3.1696 \(\int \frac{1}{(1-2 x)^3 (2+3 x)^4 (3+5 x)^2} \, dx\)

Optimal. Leaf size=97 \[ \frac{6464}{22370117 (1-2 x)}-\frac{333639}{16807 (3 x+2)}-\frac{15625}{1331 (5 x+3)}+\frac{16}{290521 (1-2 x)^2}-\frac{3078}{2401 (3 x+2)^2}-\frac{27}{343 (3 x+2)^3}-\frac{761760 \log (1-2 x)}{1722499009}+\frac{15820110 \log (3 x+2)}{117649}-\frac{1968750 \log (5 x+3)}{14641} \]

[Out]

16/(290521*(1 - 2*x)^2) + 6464/(22370117*(1 - 2*x)) - 27/(343*(2 + 3*x)^3) - 3078/(2401*(2 + 3*x)^2) - 333639/
(16807*(2 + 3*x)) - 15625/(1331*(3 + 5*x)) - (761760*Log[1 - 2*x])/1722499009 + (15820110*Log[2 + 3*x])/117649
- (1968750*Log[3 + 5*x])/14641

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Rubi [A]  time = 0.0553048, antiderivative size = 97, 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} \[ \frac{6464}{22370117 (1-2 x)}-\frac{333639}{16807 (3 x+2)}-\frac{15625}{1331 (5 x+3)}+\frac{16}{290521 (1-2 x)^2}-\frac{3078}{2401 (3 x+2)^2}-\frac{27}{343 (3 x+2)^3}-\frac{761760 \log (1-2 x)}{1722499009}+\frac{15820110 \log (3 x+2)}{117649}-\frac{1968750 \log (5 x+3)}{14641} \]

Antiderivative was successfully verified.

[In]

Int[1/((1 - 2*x)^3*(2 + 3*x)^4*(3 + 5*x)^2),x]

[Out]

16/(290521*(1 - 2*x)^2) + 6464/(22370117*(1 - 2*x)) - 27/(343*(2 + 3*x)^3) - 3078/(2401*(2 + 3*x)^2) - 333639/
(16807*(2 + 3*x)) - 15625/(1331*(3 + 5*x)) - (761760*Log[1 - 2*x])/1722499009 + (15820110*Log[2 + 3*x])/117649
- (1968750*Log[3 + 5*x])/14641

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)^4 (3+5 x)^2} \, dx &=\int \left (-\frac{64}{290521 (-1+2 x)^3}+\frac{12928}{22370117 (-1+2 x)^2}-\frac{1523520}{1722499009 (-1+2 x)}+\frac{243}{343 (2+3 x)^4}+\frac{18468}{2401 (2+3 x)^3}+\frac{1000917}{16807 (2+3 x)^2}+\frac{47460330}{117649 (2+3 x)}+\frac{78125}{1331 (3+5 x)^2}-\frac{9843750}{14641 (3+5 x)}\right ) \, dx\\ &=\frac{16}{290521 (1-2 x)^2}+\frac{6464}{22370117 (1-2 x)}-\frac{27}{343 (2+3 x)^3}-\frac{3078}{2401 (2+3 x)^2}-\frac{333639}{16807 (2+3 x)}-\frac{15625}{1331 (3+5 x)}-\frac{761760 \log (1-2 x)}{1722499009}+\frac{15820110 \log (2+3 x)}{117649}-\frac{1968750 \log (3+5 x)}{14641}\\ \end{align*}

Mathematica [A]  time = 0.122385, size = 88, normalized size = 0.91 \[ \frac{2 \left (\frac{77}{2} \left (-\frac{444073509}{3 x+2}-\frac{262609375}{5 x+3}-\frac{28677726}{(3 x+2)^2}-\frac{1760913}{(3 x+2)^3}+\frac{6464}{1-2 x}+\frac{1232}{(1-2 x)^2}\right )-380880 \log (1-2 x)[phone] \log (6 x+4)-115810734375 \log (10 x+6)\right )}{1722499009} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - 2*x)^3*(2 + 3*x)^4*(3 + 5*x)^2),x]

