Integrand size = 22, antiderivative size = 75 \[ \int \frac {1}{(1-2 x) (2+3 x)^4 (3+5 x)^2} \, dx=-\frac {3}{7 (2+3 x)^3}-\frac {324}{49 (2+3 x)^2}-\frac {34371}{343 (2+3 x)}-\frac {625}{11 (3+5 x)}-\frac {32 \log (1-2 x)}{290521}+\frac {1612242 \log (2+3 x)}{2401}-\frac {81250}{121} \log (3+5 x) \] Output:
-3/7/(2+3*x)^3-324/49/(2+3*x)^2-34371/(686+1029*x)-625/(33+55*x)-32/290521 *ln(1-2*x)+1612242/2401*ln(2+3*x)-81250/121*ln(3+5*x)
Time = 0.05 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.83 \[ \int \frac {1}{(1-2 x) (2+3 x)^4 (3+5 x)^2} \, dx=\frac {2 \left (-\frac {77 \left (6406511+29372133 x+44843517 x^2+22801770 x^3\right )}{2 (2+3 x)^3 (3+5 x)}-16 \log (1-2 x)+97540641 \log (4+6 x)-97540625 \log (6+10 x)\right )}{290521} \] Input:
Integrate[1/((1 - 2*x)*(2 + 3*x)^4*(3 + 5*x)^2),x]
Output:
(2*((-77*(6406511 + 29372133*x + 44843517*x^2 + 22801770*x^3))/(2*(2 + 3*x )^3*(3 + 5*x)) - 16*Log[1 - 2*x] + 97540641*Log[4 + 6*x] - 97540625*Log[6 + 10*x]))/290521
Time = 0.22 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {99, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{(1-2 x) (3 x+2)^4 (5 x+3)^2} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (\frac {4836726}{2401 (3 x+2)}-\frac {406250}{121 (5 x+3)}+\frac {103113}{343 (3 x+2)^2}+\frac {3125}{11 (5 x+3)^2}+\frac {1944}{49 (3 x+2)^3}+\frac {27}{7 (3 x+2)^4}-\frac {64}{290521 (2 x-1)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {34371}{343 (3 x+2)}-\frac {625}{11 (5 x+3)}-\frac {324}{49 (3 x+2)^2}-\frac {3}{7 (3 x+2)^3}-\frac {32 \log (1-2 x)}{290521}+\frac {1612242 \log (3 x+2)}{2401}-\frac {81250}{121} \log (5 x+3)\) |
Input:
Int[1/((1 - 2*x)*(2 + 3*x)^4*(3 + 5*x)^2),x]
Output:
-3/(7*(2 + 3*x)^3) - 324/(49*(2 + 3*x)^2) - 34371/(343*(2 + 3*x)) - 625/(1 1*(3 + 5*x)) - (32*Log[1 - 2*x])/290521 + (1612242*Log[2 + 3*x])/2401 - (8 1250*Log[3 + 5*x])/121
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(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]))
Time = 0.24 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.75
method | result | size |
norman | \(\frac {-\frac {44843517}{3773} x^{2}-\frac {22801770}{3773} x^{3}-\frac {4196019}{539} x -\frac {6406511}{3773}}{\left (2+3 x \right )^{3} \left (3+5 x \right )}-\frac {32 \ln \left (-1+2 x \right )}{290521}+\frac {1612242 \ln \left (2+3 x \right )}{2401}-\frac {81250 \ln \left (3+5 x \right )}{121}\) | \(56\) |
risch | \(\frac {-\frac {44843517}{3773} x^{2}-\frac {22801770}{3773} x^{3}-\frac {4196019}{539} x -\frac {6406511}{3773}}{\left (2+3 x \right )^{3} \left (3+5 x \right )}-\frac {32 \ln \left (-1+2 x \right )}{290521}+\frac {1612242 \ln \left (2+3 x \right )}{2401}-\frac {81250 \ln \left (3+5 x \right )}{121}\) | \(57\) |
default | \(-\frac {625}{11 \left (3+5 x \right )}-\frac {81250 \ln \left (3+5 x \right )}{121}-\frac {3}{7 \left (2+3 x \right )^{3}}-\frac {324}{49 \left (2+3 x \right )^{2}}-\frac {34371}{343 \left (2+3 x \right )}+\frac {1612242 \ln \left (2+3 x \right )}{2401}-\frac {32 \ln \left (-1+2 x \right )}{290521}\) | \(62\) |
