3.1708 \(\int \frac{1}{(1-2 x)^3 (2+3 x)^2 (3+5 x)^3} \, dx\)

Optimal. Leaf size=86 \[ \frac{2736}{5021863 (1-2 x)}+\frac{243}{343 (3 x+2)}+\frac{37500}{14641 (5 x+3)}+\frac{8}{65219 (1-2 x)^2}-\frac{625}{2662 (5 x+3)^2}-\frac{280752 \log (1-2 x)}{386683451}-\frac{26973 \log (3 x+2)}{2401}+\frac{1809375 \log (5 x+3)}{161051} \]

[Out]

8/(65219*(1 - 2*x)^2) + 2736/(5021863*(1 - 2*x)) + 243/(343*(2 + 3*x)) - 625/(2662*(3 + 5*x)^2) + 37500/(14641
*(3 + 5*x)) - (280752*Log[1 - 2*x])/386683451 - (26973*Log[2 + 3*x])/2401 + (1809375*Log[3 + 5*x])/161051

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Rubi [A]  time = 0.0466914, antiderivative size = 86, 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} \[ \frac{2736}{5021863 (1-2 x)}+\frac{243}{343 (3 x+2)}+\frac{37500}{14641 (5 x+3)}+\frac{8}{65219 (1-2 x)^2}-\frac{625}{2662 (5 x+3)^2}-\frac{280752 \log (1-2 x)}{386683451}-\frac{26973 \log (3 x+2)}{2401}+\frac{1809375 \log (5 x+3)}{161051} \]

Antiderivative was successfully verified.

[In]

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

[Out]

8/(65219*(1 - 2*x)^2) + 2736/(5021863*(1 - 2*x)) + 243/(343*(2 + 3*x)) - 625/(2662*(3 + 5*x)^2) + 37500/(14641
*(3 + 5*x)) - (280752*Log[1 - 2*x])/386683451 - (26973*Log[2 + 3*x])/2401 + (1809375*Log[3 + 5*x])/161051

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)^3} \, dx &=\int \left (-\frac{32}{65219 (-1+2 x)^3}+\frac{5472}{5021863 (-1+2 x)^2}-\frac{561504}{386683451 (-1+2 x)}-\frac{729}{343 (2+3 x)^2}-\frac{80919}{2401 (2+3 x)}+\frac{3125}{1331 (3+5 x)^3}-\frac{187500}{14641 (3+5 x)^2}+\frac{9046875}{161051 (3+5 x)}\right ) \, dx\\ &=\frac{8}{65219 (1-2 x)^2}+\frac{2736}{5021863 (1-2 x)}+\frac{243}{343 (2+3 x)}-\frac{625}{2662 (3+5 x)^2}+\frac{37500}{14641 (3+5 x)}-\frac{280752 \log (1-2 x)}{386683451}-\frac{26973 \log (2+3 x)}{2401}+\frac{1809375 \log (3+5 x)}{161051}\\ \end{align*}

Mathematica [A]  time = 0.0634011, size = 76, normalized size = 0.88 \[ -\frac{3 \left (-\frac{65219 (12290 x-6101)}{3 \left (10 x^2+x-3\right )^2}-\frac{154 (8570440 x-4446931)}{10 x^2+x-3}-\frac{182631834}{3 x+2}+187168 \log (3-6 x)[phone] \log (3 x+2)-2896206250 \log (-3 (5 x+3))\right )}{773366902} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(-182631834/(2 + 3*x) - (65219*(-6101 + 12290*x))/(3*(-3 + x + 10*x^2)^2) - (154*(-4446931 + 8570440*x))/(
-3 + x + 10*x^2) + 187168*Log[3 - 6*x] + 2896019082*Log[2 + 3*x] - 2896206250*Log[-3*(3 + 5*x)]))/773366902

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Maple [A]  time = 0.013, size = 71, normalized size = 0.8 \begin{align*}{\frac{8}{65219\, \left ( 2\,x-1 \right ) ^{2}}}-{\frac{2736}{10043726\,x-5021863}}-{\frac{280752\,\ln \left ( 2\,x-1 \right ) }{386683451}}+{\frac{243}{686+1029\,x}}-{\frac{26973\,\ln \left ( 2+3\,x \right ) }{2401}}-{\frac{625}{2662\, \left ( 3+5\,x \right ) ^{2}}}+{\frac{37500}{43923+73205\,x}}+{\frac{1809375\,\ln \left ( 3+5\,x \right ) }{161051}} \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)^3,x)

[Out]

8/65219/(2*x-1)^2-2736/5021863/(2*x-1)-280752/386683451*ln(2*x-1)+243/343/(2+3*x)-26973/2401*ln(2+3*x)-625/266
2/(3+5*x)^2+37500/14641/(3+5*x)+1809375/161051*ln(3+5*x)

