3.18.8 \(\int \frac {1}{(1-2 x)^3 (2+3 x)^2 (3+5 x)^3} \, dx\) [1708]

Optimal. Leaf size=86 \[ \frac {8}{65219 (1-2 x)^2}+\frac {2736}{5021863 (1-2 x)}+\frac {243}{343 (2+3 x)}-\frac {625}{2662 (3+5 x)^2}+\frac {37500}{14641 (3+5 x)}-\frac {280752 \log (1-2 x)}{386683451}-\frac {26973 \log (2+3 x)}{2401}+\frac {1809375 \log (3+5 x)}{161051} \]

[Out]

8/65219/(1-2*x)^2+2736/5021863/(1-2*x)+243/343/(2+3*x)-625/2662/(3+5*x)^2+37500/14641/(3+5*x)-280752/386683451
*ln(1-2*x)-26973/2401*ln(2+3*x)+1809375/161051*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 {2736}{5021863 (1-2 x)}+\frac {243}{343 (3 x+2)}+\frac {37500}{14641 (5 x+3)}+\frac {8}{65219 (1-2 x)^2}-\frac {625}{2662 (5 x+3)^2}-\frac {280752 \log (1-2 x)}{386683451}-\frac {26973 \log (3 x+2)}{2401}+\frac {1809375 \log (5 x+3)}{161051} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

8/(65219*(1 - 2*x)^2) + 2736/(5021863*(1 - 2*x)) + 243/(343*(2 + 3*x)) - 625/(2662*(3 + 5*x)^2) + 37500/(14641
*(3 + 5*x)) - (280752*Log[1 - 2*x])/386683451 - (26973*Log[2 + 3*x])/2401 + (1809375*Log[3 + 5*x])/161051

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)^3 (2+3 x)^2 (3+5 x)^3} \, dx &=\int \left (-\frac {32}{65219 (-1+2 x)^3}+\frac {5472}{5021863 (-1+2 x)^2}-\frac {561504}{386683451 (-1+2 x)}-\frac {729}{343 (2+3 x)^2}-\frac {80919}{2401 (2+3 x)}+\frac {3125}{1331 (3+5 x)^3}-\frac {187500}{14641 (3+5 x)^2}+\frac {9046875}{161051 (3+5 x)}\right ) \, dx\\ &=\frac {8}{65219 (1-2 x)^2}+\frac {2736}{5021863 (1-2 x)}+\frac {243}{343 (2+3 x)}-\frac {625}{2662 (3+5 x)^2}+\frac {37500}{14641 (3+5 x)}-\frac {280752 \log (1-2 x)}{386683451}-\frac {26973 \log (2+3 x)}{2401}+\frac {1809375 \log (3+5 x)}{161051}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 76, normalized size = 0.88 \begin {gather*} -\frac {3 \left (-\frac {182631834}{2+3 x}-\frac {65219 (-6101+12290 x)}{3 \left (-3+x+10 x^2\right )^2}-\frac {154 (-4446931+8570440 x)}{-3+x+10 x^2}+187168 \log (3-6 x)[phone] \log (2+3 x)-2896206250 \log (-3 (3+5 x))\right )}{773366902} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(-182631834/(2 + 3*x) - (65219*(-6101 + 12290*x))/(3*(-3 + x + 10*x^2)^2) - (154*(-4446931 + 8570440*x))/(
-3 + x + 10*x^2) + 187168*Log[3 - 6*x] + 2896019082*Log[2 + 3*x] - 2896206250*Log[-3*(3 + 5*x)]))/773366902

