\(\int \frac {(1-2 x)^3}{(2+3 x)^7 (3+5 x)^2} \, dx\) [238]

Optimal result

Integrand size = 22, antiderivative size = 88 \[ \int \frac {(1-2 x)^3}{(2+3 x)^7 (3+5 x)^2} \, dx=-\frac {343}{54 (2+3 x)^6}-\frac {3136}{45 (2+3 x)^5}-\frac {2541}{4 (2+3 x)^4}-\frac {5566}{(2+3 x)^3}-\frac {103455}{2 (2+3 x)^2}-\frac {617100}{2+3 x}-\frac {166375}{3+5 x}+3584625 \log (2+3 x)-3584625 \log (3+5 x) \] Output:

-343/54/(2+3*x)^6-3136/45/(2+3*x)^5-2541/4/(2+3*x)^4-5566/(2+3*x)^3-103455 
/2/(2+3*x)^2-617100/(2+3*x)-166375/(3+5*x)+3584625*ln(2+3*x)-3584625*ln(3+ 
5*x)
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.02 \[ \int \frac {(1-2 x)^3}{(2+3 x)^7 (3+5 x)^2} \, dx=-\frac {343}{54 (2+3 x)^6}-\frac {3136}{45 (2+3 x)^5}-\frac {2541}{4 (2+3 x)^4}-\frac {5566}{(2+3 x)^3}-\frac {103455}{2 (2+3 x)^2}-\frac {617100}{2+3 x}-\frac {166375}{3+5 x}+3584625 \log (5 (2+3 x))-3584625 \log (3+5 x) \] Input:

Integrate[(1 - 2*x)^3/((2 + 3*x)^7*(3 + 5*x)^2),x]
 

Output:

-343/(54*(2 + 3*x)^6) - 3136/(45*(2 + 3*x)^5) - 2541/(4*(2 + 3*x)^4) - 556 
6/(2 + 3*x)^3 - 103455/(2*(2 + 3*x)^2) - 617100/(2 + 3*x) - 166375/(3 + 5* 
x) + 3584625*Log[5*(2 + 3*x)] - 3584625*Log[3 + 5*x]
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 88, 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.

\(\displaystyle \int \frac {(1-2 x)^3}{(3 x+2)^7 (5 x+3)^2} \, dx\)

\(\Big \downarrow \) 99

\(\displaystyle \int \left (-\frac {17923125}{5 x+3}+\frac {831875}{(5 x+3)^2}+\frac {10753875}{3 x+2}+\frac {1851300}{(3 x+2)^2}+\frac {310365}{(3 x+2)^3}+\frac {50094}{(3 x+2)^4}+\frac {7623}{(3 x+2)^5}+\frac {3136}{3 (3 x+2)^6}+\frac {343}{3 (3 x+2)^7}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {617100}{3 x+2}-\frac {166375}{5 x+3}-\frac {103455}{2 (3 x+2)^2}-\frac {5566}{(3 x+2)^3}-\frac {2541}{4 (3 x+2)^4}-\frac {3136}{45 (3 x+2)^5}-\frac {343}{54 (3 x+2)^6}+3584625 \log (3 x+2)-3584625 \log (5 x+3)\)

Input:

Int[(1 - 2*x)^3/((2 + 3*x)^7*(3 + 5*x)^2),x]
 

Output:

-343/(54*(2 + 3*x)^6) - 3136/(45*(2 + 3*x)^5) - 2541/(4*(2 + 3*x)^4) - 556 
6/(2 + 3*x)^3 - 103455/(2*(2 + 3*x)^2) - 617100/(2 + 3*x) - 166375/(3 + 5* 
x) + 3584625*Log[2 + 3*x] - 3584625*Log[3 + 5*x]
 

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 = 0.24 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.72

