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

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

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Rubi [A]  time = 0.06, antiderivative size = 97, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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*}

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Mathematica [A]  time = 0.18, size = 88, normalized size = 0.91 \begin {gather*} \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)+115811115255 \log (6 x+4)-115810734375 \log (10 x+6)\right )}{1722499009} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(1-2 x)^3 (2+3 x)^4 (3+5 x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.29, size = 173, normalized size = 1.78 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 1.28, size = 104, normalized size = 1.07 \begin {gather*} -\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 {gather*}

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))

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maple [A]  time = 0.01, size = 80, normalized size = 0.82 \begin {gather*} -\frac {761760 \ln \left (2 x -1\right )}{1722499009}+\frac {15820110 \ln \left (3 x +2\right )}{117649}-\frac {1968750 \ln \left (5 x +3\right )}{14641}-\frac {15625}{1331 \left (5 x +3\right )}-\frac {27}{343 \left (3 x +2\right )^{3}}-\frac {3078}{2401 \left (3 x +2\right )^{2}}-\frac {333639}{16807 \left (3 x +2\right )}+\frac {16}{290521 \left (2 x -1\right )^{2}}-\frac {6464}{22370117 \left (2 x -1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.50, size = 84, normalized size = 0.87 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 0.05, size = 76, normalized size = 0.78 \begin {gather*} \frac {15820110\,\ln \left (x+\frac {2}{3}\right )}{117649}-\frac {761760\,\ln \left (x-\frac {1}{2}\right )}{1722499009}-\frac {1968750\,\ln \left (x+\frac {3}{5}\right )}{14641}-\frac {\frac {200549610\,x^5}{22370117}+\frac {27695853\,x^4}{3195731}-\frac {171864619\,x^3}{44740234}-\frac {1240661509\,x^2}{268441404}+\frac {98810707\,x}{268441404}+\frac {7606921499}{12079863180}}{x^6+\frac {8\,x^5}{5}+\frac {11\,x^4}{60}-\frac {85\,x^3}{108}-\frac {77\,x^2}{270}+\frac {13\,x}{135}+\frac {2}{45}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(15820110*log(x + 2/3))/117649 - (761760*log(x - 1/2))/1722499009 - (1968750*log(x + 3/5))/14641 - ((98810707*
x)/268441404 - (1240661509*x^2)/268441404 - (171864619*x^3)/44740234 + (27695853*x^4)/3195731 + (200549610*x^5
)/22370117 + 7606921499/12079863180)/((13*x)/135 - (77*x^2)/270 - (85*x^3)/108 + (11*x^4)/60 + (8*x^5)/5 + x^6
 + 2/45)

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sympy [A]  time = 0.29, size = 85, normalized size = 0.88 \begin {gather*} - \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 {gather*}

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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