\(\int \frac {1}{(1-2 x)^3 (2+3 x)^3 (3+5 x)} \, dx\) [1682]

Optimal result

Integrand size = 22, antiderivative size = 75 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^3 (3+5 x)} \, dx=\frac {4}{3773 (1-2 x)^2}+\frac {1072}{290521 (1-2 x)}+\frac {27}{686 (2+3 x)^2}+\frac {1107}{2401 (2+3 x)}-\frac {89792 \log (1-2 x)}{22370117}-\frac {39393 \log (2+3 x)}{16807}+\frac {3125 \log (3+5 x)}{1331} \]

[Out]

4/3773/(1-2*x)^2+1072/290521/(1-2*x)+27/686/(2+3*x)^2+1107/2401/(2+3*x)-89792/22370117*ln(1-2*x)-39393/16807*l
n(2+3*x)+3125/1331*ln(3+5*x)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {90} \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^3 (3+5 x)} \, dx=\frac {1072}{290521 (1-2 x)}+\frac {1107}{2401 (3 x+2)}+\frac {4}{3773 (1-2 x)^2}+\frac {27}{686 (3 x+2)^2}-\frac {89792 \log (1-2 x)}{22370117}-\frac {39393 \log (3 x+2)}{16807}+\frac {3125 \log (5 x+3)}{1331} \]

[In]

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

[Out]

4/(3773*(1 - 2*x)^2) + 1072/(290521*(1 - 2*x)) + 27/(686*(2 + 3*x)^2) + 1107/(2401*(2 + 3*x)) - (89792*Log[1 -
2*x])/22370117 - (39393*Log[2 + 3*x])/16807 + (3125*Log[3 + 5*x])/1331

Rule 90

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*} \text {integral}& = \int \left (-\frac {16}{3773 (-1+2 x)^3}+\frac {2144}{290521 (-1+2 x)^2}-\frac {179584}{22370117 (-1+2 x)}-\frac {81}{343 (2+3 x)^3}-\frac {3321}{2401 (2+3 x)^2}-\frac {118179}{16807 (2+3 x)}+\frac {15625}{1331 (3+5 x)}\right ) \, dx \\ & = \frac {4}{3773 (1-2 x)^2}+\frac {1072}{290521 (1-2 x)}+\frac {27}{686 (2+3 x)^2}+\frac {1107}{2401 (2+3 x)}-\frac {89792 \log (1-2 x)}{22370117}-\frac {39393 \log (2+3 x)}{16807}+\frac {3125 \log (3+5 x)}{1331} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.77 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^3 (3+5 x)} \, dx=\frac {\frac {77 \left (569697-1414978 x-1006716 x^2+3176136 x^3\right )}{\left (-2+x+6 x^2\right )^2}-179584 \log (5-10 x)-104864166 \log (5 (2+3 x))[phone] \log (3+5 x)}{44740234} \]

[In]

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

[Out]

((77*(569697 - 1414978*x - 1006716*x^2 + 3176136*x^3))/(-2 + x + 6*x^2)^2 - 179584*Log[5 - 10*x] - 104864166*L
og[5*(2 + 3*x)] + 105043750*Log[3 + 5*x])/44740234

Maple [A] (verified)

Time = 0.90 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.75

method	result	size
norman	\(\frac {-\frac {707489}{290521} x -\frac {503358}{290521} x^{2}+\frac {1588068}{290521} x^{3}+\frac {569697}{581042}}{\left (-1+2 x \right )^{2} \left (2+3 x \right )^{2}}-\frac {89792 \ln \left (-1+2 x \right )}{22370117}-\frac {39393 \ln \left (2+3 x \right )}{16807}+\frac {3125 \ln \left (3+5 x \right )}{1331}\)	\(56\)
risch	\(\frac {-\frac {707489}{290521} x -\frac {503358}{290521} x^{2}+\frac {1588068}{290521} x^{3}+\frac {569697}{581042}}{\left (-1+2 x \right )^{2} \left (2+3 x \right )^{2}}-\frac {89792 \ln \left (-1+2 x \right )}{22370117}-\frac {39393 \ln \left (2+3 x \right )}{16807}+\frac {3125 \ln \left (3+5 x \right )}{1331}\)	\(57\)
default	\(\frac {3125 \ln \left (3+5 x \right )}{1331}+\frac {4}{3773 \left (-1+2 x \right )^{2}}-\frac {1072}{290521 \left (-1+2 x \right )}-\frac {89792 \ln \left (-1+2 x \right )}{22370117}+\frac {27}{686 \left (2+3 x \right )^{2}}+\frac {1107}{2401 \left (2+3 x \right )}-\frac {39393 \ln \left (2+3 x \right )}{16807}\)	\(62\)
parallelrisch	\(-\frac {7550219952 \ln \left (\frac {2}{3}+x \right ) x^{4}-7563150000 \ln \left (x +\frac {3}{5}\right ) x^{4}[phone] \ln \left (x -\frac {1}{2}\right ) x^{4}[phone]+2516739984 \ln \left (\frac {2}{3}+x \right ) x^{3}-2521050000 \ln \left (x +\frac {3}{5}\right ) x^{3}+4310016 \ln \left (x -\frac {1}{2}\right ) x^{3}[phone] x^{4}-4823751636 \ln \left (\frac {2}{3}+x \right ) x^{2}[phone] \ln \left (x +\frac {3}{5}\right ) x^{2}-8260864 \ln \left (x -\frac {1}{2}\right ) x^{2}[phone] x^{3}-838913328 \ln \left (\frac {2}{3}+x \right ) x [phone] \ln \left (x +\frac {3}{5}\right ) x -1436672 \ln \left (x -\frac {1}{2}\right ) x -1097928755 x^{2}[phone] \ln \left (\frac {2}{3}+x \right )-840350000 \ln \left (x +\frac {3}{5}\right )+1436672 \ln \left (x -\frac {1}{2}\right )}{89480468 \left (-1+2 x \right )^{2} \left (2+3 x \right )^{2}}\)	\(154\)

