Integrand size = 22, antiderivative size = 69 \[ \int \frac {(2+3 x)^5 (3+5 x)^3}{(1-2 x)^2} \, dx=\frac {22370117}{512 (1-2 x)}+\frac {56291737 x}{256}+\frac {8881301 x^2}{64}+\frac {6179077 x^3}{64}+\frac {3724389 x^4}{64}+\frac {423009 x^5}{16}+\frac {15525 x^6}{2}+\frac {30375 x^7}{28}+\frac {39220335}{256} \log (1-2 x) \] Output:
22370117/(512-1024*x)+56291737/256*x+8881301/64*x^2+6179077/64*x^3+3724389 /64*x^4+423009/16*x^5+15525/2*x^6+30375/28*x^7+39220335/256*ln(1-2*x)
Time = 0.01 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.93 \[ \int \frac {(2+3 x)^5 (3+5 x)^3}{(1-2 x)^2} \, dx=\frac {843009185-3888550282 x+2157631560 x^2+1297354800 x^3+966981680 x^4+644755104 x^5+323374464 x^6+103507200 x^7+15552000 x^8+1098169380 (-1+2 x) \log (1-2 x)}{7168 (-1+2 x)} \] Input:
Integrate[((2 + 3*x)^5*(3 + 5*x)^3)/(1 - 2*x)^2,x]
Output:
(843009185 - 3888550282*x + 2157631560*x^2 + 1297354800*x^3 + 966981680*x^ 4 + 644755104*x^5 + 323374464*x^6 + 103507200*x^7 + 15552000*x^8 + 1098169 380*(-1 + 2*x)*Log[1 - 2*x])/(7168*(-1 + 2*x))
Time = 0.27 (sec) , antiderivative size = 69, 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.
\(\displaystyle \int \frac {(3 x+2)^5 (5 x+3)^3}{(1-2 x)^2} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (\frac {30375 x^6}{4}+46575 x^5+\frac {2115045 x^4}{16}+\frac {3724389 x^3}{16}+\frac {18537231 x^2}{64}+\frac {8881301 x}{32}+\frac {39220335}{128 (2 x-1)}+\frac {22370117}{256 (2 x-1)^2}+\frac {56291737}{256}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {30375 x^7}{28}+\frac {15525 x^6}{2}+\frac {423009 x^5}{16}+\frac {3724389 x^4}{64}+\frac {6179077 x^3}{64}+\frac {8881301 x^2}{64}+\frac {56291737 x}{256}+\frac {22370117}{512 (1-2 x)}+\frac {39220335}{256} \log (1-2 x)\) |
Input:
Int[((2 + 3*x)^5*(3 + 5*x)^3)/(1 - 2*x)^2,x]
Output:
22370117/(512*(1 - 2*x)) + (56291737*x)/256 + (8881301*x^2)/64 + (6179077* x^3)/64 + (3724389*x^4)/64 + (423009*x^5)/16 + (15525*x^6)/2 + (30375*x^7) /28 + (39220335*Log[1 - 2*x])/256
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 = 0.23 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.72
method | result | size |
risch | \(\frac {30375 x^{7}}{28}+\frac {15525 x^{6}}{2}+\frac {423009 x^{5}}{16}+\frac {3724389 x^{4}}{64}+\frac {6179077 x^{3}}{64}+\frac {8881301 x^{2}}{64}+\frac {56291737 x}{256}-\frac {22370117}{1024 \left (x -\frac {1}{2}\right )}+\frac {39220335 \ln \left (-1+2 x \right )}{256}\) | \(50\) |
default | \(\frac {30375 x^{7}}{28}+\frac {15525 x^{6}}{2}+\frac {423009 x^{5}}{16}+\frac {3724389 x^{4}}{64}+\frac {6179077 x^{3}}{64}+\frac {8881301 x^{2}}{64}+\frac {56291737 x}{256}-\frac {22370117}{512 \left (-1+2 x \right )}+\frac {39220335 \ln \left (-1+2 x \right )}{256}\) | \(52\) |