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Maple [A]
time = 0.12, size = 71, normalized size = 0.83

method	result	size
risch	\(\frac {\frac {1127115900}{5021863} x^{4}+\frac {262291830}{5021863} x^{3}-\frac {681111051}{5021863} x^{2}-\frac {159141275}{10043726} x +\frac {106897078}{5021863}}{\left (-1+2 x \right )^{2} \left (2+3 x \right ) \left (3+5 x \right )^{2}}-\frac {280752 \ln \left (-1+2 x \right )}{386683451}-\frac {26973 \ln \left (2+3 x \right )}{2401}+\frac {1809375 \ln \left (3+5 x \right )}{161051}\)	\(69\)
norman	\(\frac {-\frac {601093499}{2152227} x^{2}+\frac {1591412750}{5021863} x^{5}-\frac {2786719955}{30131178} x^{3}+\frac {683547350}{1369599} x^{4}+\frac {202381843}{5021863}}{\left (-1+2 x \right )^{2} \left (2+3 x \right ) \left (3+5 x \right )^{2}}-\frac {280752 \ln \left (-1+2 x \right )}{386683451}-\frac {26973 \ln \left (2+3 x \right )}{2401}+\frac {1809375 \ln \left (3+5 x \right )}{161051}\)	\(70\)
default	\(\frac {8}{65219 \left (-1+2 x \right )^{2}}-\frac {2736}{5021863 \left (-1+2 x \right )}-\frac {280752 \ln \left (-1+2 x \right )}{386683451}+\frac {243}{343 \left (2+3 x \right )}-\frac {26973 \ln \left (2+3 x \right )}{2401}-\frac {625}{2662 \left (3+5 x \right )^{2}}+\frac {37500}{14641 \left (3+5 x \right )}+\frac {1809375 \ln \left (3+5 x \right )}{161051}\)	\(71\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/65219/(-1+2*x)^2-2736/5021863/(-1+2*x)-280752/386683451*ln(-1+2*x)+243/343/(2+3*x)-26973/2401*ln(2+3*x)-625/
2662/(3+5*x)^2+37500/14641/(3+5*x)+1809375/161051*ln(3+5*x)

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Maxima [A]
time = 0.29, size = 74, normalized size = 0.86 \begin {gather*} \frac {2254231800 \, x^{4} + 524583660 \, x^{3} - 1362222102 \, x^{2} - 159141275 \, x + 213794156}{10043726 \, {\left (300 \, x^{5} + 260 \, x^{4} - 137 \, x^{3} - 136 \, x^{2} + 15 \, x + 18\right )}} + \frac {1809375}{161051} \, \log \left (5 \, x + 3\right ) - \frac {26973}{2401} \, \log \left (3 \, x + 2\right ) - \frac {280752}{386683451} \, \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)^2/(3+5*x)^3,x, algorithm="maxima")

[Out]

1/10043726*(2254231800*x^4 + 524583660*x^3 - 1362222102*x^2 - 159141275*x + 213794156)/(300*x^5 + 260*x^4 - 13
7*x^3 - 136*x^2 + 15*x + 18) + 1809375/161051*log(5*x + 3) - 26973/2401*log(3*x + 2) - 280752/386683451*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 = 1.17, size = 148, normalized size = 1.72 \begin {gather*} \frac {173575848600 \, x^{4} + 40392941820 \, x^{3} - 104891101854 \, x^{2} + 8688618750 \, {\left (300 \, x^{5} + 260 \, x^{4} - 137 \, x^{3} - 136 \, x^{2} + 15 \, x + 18\right )} \log \left (5 \, x + 3\right ) - 8688057246 \, {\left (300 \, x^{5} + 260 \, x^{4} - 137 \, x^{3} - 136 \, x^{2} + 15 \, x + 18\right )} \log \left (3 \, x + 2\right ) - 561504 \, {\left (300 \, x^{5} + 260 \, x^{4} - 137 \, x^{3} - 136 \, x^{2} + 15 \, x + 18\right )} \log \left (2 \, x - 1\right ) - 12253878175 \, x + 16462150012}{773366902 \, {\left (300 \, x^{5} + 260 \, x^{4} - 137 \, x^{3} - 136 \, x^{2} + 15 \, x + 18\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/773366902*(173575848600*x^4 + 40392941820*x^3 - 104891101854*x^2 + 8688618750*(300*x^5 + 260*x^4 - 137*x^3 -
136*x^2 + 15*x + 18)*log(5*x + 3) - 8688057246*(300*x^5 + 260*x^4 - 137*x^3 - 136*x^2 + 15*x + 18)*log(3*x +
2) - 561504*(300*x^5 + 260*x^4 - 137*x^3 - 136*x^2 + 15*x + 18)*log(2*x - 1) - 12253878175*x + 16462150012)/(3
00*x^5 + 260*x^4 - 137*x^3 - 136*x^2 + 15*x + 18)

