3.16.73 \(\int \frac {(2+3 x)^6 (3+5 x)^3}{(1-2 x)^2} \, dx\) [1573]

3.16.73.1 Optimal result

Integrand size = 22, antiderivative size = 76 \[ \int \frac {(2+3 x)^6 (3+5 x)^3}{(1-2 x)^2} \, dx=\frac {156590819}{1024 (1-2 x)}+\frac {230244479 x}{256}+\frac {310976027 x^2}{512}+\frac {7530189 x^3}{16}+\frac {85406805 x^4}{256}+\frac {15403257 x^5}{80}+\frac {2611845 x^6}{32}+\frac {309825 x^7}{14}+\frac {91125 x^8}{32}+\frac {616195041 \log (1-2 x)}{1024} \]

output

156590819/1024/(1-2*x)[phone]/256*x[phone]/512*x^2+7530189/16*x^3+85 
406805/256*x^4+15403257/80*x^5+2611845/32*x^6+309825/14*x^7+91125/32*x^8+6 
16195041/1024*ln(1-2*x)
 

3.16.73.2 Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.91 \[ \int \frac {(2+3 x)^6 (3+5 x)^3}{(1-2 x)^2} \, dx=\frac {153617806869-652800288858 x+341601057840 x^2+213352163360 x^3+174226352160 x^4+136105970112 x^5+87008414976 x^6+40459046400 x^7+11873952000 x^8+1632960000 x^9+172534611480 (-1+2 x) \log (1-2 x)}{286720 (-1+2 x)} \]

input

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

output

(153617806869 - 652800288858*x + 341601057840*x^2 + 213352163360*x^3 + 174 
226352160*x^4 + 136105970112*x^5 + 87008414976*x^6 + 40459046400*x^7 + 118 
73952000*x^8 + 1632960000*x^9 + 172534611480*(-1 + 2*x)*Log[1 - 2*x])/(286 
720*(-1 + 2*x))
 

3.16.73.3 Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 76, 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.

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

\(\Big \downarrow \) 99

\(\displaystyle \int \left (\frac {91125 x^7}{4}+\frac {309825 x^6}{2}+\frac {7835535 x^5}{16}+\frac {15403257 x^4}{16}+\frac {85406805 x^3}{64}+\frac {22590567 x^2}{16}+\frac {310976027 x}{256}+\frac {616195041}{512 (2 x-1)}+\frac {156590819}{512 (2 x-1)^2}+\frac {230244479}{256}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {91125 x^8}{32}+\frac {309825 x^7}{14}+\frac {2611845 x^6}{32}+\frac {15403257 x^5}{80}+\frac {85406805 x^4}{256}+\frac {7530189 x^3}{16}+\frac {310976027 x^2}{512}+\frac {230244479 x}{256}+\frac {156590819}{1024 (1-2 x)}+\frac {616195041 \log (1-2 x)}{1024}\)

input

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

output

156590819/(1024*(1 - 2*x)) + (230244479*x)/256 + (310976027*x^2)/512 + (75 
30189*x^3)/16 + (85406805*x^4)/256 + (15403257*x^5)/80 + (2611845*x^6)/32 
+ (309825*x^7)/14 + (91125*x^8)/32 + (616195041*Log[1 - 2*x])/1024
 

3.16.73.3.1 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]
 

