3.1710 \(\int \frac{1}{(1-2 x)^3 (2+3 x)^4 (3+5 x)^3} \, dx\)

Optimal. Leaf size=108 \[ \frac{15168}{246071287 (1-2 x)}+\frac{1944972}{16807 (3 x+2)}+\frac{1968750}{14641 (5 x+3)}+\frac{32}{3195731 (1-2 x)^2}+\frac{26973}{4802 (3 x+2)^2}-\frac{15625}{2662 (5 x+3)^2}+\frac{81}{343 (3 x+2)^3}-\frac{2054400 \log (1-2 x)}{18947489099}-\frac{115534350 \log (3 x+2)}{117649}+\frac{158156250 \log (5 x+3)}{161051} \]

[Out]

32/(3195731*(1 - 2*x)^2) + 15168/(246071287*(1 - 2*x)) + 81/(343*(2 + 3*x)^3) + 26973/(4802*(2 + 3*x)^2) + 194
4972/(16807*(2 + 3*x)) - 15625/(2662*(3 + 5*x)^2) + 1968750/(14641*(3 + 5*x)) - (2054400*Log[1 - 2*x])/1894748
9099 - (115534350*Log[2 + 3*x])/117649 + (158156250*Log[3 + 5*x])/161051

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Rubi [A]  time = 0.0635186, antiderivative size = 108, 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} \[ \frac{15168}{246071287 (1-2 x)}+\frac{1944972}{16807 (3 x+2)}+\frac{1968750}{14641 (5 x+3)}+\frac{32}{3195731 (1-2 x)^2}+\frac{26973}{4802 (3 x+2)^2}-\frac{15625}{2662 (5 x+3)^2}+\frac{81}{343 (3 x+2)^3}-\frac{2054400 \log (1-2 x)}{18947489099}-\frac{115534350 \log (3 x+2)}{117649}+\frac{158156250 \log (5 x+3)}{161051} \]

Antiderivative was successfully verified.

[In]

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

[Out]

32/(3195731*(1 - 2*x)^2) + 15168/(246071287*(1 - 2*x)) + 81/(343*(2 + 3*x)^3) + 26973/(4802*(2 + 3*x)^2) + 194
4972/(16807*(2 + 3*x)) - 15625/(2662*(3 + 5*x)^2) + 1968750/(14641*(3 + 5*x)) - (2054400*Log[1 - 2*x])/1894748
9099 - (115534350*Log[2 + 3*x])/117649 + (158156250*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)^4 (3+5 x)^3} \, dx &=\int \left (-\frac{128}{3195731 (-1+2 x)^3}+\frac{30336}{246071287 (-1+2 x)^2}-\frac{4108800}{18947489099 (-1+2 x)}-\frac{729}{343 (2+3 x)^4}-\frac{80919}{2401 (2+3 x)^3}-\frac{5834916}{16807 (2+3 x)^2}-\frac{346603050}{117649 (2+3 x)}+\frac{78125}{1331 (3+5 x)^3}-\frac{9843750}{14641 (3+5 x)^2}+\frac{790781250}{161051 (3+5 x)}\right ) \, dx\\ &=\frac{32}{3195731 (1-2 x)^2}+\frac{15168}{246071287 (1-2 x)}+\frac{81}{343 (2+3 x)^3}+\frac{26973}{4802 (2+3 x)^2}+\frac{1944972}{16807 (2+3 x)}-\frac{15625}{2662 (3+5 x)^2}+\frac{1968750}{14641 (3+5 x)}-\frac{2054400 \log (1-2 x)}{18947489099}-\frac{115534350 \log (2+3 x)}{117649}+\frac{158156250 \log (3+5 x)}{161051}\\ \end{align*}

