3.1611 \(\int \frac{1}{(1-2 x)^2 (3+5 x)^2} \, dx\)

Optimal. Leaf size=43 \[ \frac{2}{121 (1-2 x)}-\frac{5}{121 (5 x+3)}-\frac{20 \log (1-2 x)}{1331}+\frac{20 \log (5 x+3)}{1331} \]

[Out]

2/(121*(1 - 2*x)) - 5/(121*(3 + 5*x)) - (20*Log[1 - 2*x])/1331 + (20*Log[3 + 5*x])/1331

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Rubi [A]  time = 0.0172776, antiderivative size = 43, 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} \[ \frac{2}{121 (1-2 x)}-\frac{5}{121 (5 x+3)}-\frac{20 \log (1-2 x)}{1331}+\frac{20 \log (5 x+3)}{1331} \]

Antiderivative was successfully verified.

[In]

Int[1/((1 - 2*x)^2*(3 + 5*x)^2),x]

[Out]

2/(121*(1 - 2*x)) - 5/(121*(3 + 5*x)) - (20*Log[1 - 2*x])/1331 + (20*Log[3 + 5*x])/1331

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)^2 (3+5 x)^2} \, dx &=\int \left (\frac{4}{121 (-1+2 x)^2}-\frac{40}{1331 (-1+2 x)}+\frac{25}{121 (3+5 x)^2}+\frac{100}{1331 (3+5 x)}\right ) \, dx\\ &=\frac{2}{121 (1-2 x)}-\frac{5}{121 (3+5 x)}-\frac{20 \log (1-2 x)}{1331}+\frac{20 \log (3+5 x)}{1331}\\ \end{align*}

Mathematica [A]  time = 0.0148516, size = 40, normalized size = 0.93 \[ \frac{-20 x-1}{121 \left (10 x^2+x-3\right )}-\frac{20 \log (1-2 x)}{1331}+\frac{20 \log (5 x+3)}{1331} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1 - 20*x)/(121*(-3 + x + 10*x^2)) - (20*Log[1 - 2*x])/1331 + (20*Log[3 + 5*x])/1331

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Maple [A]  time = 0.008, size = 36, normalized size = 0.8 \begin{align*} -{\frac{2}{242\,x-121}}-{\frac{20\,\ln \left ( 2\,x-1 \right ) }{1331}}-{\frac{5}{363+605\,x}}+{\frac{20\,\ln \left ( 3+5\,x \right ) }{1331}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/121/(2*x-1)-20/1331*ln(2*x-1)-5/121/(3+5*x)+20/1331*ln(3+5*x)

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Maxima [A]  time = 1.03664, size = 46, normalized size = 1.07 \begin{align*} -\frac{20 \, x + 1}{121 \,{\left (10 \, x^{2} + x - 3\right )}} + \frac{20}{1331} \, \log \left (5 \, x + 3\right ) - \frac{20}{1331} \, \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+5*x)^2,x, algorithm="maxima")

[Out]

-1/121*(20*x + 1)/(10*x^2 + x - 3) + 20/1331*log(5*x + 3) - 20/1331*log(2*x - 1)

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Fricas [A]  time = 1.42206, size = 146, normalized size = 3.4 \begin{align*} \frac{20 \,{\left (10 \, x^{2} + x - 3\right )} \log \left (5 \, x + 3\right ) - 20 \,{\left (10 \, x^{2} + x - 3\right )} \log \left (2 \, x - 1\right ) - 220 \, x - 11}{1331 \,{\left (10 \, x^{2} + x - 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1331*(20*(10*x^2 + x - 3)*log(5*x + 3) - 20*(10*x^2 + x - 3)*log(2*x - 1) - 220*x - 11)/(10*x^2 + x - 3)

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Sympy [A]  time = 0.133551, size = 34, normalized size = 0.79 \begin{align*} - \frac{20 x + 1}{1210 x^{2} + 121 x - 363} - \frac{20 \log{\left (x - \frac{1}{2} \right )}}{1331} + \frac{20 \log{\left (x + \frac{3}{5} \right )}}{1331} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(20*x + 1)/(1210*x**2 + 121*x - 363) - 20*log(x - 1/2)/1331 + 20*log(x + 3/5)/1331

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Giac [A]  time = 2.41061, size = 54, normalized size = 1.26 \begin{align*} -\frac{5}{121 \,{\left (5 \, x + 3\right )}} + \frac{20}{1331 \,{\left (\frac{11}{5 \, x + 3} - 2\right )}} - \frac{20}{1331} \, \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+5*x)^2,x, algorithm="giac")

[Out]

-5/121/(5*x + 3) + 20/1331/(11/(5*x + 3) - 2) - 20/1331*log(abs(-11/(5*x + 3) + 2))