\(\int \frac {1}{(1-2 x)^3 (2+3 x)^2 (3+5 x)^2} \, dx\) [1694]

Optimal result

Integrand size = 22, antiderivative size = 75 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^2 (3+5 x)^2} \, dx=\frac {4}{5929 (1-2 x)^2}+\frac {1088}{456533 (1-2 x)}-\frac {81}{343 (2+3 x)}-\frac {625}{1331 (3+5 x)}-\frac {92496 \log (1-2 x)}{35153041}+\frac {6156 \log (2+3 x)}{2401}-\frac {37500 \log (3+5 x)}{14641} \]

[Out]

4/5929/(1-2*x)^2+1088/456533/(1-2*x)-81/343/(2+3*x)-625/1331/(3+5*x)-92496/35153041*ln(1-2*x)+6156/2401*ln(2+3
*x)-37500/14641*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} \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^2 (3+5 x)^2} \, dx=\frac {1088}{456533 (1-2 x)}-\frac {81}{343 (3 x+2)}-\frac {625}{1331 (5 x+3)}+\frac {4}{5929 (1-2 x)^2}-\frac {92496 \log (1-2 x)}{35153041}+\frac {6156 \log (3 x+2)}{2401}-\frac {37500 \log (5 x+3)}{14641} \]

[In]

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

[Out]

4/(5929*(1 - 2*x)^2) + 1088/(456533*(1 - 2*x)) - 81/(343*(2 + 3*x)) - 625/(1331*(3 + 5*x)) - (92496*Log[1 - 2*
x])/35153041 + (6156*Log[2 + 3*x])/2401 - (37500*Log[3 + 5*x])/14641

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}{5929 (-1+2 x)^3}+\frac {2176}{456533 (-1+2 x)^2}-\frac {184992}{35153041 (-1+2 x)}+\frac {243}{343 (2+3 x)^2}+\frac {18468}{2401 (2+3 x)}+\frac {3125}{1331 (3+5 x)^2}-\frac {187500}{14641 (3+5 x)}\right ) \, dx \\ & = \frac {4}{5929 (1-2 x)^2}+\frac {1088}{456533 (1-2 x)}-\frac {81}{343 (2+3 x)}-\frac {625}{1331 (3+5 x)}-\frac {92496 \log (1-2 x)}{35153041}+\frac {6156 \log (2+3 x)}{2401}-\frac {37500 \log (3+5 x)}{14641} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.91 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^2 (3+5 x)^2} \, dx=\frac {2 \left (77 \left (\frac {154}{(1-2 x)^2}+\frac {544}{1-2 x}-\frac {107811}{4+6 x}-\frac {214375}{6+10 x}\right )-46248 \log (1-2 x)[phone] \log (4+6 x)-45018750 \log (6+10 x)\right )}{35153041} \]

[In]

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

[Out]

(2*(77*(154/(1 - 2*x)^2 + 544/(1 - 2*x) - 107811/(4 + 6*x) - 214375/(6 + 10*x)) - 46248*Log[1 - 2*x] + 4506499
8*Log[4 + 6*x] - 45018750*Log[6 + 10*x]))/35153041

Maple [A] (verified)

