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

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

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Rubi [A]  time = 0.05, antiderivative size = 97, 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 = {88} \begin {gather*} \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} \end {gather*}

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*}

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Mathematica [A]  time = 0.07, size = 95, normalized size = 0.98 \begin {gather*} \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} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(1-2 x) (2+3 x)^5 (3+5 x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.41, size = 173, normalized size = 1.78 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.95, size = 91, normalized size = 0.94 \begin {gather*} \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 {gather*}

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))

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maple [A]  time = 0.01, size = 80, normalized size = 0.82 \begin {gather*} -\frac {128 \ln \left (2 x -1\right )}{22370117}-\frac {631722537 \ln \left (3 x +2\right )}{16807}+\frac {50028125 \ln \left (5 x +3\right )}{1331}-\frac {3125}{22 \left (5 x +3\right )^{2}}+\frac {509375}{121 \left (5 x +3\right )}+\frac {27}{28 \left (3 x +2\right )^{4}}+\frac {963}{49 \left (3 x +2\right )^{3}}+\frac {102114}{343 \left (3 x +2\right )^{2}}+\frac {11984706}{2401 \left (3 x +2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.53, size = 84, normalized size = 0.87 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.05, size = 75, normalized size = 0.77 \begin {gather*} \frac {50028125\,\ln \left (x+\frac {3}{5}\right )}{1331}-\frac {631722537\,\ln \left (x+\frac {2}{3}\right )}{16807}-\frac {128\,\ln \left (x-\frac {1}{2}\right )}{22370117}+\frac {\frac {727985017\,x^5}{290521}+\frac {70614546617\,x^4}{8715630}+\frac {76060952983\,x^3}{7263025}+\frac {5305603322807\,x^2}{784406700}+\frac {2568430130789\,x}{1176610050}+\frac {662695553413}{2353220100}}{x^6+\frac {58\,x^5}{15}+\frac {467\,x^4}{75}+\frac {3608\,x^3}{675}+\frac {5224\,x^2}{2025}+\frac {448\,x}{675}+\frac {16}{225}} \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)^3),x)

[Out]

(50028125*log(x + 3/5))/1331 - (631722537*log(x + 2/3))/16807 - (128*log(x - 1/2))/22370117 + ((2568430130789*
x)/1176610050 + (5305603322807*x^2)/784406700 + (76060952983*x^3)/7263025 + (70614546617*x^4)/8715630 + (72798
5017*x^5)/290521 + 662695553413/2353220100)/((448*x)/675 + (5224*x^2)/2025 + (3608*x^3)/675 + (467*x^4)/75 + (
58*x^5)/15 + x^6 + 16/225)

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sympy [A]  time = 0.28, size = 87, normalized size = 0.90 \begin {gather*} - \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 {gather*}

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 - 66
2695553413)/(2353220100*x**6 + 9099117720*x**5 + 14652717156*x**4 + 12578397216*x**3 + 6070726816*x**2 + 15618
40896*x + 167340096) - 128*log(x - 1/2)/22370117 + 50028125*log(x + 3/5)/1331 - 631722537*log(x + 2/3)/16807

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