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3.1498 \(\int \frac {1}{(1-2 x) (2+3 x)^5 (3+5 x)} \, dx\)

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)-32/184877*ln(1-2*x)-4774713/16807*ln(2+3
*x)+3125/11*ln(3+5*x)

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Rubi [A]  time = 0.03, antiderivative size = 75, 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 = {72} \[ \frac {136419}{2401 (3 x+2)}+\frac {3897}{686 (3 x+2)^2}+\frac {37}{49 (3 x+2)^3}+\frac {3}{28 (3 x+2)^4}-\frac {32 \log (1-2 x)}{184877}-\frac {4774713 \log (3 x+2)}{16807}+\frac {3125}{11} \log (5 x+3) \]

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*Log[3 + 5*x])/11

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {1}{(1-2 x) (2+3 x)^5 (3+5 x)} \, dx &=\int \left (-\frac {64}{184877 (-1+2 x)}-\frac {9}{7 (2+3 x)^5}-\frac {333}{49 (2+3 x)^4}-\frac {11691}{343 (2+3 x)^3}-\frac {409257}{2401 (2+3 x)^2}-\frac {14324139}{16807 (2+3 x)}+\frac {15625}{11 (3+5 x)}\right ) \, dx\\ &=\frac {3}{28 (2+3 x)^4}+\frac {37}{49 (2+3 x)^3}+\frac {3897}{686 (2+3 x)^2}+\frac {136419}{2401 (2+3 x)}-\frac {32 \log (1-2 x)}{184877}-\frac {4774713 \log (2+3 x)}{16807}+\frac {3125}{11} \log (3+5 x)\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 55, normalized size = 0.73 \[ \frac {\frac {77 \left (14733252 x^3+29957526 x^2+20320788 x+4599173\right )}{4 (3 x+2)^4}-32 \log (1-2 x)-52521843 \log (6 x+4)+52521875 \log (10 x+6)}{184877} \]

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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fricas [B]  time = 0.56, size = 123, normalized size = 1.64 \[ \frac {1134460404 \, x^{3} + 2306729502 \, x^{2} + 210087500 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (5 \, x + 3\right ) - 210087372 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (3 \, x + 2\right ) - 128 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (2 \, x - 1\right ) + 1564700676 \, x + 354136321}{739508 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)/(2+3*x)^5/(3+5*x),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) + 1564700676*x + 354136321)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

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giac [A]  time = 1.15, size = 67, normalized size = 0.89 \[ \frac {136419}{2401 \, {\left (3 \, x + 2\right )}} + \frac {3897}{686 \, {\left (3 \, x + 2\right )}^{2}} + \frac {37}{49 \, {\left (3 \, x + 2\right )}^{3}} + \frac {3}{28 \, {\left (3 \, x + 2\right )}^{4}} + \frac {3125}{11} \, \log \left ({\left | -\frac {1}{3 \, x + 2} + 5 \right |}\right ) - \frac {32}{184877} \, \log \left ({\left | -\frac {7}{3 \, x + 2} + 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)/(2+3*x)^5/(3+5*x),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*log(abs(-1/(3*x
+ 2) + 5)) - 32/184877*log(abs(-7/(3*x + 2) + 2))

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maple [A]  time = 0.01, size = 62, normalized size = 0.83 \[ -\frac {32 \ln \left (2 x -1\right )}{184877}-\frac {4774713 \ln \left (3 x +2\right )}{16807}+\frac {3125 \ln \left (5 x +3\right )}{11}+\frac {3}{28 \left (3 x +2\right )^{4}}+\frac {37}{49 \left (3 x +2\right )^{3}}+\frac {3897}{686 \left (3 x +2\right )^{2}}+\frac {136419}{2401 \left (3 x +2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.47, size = 64, normalized size = 0.85 \[ \frac {14733252 \, x^{3} + 29957526 \, x^{2} + 20320788 \, x + 4599173}{9604 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {3125}{11} \, \log \left (5 \, x + 3\right ) - \frac {4774713}{16807} \, \log \left (3 \, x + 2\right ) - \frac {32}{184877} \, \log \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)/(2+3*x)^5/(3+5*x),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) - 32/184877*log(2*x - 1)

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mupad [B]  time = 1.12, size = 55, normalized size = 0.73 \[ \frac {3125\,\ln \left (x+\frac {3}{5}\right )}{11}-\frac {4774713\,\ln \left (x+\frac {2}{3}\right )}{16807}-\frac {32\,\ln \left (x-\frac {1}{2}\right )}{184877}+\frac {\frac {45473\,x^3}{2401}+\frac {184923\,x^2}{4802}+\frac {1693399\,x}{64827}+\frac {4599173}{777924}}{x^4+\frac {8\,x^3}{3}+\frac {8\,x^2}{3}+\frac {32\,x}{27}+\frac {16}{81}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3125*log(x + 3/5))/11 - (4774713*log(x + 2/3))/16807 - (32*log(x - 1/2))/184877 + ((1693399*x)/64827 + (18492
3*x^2)/4802 + (45473*x^3)/2401 + 4599173/777924)/((32*x)/27 + (8*x^2)/3 + (8*x^3)/3 + x^4 + 16/81)

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sympy [A]  time = 0.24, size = 66, normalized size = 0.88 \[ - \frac {- 14733252 x^{3} - 29957526 x^{2} - 20320788 x - 4599173}{777924 x^{4} + 2074464 x^{3} + 2074464 x^{2} + 921984 x + 153664} - \frac {32 \log {\left (x - \frac {1}{2} \right )}}{184877} + \frac {3125 \log {\left (x + \frac {3}{5} \right )}}{11} - \frac {4774713 \log {\left (x + \frac {2}{3} \right )}}{16807} \]

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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