method	result	size
norman	\(\frac {-871063875 x^{6}-\frac {177869632819}{270} x -\frac {29917614049}{12} x^{2}-\frac {20124285489}{4} x^{3}-\frac {11419324785}{2} x^{4}-\frac {6910440075}{2} x^{5}-\frac {6526274777}{90}}{\left (2+3 x \right )^{6} \left (3+5 x \right )}+3584625 \ln \left (2+3 x \right )-3584625 \ln \left (3+5 x \right )\)	\(63\)
risch	\(\frac {-871063875 x^{6}-\frac {177869632819}{270} x -\frac {29917614049}{12} x^{2}-\frac {20124285489}{4} x^{3}-\frac {11419324785}{2} x^{4}-\frac {6910440075}{2} x^{5}-\frac {6526274777}{90}}{\left (2+3 x \right )^{6} \left (3+5 x \right )}+3584625 \ln \left (2+3 x \right )-3584625 \ln \left (3+5 x \right )\)	\(64\)
default	\(-\frac {343}{54 \left (2+3 x \right )^{6}}-\frac {3136}{45 \left (2+3 x \right )^{5}}-\frac {2541}{4 \left (2+3 x \right )^{4}}-\frac {5566}{\left (2+3 x \right )^{3}}-\frac {103455}{2 \left (2+3 x \right )^{2}}-\frac {617100}{2+3 x}-\frac {166375}{3+5 x}+3584625 \ln \left (2+3 x \right )-3584625 \ln \left (3+5 x \right )\)	\(81\)
parallelrisch	\(\frac {220239360320 x [phone] x^{7}-64420012800000 \ln \left (x +\frac {3}{5}\right ) x^{2}[phone] x^{3}[phone] x^{4}[phone] x^{2}[phone] x^{5}[phone] x^{6}[phone] \ln \left (\frac {2}{3}+x \right ) x^{7}-25086639600000 \ln \left (x +\frac {3}{5}\right ) x^{7}[phone] \ln \left (\frac {2}{3}+x \right ) x^{5}-163527724800000 \ln \left (x +\frac {3}{5}\right ) x^{3}[phone] \ln \left (\frac {2}{3}+x \right ) x^{3}[phone] \ln \left (\frac {2}{3}+x \right )[phone] \ln \left (\frac {2}{3}+x \right ) x [phone] \ln \left (\frac {2}{3}+x \right ) x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x^{6}-115398542160000 \ln \left (x +\frac {3}{5}\right ) x^{6}-14095319040000 \ln \left (x +\frac {3}{5}\right ) x [phone] \ln \left (\frac {2}{3}+x \right ) x^{4}-249008126400000 \ln \left (x +\frac {3}{5}\right ) x^{4}-227452199040000 \ln \left (x +\frac {3}{5}\right ) x^{5}-1321436160000 \ln \left (x +\frac {3}{5}\right )}{1920 \left (2+3 x \right )^{6} \left (3+5 x \right )}\)	\(185\)

Input:

int((1-2*x)^3/(2+3*x)^7/(3+5*x)^2,x,method=_RETURNVERBOSE)
 

Output:

(-871063875*x^6-177869632819/270*x-29917614049/12*x^2-20124285489/4*x^3-11 
419324785/2*x^4-6910440075/2*x^5-6526274777/90)/(2+3*x)^6/(3+5*x)+3584625* 
ln(2+3*x)-3584625*ln(3+5*x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.76 \[ \int \frac {(1-2 x)^3}{(2+3 x)^7 (3+5 x)^2} \, dx=-\frac {470374492500 \, x^{6} + 1865818820250 \, x^{5} + 3083217691950 \, x^{4} + 2716778541015 \, x^{3} + 1346292632205 \, x^{2} + 1935697500 \, {\left (3645 \, x^{7} + 16767 \, x^{6} + 33048 \, x^{5} + 36180 \, x^{4} + 23760 \, x^{3} + 9360 \, x^{2} + 2048 \, x + 192\right )} \log \left (5 \, x + 3\right ) - 1935697500 \, {\left (3645 \, x^{7} + 16767 \, x^{6} + 33048 \, x^{5} + 36180 \, x^{4} + 23760 \, x^{3} + 9360 \, x^{2} + 2048 \, x + 192\right )} \log \left (3 \, x + 2\right ) + 355739265638 \, x + 39157648662}{540 \, {\left (3645 \, x^{7} + 16767 \, x^{6} + 33048 \, x^{5} + 36180 \, x^{4} + 23760 \, x^{3} + 9360 \, x^{2} + 2048 \, x + 192\right )}} \] Input:

integrate((1-2*x)^3/(2+3*x)^7/(3+5*x)^2,x, algorithm="fricas")
 

Output:

