3.1514 \(\int \frac{1}{(1-2 x) (2+3 x)^5 (3+5 x)^2} \, dx\)

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

[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

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Rubi [A]  time = 0.0407622, antiderivative size = 86, 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} \[ -\frac{1612242}{2401 (3 x+2)}-\frac{3125}{11 (5 x+3)}-\frac{34371}{686 (3 x+2)^2}-\frac{216}{49 (3 x+2)^3}-\frac{9}{28 (3 x+2)^4}-\frac{64 \log (1-2 x)}{2033647}+\frac{70752609 \log (3 x+2)}{16807}-\frac{509375}{121} \log (5 x+3) \]

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 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) (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*}

Mathematica [A]  time = 0.033421, size = 84, normalized size = 0.98 \[ -\frac{1612242}{2401 (3 x+2)}-\frac{3125}{55 x+33}-\frac{34371}{686 (3 x+2)^2}-\frac{216}{49 (3 x+2)^3}-\frac{9}{28 (3 x+2)^4}-\frac{64 \log (1-2 x)}{2033647}+\frac{70752609 \log (6 x+4)}{16807}-\frac{509375}{121} \log (10 x+6) \]

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 = 0.01, size = 71, normalized size = 0.8 \begin{align*} -{\frac{64\,\ln \left ( 2\,x-1 \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}{4802+7203\,x}}+{\frac{70752609\,\ln \left ( 2+3\,x \right ) }{16807}}-{\frac{3125}{33+55\,x}}-{\frac{509375\,\ln \left ( 3+5\,x \right ) }{121}} \end{align*}

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)

[Out]

-64/2033647*ln(2*x-1)-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 = 1.12, size = 100, normalized size = 1.16 \begin{align*} -\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{align*}

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]  time = 1.42783, size = 541, normalized size = 6.29 \begin{align*} -\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{align*}

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 = 0.239087, size = 75, normalized size = 0.87 \begin{align*} - \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{align*}

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 = 1.46428, size = 111, normalized size = 1.29 \begin{align*} -\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{align*}

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))