3.1627 \(\int \frac{1}{(1-2 x)^2 (2+3 x)^2 (3+5 x)^3} \, dx\)

Optimal. Leaf size=75 \[ \frac{16}{65219 (1-2 x)}+\frac{81}{49 (3 x+2)}+\frac{7750}{1331 (5 x+3)}-\frac{125}{242 (5 x+3)^2}-\frac{2736 \log (1-2 x)}{5021863}-\frac{8829}{343} \log (3 x+2)+\frac{376875 \log (5 x+3)}{14641} \]

[Out]

16/(65219*(1 - 2*x)) + 81/(49*(2 + 3*x)) - 125/(242*(3 + 5*x)^2) + 7750/(1331*(3 + 5*x)) - (2736*Log[1 - 2*x])
/5021863 - (8829*Log[2 + 3*x])/343 + (376875*Log[3 + 5*x])/14641

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Rubi [A]  time = 0.0376286, 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 = {88} \[ \frac{16}{65219 (1-2 x)}+\frac{81}{49 (3 x+2)}+\frac{7750}{1331 (5 x+3)}-\frac{125}{242 (5 x+3)^2}-\frac{2736 \log (1-2 x)}{5021863}-\frac{8829}{343} \log (3 x+2)+\frac{376875 \log (5 x+3)}{14641} \]

Antiderivative was successfully verified.

[In]

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

[Out]

16/(65219*(1 - 2*x)) + 81/(49*(2 + 3*x)) - 125/(242*(3 + 5*x)^2) + 7750/(1331*(3 + 5*x)) - (2736*Log[1 - 2*x])
/5021863 - (8829*Log[2 + 3*x])/343 + (376875*Log[3 + 5*x])/14641

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 (2+3 x)^2 (3+5 x)^3} \, dx &=\int \left (\frac{32}{65219 (-1+2 x)^2}-\frac{5472}{5021863 (-1+2 x)}-\frac{243}{49 (2+3 x)^2}-\frac{26487}{343 (2+3 x)}+\frac{625}{121 (3+5 x)^3}-\frac{38750}{1331 (3+5 x)^2}+\frac{1884375}{14641 (3+5 x)}\right ) \, dx\\ &=\frac{16}{65219 (1-2 x)}+\frac{81}{49 (2+3 x)}-\frac{125}{242 (3+5 x)^2}+\frac{7750}{1331 (3+5 x)}-\frac{2736 \log (1-2 x)}{5021863}-\frac{8829}{343} \log (2+3 x)+\frac{376875 \log (3+5 x)}{14641}\\ \end{align*}

Mathematica [A]  time = 0.0623667, size = 65, normalized size = 0.87 \[ \frac{\frac{77 \left (33563700 x^3+24606540 x^2-7974123 x-6363424\right )}{(5 x+3)^2 \left (6 x^2+x-2\right )}-5472 \log (5-10 x)-258530778 \log (5 (3 x+2))[phone] \log (5 x+3)}{10043726} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((77*(-6363424 - 7974123*x + 24606540*x^2 + 33563700*x^3))/((3 + 5*x)^2*(-2 + x + 6*x^2)) - 5472*Log[5 - 10*x]
- 258530778*Log[5*(2 + 3*x)] + 258536250*Log[3 + 5*x])/10043726

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Maple [A]  time = 0.012, size = 62, normalized size = 0.8 \begin{align*} -{\frac{16}{130438\,x-65219}}-{\frac{2736\,\ln \left ( 2\,x-1 \right ) }{5021863}}+{\frac{81}{98+147\,x}}-{\frac{8829\,\ln \left ( 2+3\,x \right ) }{343}}-{\frac{125}{242\, \left ( 3+5\,x \right ) ^{2}}}+{\frac{7750}{3993+6655\,x}}+{\frac{376875\,\ln \left ( 3+5\,x \right ) }{14641}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-16/65219/(2*x-1)-2736/5021863*ln(2*x-1)+81/49/(2+3*x)-8829/343*ln(2+3*x)-125/242/(3+5*x)^2+7750/1331/(3+5*x)+
376875/14641*ln(3+5*x)

