3.1529 \(\int \frac{1}{(1-2 x) (2+3 x)^5 (3+5 x)^3} \, dx\)

Optimal. Leaf size=97 \[ \frac{11984706}{2401 (3 x+2)}+\frac{509375}{121 (5 x+3)}+\frac{102114}{343 (3 x+2)^2}-\frac{3125}{22 (5 x+3)^2}+\frac{963}{49 (3 x+2)^3}+\frac{27}{28 (3 x+2)^4}-\frac{128 \log (1-2 x)}{22370117}-\frac{631722537 \log (3 x+2)}{16807}+\frac{50028125 \log (5 x+3)}{1331} \]

[Out]

27/(28*(2 + 3*x)^4) + 963/(49*(2 + 3*x)^3) + 102114/(343*(2 + 3*x)^2) + 11984706/(2401*(2 + 3*x)) - 3125/(22*(
3 + 5*x)^2) + 509375/(121*(3 + 5*x)) - (128*Log[1 - 2*x])/22370117 - (631722537*Log[2 + 3*x])/16807 + (5002812
5*Log[3 + 5*x])/1331

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Rubi [A]  time = 0.0508522, antiderivative size = 97, 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} \[ \frac{11984706}{2401 (3 x+2)}+\frac{509375}{121 (5 x+3)}+\frac{102114}{343 (3 x+2)^2}-\frac{3125}{22 (5 x+3)^2}+\frac{963}{49 (3 x+2)^3}+\frac{27}{28 (3 x+2)^4}-\frac{128 \log (1-2 x)}{22370117}-\frac{631722537 \log (3 x+2)}{16807}+\frac{50028125 \log (5 x+3)}{1331} \]

Antiderivative was successfully verified.

[In]

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

[Out]

27/(28*(2 + 3*x)^4) + 963/(49*(2 + 3*x)^3) + 102114/(343*(2 + 3*x)^2) + 11984706/(2401*(2 + 3*x)) - 3125/(22*(
3 + 5*x)^2) + 509375/(121*(3 + 5*x)) - (128*Log[1 - 2*x])/22370117 - (631722537*Log[2 + 3*x])/16807 + (5002812
5*Log[3 + 5*x])/1331

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)^3} \, dx &=\int \left (-\frac{256}{22370117 (-1+2 x)}-\frac{81}{7 (2+3 x)^5}-\frac{8667}{49 (2+3 x)^4}-\frac{612684}{343 (2+3 x)^3}-\frac{35954118}{2401 (2+3 x)^2}-\frac{1895167611}{16807 (2+3 x)}+\frac{15625}{11 (3+5 x)^3}-\frac{2546875}{121 (3+5 x)^2}+\frac{250140625}{1331 (3+5 x)}\right ) \, dx\\ &=\frac{27}{28 (2+3 x)^4}+\frac{963}{49 (2+3 x)^3}+\frac{102114}{343 (2+3 x)^2}+\frac{11984706}{2401 (2+3 x)}-\frac{3125}{22 (3+5 x)^2}+\frac{509375}{121 (3+5 x)}-\frac{128 \log (1-2 x)}{22370117}-\frac{631722537 \log (2+3 x)}{16807}+\frac{50028125 \log (3+5 x)}{1331}\\ \end{align*}

Mathematica [A]  time = 0.0358273, size = 95, normalized size = 0.98 \[ \frac{11984706}{2401 (3 x+2)}+\frac{509375}{605 x+363}+\frac{102114}{343 (3 x+2)^2}-\frac{3125}{22 (5 x+3)^2}+\frac{963}{49 (3 x+2)^3}+\frac{27}{28 (3 x+2)^4}-\frac{128 \log (1-2 x)}{22370117}-\frac{631722537 \log (6 x+4)}{16807}+\frac{50028125 \log (10 x+6)}{1331} \]

Antiderivative was successfully verified.

[In]

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

[Out]

27/(28*(2 + 3*x)^4) + 963/(49*(2 + 3*x)^3) + 102114/(343*(2 + 3*x)^2) + 11984706/(2401*(2 + 3*x)) - 3125/(22*(
3 + 5*x)^2) + 509375/(363 + 605*x) - (128*Log[1 - 2*x])/22370117 - (631722537*Log[4 + 6*x])/16807 + (50028125*
Log[6 + 10*x])/1331

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Maple [A]  time = 0.01, size = 80, normalized size = 0.8 \begin{align*} -{\frac{128\,\ln \left ( 2\,x-1 \right ) }{22370117}}+{\frac{27}{28\, \left ( 2+3\,x \right ) ^{4}}}+{\frac{963}{49\, \left ( 2+3\,x \right ) ^{3}}}+{\frac{102114}{343\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{11984706}{4802+7203\,x}}-{\frac{631722537\,\ln \left ( 2+3\,x \right ) }{16807}}-{\frac{3125}{22\, \left ( 3+5\,x \right ) ^{2}}}+{\frac{509375}{363+605\,x}}+{\frac{50028125\,\ln \left ( 3+5\,x \right ) }{1331}} \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)^3,x)

[Out]

