3.17.15 \(\int \frac {1}{(1-2 x)^2 (2+3 x)^4 (3+5 x)^2} \, dx\) [1615]

Optimal. Leaf size=86 \[ \frac {32}{290521 (1-2 x)}-\frac {9}{49 (2+3 x)^3}-\frac {999}{343 (2+3 x)^2}-\frac {107109}{2401 (2+3 x)}-\frac {3125}{121 (3+5 x)}-\frac {6464 \log (1-2 x)}{22370117}+\frac {5050944 \log (2+3 x)}{16807}-\frac {400000 \log (3+5 x)}{1331} \]

[Out]

32/290521/(1-2*x)-9/49/(2+3*x)^3-999/343/(2+3*x)^2-107109/2401/(2+3*x)-3125/121/(3+5*x)-6464/22370117*ln(1-2*x
)+5050944/16807*ln(2+3*x)-400000/1331*ln(3+5*x)

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Rubi [A]
time = 0.03, antiderivative size = 86, 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 = {90} \begin {gather*} \frac {32}{290521 (1-2 x)}-\frac {107109}{2401 (3 x+2)}-\frac {3125}{121 (5 x+3)}-\frac {999}{343 (3 x+2)^2}-\frac {9}{49 (3 x+2)^3}-\frac {6464 \log (1-2 x)}{22370117}+\frac {5050944 \log (3 x+2)}{16807}-\frac {400000 \log (5 x+3)}{1331} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

32/(290521*(1 - 2*x)) - 9/(49*(2 + 3*x)^3) - 999/(343*(2 + 3*x)^2) - 107109/(2401*(2 + 3*x)) - 3125/(121*(3 +
5*x)) - (6464*Log[1 - 2*x])/22370117 + (5050944*Log[2 + 3*x])/16807 - (400000*Log[3 + 5*x])/1331

Rule 90

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)^2 (2+3 x)^4 (3+5 x)^2} \, dx &=\int \left (\frac {64}{290521 (-1+2 x)^2}-\frac {12928}{22370117 (-1+2 x)}+\frac {81}{49 (2+3 x)^4}+\frac {5994}{343 (2+3 x)^3}+\frac {321327}{2401 (2+3 x)^2}+\frac {15152832}{16807 (2+3 x)}+\frac {15625}{121 (3+5 x)^2}-\frac {2000000}{1331 (3+5 x)}\right ) \, dx\\ &=\frac {32}{290521 (1-2 x)}-\frac {9}{49 (2+3 x)^3}-\frac {999}{343 (2+3 x)^2}-\frac {107109}{2401 (2+3 x)}-\frac {3125}{121 (3+5 x)}-\frac {6464 \log (1-2 x)}{22370117}+\frac {5050944 \log (2+3 x)}{16807}-\frac {400000 \log (3+5 x)}{1331}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 70, normalized size = 0.81 \begin {gather*} \frac {-\frac {77 \left (-220783501-570653522 x+479067048 x^2+2305013328 x^3+1571590080 x^4\right )}{(2+3 x)^3 \left (-3+x+10 x^2\right )}-6464 \log (3-6 x)[phone] \log (2+3 x)-6722800000 \log (-3 (3+5 x))}{22370117} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-77*(-220783501 - 570653522*x + 479067048*x^2 + 2305013328*x^3 + 1571590080*x^4))/((2 + 3*x)^3*(-3 + x + 10*
x^2)) - 6464*Log[3 - 6*x] + 6722806464*Log[2 + 3*x] - 6722800000*Log[-3*(3 + 5*x)])/22370117