3.16.73.4 Maple [A] (verified)

Time = 0.86 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.72

method	result	size
risch	\(\frac {91125 x^{8}}{32}+\frac {309825 x^{7}}{14}+\frac {2611845 x^{6}}{32}+\frac {15403257 x^{5}}{80}+\frac {85406805 x^{4}}{256}+\frac {7530189 x^{3}}{16}+\frac {310976027 x^{2}}{512}+\frac {230244479 x}{256}-\frac {156590819}{2048 \left (x -\frac {1}{2}\right )}+\frac {616195041 \ln \left (-1+2 x \right )}{1024}\)	\(55\)
default	\(\frac {91125 x^{8}}{32}+\frac {309825 x^{7}}{14}+\frac {2611845 x^{6}}{32}+\frac {15403257 x^{5}}{80}+\frac {85406805 x^{4}}{256}+\frac {7530189 x^{3}}{16}+\frac {310976027 x^{2}}{512}+\frac {230244479 x}{256}+\frac {616195041 \ln \left (-1+2 x \right )}{1024}-\frac {156590819}{1024 \left (-1+2 x \right )}\)	\(57\)
norman	\(\frac {-\frac {617079777}{512} x +\frac {610001889}{512} x^{2}+\frac {190493003}{256} x^{3}+\frac {155559243}{256} x^{4}+\frac {303807969}{640} x^{5}+\frac {48553803}{160} x^{6}+\frac {15804315}{112} x^{7}+\frac {9276525}{224} x^{8}+\frac {91125}{16} x^{9}}{-1+2 x}+\frac {616195041 \ln \left (-1+2 x \right )}{1024}\)	\(62\)
parallelrisch	\(\frac {204120000 x^{9}[phone] x^{8}[phone] x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] \ln \left (x -\frac {1}{2}\right ) x [phone] x^{2}-21566826435 \ln \left (x -\frac {1}{2}\right )-43195584390 x}{-35840+71680 x}\)	\(67\)
meijerg	\(\frac {13824 x}{1-2 x}+\frac {34115 x \left (-8 x^{2}-12 x +12\right )}{2 \left (1-2 x \right )}+\frac {616195041 \ln \left (1-2 x \right )}{1024}+\frac {21207 x \left (-40 x^{3}-40 x^{2}-60 x +60\right )}{4 \left (1-2 x \right )}+\frac {18225 x \left (-8960 x^{8}-5760 x^{7}-3840 x^{6}-2688 x^{5}-2016 x^{4}-1680 x^{3}-1680 x^{2}-2520 x +2520\right )}{28672 \left (1-2 x \right )}+\frac {12540 x \left (-6 x +6\right )}{1-2 x}+\frac {2046843 x \left (-448 x^{5}-336 x^{4}-280 x^{3}-280 x^{2}-420 x +420\right )}{4480 \left (1-2 x \right )}+\frac {5148 x \left (-48 x^{4}-40 x^{3}-40 x^{2}-60 x +60\right )}{1-2 x}+\frac {90801 x \left (-1280 x^{6}-896 x^{5}-672 x^{4}-560 x^{3}-560 x^{2}-840 x +840\right )}{1024 \left (1-2 x \right )}+\frac {11745 x \left (-5760 x^{7}-3840 x^{6}-2688 x^{5}-2016 x^{4}-1680 x^{3}-1680 x^{2}-2520 x +2520\right )}{1792 \left (1-2 x \right )}\)	\(280\)

input

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

output

91125/32*x^8+309825/14*x^7+2611845/32*x^6+15403257/80*x^5+85406805/256*x^4 
+7530189/16*x^3+310976027/512*x^2+230244479/256*x-156590819/2048/(x-1/2)+6 
16195041/1024*ln(-1+2*x)
 

3.16.73.5 Fricas [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.88 \[ \int \frac {(2+3 x)^6 (3+5 x)^3}{(1-2 x)^2} \, dx=\frac {204120000 \, x^{9} + 1484244000 \, x^{8} + 5057380800 \, x^{7} + 10876051872 \, x^{6} + 17013246264 \, x^{5} + 21778294020 \, x^{4} + 26669020420 \, x^{3} + 42700132230 \, x^{2} + 21566826435 \, {\left (2 \, x - 1\right )} \log \left (2 \, x - 1\right ) - 32234227060 \, x - 5480678665}{35840 \, {\left (2 \, x - 1\right )}} \]

input

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

output

1/35840*(204120000*x^9 + 1484244000*x^8 + 5057380800*x^7 + 10876051872*x^6 
 + 17013246264*x^5 + 21778294020*x^4 + 26669020420*x^3 + 42700132230*x^2 + 
 21566826435*(2*x - 1)*log(2*x - 1) - 32234227060*x - 5480678665)/(2*x - 1 
)
 

