3.1481 \(\int \frac{1}{(1-2 x) (2+3 x)^5 (3+5 x)} \, dx\)

Optimal. Leaf size=75 \[ \frac{136419}{2401 (3 x+2)}+\frac{3897}{686 (3 x+2)^2}+\frac{37}{49 (3 x+2)^3}+\frac{3}{28 (3 x+2)^4}-\frac{32 \log (1-2 x)}{184877}-\frac{4774713 \log (3 x+2)}{16807}+\frac{3125}{11} \log (5 x+3) \]

[Out]

3/(28*(2 + 3*x)^4) + 37/(49*(2 + 3*x)^3) + 3897/(686*(2 + 3*x)^2) + 136419/(2401
*(2 + 3*x)) - (32*Log[1 - 2*x])/184877 - (4774713*Log[2 + 3*x])/16807 + (3125*Lo
g[3 + 5*x])/11

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Rubi [A]  time = 0.0827294, antiderivative size = 75, 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 \[ \frac{136419}{2401 (3 x+2)}+\frac{3897}{686 (3 x+2)^2}+\frac{37}{49 (3 x+2)^3}+\frac{3}{28 (3 x+2)^4}-\frac{32 \log (1-2 x)}{184877}-\frac{4774713 \log (3 x+2)}{16807}+\frac{3125}{11} \log (5 x+3) \]

Antiderivative was successfully verified.

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

[Out]

3/(28*(2 + 3*x)^4) + 37/(49*(2 + 3*x)^3) + 3897/(686*(2 + 3*x)^2) + 136419/(2401
*(2 + 3*x)) - (32*Log[1 - 2*x])/184877 - (4774713*Log[2 + 3*x])/16807 + (3125*Lo
g[3 + 5*x])/11

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Rubi in Sympy [A]  time = 11.1908, size = 66, normalized size = 0.88 \[ - \frac{32 \log{\left (- 2 x + 1 \right )}}{184877} - \frac{4774713 \log{\left (3 x + 2 \right )}}{16807} + \frac{3125 \log{\left (5 x + 3 \right )}}{11} + \frac{136419}{2401 \left (3 x + 2\right )} + \frac{3897}{686 \left (3 x + 2\right )^{2}} + \frac{37}{49 \left (3 x + 2\right )^{3}} + \frac{3}{28 \left (3 x + 2\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1-2*x)/(2+3*x)**5/(3+5*x),x)

[Out]

-32*log(-2*x + 1)/184877 - 4774713*log(3*x + 2)/16807 + 3125*log(5*x + 3)/11 + 1
36419/(2401*(3*x + 2)) + 3897/(686*(3*x + 2)**2) + 37/(49*(3*x + 2)**3) + 3/(28*
(3*x + 2)**4)

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Mathematica [A]  time = 0.100834, size = 55, normalized size = 0.73 \[ \frac{\frac{77 \left (14733252 x^3+29957526 x^2+20320788 x+4599173\right )}{4 (3 x+2)^4}-32 \log (1-2 x)-52521843 \log (6 x+4)[phone] \log (10 x+6)}{184877} \]

Antiderivative was successfully verified.

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

[Out]

((77*(4599173 + 20320788*x + 29957526*x^2 + 14733252*x^3))/(4*(2 + 3*x)^4) - 32*
Log[1 - 2*x] - 52521843*Log[4 + 6*x] + 52521875*Log[6 + 10*x])/184877

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Maple [A]  time = 0.013, size = 62, normalized size = 0.8 \[{\frac{3125\,\ln \left ( 3+5\,x \right ) }{11}}+{\frac{3}{28\, \left ( 2+3\,x \right ) ^{4}}}+{\frac{37}{49\, \left ( 2+3\,x \right ) ^{3}}}+{\frac{3897}{686\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{136419}{4802+7203\,x}}-{\frac{4774713\,\ln \left ( 2+3\,x \right ) }{16807}}-{\frac{32\,\ln \left ( -1+2\,x \right ) }{184877}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3125/11*ln(3+5*x)+3/28/(2+3*x)^4+37/49/(2+3*x)^3+3897/686/(2+3*x)^2+136419/2401/
(2+3*x)-4774713/16807*ln(2+3*x)-32/184877*ln(-1+2*x)

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Maxima [A]  time = 1.36999, size = 86, normalized size = 1.15 \[ \frac{14733252 \, x^{3} + 29957526 \, x^{2} + 20320788 \, x + 4599173}{9604 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{3125}{11} \, \log \left (5 \, x + 3\right ) - \frac{4774713}{16807} \, \log \left (3 \, x + 2\right ) - \frac{32}{184877} \, \log \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((5*x + 3)*(3*x + 2)^5*(2*x - 1)),x, algorithm="maxima")

[Out]

1/9604*(14733252*x^3 + 29957526*x^2 + 20320788*x + 4599173)/(81*x^4 + 216*x^3 +
216*x^2 + 96*x + 16) + 3125/11*log(5*x + 3) - 4774713/16807*log(3*x + 2) - 32/18
4877*log(2*x - 1)

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Fricas [A]  time = 0.226582, size = 166, normalized size = 2.21 \[ \frac{1134460404 \, x^{3} + 2306729502 \, x^{2} + 210087500 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (5 \, x + 3\right ) - 210087372 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (3 \, x + 2\right ) - 128 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (2 \, x - 1\right ) + 1564700676 \, x + 354136321}{739508 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/739508*(1134460404*x^3 + 2306729502*x^2 + 210087500*(81*x^4 + 216*x^3 + 216*x^
2 + 96*x + 16)*log(5*x + 3) - 210087372*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)
*log(3*x + 2) - 128*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log(2*x - 1) + 1564
700676*x + 354136321)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

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Sympy [A]  time = 0.614189, size = 65, normalized size = 0.87 \[ \frac{14733252 x^{3} + 29957526 x^{2} + 20320788 x + 4599173}{777924 x^{4} + 2074464 x^{3} + 2074464 x^{2} + 921984 x + 153664} - \frac{32 \log{\left (x - \frac{1}{2} \right )}}{184877} + \frac{3125 \log{\left (x + \frac{3}{5} \right )}}{11} - \frac{4774713 \log{\left (x + \frac{2}{3} \right )}}{16807} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(14733252*x**3 + 29957526*x**2 + 20320788*x + 4599173)/(777924*x**4 + 2074464*x*
*3 + 2074464*x**2 + 921984*x + 153664) - 32*log(x - 1/2)/184877 + 3125*log(x + 3
/5)/11 - 4774713*log(x + 2/3)/16807

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GIAC/XCAS [A]  time = 0.213445, size = 90, normalized size = 1.2 \[ \frac{136419}{2401 \,{\left (3 \, x + 2\right )}} + \frac{3897}{686 \,{\left (3 \, x + 2\right )}^{2}} + \frac{37}{49 \,{\left (3 \, x + 2\right )}^{3}} + \frac{3}{28 \,{\left (3 \, x + 2\right )}^{4}} + \frac{3125}{11} \,{\rm ln}\left ({\left | -\frac{1}{3 \, x + 2} + 5 \right |}\right ) - \frac{32}{184877} \,{\rm ln}\left ({\left | -\frac{7}{3 \, x + 2} + 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((5*x + 3)*(3*x + 2)^5*(2*x - 1)),x, algorithm="giac")

[Out]

136419/2401/(3*x + 2) + 3897/686/(3*x + 2)^2 + 37/49/(3*x + 2)^3 + 3/28/(3*x + 2
)^4 + 3125/11*ln(abs(-1/(3*x + 2) + 5)) - 32/184877*ln(abs(-7/(3*x + 2) + 2))