3.1538 \(\int \frac{(2+3 x)^8 (3+5 x)}{(1-2 x)^2} \, dx\)

Optimal. Leaf size=76 \[ \frac{32805 x^8}{32}+\frac{56862 x^7}{7}+\frac{976617 x^6}{32}+\frac{5859459 x^5}{80}+\frac{32991057 x^4}{256}+\frac{5892813 x^3}{32}+\frac{122887143 x^2}{512}+\frac{91609881 x}{256}+\frac{63412811}{1024 (1-2 x)}+\frac{246239357 \log (1-2 x)}{1024} \]

[Out]

63412811/(1024*(1 - 2*x)) + (91609881*x)/256 + (122887143*x^2)/512 + (5892813*x^3)/32 + (32991057*x^4)/256 + (
5859459*x^5)/80 + (976617*x^6)/32 + (56862*x^7)/7 + (32805*x^8)/32 + (246239357*Log[1 - 2*x])/1024

________________________________________________________________________________________

Rubi [A]  time = 0.043256, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {77} \[ \frac{32805 x^8}{32}+\frac{56862 x^7}{7}+\frac{976617 x^6}{32}+\frac{5859459 x^5}{80}+\frac{32991057 x^4}{256}+\frac{5892813 x^3}{32}+\frac{122887143 x^2}{512}+\frac{91609881 x}{256}+\frac{63412811}{1024 (1-2 x)}+\frac{246239357 \log (1-2 x)}{1024} \]

Antiderivative was successfully verified.

[In]

Int[((2 + 3*x)^8*(3 + 5*x))/(1 - 2*x)^2,x]

[Out]

63412811/(1024*(1 - 2*x)) + (91609881*x)/256 + (122887143*x^2)/512 + (5892813*x^3)/32 + (32991057*x^4)/256 + (
5859459*x^5)/80 + (976617*x^6)/32 + (56862*x^7)/7 + (32805*x^8)/32 + (246239357*Log[1 - 2*x])/1024

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{(2+3 x)^8 (3+5 x)}{(1-2 x)^2} \, dx &=\int \left (\frac{91609881}{256}+\frac{122887143 x}{256}+\frac{17678439 x^2}{32}+\frac{32991057 x^3}{64}+\frac{5859459 x^4}{16}+\frac{2929851 x^5}{16}+56862 x^6+\frac{32805 x^7}{4}+\frac{63412811}{512 (-1+2 x)^2}+\frac{246239357}{512 (-1+2 x)}\right ) \, dx\\ &=\frac{63412811}{1024 (1-2 x)}+\frac{91609881 x}{256}+\frac{122887143 x^2}{512}+\frac{5892813 x^3}{32}+\frac{32991057 x^4}{256}+\frac{5859459 x^5}{80}+\frac{976617 x^6}{32}+\frac{56862 x^7}{7}+\frac{32805 x^8}{32}+\frac{246239357 \log (1-2 x)}{1024}\\ \end{align*}

Mathematica [A]  time = 0.0157241, size = 69, normalized size = 0.91 \[ \frac{587865600 x^9+4364202240 x^8+15171909120 x^7+33250113792 x^6+52899666624 x^5+68649225120 x^4+84833995680 x^3+136389333360 x^2-259057842186 x+68947019960 (2 x-1) \log (1-2 x)[phone]}{286720 (2 x-1)} \]

Antiderivative was successfully verified.

[In]

Integrate[((2 + 3*x)^8*(3 + 5*x))/(1 - 2*x)^2,x]

[Out]

(60471800653 - 259057842186*x + 136389333360*x^2 + 84833995680*x^3 + 68649225120*x^4 + 52899666624*x^5 + 33250
113792*x^6 + 15171909120*x^7 + 4364202240*x^8 + 587865600*x^9 + 68947019960*(-1 + 2*x)*Log[1 - 2*x])/(286720*(
-1 + 2*x))

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 57, normalized size = 0.8 \begin{align*}{\frac{32805\,{x}^{8}}{32}}+{\frac{56862\,{x}^{7}}{7}}+{\frac{976617\,{x}^{6}}{32}}+{\frac{5859459\,{x}^{5}}{80}}+{\frac{32991057\,{x}^{4}}{256}}+{\frac{5892813\,{x}^{3}}{32}}+{\frac{122887143\,{x}^{2}}{512}}+{\frac{91609881\,x}{256}}+{\frac{246239357\,\ln \left ( 2\,x-1 \right ) }{1024}}-{\frac{63412811}{2048\,x-1024}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^8*(3+5*x)/(1-2*x)^2,x)

[Out]

32805/32*x^8+56862/7*x^7+976617/32*x^6+5859459/80*x^5+32991057/256*x^4+5892813/32*x^3+122887143/512*x^2+916098
81/256*x+246239357/1024*ln(2*x-1)-63412811/1024/(2*x-1)

