3.1682 \(\int \frac{1}{(1-2 x)^3 (2+3 x)^3 (3+5 x)} \, dx\)

Optimal. Leaf size=75 \[ \frac{1072}{290521 (1-2 x)}+\frac{1107}{2401 (3 x+2)}+\frac{4}{3773 (1-2 x)^2}+\frac{27}{686 (3 x+2)^2}-\frac{89792 \log (1-2 x)}{22370117}-\frac{39393 \log (3 x+2)}{16807}+\frac{3125 \log (5 x+3)}{1331} \]

[Out]

4/(3773*(1 - 2*x)^2) + 1072/(290521*(1 - 2*x)) + 27/(686*(2 + 3*x)^2) + 1107/(2401*(2 + 3*x)) - (89792*Log[1 -
2*x])/22370117 - (39393*Log[2 + 3*x])/16807 + (3125*Log[3 + 5*x])/1331

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Rubi [A]  time = 0.0379753, antiderivative size = 75, 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} \[ \frac{1072}{290521 (1-2 x)}+\frac{1107}{2401 (3 x+2)}+\frac{4}{3773 (1-2 x)^2}+\frac{27}{686 (3 x+2)^2}-\frac{89792 \log (1-2 x)}{22370117}-\frac{39393 \log (3 x+2)}{16807}+\frac{3125 \log (5 x+3)}{1331} \]

Antiderivative was successfully verified.

[In]

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

[Out]

4/(3773*(1 - 2*x)^2) + 1072/(290521*(1 - 2*x)) + 27/(686*(2 + 3*x)^2) + 1107/(2401*(2 + 3*x)) - (89792*Log[1 -
2*x])/22370117 - (39393*Log[2 + 3*x])/16807 + (3125*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)^3 (3+5 x)} \, dx &=\int \left (-\frac{16}{3773 (-1+2 x)^3}+\frac{2144}{290521 (-1+2 x)^2}-\frac{179584}{22370117 (-1+2 x)}-\frac{81}{343 (2+3 x)^3}-\frac{3321}{2401 (2+3 x)^2}-\frac{118179}{16807 (2+3 x)}+\frac{15625}{1331 (3+5 x)}\right ) \, dx\\ &=\frac{4}{3773 (1-2 x)^2}+\frac{1072}{290521 (1-2 x)}+\frac{27}{686 (2+3 x)^2}+\frac{1107}{2401 (2+3 x)}-\frac{89792 \log (1-2 x)}{22370117}-\frac{39393 \log (2+3 x)}{16807}+\frac{3125 \log (3+5 x)}{1331}\\ \end{align*}

Mathematica [A]  time = 0.0596653, size = 58, normalized size = 0.77 \[ \frac{\frac{77 \left (3176136 x^3-1006716 x^2-1414978 x+569697\right )}{\left (6 x^2+x-2\right )^2}-179584 \log (5-10 x)-104864166 \log (5 (3 x+2))[phone] \log (5 x+3)}{44740234} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((77*(569697 - 1414978*x - 1006716*x^2 + 3176136*x^3))/(-2 + x + 6*x^2)^2 - 179584*Log[5 - 10*x] - 104864166*L
og[5*(2 + 3*x)] + 105043750*Log[3 + 5*x])/44740234

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Maple [A]  time = 0.01, size = 62, normalized size = 0.8 \begin{align*}{\frac{4}{3773\, \left ( 2\,x-1 \right ) ^{2}}}-{\frac{1072}{581042\,x-290521}}-{\frac{89792\,\ln \left ( 2\,x-1 \right ) }{22370117}}+{\frac{27}{686\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{1107}{4802+7203\,x}}-{\frac{39393\,\ln \left ( 2+3\,x \right ) }{16807}}+{\frac{3125\,\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)^3/(3+5*x),x)

[Out]

4/3773/(2*x-1)^2-1072/290521/(2*x-1)-89792/22370117*ln(2*x-1)+27/686/(2+3*x)^2+1107/2401/(2+3*x)-39393/16807*l
n(2+3*x)+3125/1331*ln(3+5*x)

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Maxima [A]  time = 1.04203, size = 86, normalized size = 1.15 \begin{align*} \frac{3176136 \, x^{3} - 1006716 \, x^{2} - 1414978 \, x + 569697}{581042 \,{\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} + \frac{3125}{1331} \, \log \left (5 \, x + 3\right ) - \frac{39393}{16807} \, \log \left (3 \, x + 2\right ) - \frac{89792}{22370117} \, \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)^3/(3+5*x),x, algorithm="maxima")

[Out]

1/581042*(3176136*x^3 - 1006716*x^2 - 1414978*x + 569697)/(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4) + 3125/1331*log
(5*x + 3) - 39393/16807*log(3*x + 2) - 89792/22370117*log(2*x - 1)

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Fricas [B]  time = 1.51755, size = 389, normalized size = 5.19 \begin{align*} \frac{244562472 \, x^{3} - 77517132 \, x^{2} + 105043750 \,{\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \log \left (5 \, x + 3\right ) - 104864166 \,{\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \log \left (3 \, x + 2\right ) - 179584 \,{\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \log \left (2 \, x - 1\right ) - 108953306 \, x + 43866669}{44740234 \,{\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/44740234*(244562472*x^3 - 77517132*x^2 + 105043750*(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)*log(5*x + 3) - 10486
4166*(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)*log(3*x + 2) - 179584*(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)*log(2*x -
1) - 108953306*x + 43866669)/(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)

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Sympy [A]  time = 0.216677, size = 65, normalized size = 0.87 \begin{align*} \frac{3176136 x^{3} - 1006716 x^{2} - 1414978 x + 569697}{20917512 x^{4} + 6972504 x^{3} - 13363966 x^{2} - 2324168 x + 2324168} - \frac{89792 \log{\left (x - \frac{1}{2} \right )}}{22370117} + \frac{3125 \log{\left (x + \frac{3}{5} \right )}}{1331} - \frac{39393 \log{\left (x + \frac{2}{3} \right )}}{16807} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3176136*x**3 - 1006716*x**2 - 1414978*x + 569697)/(20917512*x**4 + 6972504*x**3 - 13363966*x**2 - 2324168*x +
2324168) - 89792*log(x - 1/2)/22370117 + 3125*log(x + 3/5)/1331 - 39393*log(x + 2/3)/16807

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Giac [A]  time = 1.49569, size = 80, normalized size = 1.07 \begin{align*} \frac{3176136 \, x^{3} - 1006716 \, x^{2} - 1414978 \, x + 569697}{581042 \,{\left (3 \, x + 2\right )}^{2}{\left (2 \, x - 1\right )}^{2}} + \frac{3125}{1331} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac{39393}{16807} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac{89792}{22370117} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/581042*(3176136*x^3 - 1006716*x^2 - 1414978*x + 569697)/((3*x + 2)^2*(2*x - 1)^2) + 3125/1331*log(abs(5*x +
3)) - 39393/16807*log(abs(3*x + 2)) - 89792/22370117*log(abs(2*x - 1))