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*Log[1 - 2*x]
)/184877 - (4774713*Log[2 + 3*x])/16807 + (3125*Log[3 + 5*x])/11

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Rubi [A]  time = 0.0349027, 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, 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*}

Mathematica [A]  time = 0.0584286, 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 = 0.01, size = 62, normalized size = 0.8 \begin{align*} -{\frac{32\,\ln \left ( 2\,x-1 \right ) }{184877}}+{\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{3125\,\ln \left ( 3+5\,x \right ) }{11}} \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),x)

[Out]

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

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Maxima [A]  time = 1.08076, size = 86, normalized size = 1.15 \begin{align*} \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 ) \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),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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Fricas [B]  time = 1.22566, size = 410, normalized size = 5.47 \begin{align*} \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 )}} \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),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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Sympy [A]  time = 0.216375, size = 65, normalized size = 0.87 \begin{align*} \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} \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),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 [A]  time = 1.75323, size = 90, normalized size = 1.2 \begin{align*} \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 ) \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),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))