________________________________________________________________________________________

Maxima [A]  time = 1.11, size = 76, normalized size = 1. \begin{align*} \frac{32805}{32} \, x^{8} + \frac{56862}{7} \, x^{7} + \frac{976617}{32} \, x^{6} + \frac{5859459}{80} \, x^{5} + \frac{32991057}{256} \, x^{4} + \frac{5892813}{32} \, x^{3} + \frac{122887143}{512} \, x^{2} + \frac{91609881}{256} \, x - \frac{63412811}{1024 \,{\left (2 \, x - 1\right )}} + \frac{246239357}{1024} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^8*(3+5*x)/(1-2*x)^2,x, algorithm="maxima")

[Out]

32805/32*x^8 + 56862/7*x^7 + 976617/32*x^6 + 5859459/80*x^5 + 32991057/256*x^4 + 5892813/32*x^3 + 122887143/51
2*x^2 + 91609881/256*x - 63412811/1024/(2*x - 1) + 246239357/1024*log(2*x - 1)

________________________________________________________________________________________

Fricas [A]  time = 1.22503, size = 296, normalized size = 3.89 \begin{align*} \frac{73483200 \, x^{9} + 545525280 \, x^{8} + 1896488640 \, x^{7} + 4156264224 \, x^{6} + 6612458328 \, x^{5} + 8581153140 \, x^{4} + 10604249460 \, x^{3} + 17048666670 \, x^{2} + 8618377495 \,{\left (2 \, x - 1\right )} \log \left (2 \, x - 1\right ) - 12825383340 \, x - 2219448385}{35840 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^8*(3+5*x)/(1-2*x)^2,x, algorithm="fricas")

[Out]

1/35840*(73483200*x^9 + 545525280*x^8 + 1896488640*x^7 + 4156264224*x^6 + 6612458328*x^5 + 8581153140*x^4 + 10
604249460*x^3 + 17048666670*x^2 + 8618377495*(2*x - 1)*log(2*x - 1) - 12825383340*x - 2219448385)/(2*x - 1)

________________________________________________________________________________________

Sympy [A]  time = 0.117976, size = 68, normalized size = 0.89 \begin{align*} \frac{32805 x^{8}}{32} + \frac{56862 x^{7}}{7} + \frac{976617 x^{6}}{32} + \frac{5859459 x^{5}}{80} + \frac{32991057 x^{4}}{256} + \frac{5892813 x^{3}}{32} + \frac{122887143 x^{2}}{512} + \frac{91609881 x}{256} + \frac{246239357 \log{\left (2 x - 1 \right )}}{1024} - \frac{63412811}{2048 x - 1024} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**8*(3+5*x)/(1-2*x)**2,x)

[Out]

32805*x**8/32 + 56862*x**7/7 + 976617*x**6/32 + 5859459*x**5/80 + 32991057*x**4/256 + 5892813*x**3/32 + 122887
143*x**2/512 + 91609881*x/256 + 246239357*log(2*x - 1)/1024 - 63412811/(2048*x - 1024)

________________________________________________________________________________________

Giac [A]  time = 2.7537, size = 138, normalized size = 1.82 \begin{align*} \frac{3}{286720} \,{\left (2 \, x - 1\right )}^{8}{\left (\frac{9127080}{2 \, x - 1} + \frac{98748720}{{\left (2 \, x - 1\right )}^{2}} + \frac{641009376}{{\left (2 \, x - 1\right )}^{3}} + \frac{2786264460}{{\left (2 \, x - 1\right )}^{4}} + \frac{8611906800}{{\left (2 \, x - 1\right )}^{5}} + \frac{19962682320}{{\left (2 \, x - 1\right )}^{6}} + \frac{39661830880}{{\left (2 \, x - 1\right )}^{7}} + 382725\right )} - \frac{63412811}{1024 \,{\left (2 \, x - 1\right )}} - \frac{246239357}{1024} \, \log \left (\frac{{\left | 2 \, x - 1 \right |}}{2 \,{\left (2 \, x - 1\right )}^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^8*(3+5*x)/(1-2*x)^2,x, algorithm="giac")

[Out]

3/286720*(2*x - 1)^8*(9127080/(2*x - 1) + 98748720/(2*x - 1)^2 + 641009376/(2*x - 1)^3 + 2786264460/(2*x - 1)^
4 + 8611906800/(2*x - 1)^5 + 19962682320/(2*x - 1)^6 + 39661830880/(2*x - 1)^7 + 382725) - 63412811/1024/(2*x
- 1) - 246239357/1024*log(1/2*abs(2*x - 1)/(2*x - 1)^2)