parallelrisch | \(\frac {632063353680 \ln \left (\frac {2}{3}+x \right ) x^{4}-103680 \ln \left (x -\frac {1}{2}\right ) x^{4}-632063250000 \ln \left (x +\frac {3}{5}\right ) x^{4}+1643364719568 \ln \left (\frac {2}{3}+x \right ) x^{3}-269568 \ln \left (x -\frac {1}{2}\right ) x^{3}-1643364450000 \ln \left (x +\frac {3}{5}\right ) x^{3}+66595681845 x^{4}+1601227162656 \ln \left (\frac {2}{3}+x \right ) x^{2}-262656 \ln \left (x -\frac {1}{2}\right ) x^{2}-1601226900000 \ln \left (x +\frac {3}{5}\right ) x^{2}+131011101837 x^{3}+692928713664 \ln \left (\frac {2}{3}+x \right ) x -113664 \ln \left (x -\frac {1}{2}\right ) x -692928600000 \ln \left (x +\frac {3}{5}\right ) x +85838241258 x^{2}+112366818432 \ln \left (\frac {2}{3}+x \right )-18432 \ln \left (x -\frac {1}{2}\right )-112366800000 \ln \left (x +\frac {3}{5}\right )+18728897572 x}{6972504 \left (2+3 x \right )^{3} \left (3+5 x \right )}\) | \(156\) |
Input:
int(1/(1-2*x)/(2+3*x)^4/(3+5*x)^2,x,method=_RETURNVERBOSE)
Output:
(-44843517/3773*x^2-22801770/3773*x^3-4196019/539*x-6406511/3773)/(2+3*x)^ 3/(3+5*x)-32/290521*ln(-1+2*x)+1612242/2401*ln(2+3*x)-81250/121*ln(3+5*x)
Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (61) = 122\).
Time = 0.10 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.64 \[ \int \frac {1}{(1-2 x) (2+3 x)^4 (3+5 x)^2} \, dx=-\frac {1755736290 \, x^{3} + 3452950809 \, x^{2} + 195081250 \, {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (5 \, x + 3\right ) - 195081282 \, {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (3 \, x + 2\right ) + 32 \, {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (2 \, x - 1\right ) + 2261654241 \, x + 493301347}{290521 \, {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )}} \] Input:
integrate(1/(1-2*x)/(2+3*x)^4/(3+5*x)^2,x, algorithm="fricas")
Output:
-1/290521*(1755736290*x^3 + 3452950809*x^2 + 195081250*(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)*log(5*x + 3) - 195081282*(135*x^4 + 351*x^3 + 342* x^2 + 148*x + 24)*log(3*x + 2) + 32*(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)*log(2*x - 1) + 2261654241*x + 493301347)/(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)
Time = 0.11 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(1-2 x) (2+3 x)^4 (3+5 x)^2} \, dx=- \frac {22801770 x^{3} + 44843517 x^{2} + 29372133 x + 6406511}{509355 x^{4} + 1324323 x^{3} + 1290366 x^{2} + 558404 x + 90552} - \frac {32 \log {\left (x - \frac {1}{2} \right )}}{290521} - \frac {81250 \log {\left (x + \frac {3}{5} \right )}}{121} + \frac {1612242 \log {\left (x + \frac {2}{3} \right )}}{2401} \] Input:
integrate(1/(1-2*x)/(2+3*x)**4/(3+5*x)**2,x)
Output:
-(22801770*x**3 + 44843517*x**2 + 29372133*x + 6406511)/(509355*x**4 + 132 4323*x**3 + 1290366*x**2 + 558404*x + 90552) - 32*log(x - 1/2)/290521 - 81 250*log(x + 3/5)/121 + 1612242*log(x + 2/3)/2401
Time = 0.03 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.85 \[ \int \frac {1}{(1-2 x) (2+3 x)^4 (3+5 x)^2} \, dx=-\frac {22801770 \, x^{3} + 44843517 \, x^{2} + 29372133 \, x + 6406511}{3773 \, {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )}} - \frac {81250}{121} \, \log \left (5 \, x + 3\right ) + \frac {1612242}{2401} \, \log \left (3 \, x + 2\right ) - \frac {32}{290521} \, \log \left (2 \, x - 1\right ) \] Input:
integrate(1/(1-2*x)/(2+3*x)^4/(3+5*x)^2,x, algorithm="maxima")
Output:
-1/3773*(22801770*x^3 + 44843517*x^2 + 29372133*x + 6406511)/(135*x^4 + 35 1*x^3 + 342*x^2 + 148*x + 24) - 81250/121*log(5*x + 3) + 1612242/2401*log( 3*x + 2) - 32/290521*log(2*x - 1)
