3.16.14 \(\int \frac {1}{(1-2 x) (2+3 x)^5 (3+5 x)^2} \, dx\) [1514]

Optimal. Leaf size=86 \[ -\frac {9}{28 (2+3 x)^4}-\frac {216}{49 (2+3 x)^3}-\frac {34371}{686 (2+3 x)^2}-\frac {1612242}{2401 (2+3 x)}-\frac {3125}{11 (3+5 x)}-\frac {64 \log (1-2 x)}{2033647}+\frac {70752609 \log (2+3 x)}{16807}-\frac {509375}{121} \log (3+5 x) \]

[Out]

-9/28/(2+3*x)^4-216/49/(2+3*x)^3-34371/686/(2+3*x)^2-1612242/2401/(2+3*x)-3125/11/(3+5*x)-64/2033647*ln(1-2*x)
[phone]/16807*ln(2+3*x)-509375/121*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 {1612242}{2401 (3 x+2)}-\frac {3125}{11 (5 x+3)}-\frac {34371}{686 (3 x+2)^2}-\frac {216}{49 (3 x+2)^3}-\frac {9}{28 (3 x+2)^4}-\frac {64 \log (1-2 x)}{2033647}+\frac {70752609 \log (3 x+2)}{16807}-\frac {509375}{121} \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-9/(28*(2 + 3*x)^4) - 216/(49*(2 + 3*x)^3) - 34371/(686*(2 + 3*x)^2) - 1612242/(2401*(2 + 3*x)) - 3125/(11*(3
+ 5*x)) - (64*Log[1 - 2*x])/2033647 + (70752609*Log[2 + 3*x])/16807 - (509375*Log[3 + 5*x])/121

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+3 x)^5 (3+5 x)^2} \, dx &=\int \left (-\frac {128}{2033647 (-1+2 x)}+\frac {27}{7 (2+3 x)^5}+\frac {1944}{49 (2+3 x)^4}+\frac {103113}{343 (2+3 x)^3}+\frac {4836726}{2401 (2+3 x)^2}+\frac {212257827}{16807 (2+3 x)}+\frac {15625}{11 (3+5 x)^2}-\frac {2546875}{121 (3+5 x)}\right ) \, dx\\ &=-\frac {9}{28 (2+3 x)^4}-\frac {216}{49 (2+3 x)^3}-\frac {34371}{686 (2+3 x)^2}-\frac {1612242}{2401 (2+3 x)}-\frac {3125}{11 (3+5 x)}-\frac {64 \log (1-2 x)}{2033647}+\frac {70752609 \log (2+3 x)}{16807}-\frac {509375}{121} \log (3+5 x)\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 84, normalized size = 0.98 \begin {gather*} -\frac {9}{28 (2+3 x)^4}-\frac {216}{49 (2+3 x)^3}-\frac {34371}{686 (2+3 x)^2}-\frac {1612242}{2401 (2+3 x)}-\frac {3125}{33+55 x}-\frac {64 \log (1-2 x)}{2033647}+\frac {70752609 \log (4+6 x)}{16807}-\frac {509375}{121} \log (6+10 x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-9/(28*(2 + 3*x)^4) - 216/(49*(2 + 3*x)^3) - 34371/(686*(2 + 3*x)^2) - 1612242/(2401*(2 + 3*x)) - 3125/(33 + 5
5*x) - (64*Log[1 - 2*x])/2033647 + (70752609*Log[4 + 6*x])/16807 - (509375*Log[6 + 10*x])/121

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Maple [A]
time = 0.11, size = 71, normalized size = 0.83

method	result	size
risch	\(\frac {-\frac {3001932495}{26411} x^{4}-\frac {15810178239}{52822} x^{3}-\frac {15605602857}{52822} x^{2}-\frac {13685553417}{105644} x -\frac {2249141207}{105644}}{\left (2+3 x \right )^{4} \left (3+5 x \right )}-\frac {64 \ln \left (-1+2 x \right )}{2033647}+\frac {70752609 \ln \left (2+3 x \right )}{16807}-\frac {509375 \ln \left (3+5 x \right )}{121}\)	\(62\)
norman	\(\frac {\frac {24674363487}{52822} x^{3}+\frac {5336793755}{158466} x +\frac {43287225301}{211288} x^{2}+\frac {303634062945}{1690304} x^{5}+\frac {799747592607}{1690304} x^{4}}{\left (2+3 x \right )^{4} \left (3+5 x \right )}-\frac {64 \ln \left (-1+2 x \right )}{2033647}+\frac {70752609 \ln \left (2+3 x \right )}{16807}-\frac {509375 \ln \left (3+5 x \right )}{121}\)	\(65\)
default	\(-\frac {64 \ln \left (-1+2 x \right )}{2033647}-\frac {9}{28 \left (2+3 x \right )^{4}}-\frac {216}{49 \left (2+3 x \right )^{3}}-\frac {34371}{686 \left (2+3 x \right )^{2}}-\frac {1612242}{2401 \left (2+3 x \right )}+\frac {70752609 \ln \left (2+3 x \right )}{16807}-\frac {3125}{11 \left (3+5 x \right )}-\frac {509375 \ln \left (3+5 x \right )}{121}\)	\(71\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-64/2033647*ln(-1+2*x)-9/28/(2+3*x)^4-216/49/(2+3*x)^3-34371/686/(2+3*x)^2-1612242/2401/(2+3*x)[phone]/16807
*ln(2+3*x)-3125/11/(3+5*x)-509375/121*ln(3+5*x)

