3.1709 \(\int \frac{1}{(1-2 x)^3 (2+3 x)^3 (3+5 x)^3} \, dx\)

Optimal. Leaf size=97 \[ \frac{6528}{35153041 (1-2 x)}+\frac{26973}{2401 (3 x+2)}+\frac{290625}{14641 (5 x+3)}+\frac{16}{456533 (1-2 x)^2}+\frac{243}{686 (3 x+2)^2}-\frac{3125}{2662 (5 x+3)^2}-\frac{776928 \log (1-2 x)}{2706784157}-\frac{1944972 \log (3 x+2)}{16807}+\frac{18637500 \log (5 x+3)}{161051} \]

[Out]

16/(456533*(1 - 2*x)^2) + 6528/(35153041*(1 - 2*x)) + 243/(686*(2 + 3*x)^2) + 26973/(2401*(2 + 3*x)) - 3125/(2
662*(3 + 5*x)^2) + 290625/(14641*(3 + 5*x)) - (776928*Log[1 - 2*x])/2706784157 - (1944972*Log[2 + 3*x])/16807
+ (18637500*Log[3 + 5*x])/161051

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Rubi [A]  time = 0.0531709, 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} \[ \frac{6528}{35153041 (1-2 x)}+\frac{26973}{2401 (3 x+2)}+\frac{290625}{14641 (5 x+3)}+\frac{16}{456533 (1-2 x)^2}+\frac{243}{686 (3 x+2)^2}-\frac{3125}{2662 (5 x+3)^2}-\frac{776928 \log (1-2 x)}{2706784157}-\frac{1944972 \log (3 x+2)}{16807}+\frac{18637500 \log (5 x+3)}{161051} \]

Antiderivative was successfully verified.

[In]

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

[Out]

16/(456533*(1 - 2*x)^2) + 6528/(35153041*(1 - 2*x)) + 243/(686*(2 + 3*x)^2) + 26973/(2401*(2 + 3*x)) - 3125/(2
662*(3 + 5*x)^2) + 290625/(14641*(3 + 5*x)) - (776928*Log[1 - 2*x])/2706784157 - (1944972*Log[2 + 3*x])/16807
+ (18637500*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 (3+5 x)^3} \, dx &=\int \left (-\frac{64}{456533 (-1+2 x)^3}+\frac{13056}{35153041 (-1+2 x)^2}-\frac{1553856}{2706784157 (-1+2 x)}-\frac{729}{343 (2+3 x)^3}-\frac{80919}{2401 (2+3 x)^2}-\frac{5834916}{16807 (2+3 x)}+\frac{15625}{1331 (3+5 x)^3}-\frac{1453125}{14641 (3+5 x)^2}+\frac{93187500}{161051 (3+5 x)}\right ) \, dx\\ &=\frac{16}{456533 (1-2 x)^2}+\frac{6528}{35153041 (1-2 x)}+\frac{243}{686 (2+3 x)^2}+\frac{26973}{2401 (2+3 x)}-\frac{3125}{2662 (3+5 x)^2}+\frac{290625}{14641 (3+5 x)}-\frac{776928 \log (1-2 x)}{2706784157}-\frac{1944972 \log (2+3 x)}{16807}+\frac{18637500 \log (3+5 x)}{161051}\\ \end{align*}

Mathematica [A]  time = 0.133297, size = 88, normalized size = 0.91 \[ \frac{2 \left (\frac{77}{4} \left (\frac{789823386}{3 x+2}+\frac{1395581250}{5 x+3}+\frac{24904341}{(3 x+2)^2}-\frac{82534375}{(5 x+3)^2}+\frac{13056}{1-2 x}+\frac{2464}{(1-2 x)^2}\right )-388464 \log (1-2 x)-156619842786 \log (6 x+4)[phone] \log (10 x+6)\right )}{2706784157} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((77*(2464/(1 - 2*x)^2 + 13056/(1 - 2*x) + 24904341/(2 + 3*x)^2 + 789823386/(2 + 3*x) - 82534375/(3 + 5*x)^
2 + 1395581250/(3 + 5*x)))/4 - 388464*Log[1 - 2*x] - 156619842786*Log[4 + 6*x] + 156620231250*Log[6 + 10*x]))/
2706784157

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Maple [A]  time = 0.013, size = 80, normalized size = 0.8 \begin{align*}{\frac{16}{456533\, \left ( 2\,x-1 \right ) ^{2}}}-{\frac{6528}{70306082\,x-35153041}}-{\frac{776928\,\ln \left ( 2\,x-1 \right ) }{2706784157}}+{\frac{243}{686\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{26973}{4802+7203\,x}}-{\frac{1944972\,\ln \left ( 2+3\,x \right ) }{16807}}-{\frac{3125}{2662\, \left ( 3+5\,x \right ) ^{2}}}+{\frac{290625}{43923+73205\,x}}+{\frac{18637500\,\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/(3+5*x)^3,x)

[Out]

16/456533/(2*x-1)^2-6528/35153041/(2*x-1)-776928/2706784157*ln(2*x-1)+243/686/(2+3*x)^2+26973/2401/(2+3*x)-194
4972/16807*ln(2+3*x)-3125/2662/(3+5*x)^2+290625/14641/(3+5*x)[phone]/161051*ln(3+5*x)