Time = 0.90 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.83

method	result	size
default	\(-\frac {625}{1331 \left (3+5 x \right )}-\frac {37500 \ln \left (3+5 x \right )}{14641}+\frac {4}{5929 \left (-1+2 x \right )^{2}}-\frac {1088}{456533 \left (-1+2 x \right )}-\frac {92496 \ln \left (-1+2 x \right )}{35153041}-\frac {81}{343 \left (2+3 x \right )}+\frac {6156 \ln \left (2+3 x \right )}{2401}\)	\(62\)
risch	\(\frac {-\frac {4761360}{456533} x^{3}+\frac {1699584}{456533} x^{2}+\frac {262860}{65219} x -\frac {743807}{456533}}{\left (-1+2 x \right )^{2} \left (2+3 x \right ) \left (3+5 x \right )}-\frac {92496 \ln \left (-1+2 x \right )}{35153041}+\frac {6156 \ln \left (2+3 x \right )}{2401}-\frac {37500 \ln \left (3+5 x \right )}{14641}\)	\(64\)
norman	\(\frac {\frac {3154320}{65219} x^{4}+\frac {1126704}{456533} x^{3}-\frac {98484}{3773} x^{2}+\frac {1464217}{456533}}{\left (-1+2 x \right )^{2} \left (2+3 x \right ) \left (3+5 x \right )}-\frac {92496 \ln \left (-1+2 x \right )}{35153041}+\frac {6156 \ln \left (2+3 x \right )}{2401}-\frac {37500 \ln \left (3+5 x \right )}{14641}\)	\(65\)
parallelrisch	\(\frac {27038998800 \ln \left (\frac {2}{3}+x \right ) x^{4}-27011250000 \ln \left (x +\frac {3}{5}\right ) x^{4}-27748800 \ln \left (x -\frac {1}{2}\right ) x^{4}[phone]+7210399680 \ln \left (\frac {2}{3}+x \right ) x^{3}-7203000000 \ln \left (x +\frac {3}{5}\right ) x^{3}-7399680 \ln \left (x -\frac {1}{2}\right ) x^{3}[phone] x^{4}-16674049260 \ln \left (\frac {2}{3}+x \right ) x^{2}[phone] \ln \left (x +\frac {3}{5}\right ) x^{2}[phone] \ln \left (x -\frac {1}{2}\right ) x^{2}[phone] x^{3}-2253249900 \ln \left (\frac {2}{3}+x \right ) x [phone] \ln \left (x +\frac {3}{5}\right ) x +2312400 \ln \left (x -\frac {1}{2}\right ) x -4587877140 x^{2}[phone] \ln \left (\frac {2}{3}+x \right )-2701125000 \ln \left (x +\frac {3}{5}\right )-2774880 \ln \left (x -\frac {1}{2}\right )}{175765205 \left (-1+2 x \right )^{2} \left (2+3 x \right ) \left (3+5 x \right )}\)	\(161\)

[In]

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

[Out]

-625/1331/(3+5*x)-37500/14641*ln(3+5*x)+4/5929/(-1+2*x)^2-1088/456533/(-1+2*x)-92496/35153041*ln(-1+2*x)-81/34
3/(2+3*x)+6156/2401*ln(2+3*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 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^2 (3+5 x)^2} \, dx=-\frac {366624720 \, x^{3} - 130867968 \, x^{2} + 90037500 \, {\left (60 \, x^{4} + 16 \, x^{3} - 37 \, x^{2} - 5 \, x + 6\right )} \log \left (5 \, x + 3\right ) - 90129996 \, {\left (60 \, x^{4} + 16 \, x^{3} - 37 \, x^{2} - 5 \, x + 6\right )} \log \left (3 \, x + 2\right ) + 92496 \, {\left (60 \, x^{4} + 16 \, x^{3} - 37 \, x^{2} - 5 \, x + 6\right )} \log \left (2 \, x - 1\right ) - 141681540 \, x + 57273139}{35153041 \, {\left (60 \, x^{4} + 16 \, x^{3} - 37 \, x^{2} - 5 \, x + 6\right )}} \]

[In]

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

[Out]

-1/35153041*(366624720*x^3 - 130867968*x^2 + 90037500*(60*x^4 + 16*x^3 - 37*x^2 - 5*x + 6)*log(5*x + 3) - 9012
9996*(60*x^4 + 16*x^3 - 37*x^2 - 5*x + 6)*log(3*x + 2) + 92496*(60*x^4 + 16*x^3 - 37*x^2 - 5*x + 6)*log(2*x -
1) - 141681540*x + 57273139)/(60*x^4 + 16*x^3 - 37*x^2 - 5*x + 6)

