3.1613 \(\int \frac{1}{(1-2 x)^2 (2+3 x)^2 (3+5 x)^2} \, dx\)

Optimal. Leaf size=64 \[ \frac{8}{5929 (1-2 x)}-\frac{27}{49 (3 x+2)}-\frac{125}{121 (5 x+3)}-\frac{1088 \log (1-2 x)}{456533}+\frac{1998}{343} \log (3 x+2)-\frac{7750 \log (5 x+3)}{1331} \]

[Out]

8/(5929*(1 - 2*x)) - 27/(49*(2 + 3*x)) - 125/(121*(3 + 5*x)) - (1088*Log[1 - 2*x])/456533 + (1998*Log[2 + 3*x]
)/343 - (7750*Log[3 + 5*x])/1331

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Rubi [A]  time = 0.0315978, antiderivative size = 64, 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} \[ \frac{8}{5929 (1-2 x)}-\frac{27}{49 (3 x+2)}-\frac{125}{121 (5 x+3)}-\frac{1088 \log (1-2 x)}{456533}+\frac{1998}{343} \log (3 x+2)-\frac{7750 \log (5 x+3)}{1331} \]

Antiderivative was successfully verified.

[In]

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

[Out]

8/(5929*(1 - 2*x)) - 27/(49*(2 + 3*x)) - 125/(121*(3 + 5*x)) - (1088*Log[1 - 2*x])/456533 + (1998*Log[2 + 3*x]
)/343 - (7750*Log[3 + 5*x])/1331

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)^2} \, dx &=\int \left (\frac{16}{5929 (-1+2 x)^2}-\frac{2176}{456533 (-1+2 x)}+\frac{81}{49 (2+3 x)^2}+\frac{5994}{343 (2+3 x)}+\frac{625}{121 (3+5 x)^2}-\frac{38750}{1331 (3+5 x)}\right ) \, dx\\ &=\frac{8}{5929 (1-2 x)}-\frac{27}{49 (2+3 x)}-\frac{125}{121 (3+5 x)}-\frac{1088 \log (1-2 x)}{456533}+\frac{1998}{343} \log (2+3 x)-\frac{7750 \log (3+5 x)}{1331}\\ \end{align*}

Mathematica [A]  time = 0.0505754, size = 59, normalized size = 0.92 \[ \frac{2 \left (-544 \log (1-2 x)+1329669 \log (6 x+4)+7 \left (\frac{44}{1-2 x}-\frac{35937}{6 x+4}-\frac{67375}{10 x+6}-189875 \log (10 x+6)\right )\right )}{456533} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-544*Log[1 - 2*x] + 1329669*Log[4 + 6*x] + 7*(44/(1 - 2*x) - 35937/(4 + 6*x) - 67375/(6 + 10*x) - 189875*L
og[6 + 10*x])))/456533

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Maple [A]  time = 0.01, size = 53, normalized size = 0.8 \begin{align*} -{\frac{8}{11858\,x-5929}}-{\frac{1088\,\ln \left ( 2\,x-1 \right ) }{456533}}-{\frac{27}{98+147\,x}}+{\frac{1998\,\ln \left ( 2+3\,x \right ) }{343}}-{\frac{125}{363+605\,x}}-{\frac{7750\,\ln \left ( 3+5\,x \right ) }{1331}} \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)^2,x)

[Out]

-8/5929/(2*x-1)-1088/456533*ln(2*x-1)-27/49/(2+3*x)+1998/343*ln(2+3*x)-125/121/(3+5*x)-7750/1331*ln(3+5*x)

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Maxima [A]  time = 1.19239, size = 73, normalized size = 1.14 \begin{align*} -\frac{69540 \, x^{2} + 9544 \, x - 22003}{5929 \,{\left (30 \, x^{3} + 23 \, x^{2} - 7 \, x - 6\right )}} - \frac{7750}{1331} \, \log \left (5 \, x + 3\right ) + \frac{1998}{343} \, \log \left (3 \, x + 2\right ) - \frac{1088}{456533} \, \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)^2,x, algorithm="maxima")

[Out]

-1/5929*(69540*x^2 + 9544*x - 22003)/(30*x^3 + 23*x^2 - 7*x - 6) - 7750/1331*log(5*x + 3) + 1998/343*log(3*x +
2) - 1088/456533*log(2*x - 1)

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Fricas [A]  time = 1.49528, size = 302, normalized size = 4.72 \begin{align*} -\frac{5354580 \, x^{2} + 2658250 \,{\left (30 \, x^{3} + 23 \, x^{2} - 7 \, x - 6\right )} \log \left (5 \, x + 3\right ) - 2659338 \,{\left (30 \, x^{3} + 23 \, x^{2} - 7 \, x - 6\right )} \log \left (3 \, x + 2\right ) + 1088 \,{\left (30 \, x^{3} + 23 \, x^{2} - 7 \, x - 6\right )} \log \left (2 \, x - 1\right ) + 734888 \, x - 1694231}{456533 \,{\left (30 \, x^{3} + 23 \, x^{2} - 7 \, x - 6\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)^2,x, algorithm="fricas")

[Out]

-1/456533*(5354580*x^2 + 2658250*(30*x^3 + 23*x^2 - 7*x - 6)*log(5*x + 3) - 2659338*(30*x^3 + 23*x^2 - 7*x - 6
)*log(3*x + 2) + 1088*(30*x^3 + 23*x^2 - 7*x - 6)*log(2*x - 1) + 734888*x - 1694231)/(30*x^3 + 23*x^2 - 7*x -
6)

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Sympy [A]  time = 0.196763, size = 54, normalized size = 0.84 \begin{align*} - \frac{69540 x^{2} + 9544 x - 22003}{177870 x^{3} + 136367 x^{2} - 41503 x - 35574} - \frac{1088 \log{\left (x - \frac{1}{2} \right )}}{456533} - \frac{7750 \log{\left (x + \frac{3}{5} \right )}}{1331} + \frac{1998 \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)**2,x)

[Out]

-(69540*x**2 + 9544*x - 22003)/(177870*x**3 + 136367*x**2 - 41503*x - 35574) - 1088*log(x - 1/2)/456533 - 7750
*log(x + 3/5)/1331 + 1998*log(x + 2/3)/343

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Giac [A]  time = 2.61517, size = 104, normalized size = 1.62 \begin{align*} -\frac{125}{121 \,{\left (5 \, x + 3\right )}} + \frac{5 \,{\left (\frac{1185937}{5 \, x + 3} - 215574\right )}}{65219 \,{\left (\frac{11}{5 \, x + 3} - 2\right )}{\left (\frac{1}{5 \, x + 3} + 3\right )}} + \frac{1998}{343} \, \log \left ({\left | -\frac{1}{5 \, x + 3} - 3 \right |}\right ) - \frac{1088}{456533} \, \log \left ({\left | -\frac{11}{5 \, x + 3} + 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)^2,x, algorithm="giac")

[Out]

-125/121/(5*x + 3) + 5/65219*(1185937/(5*x + 3) - 215574)/((11/(5*x + 3) - 2)*(1/(5*x + 3) + 3)) + 1998/343*lo
g(abs(-1/(5*x + 3) - 3)) - 1088/456533*log(abs(-11/(5*x + 3) + 2))