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Maxima [A]  time = 1.0541, size = 113, normalized size = 1.16 \begin{align*} \frac{488145765600 \, x^{5} + 439319535120 \, x^{4} - 218954328504 \, x^{3} - 231191334456 \, x^{2} + 23195310772 \, x + 30858356237}{70306082 \,{\left (900 \, x^{6} + 1380 \, x^{5} + 109 \, x^{4} - 682 \, x^{3} - 227 \, x^{2} + 84 \, x + 36\right )}} + \frac{18637500}{161051} \, \log \left (5 \, x + 3\right ) - \frac{1944972}{16807} \, \log \left (3 \, x + 2\right ) - \frac{776928}{2706784157} \, \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)^3,x, algorithm="maxima")

[Out]

1/70306082*(488145765600*x^5 + 439319535120*x^4 - 218954328504*x^3 - 231191334456*x^2 + 23195310772*x + 308583
56237)/(900*x^6 + 1380*x^5 + 109*x^4 - 682*x^3 - 227*x^2 + 84*x + 36) + 18637500/161051*log(5*x + 3) - 1944972
/16807*log(3*x + 2) - 776928/2706784157*log(2*x - 1)

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Fricas [B]  time = 1.54689, size = 625, normalized size = 6.44 \begin{align*} \frac{37587223951200 \, x^{5} + 33827604204240 \, x^{4} - 16859483294808 \, x^{3} - 17801732753112 \, x^{2} + 626480925000 \,{\left (900 \, x^{6} + 1380 \, x^{5} + 109 \, x^{4} - 682 \, x^{3} - 227 \, x^{2} + 84 \, x + 36\right )} \log \left (5 \, x + 3\right ) - 626479371144 \,{\left (900 \, x^{6} + 1380 \, x^{5} + 109 \, x^{4} - 682 \, x^{3} - 227 \, x^{2} + 84 \, x + 36\right )} \log \left (3 \, x + 2\right ) - 1553856 \,{\left (900 \, x^{6} + 1380 \, x^{5} + 109 \, x^{4} - 682 \, x^{3} - 227 \, x^{2} + 84 \, x + 36\right )} \log \left (2 \, x - 1\right ) + 1786038929444 \, x + 2376093430249}{5413568314 \,{\left (900 \, x^{6} + 1380 \, x^{5} + 109 \, x^{4} - 682 \, x^{3} - 227 \, x^{2} + 84 \, x + 36\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)^3,x, algorithm="fricas")

[Out]

1/5413568314*(37587223951200*x^5 + 33827604204240*x^4 - 16859483294808*x^3 - 17801732753112*x^2 + 626480925000
*(900*x^6 + 1380*x^5 + 109*x^4 - 682*x^3 - 227*x^2 + 84*x + 36)*log(5*x + 3) - 626479371144*(900*x^6 + 1380*x^
5 + 109*x^4 - 682*x^3 - 227*x^2 + 84*x + 36)*log(3*x + 2) - 1553856*(900*x^6 + 1380*x^5 + 109*x^4 - 682*x^3 -
227*x^2 + 84*x + 36)*log(2*x - 1) + 1786038929444*x + 2376093430249)/(900*x^6 + 1380*x^5 + 109*x^4 - 682*x^3 -
227*x^2 + 84*x + 36)

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Sympy [A]  time = 0.255539, size = 85, normalized size = 0.88 \begin{align*} \frac{488145765600 x^{5} + 439319535120 x^{4} - 218954328504 x^{3} - 231191334456 x^{2} + 23195310772 x + 30858356237}{63275473800 x^{6} + 97022393160 x^{5} + 7663362938 x^{4} - 47948747924 x^{3} - 15959480614 x^{2} + 5905710888 x + 2531018952} - \frac{776928 \log{\left (x - \frac{1}{2} \right )}}{2706784157} + \frac{18637500 \log{\left (x + \frac{3}{5} \right )}}{161051} - \frac{1944972 \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)**3,x)

[Out]

(488145765600*x**5 + 439319535120*x**4 - 218954328504*x**3 - 231191334456*x**2 + 23195310772*x + 30858356237)/
(63275473800*x**6 + 97022393160*x**5 + 7663362938*x**4 - 47948747924*x**3 - 15959480614*x**2 + 5905710888*x +
2531018952) - 776928*log(x - 1/2)/2706784157 + 18637500*log(x + 3/5)/161051 - 1944972*log(x + 2/3)/16807

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Giac [A]  time = 1.38899, size = 97, normalized size = 1. \begin{align*} \frac{488145765600 \, x^{5} + 439319535120 \, x^{4} - 218954328504 \, x^{3} - 231191334456 \, x^{2} + 23195310772 \, x + 30858356237}{70306082 \,{\left (30 \, x^{3} + 23 \, x^{2} - 7 \, x - 6\right )}^{2}} + \frac{18637500}{161051} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac{1944972}{16807} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac{776928}{2706784157} \, \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)^3,x, algorithm="giac")

[Out]

1/70306082*(488145765600*x^5 + 439319535120*x^4 - 218954328504*x^3 - 231191334456*x^2 + 23195310772*x + 308583
56237)/(30*x^3 + 23*x^2 - 7*x - 6)^2 + 18637500/161051*log(abs(5*x + 3)) - 1944972/16807*log(abs(3*x + 2)) - 7
76928/2706784157*log(abs(2*x - 1))