3.16.85 \(\int \frac {(3+5 x)^3}{(1-2 x)^2 (2+3 x)^6} \, dx\) [1585]

3.16.85.1 Optimal result

Integrand size = 22, antiderivative size = 87 \[ \int \frac {(3+5 x)^3}{(1-2 x)^2 (2+3 x)^6} \, dx=\frac {5324}{117649 (1-2 x)}+\frac {1}{2205 (2+3 x)^5}-\frac {101}{12348 (2+3 x)^4}+\frac {121}{2401 (2+3 x)^3}-\frac {3267}{33614 (2+3 x)^2}-\frac {14520}{117649 (2+3 x)}-\frac {45012 \log (1-2 x)}{823543}+\frac {45012 \log (2+3 x)}{823543} \]

output

5324/117649/(1-2*x)+1/2205/(2+3*x)^5-101/12348/(2+3*x)^4+121/2401/(2+3*x)^ 
3-3267/33614/(2+3*x)^2-14520/117649/(2+3*x)-45012/823543*ln(1-2*x)+45012/8 
23543*ln(2+3*x)
 

3.16.85.2 Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.74 \[ \int \frac {(3+5 x)^3}{(1-2 x)^2 (2+3 x)^6} \, dx=\frac {2 \left (-\frac {7 \left (-23684986+25985087 x+649342770 x^2+1747028250 x^3+1804756140 x^4+656274960 x^5\right )}{8 (-1+2 x) (2+3 x)^5}-1012770 \log (1-2 x)+1012770 \log (4+6 x)\right )}{37059435} \]

input

Integrate[(3 + 5*x)^3/((1 - 2*x)^2*(2 + 3*x)^6),x]
 

output

(2*((-7*(-23684986 + 25985087*x + 649342770*x^2 + 1747028250*x^3 + 1804756 
140*x^4 + 656274960*x^5))/(8*(-1 + 2*x)*(2 + 3*x)^5) - 1012770*Log[1 - 2*x 
] + 1012770*Log[4 + 6*x]))/37059435
 

3.16.85.3 Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 87, 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.

\(\displaystyle \int \frac {(5 x+3)^3}{(1-2 x)^2 (3 x+2)^6} \, dx\)

\(\Big \downarrow \) 99

\(\displaystyle \int \left (\frac {135036}{823543 (3 x+2)}+\frac {43560}{117649 (3 x+2)^2}+\frac {9801}{16807 (3 x+2)^3}-\frac {1089}{2401 (3 x+2)^4}+\frac {101}{1029 (3 x+2)^5}-\frac {1}{147 (3 x+2)^6}-\frac {90024}{823543 (2 x-1)}+\frac {10648}{117649 (2 x-1)^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {5324}{117649 (1-2 x)}-\frac {14520}{117649 (3 x+2)}-\frac {3267}{33614 (3 x+2)^2}+\frac {121}{2401 (3 x+2)^3}-\frac {101}{12348 (3 x+2)^4}+\frac {1}{2205 (3 x+2)^5}-\frac {45012 \log (1-2 x)}{823543}+\frac {45012 \log (3 x+2)}{823543}\)

input

Int[(3 + 5*x)^3/((1 - 2*x)^2*(2 + 3*x)^6),x]
 

output

5324/(117649*(1 - 2*x)) + 1/(2205*(2 + 3*x)^5) - 101/(12348*(2 + 3*x)^4) + 
 121/(2401*(2 + 3*x)^3) - 3267/(33614*(2 + 3*x)^2) - 14520/(117649*(2 + 3* 
x)) - (45012*Log[1 - 2*x])/823543 + (45012*Log[2 + 3*x])/823543
 

3.16.85.3.1 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]
 

