3.1630 \(\int \frac{(2+3 x)^8 (3+5 x)}{(1-2 x)^3} \, dx\)

Optimal. Leaf size=80 \[ -\frac{32805 x^7}{56}-\frac{162567 x^6}{32}-\frac{213597 x^5}{10}-\frac{7568235 x^4}{128}-\frac{16042509 x^3}{128}-\frac{118841283 x^2}{512}-\frac{120864213 x}{256}-\frac{246239357}{1024 (1-2 x)}+\frac{63412811}{2048 (1-2 x)^2}-\frac{106237047}{256} \log (1-2 x) \]

[Out]

63412811/(2048*(1 - 2*x)^2) - 246239357/(1024*(1 - 2*x)) - (120864213*x)/256 - (118841283*x^2)/512 - (16042509
*x^3)/128 - (7568235*x^4)/128 - (213597*x^5)/10 - (162567*x^6)/32 - (32805*x^7)/56 - (106237047*Log[1 - 2*x])/
256

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Rubi [A]  time = 0.0452482, antiderivative size = 80, 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} \[ -\frac{32805 x^7}{56}-\frac{162567 x^6}{32}-\frac{213597 x^5}{10}-\frac{7568235 x^4}{128}-\frac{16042509 x^3}{128}-\frac{118841283 x^2}{512}-\frac{120864213 x}{256}-\frac{246239357}{1024 (1-2 x)}+\frac{63412811}{2048 (1-2 x)^2}-\frac{106237047}{256} \log (1-2 x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

63412811/(2048*(1 - 2*x)^2) - 246239357/(1024*(1 - 2*x)) - (120864213*x)/256 - (118841283*x^2)/512 - (16042509
*x^3)/128 - (7568235*x^4)/128 - (213597*x^5)/10 - (162567*x^6)/32 - (32805*x^7)/56 - (106237047*Log[1 - 2*x])/
256

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)^3} \, dx &=\int \left (-\frac{120864213}{256}-\frac{118841283 x}{256}-\frac{48127527 x^2}{128}-\frac{7568235 x^3}{32}-\frac{213597 x^4}{2}-\frac{487701 x^5}{16}-\frac{32805 x^6}{8}-\frac{63412811}{512 (-1+2 x)^3}-\frac{246239357}{512 (-1+2 x)^2}-\frac{106237047}{128 (-1+2 x)}\right ) \, dx\\ &=\frac{63412811}{2048 (1-2 x)^2}-\frac{246239357}{1024 (1-2 x)}-\frac{120864213 x}{256}-\frac{118841283 x^2}{512}-\frac{16042509 x^3}{128}-\frac{7568235 x^4}{128}-\frac{213597 x^5}{10}-\frac{162567 x^6}{32}-\frac{32805 x^7}{56}-\frac{106237047}{256} \log (1-2 x)\\ \end{align*}

Mathematica [A]  time = 0.0198618, size = 71, normalized size = 0.89 \[ -\frac{83980800 x^9+644319360 x^8+2354821632 x^7+5596371648 x^6+10256718528 x^5+17427054960 x^4+38900302560 x^3-104409393876 x^2+44728559236 x+14873186580 (1-2 x)^2 \log (1-2 x)-3752427799}{35840 (1-2 x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-3752427799 + 44728559236*x - 104409393876*x^2 + 38900302560*x^3 + 17427054960*x^4 + 10256718528*x^5 + 55963
71648*x^6 + 2354821632*x^7 + 644319360*x^8 + 83980800*x^9 + 14873186580*(1 - 2*x)^2*Log[1 - 2*x])/(35840*(1 -
2*x)^2)

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Maple [A]  time = 0.006, size = 61, normalized size = 0.8 \begin{align*} -{\frac{32805\,{x}^{7}}{56}}-{\frac{162567\,{x}^{6}}{32}}-{\frac{213597\,{x}^{5}}{10}}-{\frac{7568235\,{x}^{4}}{128}}-{\frac{16042509\,{x}^{3}}{128}}-{\frac{118841283\,{x}^{2}}{512}}-{\frac{120864213\,x}{256}}-{\frac{106237047\,\ln \left ( 2\,x-1 \right ) }{256}}+{\frac{63412811}{2048\, \left ( 2\,x-1 \right ) ^{2}}}+{\frac{246239357}{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)^3,x)

