\(\int \frac {(2+3 x)^5}{(1-2 x)^3 (3+5 x)} \, dx\) [438]

Optimal result

Integrand size = 22, antiderivative size = 55 \[ \int \frac {(2+3 x)^5}{(1-2 x)^3 (3+5 x)} \, dx=\frac {16807}{704 (1-2 x)^2}-\frac {156065}{1936 (1-2 x)}-\frac {10287 x}{400}-\frac {243 x^2}{80}-\frac {543655 \log (1-2 x)}{10648}+\frac {\log (3+5 x)}{166375} \] Output:

16807/704/(1-2*x)^2-156065/(1936-3872*x)-10287/400*x-243/80*x^2-543655/106 
48*ln(1-2*x)+1/166375*ln(3+5*x)
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00 \[ \int \frac {(2+3 x)^5}{(1-2 x)^3 (3+5 x)} \, dx=\frac {\frac {254205875}{(1-2 x)^2}+\frac {858357500}{-1+2 x}-47005596 (3+5 x)-1293732 (3+5 x)^2-543655000 \log (5-10 x)+64 \log (3+5 x)}{10648000} \] Input:

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

Output:

(254205875/(1 - 2*x)^2 + 858357500/(-1 + 2*x) - 47005596*(3 + 5*x) - 12937 
32*(3 + 5*x)^2 - 543655000*Log[5 - 10*x] + 64*Log[3 + 5*x])/10648000
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 55, 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.

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

\(\Big \downarrow \) 99

\(\displaystyle \int \left (-\frac {243 x}{40}-\frac {543655}{5324 (2 x-1)}+\frac {1}{33275 (5 x+3)}-\frac {156065}{968 (2 x-1)^2}-\frac {16807}{176 (2 x-1)^3}-\frac {10287}{400}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {243 x^2}{80}-\frac {10287 x}{400}-\frac {156065}{1936 (1-2 x)}+\frac {16807}{704 (1-2 x)^2}-\frac {543655 \log (1-2 x)}{10648}+\frac {\log (5 x+3)}{166375}\)

Input:

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

Output:

16807/(704*(1 - 2*x)^2) - 156065/(1936*(1 - 2*x)) - (10287*x)/400 - (243*x 
^2)/80 - (543655*Log[1 - 2*x])/10648 + Log[3 + 5*x]/166375
 

Defintions of rubi rules used

rule 99

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]))
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.73

method	result	size
risch	\(-\frac {243 x^{2}}{80}-\frac {10287 x}{400}+\frac {\frac {156065 x}{968}-\frac {439383}{7744}}{\left (-1+2 x \right )^{2}}-\frac {543655 \ln \left (-1+2 x \right )}{10648}+\frac {\ln \left (3+5 x \right )}{166375}\)	\(40\)
default	\(-\frac {243 x^{2}}{80}-\frac {10287 x}{400}+\frac {\ln \left (3+5 x \right )}{166375}+\frac {16807}{704 \left (-1+2 x \right )^{2}}+\frac {156065}{1936 \left (-1+2 x \right )}-\frac {543655 \ln \left (-1+2 x \right )}{10648}\)	\(44\)
norman	\(\frac {-\frac {1106513}{12100} x +\frac {3954117}{12100} x^{2}-\frac {2268}{25} x^{3}-\frac {243}{20} x^{4}}{\left (-1+2 x \right )^{2}}-\frac {543655 \ln \left (-1+2 x \right )}{10648}+\frac {\ln \left (3+5 x \right )}{166375}\)	\(45\)
parallelrisch	\(-\frac {16171650 x^{4}[phone] \ln \left (x -\frac {1}{2}\right ) x^{2}-32 \ln \left (x +\frac {3}{5}\right ) x^{2}[phone] x^{3}-271827500 \ln \left (x -\frac {1}{2}\right ) x +32 \ln \left (x +\frac {3}{5}\right ) x -434952870 x^{2}[phone] \ln \left (x -\frac {1}{2}\right )-8 \ln \left (x +\frac {3}{5}\right )[phone] x}{1331000 \left (-1+2 x \right )^{2}}\)	\(73\)

Input:

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

Output:

-243/80*x^2-10287/400*x+4*(156065/3872*x-439383/30976)/(-1+2*x)^2-543655/1 
0648*ln(-1+2*x)+1/166375*ln(3+5*x)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.27 \[ \int \frac {(2+3 x)^5}{(1-2 x)^3 (3+5 x)} \, dx=-\frac {129373200 \, x^{4} + 965986560 \, x^{3} - 1063016460 \, x^{2} - 64 \, {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (5 \, x + 3\right ) + 543655000 \, {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (2 \, x - 1\right ) - 1442875060 \, x + 604151625}{10648000 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \] Input:

