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

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

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Rubi [A]  time = 0.05, antiderivative size = 81, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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*}

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Mathematica [A]  time = 0.03, size = 74, normalized size = 0.91 \begin {gather*} \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)} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(2+3 x)^8 (3+5 x)^2}{(1-2 x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.00, size = 72, normalized size = 0.89 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.93, size = 111, normalized size = 1.37 \begin {gather*} \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 {gather*}

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)

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maple [A]  time = 0.01, size = 62, normalized size = 0.77 \begin {gather*} \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}{2048 \left (2 x -1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x+2)^8*(5*x+3)^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-697540921/2048/(2*x-1)[phone]/1024*ln(2*x-1)

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maxima [A]  time = 0.60, size = 61, normalized size = 0.75 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.05, size = 59, normalized size = 0.73 \begin {gather*} \frac {2330515357\,x}{1024}+\frac {1512848491\,\ln \left (x-\frac {1}{2}\right )}{1024}-\frac {697540921}{4096\,\left (x-\frac {1}{2}\right )}+\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2330515357*x)/1024 + (1512848491*log(x - 1/2))/1024 - 697540921/(4096*(x - 1/2)) + (413355417*x^2)/256 + (346
239417*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

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sympy [A]  time = 0.13, size = 73, normalized size = 0.90 \begin {gather*} \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 {gather*}

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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