[Out]

-32805/56*x^7-162567/32*x^6-213597/10*x^5-7568235/128*x^4-16042509/128*x^3-118841283/512*x^2-120864213/256*x-1
06237047/256*ln(2*x-1)[phone]/2048/(2*x-1)^2+246239357/1024/(2*x-1)

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Maxima [A]  time = 2.35116, size = 82, normalized size = 1.02 \begin{align*} -\frac{32805}{56} \, x^{7} - \frac{162567}{32} \, x^{6} - \frac{213597}{10} \, x^{5} - \frac{7568235}{128} \, x^{4} - \frac{16042509}{128} \, x^{3} - \frac{118841283}{512} \, x^{2} - \frac{120864213}{256} \, x + \frac{823543 \,{\left (1196 \, x - 521\right )}}{2048 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac{106237047}{256} \, \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)^3,x, algorithm="maxima")

[Out]

-32805/56*x^7 - 162567/32*x^6 - 213597/10*x^5 - 7568235/128*x^4 - 16042509/128*x^3 - 118841283/512*x^2 - 12086
4213/256*x + 823543/2048*(1196*x - 521)/(4*x^2 - 4*x + 1) - 106237047/256*log(2*x - 1)

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Fricas [A]  time = 1.2726, size = 327, normalized size = 4.09 \begin{align*} -\frac{167961600 \, x^{9} + 1288638720 \, x^{8} + 4709643264 \, x^{7} + 11192743296 \, x^{6} + 20513437056 \, x^{5} + 34854109920 \, x^{4} + 77800605120 \, x^{3} - 118730138940 \, x^{2} + 29746373160 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (2 \, x - 1\right ) - 631530340 \, x + 15017306605}{71680 \,{\left (4 \, x^{2} - 4 \, 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)^3,x, algorithm="fricas")

[Out]

-1/71680*(167961600*x^9 + 1288638720*x^8 + 4709643264*x^7 + 11192743296*x^6 + 20513437056*x^5 + 34854109920*x^
4 + 77800605120*x^3 - 118730138940*x^2 + 29746373160*(4*x^2 - 4*x + 1)*log(2*x - 1) - 631530340*x + 1501730660
5)/(4*x^2 - 4*x + 1)

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Sympy [A]  time = 0.132354, size = 70, normalized size = 0.88 \begin{align*} - \frac{32805 x^{7}}{56} - \frac{162567 x^{6}}{32} - \frac{213597 x^{5}}{10} - \frac{7568235 x^{4}}{128} - \frac{16042509 x^{3}}{128} - \frac{118841283 x^{2}}{512} - \frac{120864213 x}{256} + \frac{984957428 x - 429065903}{8192 x^{2} - 8192 x + 2048} - \frac{106237047 \log{\left (2 x - 1 \right )}}{256} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-32805*x**7/56 - 162567*x**6/32 - 213597*x**5/10 - 7568235*x**4/128 - 16042509*x**3/128 - 118841283*x**2/512 -
120864213*x/256 + (984957428*x - 429065903)/(8192*x**2 - 8192*x + 2048) - 106237047*log(2*x - 1)/256

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Giac [A]  time = 2.50541, size = 77, normalized size = 0.96 \begin{align*} -\frac{32805}{56} \, x^{7} - \frac{162567}{32} \, x^{6} - \frac{213597}{10} \, x^{5} - \frac{7568235}{128} \, x^{4} - \frac{16042509}{128} \, x^{3} - \frac{118841283}{512} \, x^{2} - \frac{120864213}{256} \, x + \frac{823543 \,{\left (1196 \, x - 521\right )}}{2048 \,{\left (2 \, x - 1\right )}^{2}} - \frac{106237047}{256} \, \log \left ({\left | 2 \, x - 1 \right |}\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)^3,x, algorithm="giac")

[Out]

-32805/56*x^7 - 162567/32*x^6 - 213597/10*x^5 - 7568235/128*x^4 - 16042509/128*x^3 - 118841283/512*x^2 - 12086
4213/256*x + 823543/2048*(1196*x - 521)/(2*x - 1)^2 - 106237047/256*log(abs(2*x - 1))