3.16.73.6 Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.89 \[ \int \frac {(2+3 x)^6 (3+5 x)^3}{(1-2 x)^2} \, dx=\frac {91125 x^{8}}{32} + \frac {309825 x^{7}}{14} + \frac {2611845 x^{6}}{32} + \frac {15403257 x^{5}}{80} + \frac {85406805 x^{4}}{256} + \frac {7530189 x^{3}}{16} + \frac {310976027 x^{2}}{512} + \frac {230244479 x}{256} + \frac {616195041 \log {\left (2 x - 1 \right )}}{1024} - \frac {156590819}{2048 x - 1024} \]

input

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

output

91125*x**8/32 + 309825*x**7/14 + 2611845*x**6/32 + 15403257*x**5/80 + 8540 
6805*x**4/256 + 7530189*x**3/16 + 310976027*x**2/512 + 230244479*x/256 + 6 
16195041*log(2*x - 1)/1024 - 156590819/(2048*x - 1024)
 

3.16.73.7 Maxima [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.74 \[ \int \frac {(2+3 x)^6 (3+5 x)^3}{(1-2 x)^2} \, dx=\frac {91125}{32} \, x^{8} + \frac {309825}{14} \, x^{7} + \frac {2611845}{32} \, x^{6} + \frac {15403257}{80} \, x^{5} + \frac {85406805}{256} \, x^{4} + \frac {7530189}{16} \, x^{3} + \frac {310976027}{512} \, x^{2} + \frac {230244479}{256} \, x - \frac {156590819}{1024 \, {\left (2 \, x - 1\right )}} + \frac {616195041}{1024} \, \log \left (2 \, x - 1\right ) \]

input

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

output

91125/32*x^8 + 309825/14*x^7 + 2611845/32*x^6 + 15403257/80*x^5 + 85406805 
/256*x^4 + 7530189/16*x^3 + 310976027/512*x^2 + 230244479/256*x - 15659081 
9/1024/(2*x - 1) + 616195041/1024*log(2*x - 1)
 

3.16.73.8 Giac [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.34 \[ \int \frac {(2+3 x)^6 (3+5 x)^3}{(1-2 x)^2} \, dx=\frac {1}{286720} \, {\left (2 \, x - 1\right )}^{8} {\left (\frac {75087000}{2 \, x - 1} + \frac {801964800}{{\left (2 \, x - 1\right )}^{2}} + \frac {5138731584}{{\left (2 \, x - 1\right )}^{3}} + \frac {22047451020}{{\left (2 \, x - 1\right )}^{4}} + \frac {67259967600}{{\left (2 \, x - 1\right )}^{5}} + \frac {153877208800}{{\left (2 \, x - 1\right )}^{6}} + \frac {301719264000}{{\left (2 \, x - 1\right )}^{7}} + 3189375\right )} - \frac {156590819}{1024 \, {\left (2 \, x - 1\right )}} - \frac {616195041}{1024} \, \log \left (\frac {{\left | 2 \, x - 1 \right |}}{2 \, {\left (2 \, x - 1\right )}^{2}}\right ) \]

input

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

output

1/286720*(2*x - 1)^8*(75087000/(2*x - 1) + 801964800/(2*x - 1)^2 + 5138731 
584/(2*x - 1)^3 + 22047451020/(2*x - 1)^4 + 67259967600/(2*x - 1)^5 + 1538 
77208800/(2*x - 1)^6 + 301719264000/(2*x - 1)^7 + 3189375) - 156590819/102 
4/(2*x - 1) - 616195041/1024*log(1/2*abs(2*x - 1)/(2*x - 1)^2)
 

3.16.73.9 Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.71 \[ \int \frac {(2+3 x)^6 (3+5 x)^3}{(1-2 x)^2} \, dx=\frac {230244479\,x}{256}+\frac {616195041\,\ln \left (x-\frac {1}{2}\right )}{1024}-\frac {156590819}{2048\,\left (x-\frac {1}{2}\right )}+\frac {310976027\,x^2}{512}+\frac {7530189\,x^3}{16}+\frac {85406805\,x^4}{256}+\frac {15403257\,x^5}{80}+\frac {2611845\,x^6}{32}+\frac {309825\,x^7}{14}+\frac {91125\,x^8}{32} \]

input

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

output

(230244479*x)/256 + (616195041*log(x - 1/2))/1024 - 156590819/(2048*(x - 1 
/2)) + (310976027*x^2)/512 + (7530189*x^3)/16 + (85406805*x^4)/256 + (1540 
3257*x^5)/80 + (2611845*x^6)/32 + (309825*x^7)/14 + (91125*x^8)/32