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^2 (3+5 x)^2} \, dx=- \frac {4761360 x^{3} - 1699584 x^{2} - 1840020 x + 743807}{27391980 x^{4} + 7304528 x^{3} - 16891721 x^{2} - 2282665 x + 2739198} - \frac {92496 \log {\left (x - \frac {1}{2} \right )}}{35153041} - \frac {37500 \log {\left (x + \frac {3}{5} \right )}}{14641} + \frac {6156 \log {\left (x + \frac {2}{3} \right )}}{2401} \]

[In]

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

[Out]

-(4761360*x**3 - 1699584*x**2 - 1840020*x + 743807)/(27391980*x**4 + 7304528*x**3 - 16891721*x**2 - 2282665*x
+ 2739198) - 92496*log(x - 1/2)/35153041 - 37500*log(x + 3/5)/14641 + 6156*log(x + 2/3)/2401

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.85 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^2 (3+5 x)^2} \, dx=-\frac {4761360 \, x^{3} - 1699584 \, x^{2} - 1840020 \, x + 743807}{456533 \, {\left (60 \, x^{4} + 16 \, x^{3} - 37 \, x^{2} - 5 \, x + 6\right )}} - \frac {37500}{14641} \, \log \left (5 \, x + 3\right ) + \frac {6156}{2401} \, \log \left (3 \, x + 2\right ) - \frac {92496}{35153041} \, \log \left (2 \, x - 1\right ) \]

[In]

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

[Out]

-1/456533*(4761360*x^3 - 1699584*x^2 - 1840020*x + 743807)/(60*x^4 + 16*x^3 - 37*x^2 - 5*x + 6) - 37500/14641*
log(5*x + 3) + 6156/2401*log(3*x + 2) - 92496/35153041*log(2*x - 1)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.15 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^2 (3+5 x)^2} \, dx=-\frac {625}{1331 \, {\left (5 \, x + 3\right )}} - \frac {5 \, {\left (\frac {156456196}{5 \, x + 3} - \frac {430519419}{{\left (5 \, x + 3\right )}^{2}} - 14216316\right )}}{5021863 \, {\left (\frac {11}{5 \, x + 3} - 2\right )}^{2} {\left (\frac {1}{5 \, x + 3} + 3\right )}} + \frac {6156}{2401} \, \log \left ({\left | -\frac {1}{5 \, x + 3} - 3 \right |}\right ) - \frac {92496}{35153041} \, \log \left ({\left | -\frac {11}{5 \, x + 3} + 2 \right |}\right ) \]

[In]

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

[Out]

-625/1331/(5*x + 3) - 5/5021863*(156456196/(5*x + 3) - 430519419/(5*x + 3)^2 - 14216316)/((11/(5*x + 3) - 2)^2
*(1/(5*x + 3) + 3)) + 6156/2401*log(abs(-1/(5*x + 3) - 3)) - 92496/35153041*log(abs(-11/(5*x + 3) + 2))

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.73 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^2 (3+5 x)^2} \, dx=\frac {6156\,\ln \left (x+\frac {2}{3}\right )}{2401}-\frac {92496\,\ln \left (x-\frac {1}{2}\right )}{35153041}-\frac {37500\,\ln \left (x+\frac {3}{5}\right )}{14641}+\frac {-\frac {79356\,x^3}{456533}+\frac {141632\,x^2}{2282665}+\frac {4381\,x}{65219}-\frac {743807}{27391980}}{x^4+\frac {4\,x^3}{15}-\frac {37\,x^2}{60}-\frac {x}{12}+\frac {1}{10}} \]

[In]

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

[Out]

(6156*log(x + 2/3))/2401 - (92496*log(x - 1/2))/35153041 - (37500*log(x + 3/5))/14641 + ((4381*x)/65219 + (141
632*x^2)/2282665 - (79356*x^3)/456533 - 743807/27391980)/((4*x^3)/15 - (37*x^2)/60 - x/12 + x^4 + 1/10)