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Sympy [A]
time = 0.15, size = 75, normalized size = 0.87 \begin {gather*} - \frac {- 2254231800 x^{4} - 524583660 x^{3} + 1362222102 x^{2} + 159141275 x - 213794156}{3013117800 x^{5} + 2611368760 x^{4} - 1375990462 x^{3} - 1365946736 x^{2} + 150655890 x + 180787068} - \frac {280752 \log {\left (x - \frac {1}{2} \right )}}{386683451} + \frac {1809375 \log {\left (x + \frac {3}{5} \right )}}{161051} - \frac {26973 \log {\left (x + \frac {2}{3} \right )}}{2401} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-2254231800*x**4 - 524583660*x**3 + 1362222102*x**2 + 159141275*x - 213794156)/(3013117800*x**5 + 2611368760
*x**4 - 1375990462*x**3 - 1365946736*x**2 + 150655890*x + 180787068) - 280752*log(x - 1/2)/386683451 + 1809375
*log(x + 3/5)/161051 - 26973*log(x + 2/3)/2401

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Giac [A]
time = 0.58, size = 95, normalized size = 1.10 \begin {gather*} \frac {243}{343 \, {\left (3 \, x + 2\right )}} - \frac {9 \, {\left (\frac {55432245900}{3 \, x + 2} - \frac {106776659235}{{\left (3 \, x + 2\right )}^{2}} + \frac {22794463286}{{\left (3 \, x + 2\right )}^{3}} - 7652987500\right )}}{70306082 \, {\left (\frac {7}{3 \, x + 2} - 2\right )}^{2} {\left (\frac {1}{3 \, x + 2} - 5\right )}^{2}} + \frac {1809375}{161051} \, \log \left ({\left | -\frac {1}{3 \, x + 2} + 5 \right |}\right ) - \frac {280752}{386683451} \, \log \left ({\left | -\frac {7}{3 \, x + 2} + 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)^2/(3+5*x)^3,x, algorithm="giac")

[Out]

243/343/(3*x + 2) - 9/70306082*(55432245900/(3*x + 2) - 106776659235/(3*x + 2)^2 + 22794463286/(3*x + 2)^3 - 7
652987500)/((7/(3*x + 2) - 2)^2*(1/(3*x + 2) - 5)^2) + 1809375/161051*log(abs(-1/(3*x + 2) + 5)) - 280752/3866
83451*log(abs(-7/(3*x + 2) + 2))

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Mupad [B]
time = 0.04, size = 65, normalized size = 0.76 \begin {gather*} \frac {1809375\,\ln \left (x+\frac {3}{5}\right )}{161051}-\frac {26973\,\ln \left (x+\frac {2}{3}\right )}{2401}-\frac {280752\,\ln \left (x-\frac {1}{2}\right )}{386683451}+\frac {\frac {3757053\,x^4}{5021863}+\frac {8743061\,x^3}{50218630}-\frac {227037017\,x^2}{502186300}-\frac {6365651\,x}{120524712}+\frac {53448539}{753279450}}{x^5+\frac {13\,x^4}{15}-\frac {137\,x^3}{300}-\frac {34\,x^2}{75}+\frac {x}{20}+\frac {3}{50}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1809375*log(x + 3/5))/161051 - (26973*log(x + 2/3))/2401 - (280752*log(x - 1/2))/386683451 + ((8743061*x^3)/5
0218630 - (227037017*x^2)/502186300 - (6365651*x)/120524712 + (3757053*x^4)/5021863 + 53448539/753279450)/(x/2
0 - (34*x^2)/75 - (137*x^3)/300 + (13*x^4)/15 + x^5 + 3/50)

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