3.1410 \(\int \frac{(1-2 x)^3}{(2+3 x) (3+5 x)^2} \, dx\)

Optimal. Leaf size=37 \[ -\frac{8 x}{75}-\frac{1331}{125 (5 x+3)}+\frac{343}{9} \log (3 x+2)-\frac{4719}{125} \log (5 x+3) \]

[Out]

(-8*x)/75 - 1331/(125*(3 + 5*x)) + (343*Log[2 + 3*x])/9 - (4719*Log[3 + 5*x])/125

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Rubi [A]  time = 0.0189843, antiderivative size = 37, 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 = {88} \[ -\frac{8 x}{75}-\frac{1331}{125 (5 x+3)}+\frac{343}{9} \log (3 x+2)-\frac{4719}{125} \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*x)/75 - 1331/(125*(3 + 5*x)) + (343*Log[2 + 3*x])/9 - (4719*Log[3 + 5*x])/125

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \frac{(1-2 x)^3}{(2+3 x) (3+5 x)^2} \, dx &=\int \left (-\frac{8}{75}+\frac{343}{3 (2+3 x)}+\frac{1331}{25 (3+5 x)^2}-\frac{4719}{25 (3+5 x)}\right ) \, dx\\ &=-\frac{8 x}{75}-\frac{1331}{125 (3+5 x)}+\frac{343}{9} \log (2+3 x)-\frac{4719}{125} \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0203159, size = 36, normalized size = 0.97 \[ \frac{-120 x-\frac{11979}{5 x+3}+42875 \log (3 x+2)-42471 \log (-3 (5 x+3))-80}{1125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-80 - 120*x - 11979/(3 + 5*x) + 42875*Log[2 + 3*x] - 42471*Log[-3*(3 + 5*x)])/1125

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Maple [A]  time = 0.006, size = 30, normalized size = 0.8 \begin{align*} -{\frac{8\,x}{75}}-{\frac{1331}{375+625\,x}}+{\frac{343\,\ln \left ( 2+3\,x \right ) }{9}}-{\frac{4719\,\ln \left ( 3+5\,x \right ) }{125}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8/75*x-1331/125/(3+5*x)+343/9*ln(2+3*x)-4719/125*ln(3+5*x)

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Maxima [A]  time = 1.00039, size = 39, normalized size = 1.05 \begin{align*} -\frac{8}{75} \, x - \frac{1331}{125 \,{\left (5 \, x + 3\right )}} - \frac{4719}{125} \, \log \left (5 \, x + 3\right ) + \frac{343}{9} \, \log \left (3 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8/75*x - 1331/125/(5*x + 3) - 4719/125*log(5*x + 3) + 343/9*log(3*x + 2)

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Fricas [A]  time = 1.43697, size = 144, normalized size = 3.89 \begin{align*} -\frac{600 \, x^{2} + 42471 \,{\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) - 42875 \,{\left (5 \, x + 3\right )} \log \left (3 \, x + 2\right ) + 360 \, x + 11979}{1125 \,{\left (5 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1125*(600*x^2 + 42471*(5*x + 3)*log(5*x + 3) - 42875*(5*x + 3)*log(3*x + 2) + 360*x + 11979)/(5*x + 3)

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Sympy [A]  time = 0.132685, size = 31, normalized size = 0.84 \begin{align*} - \frac{8 x}{75} - \frac{4719 \log{\left (x + \frac{3}{5} \right )}}{125} + \frac{343 \log{\left (x + \frac{2}{3} \right )}}{9} - \frac{1331}{625 x + 375} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8*x/75 - 4719*log(x + 3/5)/125 + 343*log(x + 2/3)/9 - 1331/(625*x + 375)

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Giac [A]  time = 3.53896, size = 63, normalized size = 1.7 \begin{align*} -\frac{8}{75} \, x - \frac{1331}{125 \,{\left (5 \, x + 3\right )}} - \frac{404}{1125} \, \log \left (\frac{{\left | 5 \, x + 3 \right |}}{5 \,{\left (5 \, x + 3\right )}^{2}}\right ) + \frac{343}{9} \, \log \left ({\left | -\frac{1}{5 \, x + 3} - 3 \right |}\right ) - \frac{8}{125} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8/75*x - 1331/125/(5*x + 3) - 404/1125*log(1/5*abs(5*x + 3)/(5*x + 3)^2) + 343/9*log(abs(-1/(5*x + 3) - 3)) -
8/125