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

Output:

-1/10648000*(129373200*x^4 + 965986560*x^3 - 1063016460*x^2 - 64*(4*x^2 - 
4*x + 1)*log(5*x + 3) + 543655000*(4*x^2 - 4*x + 1)*log(2*x - 1) - 1442875 
060*x + 604151625)/(4*x^2 - 4*x + 1)
 

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.80 \[ \int \frac {(2+3 x)^5}{(1-2 x)^3 (3+5 x)} \, dx=- \frac {243 x^{2}}{80} - \frac {10287 x}{400} - \frac {439383 - 1248520 x}{30976 x^{2} - 30976 x + 7744} - \frac {543655 \log {\left (x - \frac {1}{2} \right )}}{10648} + \frac {\log {\left (x + \frac {3}{5} \right )}}{166375} \] Input:

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

Output:

-243*x**2/80 - 10287*x/400 - (439383 - 1248520*x)/(30976*x**2 - 30976*x + 
7744) - 543655*log(x - 1/2)/10648 + log(x + 3/5)/166375
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.80 \[ \int \frac {(2+3 x)^5}{(1-2 x)^3 (3+5 x)} \, dx=-\frac {243}{80} \, x^{2} - \frac {10287}{400} \, x + \frac {2401 \, {\left (520 \, x - 183\right )}}{7744 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {1}{166375} \, \log \left (5 \, x + 3\right ) - \frac {543655}{10648} \, \log \left (2 \, x - 1\right ) \] Input:

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

Output:

-243/80*x^2 - 10287/400*x + 2401/7744*(520*x - 183)/(4*x^2 - 4*x + 1) + 1/ 
166375*log(5*x + 3) - 543655/10648*log(2*x - 1)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.75 \[ \int \frac {(2+3 x)^5}{(1-2 x)^3 (3+5 x)} \, dx=-\frac {243}{80} \, x^{2} - \frac {10287}{400} \, x + \frac {2401 \, {\left (520 \, x - 183\right )}}{7744 \, {\left (2 \, x - 1\right )}^{2}} + \frac {1}{166375} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {543655}{10648} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \] Input:

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

Output:

-243/80*x^2 - 10287/400*x + 2401/7744*(520*x - 183)/(2*x - 1)^2 + 1/166375 
*log(abs(5*x + 3)) - 543655/10648*log(abs(2*x - 1))
 

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.67 \[ \int \frac {(2+3 x)^5}{(1-2 x)^3 (3+5 x)} \, dx=\frac {\ln \left (x+\frac {3}{5}\right )}{166375}-\frac {543655\,\ln \left (x-\frac {1}{2}\right )}{10648}-\frac {10287\,x}{400}+\frac {\frac {156065\,x}{3872}-\frac {439383}{30976}}{x^2-x+\frac {1}{4}}-\frac {243\,x^2}{80} \] Input:

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

Output:

log(x + 3/5)/166375 - (543655*log(x - 1/2))/10648 - (10287*x)/400 + ((1560 
65*x)/3872 - 439383/30976)/(x^2 - x + 1/4) - (243*x^2)/80
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.56 \[ \int \frac {(2+3 x)^5}{(1-2 x)^3 (3+5 x)} \, dx=\frac {64 \,\mathrm {log}\left (5 x +3\right ) x^{2}-64 \,\mathrm {log}\left (5 x +3\right ) x +16 \,\mathrm {log}\left (5 x +3\right )-543655000 \,\mathrm {log}\left (2 x -1\right ) x^{2}[phone] \,\mathrm {log}\left (2 x -1\right ) x -135913750 \,\mathrm {log}\left (2 x -1\right )-32343300 x^{4}-241496640 x^{3}[phone] x^{2}-60858215}{10648000 x^{2}-10648000 x +2662000} \] Input:

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

Output:

(64*log(5*x + 3)*x**2 - 64*log(5*x + 3)*x + 16*log(5*x + 3) - 543655000*lo 
g(2*x - 1)*x**2 + 543655000*log(2*x - 1)*x - 135913750*log(2*x - 1) - 3234 
3300*x**4 - 241496640*x**3 + 626472880*x**2 - 60858215)/(2662000*(4*x**2 - 
 4*x + 1))