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Maxima [A]  time = 1.78432, size = 86, normalized size = 1.15 \begin{align*} \frac{33563700 \, x^{3} + 24606540 \, x^{2} - 7974123 \, x - 6363424}{130438 \,{\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )}} + \frac{376875}{14641} \, \log \left (5 \, x + 3\right ) - \frac{8829}{343} \, \log \left (3 \, x + 2\right ) - \frac{2736}{5021863} \, \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/(2+3*x)^2/(3+5*x)^3,x, algorithm="maxima")

[Out]

1/130438*(33563700*x^3 + 24606540*x^2 - 7974123*x - 6363424)/(150*x^4 + 205*x^3 + 34*x^2 - 51*x - 18) + 376875
/14641*log(5*x + 3) - 8829/343*log(3*x + 2) - 2736/5021863*log(2*x - 1)

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Fricas [B]  time = 1.19796, size = 413, normalized size = 5.51 \begin{align*} \frac{2584404900 \, x^{3} + 1894703580 \, x^{2} + 258536250 \,{\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )} \log \left (5 \, x + 3\right ) - 258530778 \,{\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )} \log \left (3 \, x + 2\right ) - 5472 \,{\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )} \log \left (2 \, x - 1\right ) - 614007471 \, x - 489983648}{10043726 \,{\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10043726*(2584404900*x^3 + 1894703580*x^2 + 258536250*(150*x^4 + 205*x^3 + 34*x^2 - 51*x - 18)*log(5*x + 3)
- 258530778*(150*x^4 + 205*x^3 + 34*x^2 - 51*x - 18)*log(3*x + 2) - 5472*(150*x^4 + 205*x^3 + 34*x^2 - 51*x -
18)*log(2*x - 1) - 614007471*x - 489983648)/(150*x^4 + 205*x^3 + 34*x^2 - 51*x - 18)

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Sympy [A]  time = 0.217402, size = 65, normalized size = 0.87 \begin{align*} \frac{33563700 x^{3} + 24606540 x^{2} - 7974123 x - 6363424}{19565700 x^{4} + 26739790 x^{3} + 4434892 x^{2} - 6652338 x - 2347884} - \frac{2736 \log{\left (x - \frac{1}{2} \right )}}{5021863} + \frac{376875 \log{\left (x + \frac{3}{5} \right )}}{14641} - \frac{8829 \log{\left (x + \frac{2}{3} \right )}}{343} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(33563700*x**3 + 24606540*x**2 - 7974123*x - 6363424)/(19565700*x**4 + 26739790*x**3 + 4434892*x**2 - 6652338*
x - 2347884) - 2736*log(x - 1/2)/5021863 + 376875*log(x + 3/5)/14641 - 8829*log(x + 2/3)/343

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Giac [A]  time = 2.52825, size = 116, normalized size = 1.55 \begin{align*} \frac{81}{49 \,{\left (3 \, x + 2\right )}} + \frac{27 \,{\left (\frac{139939165}{3 \, x + 2} - \frac{31679854}{{\left (3 \, x + 2\right )}^{2}} - 37396350\right )}}{913066 \,{\left (\frac{7}{3 \, x + 2} - 2\right )}{\left (\frac{1}{3 \, x + 2} - 5\right )}^{2}} + \frac{376875}{14641} \, \log \left ({\left | -\frac{1}{3 \, x + 2} + 5 \right |}\right ) - \frac{2736}{5021863} \, \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/(2+3*x)^2/(3+5*x)^3,x, algorithm="giac")

[Out]

81/49/(3*x + 2) + 27/913066*(139939165/(3*x + 2) - 31679854/(3*x + 2)^2 - 37396350)/((7/(3*x + 2) - 2)*(1/(3*x
+ 2) - 5)^2) + 376875/14641*log(abs(-1/(3*x + 2) + 5)) - 2736/5021863*log(abs(-7/(3*x + 2) + 2))