3.1598 \(\int \frac{1}{(1-2 x)^2 (2+3 x)^2 (3+5 x)} \, dx\)

Optimal. Leaf size=53 \[ \frac{4}{539 (1-2 x)}+\frac{9}{49 (3 x+2)}-\frac{404 \log (1-2 x)}{41503}-\frac{351}{343} \log (3 x+2)+\frac{125}{121} \log (5 x+3) \]

[Out]

4/(539*(1 - 2*x)) + 9/(49*(2 + 3*x)) - (404*Log[1 - 2*x])/41503 - (351*Log[2 + 3*x])/343 + (125*Log[3 + 5*x])/
121

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Rubi [A]  time = 0.0237716, antiderivative size = 53, 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} \[ \frac{4}{539 (1-2 x)}+\frac{9}{49 (3 x+2)}-\frac{404 \log (1-2 x)}{41503}-\frac{351}{343} \log (3 x+2)+\frac{125}{121} \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

4/(539*(1 - 2*x)) + 9/(49*(2 + 3*x)) - (404*Log[1 - 2*x])/41503 - (351*Log[2 + 3*x])/343 + (125*Log[3 + 5*x])/
121

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)} \, dx &=\int \left (\frac{8}{539 (-1+2 x)^2}-\frac{808}{41503 (-1+2 x)}-\frac{27}{49 (2+3 x)^2}-\frac{1053}{343 (2+3 x)}+\frac{625}{121 (3+5 x)}\right ) \, dx\\ &=\frac{4}{539 (1-2 x)}+\frac{9}{49 (2+3 x)}-\frac{404 \log (1-2 x)}{41503}-\frac{351}{343} \log (2+3 x)+\frac{125}{121} \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0290745, size = 56, normalized size = 1.06 \[ \frac{\frac{14322 x}{6 x^2+x-2}-\frac{8239}{6 x^2+x-2}-404 \log (5-10 x)-42471 \log (5 (3 x+2))+42875 \log (5 x+3)}{41503} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8239/(-2 + x + 6*x^2) + (14322*x)/(-2 + x + 6*x^2) - 404*Log[5 - 10*x] - 42471*Log[5*(2 + 3*x)] + 42875*Log[
3 + 5*x])/41503

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Maple [A]  time = 0.009, size = 44, normalized size = 0.8 \begin{align*} -{\frac{4}{1078\,x-539}}-{\frac{404\,\ln \left ( 2\,x-1 \right ) }{41503}}+{\frac{9}{98+147\,x}}-{\frac{351\,\ln \left ( 2+3\,x \right ) }{343}}+{\frac{125\,\ln \left ( 3+5\,x \right ) }{121}} \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),x)

[Out]

-4/539/(2*x-1)-404/41503*ln(2*x-1)+9/49/(2+3*x)-351/343*ln(2+3*x)+125/121*ln(3+5*x)

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Maxima [A]  time = 1.14267, size = 57, normalized size = 1.08 \begin{align*} \frac{186 \, x - 107}{539 \,{\left (6 \, x^{2} + x - 2\right )}} + \frac{125}{121} \, \log \left (5 \, x + 3\right ) - \frac{351}{343} \, \log \left (3 \, x + 2\right ) - \frac{404}{41503} \, \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),x, algorithm="maxima")

[Out]

1/539*(186*x - 107)/(6*x^2 + x - 2) + 125/121*log(5*x + 3) - 351/343*log(3*x + 2) - 404/41503*log(2*x - 1)

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Fricas [A]  time = 1.20758, size = 204, normalized size = 3.85 \begin{align*} \frac{42875 \,{\left (6 \, x^{2} + x - 2\right )} \log \left (5 \, x + 3\right ) - 42471 \,{\left (6 \, x^{2} + x - 2\right )} \log \left (3 \, x + 2\right ) - 404 \,{\left (6 \, x^{2} + x - 2\right )} \log \left (2 \, x - 1\right ) + 14322 \, x - 8239}{41503 \,{\left (6 \, x^{2} + x - 2\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),x, algorithm="fricas")

[Out]

1/41503*(42875*(6*x^2 + x - 2)*log(5*x + 3) - 42471*(6*x^2 + x - 2)*log(3*x + 2) - 404*(6*x^2 + x - 2)*log(2*x
- 1) + 14322*x - 8239)/(6*x^2 + x - 2)

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Sympy [A]  time = 0.175219, size = 44, normalized size = 0.83 \begin{align*} \frac{186 x - 107}{3234 x^{2} + 539 x - 1078} - \frac{404 \log{\left (x - \frac{1}{2} \right )}}{41503} + \frac{125 \log{\left (x + \frac{3}{5} \right )}}{121} - \frac{351 \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),x)

[Out]

(186*x - 107)/(3234*x**2 + 539*x - 1078) - 404*log(x - 1/2)/41503 + 125*log(x + 3/5)/121 - 351*log(x + 2/3)/34
3

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Giac [A]  time = 1.77232, size = 74, normalized size = 1.4 \begin{align*} \frac{9}{49 \,{\left (3 \, x + 2\right )}} + \frac{24}{3773 \,{\left (\frac{7}{3 \, x + 2} - 2\right )}} + \frac{125}{121} \, \log \left ({\left | -\frac{1}{3 \, x + 2} + 5 \right |}\right ) - \frac{404}{41503} \, \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),x, algorithm="giac")

[Out]

9/49/(3*x + 2) + 24/3773/(7/(3*x + 2) - 2) + 125/121*log(abs(-1/(3*x + 2) + 5)) - 404/41503*log(abs(-7/(3*x +
2) + 2))