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Maple [A]
time = 0.10, size = 71, normalized size = 0.83

method	result	size
risch	\(\frac {-\frac {1571590080}{290521} x^{4}-\frac {2305013328}{290521} x^{3}-\frac {479067048}{290521} x^{2}+\frac {570653522}{290521} x +\frac {220783501}{290521}}{\left (-1+2 x \right ) \left (2+3 x \right )^{3} \left (3+5 x \right )}-\frac {6464 \ln \left (-1+2 x \right )}{22370117}+\frac {5050944 \ln \left (2+3 x \right )}{16807}-\frac {400000 \ln \left (3+5 x \right )}{1331}\)	\(69\)
default	\(-\frac {32}{290521 \left (-1+2 x \right )}-\frac {6464 \ln \left (-1+2 x \right )}{22370117}-\frac {9}{49 \left (2+3 x \right )^{3}}-\frac {999}{343 \left (2+3 x \right )^{2}}-\frac {107109}{2401 \left (2+3 x \right )}+\frac {5050944 \ln \left (2+3 x \right )}{16807}-\frac {3125}{121 \left (3+5 x \right )}-\frac {400000 \ln \left (3+5 x \right )}{1331}\)	\(71\)
norman	\(\frac {\frac {29155361049}{2324168} x^{4}+\frac {6066861987}{2324168} x^{3}+\frac {9935257545}{1162084} x^{5}-\frac {10826825099}{3486252} x^{2}-\frac {2095666393}{1743126} x}{\left (-1+2 x \right ) \left (2+3 x \right )^{3} \left (3+5 x \right )}-\frac {6464 \ln \left (-1+2 x \right )}{22370117}+\frac {5050944 \ln \left (2+3 x \right )}{16807}-\frac {400000 \ln \left (3+5 x \right )}{1331}\)	\(72\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-32/290521/(-1+2*x)-6464/22370117*ln(-1+2*x)-9/49/(2+3*x)^3-999/343/(2+3*x)^2-107109/2401/(2+3*x)+5050944/1680
7*ln(2+3*x)-3125/121/(3+5*x)-400000/1331*ln(3+5*x)

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Maxima [A]
time = 0.33, size = 74, normalized size = 0.86 \begin {gather*} -\frac {1571590080 \, x^{4} + 2305013328 \, x^{3} + 479067048 \, x^{2} - 570653522 \, x - 220783501}{290521 \, {\left (270 \, x^{5} + 567 \, x^{4} + 333 \, x^{3} - 46 \, x^{2} - 100 \, x - 24\right )}} - \frac {400000}{1331} \, \log \left (5 \, x + 3\right ) + \frac {5050944}{16807} \, \log \left (3 \, x + 2\right ) - \frac {6464}{22370117} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/290521*(1571590080*x^4 + 2305013328*x^3 + 479067048*x^2 - 570653522*x - 220783501)/(270*x^5 + 567*x^4 + 333
*x^3 - 46*x^2 - 100*x - 24) - 400000/1331*log(5*x + 3) + 5050944/16807*log(3*x + 2) - 6464/22370117*log(2*x -
1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (70) = 140\).
time = 0.59, size = 148, normalized size = 1.72 \begin {gather*} -\frac {121012436160 \, x^{4} + 177486026256 \, x^{3} + 36888162696 \, x^{2} + 6722800000 \, {\left (270 \, x^{5} + 567 \, x^{4} + 333 \, x^{3} - 46 \, x^{2} - 100 \, x - 24\right )} \log \left (5 \, x + 3\right ) - 6722806464 \, {\left (270 \, x^{5} + 567 \, x^{4} + 333 \, x^{3} - 46 \, x^{2} - 100 \, x - 24\right )} \log \left (3 \, x + 2\right ) + 6464 \, {\left (270 \, x^{5} + 567 \, x^{4} + 333 \, x^{3} - 46 \, x^{2} - 100 \, x - 24\right )} \log \left (2 \, x - 1\right ) - 43940321194 \, x - 17000329577}{22370117 \, {\left (270 \, x^{5} + 567 \, x^{4} + 333 \, x^{3} - 46 \, x^{2} - 100 \, x - 24\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/22370117*(121012436160*x^4 + 177486026256*x^3 + 36888162696*x^2 + 6722800000*(270*x^5 + 567*x^4 + 333*x^3 -
46*x^2 - 100*x - 24)*log(5*x + 3) - 6722806464*(270*x^5 + 567*x^4 + 333*x^3 - 46*x^2 - 100*x - 24)*log(3*x +
2) + 6464*(270*x^5 + 567*x^4 + 333*x^3 - 46*x^2 - 100*x - 24)*log(2*x - 1) - 43940321194*x - 17000329577)/(270
*x^5 + 567*x^4 + 333*x^3 - 46*x^2 - 100*x - 24)

