 2*x])/121 + (2*Log[3 + 5*x])/121

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Rubi [A]  time = 0.0111126, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {44} \[ -\frac{1}{11 (5 x+3)}-\frac{2}{121} \log (1-2 x)+\frac{2}{121} \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

Int[1/((1 - 2*x)*(3 + 5*x)^2),x]

[Out]

-1/(11*(3 + 5*x)) - (2*Log[1 - 2*x])/121 + (2*Log[3 + 5*x])/121

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{(1-2 x) (3+5 x)^2} \, dx &=\int \left (-\frac{4}{121 (-1+2 x)}+\frac{5}{11 (3+5 x)^2}+\frac{10}{121 (3+5 x)}\right ) \, dx\\ &=-\frac{1}{11 (3+5 x)}-\frac{2}{121} \log (1-2 x)+\frac{2}{121} \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0109573, size = 30, normalized size = 0.94 \[ \frac{1}{121} \left (-\frac{11}{5 x+3}-2 \log (5-10 x)+2 \log (5 x+3)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-11/(3 + 5*x) - 2*Log[5 - 10*x] + 2*Log[3 + 5*x])/121

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Maple [A]  time = 0.006, size = 27, normalized size = 0.8 \begin{align*} -{\frac{2\,\ln \left ( 2\,x-1 \right ) }{121}}-{\frac{1}{33+55\,x}}+{\frac{2\,\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)/(3+5*x)^2,x)

[Out]

-2/121*ln(2*x-1)-1/11/(3+5*x)+2/121*ln(3+5*x)

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Maxima [A]  time = 2.06305, size = 35, normalized size = 1.09 \begin{align*} -\frac{1}{11 \,{\left (5 \, x + 3\right )}} + \frac{2}{121} \, \log \left (5 \, x + 3\right ) - \frac{2}{121} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/11/(5*x + 3) + 2/121*log(5*x + 3) - 2/121*log(2*x - 1)

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Fricas [A]  time = 1.59599, size = 103, normalized size = 3.22 \begin{align*} \frac{2 \,{\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) - 2 \,{\left (5 \, x + 3\right )} \log \left (2 \, x - 1\right ) - 11}{121 \,{\left (5 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/121*(2*(5*x + 3)*log(5*x + 3) - 2*(5*x + 3)*log(2*x - 1) - 11)/(5*x + 3)

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Sympy [A]  time = 0.117526, size = 26, normalized size = 0.81 \begin{align*} - \frac{2 \log{\left (x - \frac{1}{2} \right )}}{121} + \frac{2 \log{\left (x + \frac{3}{5} \right )}}{121} - \frac{1}{55 x + 33} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*log(x - 1/2)/121 + 2*log(x + 3/5)/121 - 1/(55*x + 33)

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Giac [A]  time = 1.61284, size = 34, normalized size = 1.06 \begin{align*} -\frac{1}{11 \,{\left (5 \, x + 3\right )}} - \frac{2}{121} \, \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)/(3+5*x)^2,x, algorithm="giac")

[Out]

-1/11/(5*x + 3) - 2/121*log(abs(-11/(5*x + 3) + 2))