3.16.85.4 Maple [A] (verified)

Time = 2.70 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.67

method	result	size
norman	\(\frac {-\frac {25985087}{21176820} x -\frac {21644759}{705894} x^{2}-\frac {19411425}{235298} x^{3}-\frac {10026423}{117649} x^{4}-\frac {3645972}{117649} x^{5}+\frac {11842493}{10588410}}{\left (-1+2 x \right ) \left (2+3 x \right )^{5}}-\frac {45012 \ln \left (-1+2 x \right )}{823543}+\frac {45012 \ln \left (2+3 x \right )}{823543}\)	\(58\)
risch	\(\frac {-\frac {25985087}{21176820} x -\frac {21644759}{705894} x^{2}-\frac {19411425}{235298} x^{3}-\frac {10026423}{117649} x^{4}-\frac {3645972}{117649} x^{5}+\frac {11842493}{10588410}}{\left (-1+2 x \right ) \left (2+3 x \right )^{5}}-\frac {45012 \ln \left (-1+2 x \right )}{823543}+\frac {45012 \ln \left (2+3 x \right )}{823543}\)	\(59\)
default	\(-\frac {5324}{117649 \left (-1+2 x \right )}-\frac {45012 \ln \left (-1+2 x \right )}{823543}+\frac {1}{2205 \left (2+3 x \right )^{5}}-\frac {101}{12348 \left (2+3 x \right )^{4}}+\frac {121}{2401 \left (2+3 x \right )^{3}}-\frac {3267}{33614 \left (2+3 x \right )^{2}}-\frac {14520}{117649 \left (2+3 x \right )}+\frac {45012 \ln \left (2+3 x \right )}{823543}\)	\(72\)
parallelrisch	\(\frac {-1944475680 x [phone] \ln \left (\frac {2}{3}+x \right ) x^{3}-3456921600 \ln \left (\frac {2}{3}+x \right ) x^{2}-2535075840 \ln \left (\frac {2}{3}+x \right ) x [phone] x^{5}[phone] x^{6}-18424897960 x^{3}-10024569870 x^{4}-10291308720 x^{2}-19445184000 \ln \left (x -\frac {1}{2}\right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right ) x^{4}-460922880 \ln \left (\frac {2}{3}+x \right )-5185382400 \ln \left (x -\frac {1}{2}\right ) x^{3}[phone] \ln \left (x -\frac {1}{2}\right ) x^{2}[phone] \ln \left (x -\frac {1}{2}\right ) x [phone] \ln \left (\frac {2}{3}+x \right ) x^{5}[phone] \ln \left (\frac {2}{3}+x \right ) x^{6}[phone] \ln \left (x -\frac {1}{2}\right )-7000266240 \ln \left (x -\frac {1}{2}\right ) x^{6}-19834087680 \ln \left (x -\frac {1}{2}\right ) x^{5}}{263533760 \left (-1+2 x \right ) \left (2+3 x \right )^{5}}\)	\(162\)

input

int((3+5*x)^3/(1-2*x)^2/(2+3*x)^6,x,method=_RETURNVERBOSE)
 

output

(-25985087/21176820*x-21644759/705894*x^2-19411425/235298*x^3-10026423/117 
649*x^4-3645972/117649*x^5+11842493/10588410)/(-1+2*x)/(2+3*x)^5-45012/823 
543*ln(-1+2*x)+45012/823543*ln(2+3*x)
 

3.16.85.5 Fricas [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.55 \[ \int \frac {(3+5 x)^3}{(1-2 x)^2 (2+3 x)^6} \, dx=-\frac {4593924720 \, x^{5} + 12633292980 \, x^{4} + 12229197750 \, x^{3} + 4545399390 \, x^{2} - 8102160 \, {\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )} \log \left (3 \, x + 2\right ) + 8102160 \, {\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )} \log \left (2 \, x - 1\right ) + 181895609 \, x - 165794902}{148237740 \, {\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )}} \]

input

integrate((3+5*x)^3/(1-2*x)^2/(2+3*x)^6,x, algorithm="fricas")
 

output

-1/148237740*(4593924720*x^5 + 12633292980*x^4 + 12229197750*x^3 + 4545399 
390*x^2 - 8102160*(486*x^6 + 1377*x^5 + 1350*x^4 + 360*x^3 - 240*x^2 - 176 
*x - 32)*log(3*x + 2) + 8102160*(486*x^6 + 1377*x^5 + 1350*x^4 + 360*x^3 - 
 240*x^2 - 176*x - 32)*log(2*x - 1) + 181895609*x - 165794902)/(486*x^6 + 
1377*x^5 + 1350*x^4 + 360*x^3 - 240*x^2 - 176*x - 32)
 

