3.1588 \(\int \frac{(2+3 x)^8}{(1-2 x)^2 (3+5 x)} \, dx\)

Optimal. Leaf size=72 \[ \frac{2187 x^6}{40}+\frac{94041 x^5}{250}+\frac{9899091 x^4}{8000}+\frac{26773659 x^3}{10000}+\frac{1839811401 x^2}{400000}+\frac{2041906293 x}{250000}+\frac{5764801}{2816 (1-2 x)}+\frac{188591347 \log (1-2 x)}{30976}+\frac{\log (5 x+3)}{9453125} \]

[Out]

5764801/(2816*(1 - 2*x)) + (2041906293*x)/250000 + (1839811401*x^2)/400000 + (26773659*x^3)/10000 + (9899091*x
^4)/8000 + (94041*x^5)/250 + (2187*x^6)/40 + (188591347*Log[1 - 2*x])/30976 + Log[3 + 5*x]/9453125

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Rubi [A]  time = 0.0378205, antiderivative size = 72, 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} \[ \frac{2187 x^6}{40}+\frac{94041 x^5}{250}+\frac{9899091 x^4}{8000}+\frac{26773659 x^3}{10000}+\frac{1839811401 x^2}{400000}+\frac{2041906293 x}{250000}+\frac{5764801}{2816 (1-2 x)}+\frac{188591347 \log (1-2 x)}{30976}+\frac{\log (5 x+3)}{9453125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

5764801/(2816*(1 - 2*x)) + (2041906293*x)/250000 + (1839811401*x^2)/400000 + (26773659*x^3)/10000 + (9899091*x
^4)/8000 + (94041*x^5)/250 + (2187*x^6)/40 + (188591347*Log[1 - 2*x])/30976 + Log[3 + 5*x]/9453125

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}{(1-2 x)^2 (3+5 x)} \, dx &=\int \left (\frac{2041906293}{250000}+\frac{1839811401 x}{200000}+\frac{80320977 x^2}{10000}+\frac{9899091 x^3}{2000}+\frac{94041 x^4}{50}+\frac{6561 x^5}{20}+\frac{5764801}{1408 (-1+2 x)^2}+\frac{188591347}{15488 (-1+2 x)}+\frac{1}{1890625 (3+5 x)}\right ) \, dx\\ &=\frac{5764801}{2816 (1-2 x)}+\frac{2041906293 x}{250000}+\frac{1839811401 x^2}{400000}+\frac{26773659 x^3}{10000}+\frac{9899091 x^4}{8000}+\frac{94041 x^5}{250}+\frac{2187 x^6}{40}+\frac{188591347 \log (1-2 x)}{30976}+\frac{\log (3+5 x)}{9453125}\\ \end{align*}

Mathematica [A]  time = 0.0525396, size = 66, normalized size = 0.92 \[ \frac{109350000 x^6+752328000 x^5+2474772750 x^4+5354731800 x^3+9199057005 x^2+16335250344 x+\frac{90075015625}{22-44 x}[phone]}{2000000}+\frac{188591347 \log (3-6 x)}{30976}+\frac{\log (-3 (5 x+3))}{9453125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(7988912316 + 90075015625/(22 - 44*x) + 16335250344*x + 9199057005*x^2 + 5354731800*x^3 + 2474772750*x^4 + 752
328000*x^5 + 109350000*x^6)/2000000 + (188591347*Log[3 - 6*x])/30976 + Log[-3*(3 + 5*x)]/9453125

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Maple [A]  time = 0.007, size = 55, normalized size = 0.8 \begin{align*}{\frac{2187\,{x}^{6}}{40}}+{\frac{94041\,{x}^{5}}{250}}+{\frac{9899091\,{x}^{4}}{8000}}+{\frac{26773659\,{x}^{3}}{10000}}+{\frac{1839811401\,{x}^{2}}{400000}}+{\frac{2041906293\,x}{250000}}-{\frac{5764801}{5632\,x-2816}}+{\frac{188591347\,\ln \left ( 2\,x-1 \right ) }{30976}}+{\frac{\ln \left ( 3+5\,x \right ) }{9453125}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2187/40*x^6+94041/250*x^5+9899091/8000*x^4+26773659/10000*x^3+1839811401/400000*x^2+2041906293/250000*x-576480
1/2816/(2*x-1)[phone]/30976*ln(2*x-1)+1/9453125*ln(3+5*x)

