3.1554 \(\int \frac{(2+3 x)^8 (3+5 x)^2}{(1-2 x)^2} \, dx\)

Optimal. Leaf size=81 \[ \frac{18225 x^9}{4}+\frac{1235655 x^8}{32}+\frac{17378631 x^7}{112}+396738 x^6+\frac{235268793 x^5}{320}+\frac{275757561 x^4}{256}+\frac{346239417 x^3}{256}+\frac{413355417 x^2}{256}+\frac{2330515357 x}{1024}+\frac{697540921}{2048 (1-2 x)}+\frac{1512848491 \log (1-2 x)}{1024} \]

[Out]

697540921/(2048*(1 - 2*x)) + (2330515357*x)/1024 + (413355417*x^2)/256 + (346239417*x^3)/256 + (275757561*x^4)
/256 + (235268793*x^5)/320 + 396738*x^6 + (17378631*x^7)/112 + (1235655*x^8)/32 + (18225*x^9)/4 + (1512848491*
Log[1 - 2*x])/1024

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Rubi [A]  time = 0.0457039, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {88} \[ \frac{18225 x^9}{4}+\frac{1235655 x^8}{32}+\frac{17378631 x^7}{112}+396738 x^6+\frac{235268793 x^5}{320}+\frac{275757561 x^4}{256}+\frac{346239417 x^3}{256}+\frac{413355417 x^2}{256}+\frac{2330515357 x}{1024}+\frac{697540921}{2048 (1-2 x)}+\frac{1512848491 \log (1-2 x)}{1024} \]

Antiderivative was successfully verified.

[In]

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

[Out]

697540921/(2048*(1 - 2*x)) + (2330515357*x)/1024 + (413355417*x^2)/256 + (346239417*x^3)/256 + (275757561*x^4)
/256 + (235268793*x^5)/320 + 396738*x^6 + (17378631*x^7)/112 + (1235655*x^8)/32 + (18225*x^9)/4 + (1512848491*
Log[1 - 2*x])/1024

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(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]))

Rubi steps

\begin{align*} \int \frac{(2+3 x)^8 (3+5 x)^2}{(1-2 x)^2} \, dx &=\int \left (\frac{2330515357}{1024}+\frac{413355417 x}{128}+\frac{1038718251 x^2}{256}+\frac{275757561 x^3}{64}+\frac{235268793 x^4}{64}+2380428 x^5+\frac{17378631 x^6}{16}+\frac{1235655 x^7}{4}+\frac{164025 x^8}{4}+\frac{697540921}{1024 (-1+2 x)^2}+\frac{1512848491}{512 (-1+2 x)}\right ) \, dx\\ &=\frac{697540921}{2048 (1-2 x)}+\frac{2330515357 x}{1024}+\frac{413355417 x^2}{256}+\frac{346239417 x^3}{256}+\frac{275757561 x^4}{256}+\frac{235268793 x^5}{320}+396738 x^6+\frac{17378631 x^7}{112}+\frac{1235655 x^8}{32}+\frac{18225 x^9}{4}+\frac{1512848491 \log (1-2 x)}{1024}\\ \end{align*}

Mathematica [A]  time = 0.0221772, size = 74, normalized size = 0.91 \[ \frac{2612736000 x^{10}[phone] x^9+77907121920 x^8+183016143360 x^7+307848957696 x^6+406896098112 x^5+466727825760 x^4+538127987040 x^3+842130532880 x^2-1689637297718 x+423597577480 (2 x-1) \log (1-2 x)[phone]}{286720 (2 x-1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(420890769939 - 1689637297718*x + 842130532880*x^2 + 538127987040*x^3 + 466727825760*x^4 + 406896098112*x^5 +
307848957696*x^6 + 183016143360*x^7 + 77907121920*x^8 + 20836569600*x^9 + 2612736000*x^10 + 423597577480*(-1 +
2*x)*Log[1 - 2*x])/(286720*(-1 + 2*x))

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Maple [A]  time = 0.006, size = 62, normalized size = 0.8 \begin{align*}{\frac{18225\,{x}^{9}}{4}}+{\frac{1235655\,{x}^{8}}{32}}+{\frac{17378631\,{x}^{7}}{112}}+396738\,{x}^{6}+{\frac{235268793\,{x}^{5}}{320}}+{\frac{275757561\,{x}^{4}}{256}}+{\frac{346239417\,{x}^{3}}{256}}+{\frac{413355417\,{x}^{2}}{256}}+{\frac{2330515357\,x}{1024}}+{\frac{1512848491\,\ln \left ( 2\,x-1 \right ) }{1024}}-{\frac{697540921}{4096\,x-2048}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