[In]

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

[Out]

(-707489/290521*x-503358/290521*x^2+1588068/290521*x^3+569697/581042)/(-1+2*x)^2/(2+3*x)^2-89792/22370117*ln(-
1+2*x)-39393/16807*ln(2+3*x)+3125/1331*ln(3+5*x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (61) = 122\).

Time = 0.23 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.64 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^3 (3+5 x)} \, dx=\frac {244562472 \, x^{3} - 77517132 \, x^{2} + 105043750 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \log \left (5 \, x + 3\right ) - 104864166 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \log \left (3 \, x + 2\right ) - 179584 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \log \left (2 \, x - 1\right ) - 108953306 \, x + 43866669}{44740234 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} \]

[In]

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

[Out]

1/44740234*(244562472*x^3 - 77517132*x^2 + 105043750*(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)*log(5*x + 3) - 10486
4166*(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)*log(3*x + 2) - 179584*(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)*log(2*x -
1) - 108953306*x + 43866669)/(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^3 (3+5 x)} \, dx=- \frac {- 3176136 x^{3} + 1006716 x^{2} + 1414978 x - 569697}{20917512 x^{4} + 6972504 x^{3} - 13363966 x^{2} - 2324168 x + 2324168} - \frac {89792 \log {\left (x - \frac {1}{2} \right )}}{22370117} + \frac {3125 \log {\left (x + \frac {3}{5} \right )}}{1331} - \frac {39393 \log {\left (x + \frac {2}{3} \right )}}{16807} \]

[In]

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

[Out]

-(-3176136*x**3 + 1006716*x**2 + 1414978*x - 569697)/(20917512*x**4 + 6972504*x**3 - 13363966*x**2 - 2324168*x
+ 2324168) - 89792*log(x - 1/2)/22370117 + 3125*log(x + 3/5)/1331 - 39393*log(x + 2/3)/16807

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.85 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^3 (3+5 x)} \, dx=\frac {3176136 \, x^{3} - 1006716 \, x^{2} - 1414978 \, x + 569697}{581042 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} + \frac {3125}{1331} \, \log \left (5 \, x + 3\right ) - \frac {39393}{16807} \, \log \left (3 \, x + 2\right ) - \frac {89792}{22370117} \, \log \left (2 \, x - 1\right ) \]

[In]

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

[Out]

1/581042*(3176136*x^3 - 1006716*x^2 - 1414978*x + 569697)/(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4) + 3125/1331*log
(5*x + 3) - 39393/16807*log(3*x + 2) - 89792/22370117*log(2*x - 1)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.79 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^3 (3+5 x)} \, dx=\frac {3176136 \, x^{3} - 1006716 \, x^{2} - 1414978 \, x + 569697}{581042 \, {\left (3 \, x + 2\right )}^{2} {\left (2 \, x - 1\right )}^{2}} + \frac {3125}{1331} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {39393}{16807} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {89792}{22370117} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]

[In]

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

[Out]

1/581042*(3176136*x^3 - 1006716*x^2 - 1414978*x + 569697)/((3*x + 2)^2*(2*x - 1)^2) + 3125/1331*log(abs(5*x +
3)) - 39393/16807*log(abs(3*x + 2)) - 89792/22370117*log(abs(2*x - 1))

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.75 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^3 (3+5 x)} \, dx=\frac {3125\,\ln \left (x+\frac {3}{5}\right )}{1331}-\frac {39393\,\ln \left (x+\frac {2}{3}\right )}{16807}-\frac {89792\,\ln \left (x-\frac {1}{2}\right )}{22370117}-\frac {-\frac {44113\,x^3}{290521}+\frac {83893\,x^2}{1743126}+\frac {707489\,x}{10458756}-\frac {189899}{6972504}}{x^4+\frac {x^3}{3}-\frac {23\,x^2}{36}-\frac {x}{9}+\frac {1}{9}} \]

[In]

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

[Out]

(3125*log(x + 3/5))/1331 - (39393*log(x + 2/3))/16807 - (89792*log(x - 1/2))/22370117 - ((707489*x)/10458756 +
(83893*x^2)/1743126 - (44113*x^3)/290521 - 189899/6972504)/(x^3/3 - (23*x^2)/36 - x/9 + x^4 + 1/9)