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Maxima [A]  time = 2.40309, size = 73, normalized size = 1.01 \begin{align*} \frac{2187}{40} \, x^{6} + \frac{94041}{250} \, x^{5} + \frac{9899091}{8000} \, x^{4} + \frac{26773659}{10000} \, x^{3} + \frac{1839811401}{400000} \, x^{2} + \frac{2041906293}{250000} \, x - \frac{5764801}{2816 \,{\left (2 \, x - 1\right )}} + \frac{1}{9453125} \, \log \left (5 \, x + 3\right ) + \frac{188591347}{30976} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2187/40*x^6 + 94041/250*x^5 + 9899091/8000*x^4 + 26773659/10000*x^3 + 1839811401/400000*x^2 + 2041906293/25000
0*x - 5764801/2816/(2*x - 1) + 1/9453125*log(5*x + 3) + 188591347/30976*log(2*x - 1)

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Fricas [A]  time = 1.27579, size = 336, normalized size = 4.67 \begin{align*} \frac{264627000000 \, x^{7} + 1688320260000 \, x^{6} + 5078633175000 \, x^{5} + 9963975928500 \, x^{4} + 15782492474100 \, x^{3} + 28400446856430 \, x^{2} + 256 \,{\left (2 \, x - 1\right )} \log \left (5 \, x + 3\right ) + 14733698984375 \,{\left (2 \, x - 1\right )} \log \left (2 \, x - 1\right ) - 19765652916240 \, x - 4954125859375}{2420000000 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2420000000*(264627000000*x^7 + 1688320260000*x^6 + 5078633175000*x^5 + 9963975928500*x^4 + 15782492474100*x^
3 + 28400446856430*x^2 + 256*(2*x - 1)*log(5*x + 3) + 14733698984375*(2*x - 1)*log(2*x - 1) - 19765652916240*x
- 4954125859375)/(2*x - 1)

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Sympy [A]  time = 0.141709, size = 63, normalized size = 0.88 \begin{align*} \frac{2187 x^{6}}{40} + \frac{94041 x^{5}}{250} + \frac{9899091 x^{4}}{8000} + \frac{26773659 x^{3}}{10000} + \frac{1839811401 x^{2}}{400000} + \frac{2041906293 x}{250000} + \frac{188591347 \log{\left (x - \frac{1}{2} \right )}}{30976} + \frac{\log{\left (x + \frac{3}{5} \right )}}{9453125} - \frac{5764801}{5632 x - 2816} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2187*x**6/40 + 94041*x**5/250 + 9899091*x**4/8000 + 26773659*x**3/10000 + 1839811401*x**2/400000 + 2041906293*
x/250000 + 188591347*log(x - 1/2)/30976 + log(x + 3/5)/9453125 - 5764801/(5632*x - 2816)

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Giac [A]  time = 3.08027, size = 134, normalized size = 1.86 \begin{align*} \frac{27}{16000000} \,{\left (2 \, x - 1\right )}^{6}{\left (\frac{10003500}{2 \, x - 1} + \frac{88252875}{{\left (2 \, x - 1\right )}^{2}} + \frac{461424900}{{\left (2 \, x - 1\right )}^{3}} + \frac{1628610330}{{\left (2 \, x - 1\right )}^{4}} + \frac{4599014548}{{\left (2 \, x - 1\right )}^{5}} + 506250\right )} - \frac{5764801}{2816 \,{\left (2 \, x - 1\right )}} - \frac{121766107311}{20000000} \, \log \left (\frac{{\left | 2 \, x - 1 \right |}}{2 \,{\left (2 \, x - 1\right )}^{2}}\right ) + \frac{1}{9453125} \, \log \left ({\left | -\frac{11}{2 \, x - 1} - 5 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

27/16000000*(2*x - 1)^6*(10003500/(2*x - 1) + 88252875/(2*x - 1)^2 + 461424900/(2*x - 1)^3 + 1628610330/(2*x -
1)^4 + 4599014548/(2*x - 1)^5 + 506250) - 5764801/2816/(2*x - 1) - 121766107311/20000000*log(1/2*abs(2*x - 1)
/(2*x - 1)^2) + 1/9453125*log(abs(-11/(2*x - 1) - 5))