3.1707 \(\int \frac{1}{(1-2 x)^3 (2+3 x) (3+5 x)^3} \, dx\)

Optimal. Leaf size=75 \[ \frac{1104}{717409 (1-2 x)}+\frac{3375}{14641 (5 x+3)}+\frac{4}{9317 (1-2 x)^2}-\frac{125}{2662 (5 x+3)^2}-\frac{95232 \log (1-2 x)}{55240493}-\frac{243}{343} \log (3 x+2)+\frac{114375 \log (5 x+3)}{161051} \]

[Out]

4/(9317*(1 - 2*x)^2) + 1104/(717409*(1 - 2*x)) - 125/(2662*(3 + 5*x)^2) + 3375/(14641*(3 + 5*x)) - (95232*Log[
1 - 2*x])/55240493 - (243*Log[2 + 3*x])/343 + (114375*Log[3 + 5*x])/161051

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Rubi [A]  time = 0.0385246, 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} \[ \frac{1104}{717409 (1-2 x)}+\frac{3375}{14641 (5 x+3)}+\frac{4}{9317 (1-2 x)^2}-\frac{125}{2662 (5 x+3)^2}-\frac{95232 \log (1-2 x)}{55240493}-\frac{243}{343} \log (3 x+2)+\frac{114375 \log (5 x+3)}{161051} \]

Antiderivative was successfully verified.

[In]

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

[Out]

4/(9317*(1 - 2*x)^2) + 1104/(717409*(1 - 2*x)) - 125/(2662*(3 + 5*x)^2) + 3375/(14641*(3 + 5*x)) - (95232*Log[
1 - 2*x])/55240493 - (243*Log[2 + 3*x])/343 + (114375*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) (3+5 x)^3} \, dx &=\int \left (-\frac{16}{9317 (-1+2 x)^3}+\frac{2208}{717409 (-1+2 x)^2}-\frac{190464}{55240493 (-1+2 x)}-\frac{729}{343 (2+3 x)}+\frac{625}{1331 (3+5 x)^3}-\frac{16875}{14641 (3+5 x)^2}+\frac{571875}{161051 (3+5 x)}\right ) \, dx\\ &=\frac{4}{9317 (1-2 x)^2}+\frac{1104}{717409 (1-2 x)}-\frac{125}{2662 (3+5 x)^2}+\frac{3375}{14641 (3+5 x)}-\frac{95232 \log (1-2 x)}{55240493}-\frac{243}{343} \log (2+3 x)+\frac{114375 \log (3+5 x)}{161051}\\ \end{align*}

Mathematica [A]  time = 0.0604798, size = 60, normalized size = 0.8 \[ -\frac{3 \left (-\frac{77 \left (6504600 x^3-2977380 x^2-2000774 x+950291\right )}{3 \left (10 x^2+x-3\right )^2}+63488 \log (3-6 x)[phone] \log (3 x+2)-26153750 \log (-3 (5 x+3))\right )}{110480986} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*((-77*(950291 - 2000774*x - 2977380*x^2 + 6504600*x^3))/(3*(-3 + x + 10*x^2)^2) + 63488*Log[3 - 6*x] + 260
90262*Log[2 + 3*x] - 26153750*Log[-3*(3 + 5*x)]))/110480986

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Maple [A]  time = 0.01, size = 62, normalized size = 0.8 \begin{align*}{\frac{4}{9317\, \left ( 2\,x-1 \right ) ^{2}}}-{\frac{1104}{1434818\,x-717409}}-{\frac{95232\,\ln \left ( 2\,x-1 \right ) }{55240493}}-{\frac{243\,\ln \left ( 2+3\,x \right ) }{343}}-{\frac{125}{2662\, \left ( 3+5\,x \right ) ^{2}}}+{\frac{3375}{43923+73205\,x}}+{\frac{114375\,\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)/(3+5*x)^3,x)

[Out]

4/9317/(2*x-1)^2-1104/717409/(2*x-1)-95232/55240493*ln(2*x-1)-243/343*ln(2+3*x)-125/2662/(3+5*x)^2+3375/14641/
(3+5*x)+114375/161051*ln(3+5*x)

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Maxima [A]  time = 1.51788, size = 86, normalized size = 1.15 \begin{align*} \frac{6504600 \, x^{3} - 2977380 \, x^{2} - 2000774 \, x + 950291}{1434818 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} + \frac{114375}{161051} \, \log \left (5 \, x + 3\right ) - \frac{243}{343} \, \log \left (3 \, x + 2\right ) - \frac{95232}{55240493} \, \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+5*x)^3,x, algorithm="maxima")

[Out]

1/1434818*(6504600*x^3 - 2977380*x^2 - 2000774*x + 950291)/(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9) + 114375/1610
51*log(5*x + 3) - 243/343*log(3*x + 2) - 95232/55240493*log(2*x - 1)

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Fricas [B]  time = 1.55965, size = 394, normalized size = 5.25 \begin{align*} \frac{500854200 \, x^{3} - 229258260 \, x^{2} + 78461250 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (5 \, x + 3\right ) - 78270786 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (3 \, x + 2\right ) - 190464 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (2 \, x - 1\right ) - 154059598 \, x + 73172407}{110480986 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/110480986*(500854200*x^3 - 229258260*x^2 + 78461250*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*log(5*x + 3) - 782
70786*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*log(3*x + 2) - 190464*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*log(2*
x - 1) - 154059598*x + 73172407)/(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)

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Sympy [A]  time = 0.217213, size = 65, normalized size = 0.87 \begin{align*} \frac{6504600 x^{3} - 2977380 x^{2} - 2000774 x + 950291}{143481800 x^{4} + 28696360 x^{3} - 84654262 x^{2} - 8608908 x + 12913362} - \frac{95232 \log{\left (x - \frac{1}{2} \right )}}{55240493} + \frac{114375 \log{\left (x + \frac{3}{5} \right )}}{161051} - \frac{243 \log{\left (x + \frac{2}{3} \right )}}{343} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(6504600*x**3 - 2977380*x**2 - 2000774*x + 950291)/(143481800*x**4 + 28696360*x**3 - 84654262*x**2 - 8608908*x
+ 12913362) - 95232*log(x - 1/2)/55240493 + 114375*log(x + 3/5)/161051 - 243*log(x + 2/3)/343

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Giac [A]  time = 2.09771, size = 80, normalized size = 1.07 \begin{align*} \frac{6504600 \, x^{3} - 2977380 \, x^{2} - 2000774 \, x + 950291}{1434818 \,{\left (5 \, x + 3\right )}^{2}{\left (2 \, x - 1\right )}^{2}} + \frac{114375}{161051} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac{243}{343} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac{95232}{55240493} \, \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+5*x)^3,x, algorithm="giac")

[Out]

1/1434818*(6504600*x^3 - 2977380*x^2 - 2000774*x + 950291)/((5*x + 3)^2*(2*x - 1)^2) + 114375/161051*log(abs(5
*x + 3)) - 243/343*log(abs(3*x + 2)) - 95232/55240493*log(abs(2*x - 1))