Time = 0.12 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.97 \[ \int \frac {1}{(1-2 x) (2+3 x)^4 (3+5 x)^2} \, dx=-\frac {625}{11 \, {\left (5 \, x + 3\right )}} + \frac {135 \, {\left (\frac {37929}{5 \, x + 3} + \frac {7564}{{\left (5 \, x + 3\right )}^{2}} + 49386\right )}}{343 \, {\left (\frac {1}{5 \, x + 3} + 3\right )}^{3}} + \frac {1612242}{2401} \, \log \left ({\left | -\frac {1}{5 \, x + 3} - 3 \right |}\right ) - \frac {32}{290521} \, \log \left ({\left | -\frac {11}{5 \, x + 3} + 2 \right |}\right ) \] Input:
integrate(1/(1-2*x)/(2+3*x)^4/(3+5*x)^2,x, algorithm="giac")
Output:
-625/11/(5*x + 3) + 135/343*(37929/(5*x + 3) + 7564/(5*x + 3)^2 + 49386)/( 1/(5*x + 3) + 3)^3 + 1612242/2401*log(abs(-1/(5*x + 3) - 3)) - 32/290521*l og(abs(-11/(5*x + 3) + 2))
Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.75 \[ \int \frac {1}{(1-2 x) (2+3 x)^4 (3+5 x)^2} \, dx=\frac {1612242\,\ln \left (x+\frac {2}{3}\right )}{2401}-\frac {32\,\ln \left (x-\frac {1}{2}\right )}{290521}-\frac {81250\,\ln \left (x+\frac {3}{5}\right )}{121}-\frac {\frac {168902\,x^3}{3773}+\frac {1660871\,x^2}{18865}+\frac {1398673\,x}{24255}+\frac {6406511}{509355}}{x^4+\frac {13\,x^3}{5}+\frac {38\,x^2}{15}+\frac {148\,x}{135}+\frac {8}{45}} \] Input:
int(-1/((2*x - 1)*(3*x + 2)^4*(5*x + 3)^2),x)
Output:
(1612242*log(x + 2/3))/2401 - (32*log(x - 1/2))/290521 - (81250*log(x + 3/ 5))/121 - ((1398673*x)/24255 + (1660871*x^2)/18865 + (168902*x^3)/3773 + 6 406511/509355)/((148*x)/135 + (38*x^2)/15 + (13*x^3)/5 + x^4 + 8/45)
Time = 0.19 (sec) , antiderivative size = 188, normalized size of antiderivative = 2.51 \[ \int \frac {1}{(1-2 x) (2+3 x)^4 (3+5 x)^2} \, dx=\frac {-342367593750 \,\mathrm {log}\left (5 x +3\right ) x^{4}-890155743750 \,\mathrm {log}\left (5 x +3\right ) x^{3}-867331237500 \,\mathrm {log}\left (5 x +3\right ) x^{2}-375336325000 \,\mathrm {log}\left (5 x +3\right ) x -60865350000 \,\mathrm {log}\left (5 x +3\right )+342367649910 \,\mathrm {log}\left (3 x +2\right ) x^{4}+890155889766 \,\mathrm {log}\left (3 x +2\right ) x^{3}+867331379772 \,\mathrm {log}\left (3 x +2\right ) x^{2}+375336386568 \,\mathrm {log}\left (3 x +2\right ) x +60865359984 \,\mathrm {log}\left (3 x +2\right )-56160 \,\mathrm {log}\left (2 x -1\right ) x^{4}-146016 \,\mathrm {log}\left (2 x -1\right ) x^{3}-142272 \,\mathrm {log}\left (2 x -1\right ) x^{2}-61568 \,\mathrm {log}\left (2 x -1\right ) x -9984 \,\mathrm {log}\left (2 x -1\right )+8778681450 x^{4}-22649034177 x^{2}-19777469173 x -4852263031}{509864355 x^{4}+1325647323 x^{3}+1291656366 x^{2}+558962404 x +90642552} \] Input:
int(1/(1-2*x)/(2+3*x)^4/(3+5*x)^2,x)
Output:
( - 342367593750*log(5*x + 3)*x**4 - 890155743750*log(5*x + 3)*x**3 - 8673 31237500*log(5*x + 3)*x**2 - 375336325000*log(5*x + 3)*x - 60865350000*log (5*x + 3) + 342367649910*log(3*x + 2)*x**4 + 890155889766*log(3*x + 2)*x** 3 + 867331379772*log(3*x + 2)*x**2 + 375336386568*log(3*x + 2)*x + 6086535 9984*log(3*x + 2) - 56160*log(2*x - 1)*x**4 - 146016*log(2*x - 1)*x**3 - 1 42272*log(2*x - 1)*x**2 - 61568*log(2*x - 1)*x - 9984*log(2*x - 1) + 87786 81450*x**4 - 22649034177*x**2 - 19777469173*x - 4852263031)/(3776773*(135* x**4 + 351*x**3 + 342*x**2 + 148*x + 24))