[Out]

(2*((77*(1232/(1 - 2*x)^2 + 6464/(1 - 2*x) - 1760913/(2 + 3*x)^3 - 28677726/(2 + 3*x)^2 - 444073509/(2 + 3*x)
- 262609375/(3 + 5*x)))/2 - 380880*Log[1 - 2*x] + 115811115255*Log[4 + 6*x] - 115810734375*Log[6 + 10*x]))/172
2499009

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Maple [A]  time = 0.013, size = 80, normalized size = 0.8 \begin{align*}{\frac{16}{290521\, \left ( 2\,x-1 \right ) ^{2}}}-{\frac{6464}{44740234\,x-22370117}}-{\frac{761760\,\ln \left ( 2\,x-1 \right ) }{1722499009}}-{\frac{27}{343\, \left ( 2+3\,x \right ) ^{3}}}-{\frac{3078}{2401\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{333639}{33614+50421\,x}}+{\frac{15820110\,\ln \left ( 2+3\,x \right ) }{117649}}-{\frac{15625}{3993+6655\,x}}-{\frac{1968750\,\ln \left ( 3+5\,x \right ) }{14641}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-2*x)^3/(2+3*x)^4/(3+5*x)^2,x)

[Out]

16/290521/(2*x-1)^2-6464/22370117/(2*x-1)-761760/1722499009*ln(2*x-1)-27/343/(2+3*x)^3-3078/2401/(2+3*x)^2-333
639/16807/(2+3*x)[phone]/117649*ln(2+3*x)-15625/1331/(3+5*x)-1968750/14641*ln(3+5*x)

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Maxima [A]  time = 1.02425, size = 113, normalized size = 1.16 \begin{align*} -\frac{108296789400 \, x^{5} + 104690324340 \, x^{4} - 46403447130 \, x^{3} - 55829767905 \, x^{2} + 4446481815 \, x + 7606921499}{22370117 \,{\left (540 \, x^{6} + 864 \, x^{5} + 99 \, x^{4} - 425 \, x^{3} - 154 \, x^{2} + 52 \, x + 24\right )}} - \frac{1968750}{14641} \, \log \left (5 \, x + 3\right ) + \frac{15820110}{117649} \, \log \left (3 \, x + 2\right ) - \frac{761760}{1722499009} \, \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)^4/(3+5*x)^2,x, algorithm="maxima")

[Out]

-1/22370117*(108296789400*x^5 + 104690324340*x^4 - 46403447130*x^3 - 55829767905*x^2 + 4446481815*x + 76069214
99)/(540*x^6 + 864*x^5 + 99*x^4 - 425*x^3 - 154*x^2 + 52*x + 24) - 1968750/14641*log(5*x + 3) + 15820110/11764
9*log(3*x + 2) - 761760/1722499009*log(2*x - 1)

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Fricas [B]  time = 1.54411, size = 606, normalized size = 6.25 \begin{align*} -\frac{8338852783800 \, x^{5} + 8061154974180 \, x^{4} - 3573065429010 \, x^{3} - 4298892128685 \, x^{2} + 231621468750 \,{\left (540 \, x^{6} + 864 \, x^{5} + 99 \, x^{4} - 425 \, x^{3} - 154 \, x^{2} + 52 \, x + 24\right )} \log \left (5 \, x + 3\right ) - 231622230510 \,{\left (540 \, x^{6} + 864 \, x^{5} + 99 \, x^{4} - 425 \, x^{3} - 154 \, x^{2} + 52 \, x + 24\right )} \log \left (3 \, x + 2\right ) + 761760 \,{\left (540 \, x^{6} + 864 \, x^{5} + 99 \, x^{4} - 425 \, x^{3} - 154 \, x^{2} + 52 \, x + 24\right )} \log \left (2 \, x - 1\right ) + 342379099755 \, x + 585732955423}{1722499009 \,{\left (540 \, x^{6} + 864 \, x^{5} + 99 \, x^{4} - 425 \, x^{3} - 154 \, x^{2} + 52 \, x + 24\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)^3/(2+3*x)^4/(3+5*x)^2,x, algorithm="fricas")