Mathematica [A]  time = 0.10233, size = 82, normalized size = 0.76 \[ -\frac{3 \left (-\frac{77 \left (86993245890000 x^6+136289326113000 x^5+13177709631900 x^4-67213599053550 x^3-23334840827100 x^2+8254486652965 x+3666255393392\right )}{3 (3 x+2)^3 \left (10 x^2+x-3\right )^2}+1369600 \log (3-6 x)[phone] \log (3 x+2)-12404616437500 \log (-3 (5 x+3))\right )}{37894978198} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*((-77*(3666255393392 + 8254486652965*x - 23334840827100*x^2 - 67213599053550*x^3 + 13177709631900*x^4 + 13
6289326113000*x^5 + 86993245890000*x^6))/(3*(2 + 3*x)^3*(-3 + x + 10*x^2)^2) + 1369600*Log[3 - 6*x] + 12404615
067900*Log[2 + 3*x] - 12404616437500*Log[-3*(3 + 5*x)]))/37894978198

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Maple [A]  time = 0.013, size = 89, normalized size = 0.8 \begin{align*}{\frac{32}{3195731\, \left ( 2\,x-1 \right ) ^{2}}}-{\frac{15168}{492142574\,x-246071287}}-{\frac{2054400\,\ln \left ( 2\,x-1 \right ) }{18947489099}}+{\frac{81}{343\, \left ( 2+3\,x \right ) ^{3}}}+{\frac{26973}{4802\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{1944972}{33614+50421\,x}}-{\frac{115534350\,\ln \left ( 2+3\,x \right ) }{117649}}-{\frac{15625}{2662\, \left ( 3+5\,x \right ) ^{2}}}+{\frac{1968750}{43923+73205\,x}}+{\frac{158156250\,\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)^4/(3+5*x)^3,x)

[Out]

32/3195731/(2*x-1)^2-15168/246071287/(2*x-1)-2054400/18947489099*ln(2*x-1)+81/343/(2+3*x)^3+26973/4802/(2+3*x)
^2+1944972/16807/(2+3*x)-115534350/117649*ln(2+3*x)-15625/2662/(3+5*x)^2+1968750/14641/(3+5*x)[phone]/16105
1*ln(3+5*x)

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Maxima [A]  time = 2.91492, size = 127, normalized size = 1.18 \begin{align*} \frac{86993245890000 \, x^{6} + 136289326113000 \, x^{5} + 13177709631900 \, x^{4} - 67213599053550 \, x^{3} - 23334840827100 \, x^{2} + 8254486652965 \, x + 3666255393392}{492142574 \,{\left (2700 \, x^{7} + 5940 \, x^{6} + 3087 \, x^{5} - 1828 \, x^{4} - 2045 \, x^{3} - 202 \, x^{2} + 276 \, x + 72\right )}} + \frac{158156250}{161051} \, \log \left (5 \, x + 3\right ) - \frac{115534350}{117649} \, \log \left (3 \, x + 2\right ) - \frac{2054400}{18947489099} \, \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)^3,x, algorithm="maxima")

[Out]

1/492142574*(86993245890000*x^6 + 136289326113000*x^5 + 13177709631900*x^4 - 67213599053550*x^3 - 233348408271
00*x^2 + 8254486652965*x + 3666255393392)/(2700*x^7 + 5940*x^6 + 3087*x^5 - 1828*x^4 - 2045*x^3 - 202*x^2 + 27
6*x + 72) + 158156250/161051*log(5*x + 3) - 115534350/117649*log(3*x + 2) - 2054400/18947489099*log(2*x - 1)

________________________________________________________________________________________

Fricas [B]  time = 1.53648, size = 761, normalized size = 7.05 \begin{align*} \frac{6698479933530000 \, x^{6} + 10494278110701000 \, x^{5} + 1014683641656300 \, x^{4} - 5175447127123350 \, x^{3} - 1796782743686700 \, x^{2} + 37213849312500 \,{\left (2700 \, x^{7} + 5940 \, x^{6} + 3087 \, x^{5} - 1828 \, x^{4} - 2045 \, x^{3} - 202 \, x^{2} + 276 \, x + 72\right )} \log \left (5 \, x + 3\right ) - 37213845203700 \,{\left (2700 \, x^{7} + 5940 \, x^{6} + 3087 \, x^{5} - 1828 \, x^{4} - 2045 \, x^{3} - 202 \, x^{2} + 276 \, x + 72\right )} \log \left (3 \, x + 2\right ) - 4108800 \,{\left (2700 \, x^{7} + 5940 \, x^{6} + 3087 \, x^{5} - 1828 \, x^{4} - 2045 \, x^{3} - 202 \, x^{2} + 276 \, x + 72\right )} \log \left (2 \, x - 1\right ) + 635595472278305 \, x + 282301665291184}{37894978198 \,{\left (2700 \, x^{7} + 5940 \, x^{6} + 3087 \, x^{5} - 1828 \, x^{4} - 2045 \, x^{3} - 202 \, x^{2} + 276 \, x + 72\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)^3,x, algorithm="fricas")