18225/4*x^9+1235655/32*x^8+17378631/112*x^7+396738*x^6+235268793/320*x^5+275757561/256*x^4+346239417/256*x^3+4
13355417/256*x^2+2330515357/1024*x+1512848491/1024*ln(2*x-1)-697540921/2048/(2*x-1)

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Maxima [A]  time = 1.18505, size = 82, normalized size = 1.01 \begin{align*} \frac{18225}{4} \, x^{9} + \frac{1235655}{32} \, x^{8} + \frac{17378631}{112} \, x^{7} + 396738 \, x^{6} + \frac{235268793}{320} \, x^{5} + \frac{275757561}{256} \, x^{4} + \frac{346239417}{256} \, x^{3} + \frac{413355417}{256} \, x^{2} + \frac{2330515357}{1024} \, x - \frac{697540921}{2048 \,{\left (2 \, x - 1\right )}} + \frac{1512848491}{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)^2/(1-2*x)^2,x, algorithm="maxima")

[Out]

18225/4*x^9 + 1235655/32*x^8 + 17378631/112*x^7 + 396738*x^6 + 235268793/320*x^5 + 275757561/256*x^4 + 3462394
17/256*x^3 + 413355417/256*x^2 + 2330515357/1024*x - 697540921/2048/(2*x - 1) + 1512848491/1024*log(2*x - 1)

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Fricas [A]  time = 1.61687, size = 340, normalized size = 4.2 \begin{align*} \frac{653184000 \, x^{10} + 5209142400 \, x^{9} + 19476780480 \, x^{8} + 45754035840 \, x^{7} + 76962239424 \, x^{6} + 101724024528 \, x^{5} + 116681956440 \, x^{4} + 134531996760 \, x^{3} + 210532633220 \, x^{2} + 105899394370 \,{\left (2 \, x - 1\right )} \log \left (2 \, x - 1\right ) - 163136074990 \, x - 24413932235}{71680 \,{\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)^2/(1-2*x)^2,x, algorithm="fricas")

[Out]

1/71680*(653184000*x^10 + 5209142400*x^9 + 19476780480*x^8 + 45754035840*x^7 + 76962239424*x^6 + 101724024528*
x^5 + 116681956440*x^4 + 134531996760*x^3 + 210532633220*x^2 + 105899394370*(2*x - 1)*log(2*x - 1) - 163136074
990*x - 24413932235)/(2*x - 1)

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Sympy [A]  time = 0.120125, size = 73, normalized size = 0.9 \begin{align*} \frac{18225 x^{9}}{4} + \frac{1235655 x^{8}}{32} + \frac{17378631 x^{7}}{112} + 396738 x^{6} + \frac{235268793 x^{5}}{320} + \frac{275757561 x^{4}}{256} + \frac{346239417 x^{3}}{256} + \frac{413355417 x^{2}}{256} + \frac{2330515357 x}{1024} + \frac{1512848491 \log{\left (2 x - 1 \right )}}{1024} - \frac{697540921}{4096 x - 2048} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

18225*x**9/4 + 1235655*x**8/32 + 17378631*x**7/112 + 396738*x**6 + 235268793*x**5/320 + 275757561*x**4/256 + 3
46239417*x**3/256 + 413355417*x**2/256 + 2330515357*x/1024 + 1512848491*log(2*x - 1)/1024 - 697540921/(4096*x
- 2048)

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Giac [A]  time = 1.73584, size = 150, normalized size = 1.85 \begin{align*} \frac{1}{286720} \,{\left (2 \, x - 1\right )}^{9}{\left (\frac{66211425}{2 \, x - 1} + \frac{785410020}{{\left (2 \, x - 1\right )}^{2}} + \frac{5635662480}{{\left (2 \, x - 1\right )}^{3}} + \frac{27294241464}{{\left (2 \, x - 1\right )}^{4}} + \frac{94415339340}{{\left (2 \, x - 1\right )}^{5}} + \frac{241909873800}{{\left (2 \, x - 1\right )}^{6}} + \frac{478116124080}{{\left (2 \, x - 1\right )}^{7}} + \frac{826787759420}{{\left (2 \, x - 1\right )}^{8}} + 2551500\right )} - \frac{697540921}{2048 \,{\left (2 \, x - 1\right )}} - \frac{1512848491}{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)^2/(1-2*x)^2,x, algorithm="giac")

[Out]

1/286720*(2*x - 1)^9*(66211425/(2*x - 1) + 785410020/(2*x - 1)^2 + 5635662480/(2*x - 1)^3 + 27294241464/(2*x -
1)^4 + 94415339340/(2*x - 1)^5 + 241909873800/(2*x - 1)^6 + 478116124080/(2*x - 1)^7 + 826787759420/(2*x - 1)
^8 + 2551500) - 697540921/2048/(2*x - 1) - 1512848491/1024*log(1/2*abs(2*x - 1)/(2*x - 1)^2)