norman | \(\frac {-\frac {39330927}{128} x +\frac {38529135}{128} x^{2}+\frac {11583525}{64} x^{3}+\frac {8633765}{64} x^{4}+\frac {2878371}{32} x^{5}+\frac {360909}{8} x^{6}+\frac {404325}{28} x^{7}+\frac {30375}{14} x^{8}}{-1+2 x}+\frac {39220335 \ln \left (-1+2 x \right )}{256}\) | \(57\) |
parallelrisch | \(\frac {3888000 x^{8}+25876800 x^{7}+80843616 x^{6}+161188776 x^{5}+241745420 x^{4}+324338700 x^{3}+549084690 \ln \left (x -\frac {1}{2}\right ) x +539407890 x^{2}-274542345 \ln \left (x -\frac {1}{2}\right )-550632978 x}{-1792+3584 x}\) | \(62\) |
meijerg | \(\frac {6264 x}{1-2 x}+\frac {39220335 \ln \left (1-2 x \right )}{256}+\frac {4920 x \left (-6 x +6\right )}{1-2 x}+\frac {23045 x \left (-8 x^{2}-12 x +12\right )}{4 \left (1-2 x \right )}+\frac {11989 x \left (-40 x^{3}-40 x^{2}-60 x +60\right )}{8 \left (1-2 x \right )}+\frac {149637 x \left (-48 x^{4}-40 x^{3}-40 x^{2}-60 x +60\right )}{128 \left (1-2 x \right )}+\frac {70011 x \left (-448 x^{5}-336 x^{4}-280 x^{3}-280 x^{2}-420 x +420\right )}{896 \left (1-2 x \right )}+\frac {10395 x \left (-1280 x^{6}-896 x^{5}-672 x^{4}-560 x^{3}-560 x^{2}-840 x +840\right )}{1024 \left (1-2 x \right )}+\frac {675 x \left (-5760 x^{7}-3840 x^{6}-2688 x^{5}-2016 x^{4}-1680 x^{3}-1680 x^{2}-2520 x +2520\right )}{1792 \left (1-2 x \right )}\) | \(230\) |
Input:
int((2+3*x)^5*(3+5*x)^3/(1-2*x)^2,x,method=_RETURNVERBOSE)
Output:
30375/28*x^7+15525/2*x^6+423009/16*x^5+3724389/64*x^4+6179077/64*x^3+88813 01/64*x^2+56291737/256*x-22370117/1024/(x-1/2)+39220335/256*ln(-1+2*x)
Time = 0.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.90 \[ \int \frac {(2+3 x)^5 (3+5 x)^3}{(1-2 x)^2} \, dx=\frac {7776000 \, x^{8} + 51753600 \, x^{7} + 161687232 \, x^{6} + 322377552 \, x^{5} + 483490840 \, x^{4} + 648677400 \, x^{3} + 1078815780 \, x^{2} + 549084690 \, {\left (2 \, x - 1\right )} \log \left (2 \, x - 1\right ) - 788084318 \, x - 156590819}{3584 \, {\left (2 \, x - 1\right )}} \] Input:
integrate((2+3*x)^5*(3+5*x)^3/(1-2*x)^2,x, algorithm="fricas")
Output:
1/3584*(7776000*x^8 + 51753600*x^7 + 161687232*x^6 + 322377552*x^5 + 48349 0840*x^4 + 648677400*x^3 + 1078815780*x^2 + 549084690*(2*x - 1)*log(2*x - 1) - 788084318*x - 156590819)/(2*x - 1)
Time = 0.07 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.88 \[ \int \frac {(2+3 x)^5 (3+5 x)^3}{(1-2 x)^2} \, dx=\frac {30375 x^{7}}{28} + \frac {15525 x^{6}}{2} + \frac {423009 x^{5}}{16} + \frac {3724389 x^{4}}{64} + \frac {6179077 x^{3}}{64} + \frac {8881301 x^{2}}{64} + \frac {56291737 x}{256} + \frac {39220335 \log {\left (2 x - 1 \right )}}{256} - \frac {22370117}{1024 x - 512} \] Input:
integrate((2+3*x)**5*(3+5*x)**3/(1-2*x)**2,x)
Output:
30375*x**7/28 + 15525*x**6/2 + 423009*x**5/16 + 3724389*x**4/64 + 6179077* x**3/64 + 8881301*x**2/64 + 56291737*x/256 + 39220335*log(2*x - 1)/256 - 2 2370117/(1024*x - 512)