3.16.85.6 Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int \frac {(3+5 x)^3}{(1-2 x)^2 (2+3 x)^6} \, dx=\frac {- 656274960 x^{5} - 1804756140 x^{4} - 1747028250 x^{3} - 649342770 x^{2} - 25985087 x + 23684986}{10291934520 x^{6} + 29160481140 x^{5} + 28588707000 x^{4} + 7623655200 x^{3} - 5082436800 x^{2} - 3727120320 x - 677658240} - \frac {45012 \log {\left (x - \frac {1}{2} \right )}}{823543} + \frac {45012 \log {\left (x + \frac {2}{3} \right )}}{823543} \]

input

integrate((3+5*x)**3/(1-2*x)**2/(2+3*x)**6,x)
 

output

(-656274960*x**5 - 1804756140*x**4 - 1747028250*x**3 - 649342770*x**2 - 25 
985087*x + 23684986)/(10291934520*x**6 + 29160481140*x**5 + 28588707000*x* 
*4 + 7623655200*x**3 - 5082436800*x**2 - 3727120320*x - 677658240) - 45012 
*log(x - 1/2)/823543 + 45012*log(x + 2/3)/823543
 

3.16.85.7 Maxima [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.87 \[ \int \frac {(3+5 x)^3}{(1-2 x)^2 (2+3 x)^6} \, dx=-\frac {656274960 \, x^{5} + 1804756140 \, x^{4} + 1747028250 \, x^{3} + 649342770 \, x^{2} + 25985087 \, x - 23684986}{21176820 \, {\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )}} + \frac {45012}{823543} \, \log \left (3 \, x + 2\right ) - \frac {45012}{823543} \, \log \left (2 \, x - 1\right ) \]

input

integrate((3+5*x)^3/(1-2*x)^2/(2+3*x)^6,x, algorithm="maxima")
 

output

-1/21176820*(656274960*x^5 + 1804756140*x^4 + 1747028250*x^3 + 649342770*x 
^2 + 25985087*x - 23684986)/(486*x^6 + 1377*x^5 + 1350*x^4 + 360*x^3 - 240 
*x^2 - 176*x - 32) + 45012/823543*log(3*x + 2) - 45012/823543*log(2*x - 1)
 

3.16.85.8 Giac [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.90 \[ \int \frac {(3+5 x)^3}{(1-2 x)^2 (2+3 x)^6} \, dx=-\frac {5324}{117649 \, {\left (2 \, x - 1\right )}} + \frac {2 \, {\left (\frac {204418935}{2 \, x - 1} + \frac {740244225}{{\left (2 \, x - 1\right )}^{2}} + \frac {1185622375}{{\left (2 \, x - 1\right )}^{3}} + \frac {709135350}{{\left (2 \, x - 1\right )}^{4}} + 21049983\right )}}{4117715 \, {\left (\frac {7}{2 \, x - 1} + 3\right )}^{5}} + \frac {45012}{823543} \, \log \left ({\left | -\frac {7}{2 \, x - 1} - 3 \right |}\right ) \]

input

integrate((3+5*x)^3/(1-2*x)^2/(2+3*x)^6,x, algorithm="giac")
 

output

-5324/117649/(2*x - 1) + 2/4117715*(204418935/(2*x - 1) + 740244225/(2*x - 
 1)^2 + 1185622375/(2*x - 1)^3 + 709135350/(2*x - 1)^4 + 21049983)/(7/(2*x 
 - 1) + 3)^5 + 45012/823543*log(abs(-7/(2*x - 1) - 3))
 

3.16.85.9 Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.76 \[ \int \frac {(3+5 x)^3}{(1-2 x)^2 (2+3 x)^6} \, dx=\frac {90024\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{823543}-\frac {\frac {7502\,x^5}{117649}+\frac {41261\,x^4}{235298}+\frac {2156825\,x^3}{12706092}+\frac {21644759\,x^2}{343064484}+\frac {25985087\,x}{10291934520}-\frac {11842493}{5145967260}}{x^6+\frac {17\,x^5}{6}+\frac {25\,x^4}{9}+\frac {20\,x^3}{27}-\frac {40\,x^2}{81}-\frac {88\,x}{243}-\frac {16}{243}} \]

input

int((5*x + 3)^3/((2*x - 1)^2*(3*x + 2)^6),x)
 

output

(90024*atanh((12*x)/7 + 1/7))/823543 - ((25985087*x)/10291934520 + (216447 
59*x^2)/343064484 + (2156825*x^3)/12706092 + (41261*x^4)/235298 + (7502*x^ 
5)/117649 - 11842493/5145967260)/((20*x^3)/27 - (40*x^2)/81 - (88*x)/243 + 
 (25*x^4)/9 + (17*x^5)/6 + x^6 - 16/243)