3.1411 \(\int \frac{(1-2 x)^3}{(2+3 x)^2 (3+5 x)^2} \, dx\)

Optimal. Leaf size=43 \[ -\frac{343}{9 (3 x+2)}-\frac{1331}{25 (5 x+3)}+\frac{3136}{9} \log (3 x+2)-\frac{8712}{25} \log (5 x+3) \]

[Out]

-343/(9*(2 + 3*x)) - 1331/(25*(3 + 5*x)) + (3136*Log[2 + 3*x])/9 - (8712*Log[3 + 5*x])/25

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Rubi [A]  time = 0.019281, antiderivative size = 43, 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} \[ -\frac{343}{9 (3 x+2)}-\frac{1331}{25 (5 x+3)}+\frac{3136}{9} \log (3 x+2)-\frac{8712}{25} \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-343/(9*(2 + 3*x)) - 1331/(25*(3 + 5*x)) + (3136*Log[2 + 3*x])/9 - (8712*Log[3 + 5*x])/25

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)^2 (3+5 x)^2} \, dx &=\int \left (\frac{343}{3 (2+3 x)^2}+\frac{3136}{3 (2+3 x)}+\frac{1331}{5 (3+5 x)^2}-\frac{8712}{5 (3+5 x)}\right ) \, dx\\ &=-\frac{343}{9 (2+3 x)}-\frac{1331}{25 (3+5 x)}+\frac{3136}{9} \log (2+3 x)-\frac{8712}{25} \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0269961, size = 61, normalized size = 1.42 \[ -\frac{-78400 \left (15 x^2+19 x+6\right ) \log (5 (3 x+2))+78408 \left (15 x^2+19 x+6\right ) \log (5 x+3)+78812 x+49683}{225 (3 x+2) (5 x+3)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(49683 + 78812*x - 78400*(6 + 19*x + 15*x^2)*Log[5*(2 + 3*x)] + 78408*(6 + 19*x + 15*x^2)*Log[3 + 5*x])/(225*
(2 + 3*x)*(3 + 5*x))

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Maple [A]  time = 0.009, size = 36, normalized size = 0.8 \begin{align*} -{\frac{343}{18+27\,x}}-{\frac{1331}{75+125\,x}}+{\frac{3136\,\ln \left ( 2+3\,x \right ) }{9}}-{\frac{8712\,\ln \left ( 3+5\,x \right ) }{25}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-343/9/(2+3*x)-1331/25/(3+5*x)+3136/9*ln(2+3*x)-8712/25*ln(3+5*x)

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Maxima [A]  time = 0.998278, size = 49, normalized size = 1.14 \begin{align*} -\frac{78812 \, x + 49683}{225 \,{\left (15 \, x^{2} + 19 \, x + 6\right )}} - \frac{8712}{25} \, \log \left (5 \, x + 3\right ) + \frac{3136}{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)^2/(3+5*x)^2,x, algorithm="maxima")

[Out]

-1/225*(78812*x + 49683)/(15*x^2 + 19*x + 6) - 8712/25*log(5*x + 3) + 3136/9*log(3*x + 2)

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Fricas [A]  time = 1.29829, size = 173, normalized size = 4.02 \begin{align*} -\frac{78408 \,{\left (15 \, x^{2} + 19 \, x + 6\right )} \log \left (5 \, x + 3\right ) - 78400 \,{\left (15 \, x^{2} + 19 \, x + 6\right )} \log \left (3 \, x + 2\right ) + 78812 \, x + 49683}{225 \,{\left (15 \, x^{2} + 19 \, x + 6\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/225*(78408*(15*x^2 + 19*x + 6)*log(5*x + 3) - 78400*(15*x^2 + 19*x + 6)*log(3*x + 2) + 78812*x + 49683)/(15
*x^2 + 19*x + 6)

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Sympy [A]  time = 0.150202, size = 34, normalized size = 0.79 \begin{align*} - \frac{78812 x + 49683}{3375 x^{2} + 4275 x + 1350} - \frac{8712 \log{\left (x + \frac{3}{5} \right )}}{25} + \frac{3136 \log{\left (x + \frac{2}{3} \right )}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(78812*x + 49683)/(3375*x**2 + 4275*x + 1350) - 8712*log(x + 3/5)/25 + 3136*log(x + 2/3)/9

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Giac [A]  time = 2.71365, size = 76, normalized size = 1.77 \begin{align*} -\frac{1331}{25 \,{\left (5 \, x + 3\right )}} + \frac{1715}{3 \,{\left (\frac{1}{5 \, x + 3} + 3\right )}} + \frac{8}{225} \, \log \left (\frac{{\left | 5 \, x + 3 \right |}}{5 \,{\left (5 \, x + 3\right )}^{2}}\right ) + \frac{3136}{9} \, \log \left ({\left | -\frac{1}{5 \, x + 3} - 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1331/25/(5*x + 3) + 1715/3/(1/(5*x + 3) + 3) + 8/225*log(1/5*abs(5*x + 3)/(5*x + 3)^2) + 3136/9*log(abs(-1/(5
*x + 3) - 3))