-128/22370117*ln(2*x-1)+27/28/(2+3*x)^4+963/49/(2+3*x)^3+102114/343/(2+3*x)^2+11984706/2401/(2+3*x)-631722537/
16807*ln(2+3*x)-3125/22/(3+5*x)^2+509375/121/(3+5*x)[phone]/1331*ln(3+5*x)

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Maxima [A]  time = 1.79435, size = 113, normalized size = 1.16 \begin{align*} \frac{5896678637700 \, x^{5} + 19065927586590 \, x^{4} + 24643748766492 \, x^{3} + 15916809968421 \, x^{2} + 5136860261578 \, x + 662695553413}{1162084 \,{\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )}} + \frac{50028125}{1331} \, \log \left (5 \, x + 3\right ) - \frac{631722537}{16807} \, \log \left (3 \, x + 2\right ) - \frac{128}{22370117} \, \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)^3,x, algorithm="maxima")

[Out]

1/1162084*(5896678637700*x^5 + 19065927586590*x^4 + 24643748766492*x^3 + 15916809968421*x^2 + 5136860261578*x
+ 662695553413)/(2025*x^6 + 7830*x^5 + 12609*x^4 + 10824*x^3 + 5224*x^2 + 1344*x + 144) + 50028125/1331*log(5*
x + 3) - 631722537/16807*log(3*x + 2) - 128/22370117*log(2*x - 1)

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Fricas [B]  time = 1.35472, size = 682, normalized size = 7.03 \begin{align*} \frac{454044255102900 \, x^{5} + 1468076424167430 \, x^{4} + 1897568655019884 \, x^{3} + 1225594367568417 \, x^{2} + 3363290787500 \,{\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )} \log \left (5 \, x + 3\right ) - 3363290786988 \,{\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )} \log \left (3 \, x + 2\right ) - 512 \,{\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )} \log \left (2 \, x - 1\right ) + 395538240141506 \, x + 51027557612801}{89480468 \,{\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\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)^3,x, algorithm="fricas")

[Out]

1/89480468*(454044255102900*x^5 + 1468076424167430*x^4 + 1897568655019884*x^3 + 1225594367568417*x^2 + 3363290
787500*(2025*x^6 + 7830*x^5 + 12609*x^4 + 10824*x^3 + 5224*x^2 + 1344*x + 144)*log(5*x + 3) - 3363290786988*(2
025*x^6 + 7830*x^5 + 12609*x^4 + 10824*x^3 + 5224*x^2 + 1344*x + 144)*log(3*x + 2) - 512*(2025*x^6 + 7830*x^5
+ 12609*x^4 + 10824*x^3 + 5224*x^2 + 1344*x + 144)*log(2*x - 1) + 395538240141506*x + 51027557612801)/(2025*x^
6 + 7830*x^5 + 12609*x^4 + 10824*x^3 + 5224*x^2 + 1344*x + 144)

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Sympy [A]  time = 0.262944, size = 85, normalized size = 0.88 \begin{align*} \frac{5896678637700 x^{5} + 19065927586590 x^{4} + 24643748766492 x^{3} + 15916809968421 x^{2} + 5136860261578 x + 662695553413}{2353220100 x^{6} + 9099117720 x^{5} + 14652717156 x^{4} + 12578397216 x^{3} + 6070726816 x^{2} + 1561840896 x + 167340096} - \frac{128 \log{\left (x - \frac{1}{2} \right )}}{22370117} + \frac{50028125 \log{\left (x + \frac{3}{5} \right )}}{1331} - \frac{631722537 \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)**3,x)

[Out]

(5896678637700*x**5 + 19065927586590*x**4 + 24643748766492*x**3 + 15916809968421*x**2 + 5136860261578*x + 6626
95553413)/(2353220100*x**6 + 9099117720*x**5 + 14652717156*x**4 + 12578397216*x**3 + 6070726816*x**2 + 1561840
896*x + 167340096) - 128*log(x - 1/2)/22370117 + 50028125*log(x + 3/5)/1331 - 631722537*log(x + 2/3)/16807

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Giac [A]  time = 1.69823, size = 123, normalized size = 1.27 \begin{align*} \frac{11984706}{2401 \,{\left (3 \, x + 2\right )}} - \frac{46875 \,{\left (\frac{392}{3 \, x + 2} - 1795\right )}}{242 \,{\left (\frac{1}{3 \, x + 2} - 5\right )}^{2}} + \frac{102114}{343 \,{\left (3 \, x + 2\right )}^{2}} + \frac{963}{49 \,{\left (3 \, x + 2\right )}^{3}} + \frac{27}{28 \,{\left (3 \, x + 2\right )}^{4}} + \frac{50028125}{1331} \, \log \left ({\left | -\frac{1}{3 \, x + 2} + 5 \right |}\right ) - \frac{128}{22370117} \, \log \left ({\left | -\frac{7}{3 \, x + 2} + 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)^3,x, algorithm="giac")

[Out]

11984706/2401/(3*x + 2) - 46875/242*(392/(3*x + 2) - 1795)/(1/(3*x + 2) - 5)^2 + 102114/343/(3*x + 2)^2 + 963/
49/(3*x + 2)^3 + 27/28/(3*x + 2)^4 + 50028125/1331*log(abs(-1/(3*x + 2) + 5)) - 128/22370117*log(abs(-7/(3*x +
2) + 2))