[Out]

1/37894978198*(6698479933530000*x^6 + 10494278110701000*x^5 + 1014683641656300*x^4 - 5175447127123350*x^3 - 17
96782743686700*x^2 + 37213849312500*(2700*x^7 + 5940*x^6 + 3087*x^5 - 1828*x^4 - 2045*x^3 - 202*x^2 + 276*x +
72)*log(5*x + 3) - 37213845203700*(2700*x^7 + 5940*x^6 + 3087*x^5 - 1828*x^4 - 2045*x^3 - 202*x^2 + 276*x + 72
)*log(3*x + 2) - 4108800*(2700*x^7 + 5940*x^6 + 3087*x^5 - 1828*x^4 - 2045*x^3 - 202*x^2 + 276*x + 72)*log(2*x
- 1) + 635595472278305*x + 282301665291184)/(2700*x^7 + 5940*x^6 + 3087*x^5 - 1828*x^4 - 2045*x^3 - 202*x^2 +
276*x + 72)

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Sympy [A]  time = 0.288927, size = 95, normalized size = 0.88 \begin{align*} \frac{86993245890000 x^{6} + 136289326113000 x^{5} + 13177709631900 x^{4} - 67213599053550 x^{3} - 23334840827100 x^{2} + 8254486652965 x + 3666255393392}{1328784949800 x^{7} + 2923326889560 x^{6} + 1519244125938 x^{5} - 899636625272 x^{4} - 1006431563830 x^{3} - 99412799948 x^{2} + 135831350424 x + 35434265328} - \frac{2054400 \log{\left (x - \frac{1}{2} \right )}}{18947489099} + \frac{158156250 \log{\left (x + \frac{3}{5} \right )}}{161051} - \frac{115534350 \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)**3,x)

[Out]

(86993245890000*x**6 + 136289326113000*x**5 + 13177709631900*x**4 - 67213599053550*x**3 - 23334840827100*x**2
+ 8254486652965*x + 3666255393392)/(1328784949800*x**7 + 2923326889560*x**6 + 1519244125938*x**5 - 89963662527
2*x**4 - 1006431563830*x**3 - 99412799948*x**2 + 135831350424*x + 35434265328) - 2054400*log(x - 1/2)/18947489
099 + 158156250*log(x + 3/5)/161051 - 115534350*log(x + 2/3)/117649

________________________________________________________________________________________

Giac [A]  time = 1.99857, size = 109, normalized size = 1.01 \begin{align*} \frac{86993245890000 \, x^{6} + 136289326113000 \, x^{5} + 13177709631900 \, x^{4} - 67213599053550 \, x^{3} - 23334840827100 \, x^{2} + 8254486652965 \, x + 3666255393392}{492142574 \,{\left (5 \, x + 3\right )}^{2}{\left (3 \, x + 2\right )}^{3}{\left (2 \, x - 1\right )}^{2}} + \frac{158156250}{161051} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac{115534350}{117649} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac{2054400}{18947489099} \, \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)^4/(3+5*x)^3,x, algorithm="giac")

[Out]

1/492142574*(86993245890000*x^6 + 136289326113000*x^5 + 13177709631900*x^4 - 67213599053550*x^3 - 233348408271
00*x^2 + 8254486652965*x + 3666255393392)/((5*x + 3)^2*(3*x + 2)^3*(2*x - 1)^2) + 158156250/161051*log(abs(5*x
+ 3)) - 115534350/117649*log(abs(3*x + 2)) - 2054400/18947489099*log(abs(2*x - 1))