3.1681 \(\int \frac{1}{(1-2 x)^3 (2+3 x)^2 (3+5 x)} \, dx\)

Optimal. Leaf size=64 \[ \frac{404}{41503 (1-2 x)}+\frac{27}{343 (3 x+2)}+\frac{2}{539 (1-2 x)^2}-\frac{27208 \log (1-2 x)}{3195731}-\frac{1107 \log (3 x+2)}{2401}+\frac{625 \log (5 x+3)}{1331} \]

[Out]

2/(539*(1 - 2*x)^2) + 404/(41503*(1 - 2*x)) + 27/(343*(2 + 3*x)) - (27208*Log[1 - 2*x])/3195731 - (1107*Log[2
+ 3*x])/2401 + (625*Log[3 + 5*x])/1331

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Rubi [A]  time = 0.0323617, antiderivative size = 64, 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} \[ \frac{404}{41503 (1-2 x)}+\frac{27}{343 (3 x+2)}+\frac{2}{539 (1-2 x)^2}-\frac{27208 \log (1-2 x)}{3195731}-\frac{1107 \log (3 x+2)}{2401}+\frac{625 \log (5 x+3)}{1331} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2/(539*(1 - 2*x)^2) + 404/(41503*(1 - 2*x)) + 27/(343*(2 + 3*x)) - (27208*Log[1 - 2*x])/3195731 - (1107*Log[2
+ 3*x])/2401 + (625*Log[3 + 5*x])/1331

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}{(1-2 x)^3 (2+3 x)^2 (3+5 x)} \, dx &=\int \left (-\frac{8}{539 (-1+2 x)^3}+\frac{808}{41503 (-1+2 x)^2}-\frac{54416}{3195731 (-1+2 x)}-\frac{81}{343 (2+3 x)^2}-\frac{3321}{2401 (2+3 x)}+\frac{3125}{1331 (3+5 x)}\right ) \, dx\\ &=\frac{2}{539 (1-2 x)^2}+\frac{404}{41503 (1-2 x)}+\frac{27}{343 (2+3 x)}-\frac{27208 \log (1-2 x)}{3195731}-\frac{1107 \log (2+3 x)}{2401}+\frac{625 \log (3+5 x)}{1331}\\ \end{align*}

Mathematica [A]  time = 0.0552446, size = 57, normalized size = 0.89 \[ \frac{\frac{77 \left (10644 x^2-13010 x+4383\right )}{(1-2 x)^2 (3 x+2)}-27208 \log (5-10 x)-1473417 \log (5 (3 x+2))+1500625 \log (5 x+3)}{3195731} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((77*(4383 - 13010*x + 10644*x^2))/((1 - 2*x)^2*(2 + 3*x)) - 27208*Log[5 - 10*x] - 1473417*Log[5*(2 + 3*x)] +
1500625*Log[3 + 5*x])/3195731

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Maple [A]  time = 0.01, size = 53, normalized size = 0.8 \begin{align*}{\frac{2}{539\, \left ( 2\,x-1 \right ) ^{2}}}-{\frac{404}{83006\,x-41503}}-{\frac{27208\,\ln \left ( 2\,x-1 \right ) }{3195731}}+{\frac{27}{686+1029\,x}}-{\frac{1107\,\ln \left ( 2+3\,x \right ) }{2401}}+{\frac{625\,\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)^3/(2+3*x)^2/(3+5*x),x)

[Out]

2/539/(2*x-1)^2-404/41503/(2*x-1)-27208/3195731*ln(2*x-1)+27/343/(2+3*x)-1107/2401*ln(2+3*x)+625/1331*ln(3+5*x
)

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Maxima [A]  time = 1.07659, size = 73, normalized size = 1.14 \begin{align*} \frac{10644 \, x^{2} - 13010 \, x + 4383}{41503 \,{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )}} + \frac{625}{1331} \, \log \left (5 \, x + 3\right ) - \frac{1107}{2401} \, \log \left (3 \, x + 2\right ) - \frac{27208}{3195731} \, \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/(2+3*x)^2/(3+5*x),x, algorithm="maxima")

[Out]

1/41503*(10644*x^2 - 13010*x + 4383)/(12*x^3 - 4*x^2 - 5*x + 2) + 625/1331*log(5*x + 3) - 1107/2401*log(3*x +
2) - 27208/3195731*log(2*x - 1)

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Fricas [A]  time = 1.69795, size = 297, normalized size = 4.64 \begin{align*} \frac{819588 \, x^{2} + 1500625 \,{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \log \left (5 \, x + 3\right ) - 1473417 \,{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \log \left (3 \, x + 2\right ) - 27208 \,{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \log \left (2 \, x - 1\right ) - 1001770 \, x + 337491}{3195731 \,{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3195731*(819588*x^2 + 1500625*(12*x^3 - 4*x^2 - 5*x + 2)*log(5*x + 3) - 1473417*(12*x^3 - 4*x^2 - 5*x + 2)*l
og(3*x + 2) - 27208*(12*x^3 - 4*x^2 - 5*x + 2)*log(2*x - 1) - 1001770*x + 337491)/(12*x^3 - 4*x^2 - 5*x + 2)

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Sympy [A]  time = 0.198533, size = 54, normalized size = 0.84 \begin{align*} \frac{10644 x^{2} - 13010 x + 4383}{498036 x^{3} - 166012 x^{2} - 207515 x + 83006} - \frac{27208 \log{\left (x - \frac{1}{2} \right )}}{3195731} + \frac{625 \log{\left (x + \frac{3}{5} \right )}}{1331} - \frac{1107 \log{\left (x + \frac{2}{3} \right )}}{2401} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(10644*x**2 - 13010*x + 4383)/(498036*x**3 - 166012*x**2 - 207515*x + 83006) - 27208*log(x - 1/2)/3195731 + 62
5*log(x + 3/5)/1331 - 1107*log(x + 2/3)/2401

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Giac [A]  time = 2.31596, size = 89, normalized size = 1.39 \begin{align*} \frac{27}{343 \,{\left (3 \, x + 2\right )}} + \frac{24 \,{\left (\frac{938}{3 \, x + 2} - 235\right )}}{290521 \,{\left (\frac{7}{3 \, x + 2} - 2\right )}^{2}} + \frac{625}{1331} \, \log \left ({\left | -\frac{1}{3 \, x + 2} + 5 \right |}\right ) - \frac{27208}{3195731} \, \log \left ({\left | -\frac{7}{3 \, x + 2} + 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

27/343/(3*x + 2) + 24/290521*(938/(3*x + 2) - 235)/(7/(3*x + 2) - 2)^2 + 625/1331*log(abs(-1/(3*x + 2) + 5)) -
27208/3195731*log(abs(-7/(3*x + 2) + 2))