-1/540*(470374492500*x^6 + 1865818820250*x^5 + 3083217691950*x^4 + 2716778 
541015*x^3 + 1346292632205*x^2 + 1935697500*(3645*x^7 + 16767*x^6 + 33048* 
x^5 + 36180*x^4 + 23760*x^3 + 9360*x^2 + 2048*x + 192)*log(5*x + 3) - 1935 
697500*(3645*x^7 + 16767*x^6 + 33048*x^5 + 36180*x^4 + 23760*x^3 + 9360*x^ 
2 + 2048*x + 192)*log(3*x + 2) + 355739265638*x + 39157648662)/(3645*x^7 + 
 16767*x^6 + 33048*x^5 + 36180*x^4 + 23760*x^3 + 9360*x^2 + 2048*x + 192)
 

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.93 \[ \int \frac {(1-2 x)^3}{(2+3 x)^7 (3+5 x)^2} \, dx=- \frac {470374492500 x^{6} + 1865818820250 x^{5} + 3083217691950 x^{4} + 2716778541015 x^{3} + 1346292632205 x^{2} + 355739265638 x + 39157648662}{1968300 x^{7} + 9054180 x^{6} + 17845920 x^{5} + 19537200 x^{4} + 12830400 x^{3} + 5054400 x^{2} + 1105920 x + 103680} - 3584625 \log {\left (x + \frac {3}{5} \right )} + 3584625 \log {\left (x + \frac {2}{3} \right )} \] Input:

integrate((1-2*x)**3/(2+3*x)**7/(3+5*x)**2,x)
                                                                                    
                                                                                    
 

Output:

-(470374492500*x**6 + 1865818820250*x**5 + 3083217691950*x**4 + 2716778541 
015*x**3 + 1346292632205*x**2 + 355739265638*x + 39157648662)/(1968300*x** 
7 + 9054180*x**6 + 17845920*x**5 + 19537200*x**4 + 12830400*x**3 + 5054400 
*x**2 + 1105920*x + 103680) - 3584625*log(x + 3/5) + 3584625*log(x + 2/3)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.98 \[ \int \frac {(1-2 x)^3}{(2+3 x)^7 (3+5 x)^2} \, dx=-\frac {470374492500 \, x^{6} + 1865818820250 \, x^{5} + 3083217691950 \, x^{4} + 2716778541015 \, x^{3} + 1346292632205 \, x^{2} + 355739265638 \, x + 39157648662}{540 \, {\left (3645 \, x^{7} + 16767 \, x^{6} + 33048 \, x^{5} + 36180 \, x^{4} + 23760 \, x^{3} + 9360 \, x^{2} + 2048 \, x + 192\right )}} - 3584625 \, \log \left (5 \, x + 3\right ) + 3584625 \, \log \left (3 \, x + 2\right ) \] Input:

integrate((1-2*x)^3/(2+3*x)^7/(3+5*x)^2,x, algorithm="maxima")
 

Output:

-1/540*(470374492500*x^6 + 1865818820250*x^5 + 3083217691950*x^4 + 2716778 
541015*x^3 + 1346292632205*x^2 + 355739265638*x + 39157648662)/(3645*x^7 + 
 16767*x^6 + 33048*x^5 + 36180*x^4 + 23760*x^3 + 9360*x^2 + 2048*x + 192) 
- 3584625*log(5*x + 3) + 3584625*log(3*x + 2)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.97 \[ \int \frac {(1-2 x)^3}{(2+3 x)^7 (3+5 x)^2} \, dx=-\frac {166375}{5 \, x + 3} + \frac {125 \, {\left (\frac {246075138}{5 \, x + 3} + \frac {181716633}{{\left (5 \, x + 3\right )}^{2}} + \frac {68296076}{{\left (5 \, x + 3\right )}^{3}} + \frac {13169954}{{\left (5 \, x + 3\right )}^{4}} + \frac {1059036}{{\left (5 \, x + 3\right )}^{5}} + 135033993\right )}}{4 \, {\left (\frac {1}{5 \, x + 3} + 3\right )}^{6}} + 3584625 \, \log \left ({\left | -\frac {1}{5 \, x + 3} - 3 \right |}\right ) \] Input:

integrate((1-2*x)^3/(2+3*x)^7/(3+5*x)^2,x, algorithm="giac")
 

Output:

-166375/(5*x + 3) + 125/4*(246075138/(5*x + 3) + 181716633/(5*x + 3)^2 + 6 
8296076/(5*x + 3)^3 + 13169954/(5*x + 3)^4 + 1059036/(5*x + 3)^5 + 1350339 
93)/(1/(5*x + 3) + 3)^6 + 3584625*log(abs(-1/(5*x + 3) - 3))
 