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Maxima [A]
time = 0.33, size = 74, normalized size = 0.86 \begin {gather*} -\frac {12007729980 \, x^{4} + 31620356478 \, x^{3} + 31211205714 \, x^{2} + 13685553417 \, x + 2249141207}{105644 \, {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )}} - \frac {509375}{121} \, \log \left (5 \, x + 3\right ) + \frac {70752609}{16807} \, \log \left (3 \, x + 2\right ) - \frac {64}{2033647} \, \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+3*x)^5/(3+5*x)^2,x, algorithm="maxima")

[Out]

-1/105644*(12007729980*x^4 + 31620356478*x^3 + 31211205714*x^2 + 13685553417*x + 2249141207)/(405*x^5 + 1323*x
^4 + 1728*x^3 + 1128*x^2 + 368*x + 48) - 509375/121*log(5*x + 3) + 70752609/16807*log(3*x + 2) - 64/2033647*lo
g(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 = 1.00, size = 148, normalized size = 1.72 \begin {gather*} -\frac {924595208460 \, x^{4} + 2434767448806 \, x^{3} + 2403262839978 \, x^{2} + 34244262500 \, {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} \log \left (5 \, x + 3\right ) - 34244262756 \, {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} \log \left (3 \, x + 2\right ) + 256 \, {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} \log \left (2 \, x - 1\right ) + 1053787613109 \, x + 173183872939}{8134588 \, {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8134588*(924595208460*x^4 + 2434767448806*x^3 + 2403262839978*x^2 + 34244262500*(405*x^5 + 1323*x^4 + 1728*
x^3 + 1128*x^2 + 368*x + 48)*log(5*x + 3) - 34244262756*(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 + 368*x + 48
)*log(3*x + 2) + 256*(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 + 368*x + 48)*log(2*x - 1) + 1053787613109*x +
173183872939)/(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 + 368*x + 48)

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Sympy [A]
time = 0.14, size = 75, normalized size = 0.87 \begin {gather*} - \frac {12007729980 x^{4} + 31620356478 x^{3} + 31211205714 x^{2} + 13685553417 x + 2249141207}{42785820 x^{5} + 139767012 x^{4} + 182552832 x^{3} + 119166432 x^{2} + 38876992 x + 5070912} - \frac {64 \log {\left (x - \frac {1}{2} \right )}}{2033647} - \frac {509375 \log {\left (x + \frac {3}{5} \right )}}{121} + \frac {70752609 \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+3*x)**5/(3+5*x)**2,x)

[Out]

-(12007729980*x**4 + 31620356478*x**3 + 31211205714*x**2 + 13685553417*x + 2249141207)/(42785820*x**5 + 139767
012*x**4 + 182552832*x**3 + 119166432*x**2 + 38876992*x + 5070912) - 64*log(x - 1/2)/2033647 - 509375*log(x +
3/5)/121 + 70752609*log(x + 2/3)/16807

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Giac [A]
time = 1.10, size = 82, normalized size = 0.95 \begin {gather*} -\frac {3125}{11 \, {\left (5 \, x + 3\right )}} + \frac {135 \, {\left (\frac {34747884}{5 \, x + 3} + \frac {13347468}{{\left (5 \, x + 3\right )}^{2}} + \frac {1775512}{{\left (5 \, x + 3\right )}^{3}} + 30897639\right )}}{9604 \, {\left (\frac {1}{5 \, x + 3} + 3\right )}^{4}} + \frac {70752609}{16807} \, \log \left ({\left | -\frac {1}{5 \, x + 3} - 3 \right |}\right ) - \frac {64}{2033647} \, \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+3*x)^5/(3+5*x)^2,x, algorithm="giac")

[Out]

-3125/11/(5*x + 3) + 135/9604*(34747884/(5*x + 3) + 13347468/(5*x + 3)^2 + 1775512/(5*x + 3)^3 + 30897639)/(1/
(5*x + 3) + 3)^4 + 70752609/16807*log(abs(-1/(5*x + 3) - 3)) - 64/2033647*log(abs(-11/(5*x + 3) + 2))

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Mupad [B]
time = 0.05, size = 66, normalized size = 0.77 \begin {gather*} \frac {70752609\,\ln \left (x+\frac {2}{3}\right )}{16807}-\frac {64\,\ln \left (x-\frac {1}{2}\right )}{2033647}-\frac {509375\,\ln \left (x+\frac {3}{5}\right )}{121}-\frac {\frac {7412179\,x^4}{26411}+\frac {585562157\,x^3}{792330}+\frac {577985291\,x^2}{792330}+\frac {4561851139\,x}{14261940}+\frac {2249141207}{42785820}}{x^5+\frac {49\,x^4}{15}+\frac {64\,x^3}{15}+\frac {376\,x^2}{135}+\frac {368\,x}{405}+\frac {16}{135}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(70752609*log(x + 2/3))/16807 - (64*log(x - 1/2))/2033647 - (509375*log(x + 3/5))/121 - ((4561851139*x)/142619
40 + (577985291*x^2)/792330 + (585562157*x^3)/792330 + (7412179*x^4)/26411 + 2249141207/42785820)/((368*x)/405
+ (376*x^2)/135 + (64*x^3)/15 + (49*x^4)/15 + x^5 + 16/135)

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