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Maxima [A]  time = 1.05993, size = 100, normalized size = 1.16 \begin{align*} \frac{2254231800 \, x^{4} + 524583660 \, x^{3} - 1362222102 \, x^{2} - 159141275 \, x + 213794156}{10043726 \,{\left (300 \, x^{5} + 260 \, x^{4} - 137 \, x^{3} - 136 \, x^{2} + 15 \, x + 18\right )}} + \frac{1809375}{161051} \, \log \left (5 \, x + 3\right ) - \frac{26973}{2401} \, \log \left (3 \, x + 2\right ) - \frac{280752}{386683451} \, \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)^3,x, algorithm="maxima")

[Out]

1/10043726*(2254231800*x^4 + 524583660*x^3 - 1362222102*x^2 - 159141275*x + 213794156)/(300*x^5 + 260*x^4 - 13
7*x^3 - 136*x^2 + 15*x + 18) + 1809375/161051*log(5*x + 3) - 26973/2401*log(3*x + 2) - 280752/386683451*log(2*
x - 1)

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Fricas [B]  time = 1.59973, size = 514, normalized size = 5.98 \begin{align*} \frac{173575848600 \, x^{4} + 40392941820 \, x^{3} - 104891101854 \, x^{2} + 8688618750 \,{\left (300 \, x^{5} + 260 \, x^{4} - 137 \, x^{3} - 136 \, x^{2} + 15 \, x + 18\right )} \log \left (5 \, x + 3\right ) - 8688057246 \,{\left (300 \, x^{5} + 260 \, x^{4} - 137 \, x^{3} - 136 \, x^{2} + 15 \, x + 18\right )} \log \left (3 \, x + 2\right ) - 561504 \,{\left (300 \, x^{5} + 260 \, x^{4} - 137 \, x^{3} - 136 \, x^{2} + 15 \, x + 18\right )} \log \left (2 \, x - 1\right ) - 12253878175 \, x + 16462150012}{773366902 \,{\left (300 \, x^{5} + 260 \, x^{4} - 137 \, x^{3} - 136 \, x^{2} + 15 \, x + 18\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)^3,x, algorithm="fricas")

[Out]

1/773366902*(173575848600*x^4 + 40392941820*x^3 - 104891101854*x^2 + 8688618750*(300*x^5 + 260*x^4 - 137*x^3 -
136*x^2 + 15*x + 18)*log(5*x + 3) - 8688057246*(300*x^5 + 260*x^4 - 137*x^3 - 136*x^2 + 15*x + 18)*log(3*x +
2) - 561504*(300*x^5 + 260*x^4 - 137*x^3 - 136*x^2 + 15*x + 18)*log(2*x - 1) - 12253878175*x + 16462150012)/(3
00*x^5 + 260*x^4 - 137*x^3 - 136*x^2 + 15*x + 18)

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Sympy [A]  time = 0.236783, size = 75, normalized size = 0.87 \begin{align*} \frac{2254231800 x^{4} + 524583660 x^{3} - 1362222102 x^{2} - 159141275 x + 213794156}{3013117800 x^{5} + 2611368760 x^{4} - 1375990462 x^{3} - 1365946736 x^{2} + 150655890 x + 180787068} - \frac{280752 \log{\left (x - \frac{1}{2} \right )}}{386683451} + \frac{1809375 \log{\left (x + \frac{3}{5} \right )}}{161051} - \frac{26973 \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)**3,x)

[Out]

(2254231800*x**4 + 524583660*x**3 - 1362222102*x**2 - 159141275*x + 213794156)/(3013117800*x**5 + 2611368760*x
**4 - 1375990462*x**3 - 1365946736*x**2 + 150655890*x + 180787068) - 280752*log(x - 1/2)/386683451 + 1809375*l
og(x + 3/5)/161051 - 26973*log(x + 2/3)/2401

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Giac [A]  time = 1.75341, size = 128, normalized size = 1.49 \begin{align*} \frac{243}{343 \,{\left (3 \, x + 2\right )}} - \frac{9 \,{\left (\frac{55432245900}{3 \, x + 2} - \frac{106776659235}{{\left (3 \, x + 2\right )}^{2}} + \frac{22794463286}{{\left (3 \, x + 2\right )}^{3}} - 7652987500\right )}}{70306082 \,{\left (\frac{7}{3 \, x + 2} - 2\right )}^{2}{\left (\frac{1}{3 \, x + 2} - 5\right )}^{2}} + \frac{1809375}{161051} \, \log \left ({\left | -\frac{1}{3 \, x + 2} + 5 \right |}\right ) - \frac{280752}{386683451} \, \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)^3,x, algorithm="giac")

[Out]

243/343/(3*x + 2) - 9/70306082*(55432245900/(3*x + 2) - 106776659235/(3*x + 2)^2 + 22794463286/(3*x + 2)^3 - 7
652987500)/((7/(3*x + 2) - 2)^2*(1/(3*x + 2) - 5)^2) + 1809375/161051*log(abs(-1/(3*x + 2) + 5)) - 280752/3866
83451*log(abs(-7/(3*x + 2) + 2))