Time = 0.03 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.74 \[ \int \frac {(2+3 x)^5 (3+5 x)^3}{(1-2 x)^2} \, dx=\frac {30375}{28} \, x^{7} + \frac {15525}{2} \, x^{6} + \frac {423009}{16} \, x^{5} + \frac {3724389}{64} \, x^{4} + \frac {6179077}{64} \, x^{3} + \frac {8881301}{64} \, x^{2} + \frac {56291737}{256} \, x - \frac {22370117}{512 \, {\left (2 \, x - 1\right )}} + \frac {39220335}{256} \, \log \left (2 \, x - 1\right ) \] Input:
integrate((2+3*x)^5*(3+5*x)^3/(1-2*x)^2,x, algorithm="maxima")
Output:
30375/28*x^7 + 15525/2*x^6 + 423009/16*x^5 + 3724389/64*x^4 + 6179077/64*x ^3 + 8881301/64*x^2 + 56291737/256*x - 22370117/512/(2*x - 1) + 39220335/2 56*log(2*x - 1)
Time = 0.12 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.35 \[ \int \frac {(2+3 x)^5 (3+5 x)^3}{(1-2 x)^2} \, dx=\frac {1}{7168} \, {\left (2 \, x - 1\right )}^{7} {\left (\frac {1294650}{2 \, x - 1} + \frac {12414276}{{\left (2 \, x - 1\right )}^{2}} + \frac {70848603}{{\left (2 \, x - 1\right )}^{3}} + \frac {269525480}{{\left (2 \, x - 1\right )}^{4}} + \frac {738160010}{{\left (2 \, x - 1\right )}^{5}} + \frac {1684493580}{{\left (2 \, x - 1\right )}^{6}} + 60750\right )} - \frac {22370117}{512 \, {\left (2 \, x - 1\right )}} - \frac {39220335}{256} \, \log \left (\frac {{\left | 2 \, x - 1 \right |}}{2 \, {\left (2 \, x - 1\right )}^{2}}\right ) \] Input:
integrate((2+3*x)^5*(3+5*x)^3/(1-2*x)^2,x, algorithm="giac")
Output:
1/7168*(2*x - 1)^7*(1294650/(2*x - 1) + 12414276/(2*x - 1)^2 + 70848603/(2 *x - 1)^3 + 269525480/(2*x - 1)^4 + 738160010/(2*x - 1)^5 + 1684493580/(2* x - 1)^6 + 60750) - 22370117/512/(2*x - 1) - 39220335/256*log(1/2*abs(2*x - 1)/(2*x - 1)^2)
Time = 0.03 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.71 \[ \int \frac {(2+3 x)^5 (3+5 x)^3}{(1-2 x)^2} \, dx=\frac {56291737\,x}{256}+\frac {39220335\,\ln \left (x-\frac {1}{2}\right )}{256}-\frac {22370117}{1024\,\left (x-\frac {1}{2}\right )}+\frac {8881301\,x^2}{64}+\frac {6179077\,x^3}{64}+\frac {3724389\,x^4}{64}+\frac {423009\,x^5}{16}+\frac {15525\,x^6}{2}+\frac {30375\,x^7}{28} \] Input:
int(((3*x + 2)^5*(5*x + 3)^3)/(2*x - 1)^2,x)
Output:
(56291737*x)/256 + (39220335*log(x - 1/2))/256 - 22370117/(1024*(x - 1/2)) + (8881301*x^2)/64 + (6179077*x^3)/64 + (3724389*x^4)/64 + (423009*x^5)/1 6 + (15525*x^6)/2 + (30375*x^7)/28
Time = 0.16 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.93 \[ \int \frac {(2+3 x)^5 (3+5 x)^3}{(1-2 x)^2} \, dx=\frac {549084690 \,\mathrm {log}\left (2 x -1\right ) x -274542345 \,\mathrm {log}\left (2 x -1\right )+3888000 x^{8}+25876800 x^{7}+80843616 x^{6}+161188776 x^{5}+241745420 x^{4}+324338700 x^{3}+539407890 x^{2}-550632978 x}{3584 x -1792} \] Input:
int((2+3*x)^5*(3+5*x)^3/(1-2*x)^2,x)
Output:
(549084690*log(2*x - 1)*x - 274542345*log(2*x - 1) + 3888000*x**8 + 258768 00*x**7 + 80843616*x**6 + 161188776*x**5 + 241745420*x**4 + 324338700*x**3 + 539407890*x**2 - 550632978*x)/(1792*(2*x - 1))