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.86 \[ \int \frac {(1-2 x)^3}{(2+3 x)^7 (3+5 x)^2} \, dx=7169250\,\mathrm {atanh}\left (30\,x+19\right )-\frac {238975\,x^6+\frac {5687605\,x^5}{6}+\frac {84587591\,x^4}{54}+\frac {248447969\,x^3}{180}+\frac {29917614049\,x^2}{43740}+\frac {177869632819\,x}{984150}+\frac {6526274777}{328050}}{x^7+\frac {23\,x^6}{5}+\frac {136\,x^5}{15}+\frac {268\,x^4}{27}+\frac {176\,x^3}{27}+\frac {208\,x^2}{81}+\frac {2048\,x}{3645}+\frac {64}{1215}} \] Input:

int(-(2*x - 1)^3/((3*x + 2)^7*(5*x + 3)^2),x)
 

Output:

7169250*atanh(30*x + 19) - ((177869632819*x)/984150 + (29917614049*x^2)/43 
740 + (248447969*x^3)/180 + (84587591*x^4)/54 + (5687605*x^5)/6 + 238975*x 
^6 + 6526274777/328050)/((2048*x)/3645 + (208*x^2)/81 + (176*x^3)/27 + (26 
8*x^4)/27 + (136*x^5)/15 + (23*x^6)/5 + x^7 + 64/1215)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 234, normalized size of antiderivative = 2.66 \[ \int \frac {(1-2 x)^3}{(2+3 x)^7 (3+5 x)^2} \, dx=\frac {-162279199912500 \,\mathrm {log}\left (5 x +3\right ) x^{7}-746484319597500 \,\mathrm {log}\left (5 x +3\right ) x^{6}-1471331412540000 \,\mathrm {log}\left (5 x +3\right ) x^{5}-1610771317650000 \,\mathrm {log}\left (5 x +3\right ) x^{4}-1057819969800000 \,\mathrm {log}\left (5 x +3\right ) x^{3}-416716957800000 \,\mathrm {log}\left (5 x +3\right ) x^{2}-91179095040000 \,\mathrm {log}\left (5 x +3\right ) x -8548040160000 \,\mathrm {log}\left (5 x +3\right )[phone] \,\mathrm {log}\left (3 x +2\right ) x^{7}[phone] \,\mathrm {log}\left (3 x +2\right ) x^{6}+1471331412540000 \,\mathrm {log}\left (3 x +2\right ) x^{5}+1610771317650000 \,\mathrm {log}\left (3 x +2\right ) x^{4}+1057819969800000 \,\mathrm {log}\left (3 x +2\right ) x^{3}[phone] \,\mathrm {log}\left (3 x +2\right ) x^{2}[phone] \,\mathrm {log}\left (3 x +2\right ) x [phone] \,\mathrm {log}\left (3 x +2\right )[phone] x^{7}-21590189205750 x^{5}-47569495064850 x^{4}-47155182243345 x^{3}-24925354340715 x^{2}-6860566949674 x -776741279226}{45270900 x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x +2384640} \] Input:

int((1-2*x)^3/(2+3*x)^7/(3+5*x)^2,x)
 

Output:

( - 162279199912500*log(5*x + 3)*x**7 - 746484319597500*log(5*x + 3)*x**6 
- 1471331412540000*log(5*x + 3)*x**5 - 1610771317650000*log(5*x + 3)*x**4 
- 1057819969800000*log(5*x + 3)*x**3 - 416716957800000*log(5*x + 3)*x**2 - 
 91179095040000*log(5*x + 3)*x - 8548040160000*log(5*x + 3) + 162279199912 
500*log(3*x + 2)*x**7 + 746484319597500*log(3*x + 2)*x**6 + 14713314125400 
00*log(3*x + 2)*x**5 + 1610771317650000*log(3*x + 2)*x**4 + 10578199698000 
00*log(3*x + 2)*x**3 + 416716957800000*log(3*x + 2)*x**2 + 91179095040000* 
log(3*x + 2)*x + 8548040160000*log(3*x + 2) + 2351872462500*x**7 - 2159018 
9205750*x**5 - 47569495064850*x**4 - 47155182243345*x**3 - 24925354340715* 
x**2 - 6860566949674*x - 776741279226)/(12420*(3645*x**7 + 16767*x**6 + 33 
048*x**5 + 36180*x**4 + 23760*x**3 + 9360*x**2 + 2048*x + 192))