[Out]

-1/1722499009*(8338852783800*x^5 + 8061154974180*x^4 - 3573065429010*x^3 - 4298892128685*x^2 + 231621468750*(5
40*x^6 + 864*x^5 + 99*x^4 - 425*x^3 - 154*x^2 + 52*x + 24)*log(5*x + 3) - 231622230510*(540*x^6 + 864*x^5 + 99
*x^4 - 425*x^3 - 154*x^2 + 52*x + 24)*log(3*x + 2) + 761760*(540*x^6 + 864*x^5 + 99*x^4 - 425*x^3 - 154*x^2 +
52*x + 24)*log(2*x - 1) + 342379099755*x + 585732955423)/(540*x^6 + 864*x^5 + 99*x^4 - 425*x^3 - 154*x^2 + 52*
x + 24)

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Sympy [A]  time = 0.256681, size = 85, normalized size = 0.88 \begin{align*} - \frac{108296789400 x^{5} + 104690324340 x^{4} - 46403447130 x^{3} - 55829767905 x^{2} + 4446481815 x + 7606921499}{12079863180 x^{6} + 19327781088 x^{5} + 2214641583 x^{4} - 9507299725 x^{3} - 3444998018 x^{2} + 1163246084 x + 536882808} - \frac{761760 \log{\left (x - \frac{1}{2} \right )}}{1722499009} - \frac{1968750 \log{\left (x + \frac{3}{5} \right )}}{14641} + \frac{15820110 \log{\left (x + \frac{2}{3} \right )}}{117649} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)**3/(2+3*x)**4/(3+5*x)**2,x)

[Out]

-(108296789400*x**5 + 104690324340*x**4 - 46403447130*x**3 - 55829767905*x**2 + 4446481815*x + 7606921499)/(12
079863180*x**6 + 19327781088*x**5 + 2214641583*x**4 - 9507299725*x**3 - 3444998018*x**2 + 1163246084*x + 53688
2808) - 761760*log(x - 1/2)/1722499009 - 1968750*log(x + 3/5)/14641 + 15820110*log(x + 2/3)/117649

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Giac [A]  time = 2.13175, size = 140, normalized size = 1.44 \begin{align*} -\frac{15625}{1331 \,{\left (5 \, x + 3\right )}} - \frac{25 \,{\left (\frac{1535578147116}{5 \, x + 3} - \frac{3297944687832}{{\left (5 \, x + 3\right )}^{2}} - \frac{3224232263641}{{\left (5 \, x + 3\right )}^{3}} - \frac{689127341628}{{\left (5 \, x + 3\right )}^{4}} - 150040675728\right )}}{246071287 \,{\left (\frac{11}{5 \, x + 3} - 2\right )}^{2}{\left (\frac{1}{5 \, x + 3} + 3\right )}^{3}} + \frac{15820110}{117649} \, \log \left ({\left | -\frac{1}{5 \, x + 3} - 3 \right |}\right ) - \frac{761760}{1722499009} \, \log \left ({\left | -\frac{11}{5 \, x + 3} + 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)^4/(3+5*x)^2,x, algorithm="giac")

[Out]

-15625/1331/(5*x + 3) - 25/246071287*(1535578147116/(5*x + 3) - 3297944687832/(5*x + 3)^2 - 3224232263641/(5*x
+ 3)^3 - 689127341628/(5*x + 3)^4 - 150040675728)/((11/(5*x + 3) - 2)^2*(1/(5*x + 3) + 3)^3) + 15820110/11764
9*log(abs(-1/(5*x + 3) - 3)) - 761760/1722499009*log(abs(-11/(5*x + 3) + 2))