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Sympy [A]
time = 0.16, size = 75, normalized size = 0.87 \begin {gather*} \frac {- 1571590080 x^{4} - 2305013328 x^{3} - 479067048 x^{2} + 570653522 x + 220783501}{78440670 x^{5} + 164725407 x^{4} + 96743493 x^{3} - 13363966 x^{2} - 29052100 x - 6972504} - \frac {6464 \log {\left (x - \frac {1}{2} \right )}}{22370117} - \frac {400000 \log {\left (x + \frac {3}{5} \right )}}{1331} + \frac {5050944 \log {\left (x + \frac {2}{3} \right )}}{16807} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-1571590080*x**4 - 2305013328*x**3 - 479067048*x**2 + 570653522*x + 220783501)/(78440670*x**5 + 164725407*x**
4 + 96743493*x**3 - 13363966*x**2 - 29052100*x - 6972504) - 6464*log(x - 1/2)/22370117 - 400000*log(x + 3/5)/1
331 + 5050944*log(x + 2/3)/16807

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Giac [A]
time = 0.71, size = 95, normalized size = 1.10 \begin {gather*} -\frac {3125}{121 \, {\left (5 \, x + 3\right )}} + \frac {5 \, {\left (\frac {52083388017}{5 \, x + 3} + \frac {44729490744}{{\left (5 \, x + 3\right )}^{2}} + \frac {9228837286}{{\left (5 \, x + 3\right )}^{3}} - 11003835798\right )}}{3195731 \, {\left (\frac {11}{5 \, x + 3} - 2\right )} {\left (\frac {1}{5 \, x + 3} + 3\right )}^{3}} + \frac {5050944}{16807} \, \log \left ({\left | -\frac {1}{5 \, x + 3} - 3 \right |}\right ) - \frac {6464}{22370117} \, \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)^2/(2+3*x)^4/(3+5*x)^2,x, algorithm="giac")

[Out]

-3125/121/(5*x + 3) + 5/3195731*(52083388017/(5*x + 3) + 44729490744/(5*x + 3)^2 + 9228837286/(5*x + 3)^3 - 11
003835798)/((11/(5*x + 3) - 2)*(1/(5*x + 3) + 3)^3) + 5050944/16807*log(abs(-1/(5*x + 3) - 3)) - 6464/22370117
*log(abs(-11/(5*x + 3) + 2))

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Mupad [B]
time = 1.07, size = 67, normalized size = 0.78 \begin {gather*} \frac {5050944\,\ln \left (x+\frac {2}{3}\right )}{16807}-\frac {6464\,\ln \left (x-\frac {1}{2}\right )}{22370117}-\frac {400000\,\ln \left (x+\frac {3}{5}\right )}{1331}+\frac {\frac {5820704\,x^4}{290521}+\frac {42685432\,x^3}{1452605}+\frac {8871612\,x^2}{1452605}-\frac {285326761\,x}{39220335}-\frac {220783501}{78440670}}{-x^5-\frac {21\,x^4}{10}-\frac {37\,x^3}{30}+\frac {23\,x^2}{135}+\frac {10\,x}{27}+\frac {4}{45}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(5050944*log(x + 2/3))/16807 - (6464*log(x - 1/2))/22370117 - (400000*log(x + 3/5))/1331 + ((8871612*x^2)/1452
605 - (285326761*x)/39220335 + (42685432*x^3)/1452605 + (5820704*x^4)/290521 - 220783501/78440670)/((10*x)/27
+ (23*x^2)/135 - (37*x^3)/30 - (21*x^4)/10 - x^5 + 4/45)

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