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

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Rubi [A]  time = 0.04, antiderivative size = 72, 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} \[ \frac {2187 x^6}{40}+\frac {94041 x^5}{250}+\frac {9899091 x^4}{8000}+\frac {26773659 x^3}{10000}+\frac {1839811401 x^2}{400000}+\frac {2041906293 x}{250000}+\frac {5764801}{2816 (1-2 x)}+\frac {188591347 \log (1-2 x)}{30976}+\frac {\log (5 x+3)}{9453125} \]

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*}

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Mathematica [A]  time = 0.07, 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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fricas [A]  time = 0.63, size = 70, normalized size = 0.97 \[ \frac {264627000000 \, x^{7} + 1688320260000 \, x^{6} + 5078633175000 \, x^{5} + 9963975928500 \, x^{4} + 15782492474100 \, x^{3} + 28400446856430 \, x^{2} + 256 \, {\left (2 \, x - 1\right )} \log \left (5 \, x + 3\right ) + 14733698984375 \, {\left (2 \, x - 1\right )} \log \left (2 \, x - 1\right ) - 19765652916240 \, x - 4954125859375}{2420000000 \, {\left (2 \, x - 1\right )}} \]

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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giac [A]  time = 0.94, size = 99, normalized size = 1.38 \[ \frac {27}{16000000} \, {\left (2 \, x - 1\right )}^{6} {\left (\frac {10003500}{2 \, x - 1} + \frac {88252875}{{\left (2 \, x - 1\right )}^{2}} + \frac {461424900}{{\left (2 \, x - 1\right )}^{3}} + \frac {1628610330}{{\left (2 \, x - 1\right )}^{4}} + \frac {4599014548}{{\left (2 \, x - 1\right )}^{5}} + 506250\right )} - \frac {5764801}{2816 \, {\left (2 \, x - 1\right )}} - \frac {121766107311}{20000000} \, \log \left (\frac {{\left | 2 \, x - 1 \right |}}{2 \, {\left (2 \, x - 1\right )}^{2}}\right ) + \frac {1}{9453125} \, \log \left ({\left | -\frac {11}{2 \, x - 1} - 5 \right |}\right ) \]

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))

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maple [A]  time = 0.01, size = 55, normalized size = 0.76 \[ \frac {2187 x^{6}}{40}+\frac {94041 x^{5}}{250}+\frac {9899091 x^{4}}{8000}+\frac {26773659 x^{3}}{10000}+\frac {1839811401 x^{2}}{400000}+\frac {2041906293 x}{250000}+\frac {188591347 \ln \left (2 x -1\right )}{30976}+\frac {\ln \left (5 x +3\right )}{9453125}-\frac {5764801}{2816 \left (2 x -1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x+2)^8/(1-2*x)^2/(5*x+3),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+1/9453
125*ln(5*x+3)-5764801/2816/(2*x-1)[phone]/30976*ln(2*x-1)

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maxima [A]  time = 0.45, size = 54, normalized size = 0.75 \[ \frac {2187}{40} \, x^{6} + \frac {94041}{250} \, x^{5} + \frac {9899091}{8000} \, x^{4} + \frac {26773659}{10000} \, x^{3} + \frac {1839811401}{400000} \, x^{2} + \frac {2041906293}{250000} \, x - \frac {5764801}{2816 \, {\left (2 \, x - 1\right )}} + \frac {1}{9453125} \, \log \left (5 \, x + 3\right ) + \frac {188591347}{30976} \, \log \left (2 \, x - 1\right ) \]

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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mupad [B]  time = 1.09, size = 50, normalized size = 0.69 \[ \frac {2041906293\,x}{250000}+\frac {188591347\,\ln \left (x-\frac {1}{2}\right )}{30976}+\frac {\ln \left (x+\frac {3}{5}\right )}{9453125}-\frac {5764801}{5632\,\left (x-\frac {1}{2}\right )}+\frac {1839811401\,x^2}{400000}+\frac {26773659\,x^3}{10000}+\frac {9899091\,x^4}{8000}+\frac {94041\,x^5}{250}+\frac {2187\,x^6}{40} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2041906293*x)/250000 + (188591347*log(x - 1/2))/30976 + log(x + 3/5)/9453125 - 5764801/(5632*(x - 1/2)) + (18
39811401*x^2)/400000 + (26773659*x^3)/10000 + (9899091*x^4)/8000 + (94041*x^5)/250 + (2187*x^6)/40

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sympy [A]  time = 0.16, size = 63, normalized size = 0.88 \[ \frac {2187 x^{6}}{40} + \frac {94041 x^{5}}{250} + \frac {9899091 x^{4}}{8000} + \frac {26773659 x^{3}}{10000} + \frac {1839811401 x^{2}}{400000} + \frac {2041906293 x}{250000} + \frac {188591347 \log {\left (x - \frac {1}{2} \right )}}{30976} + \frac {\log {\left (x + \frac {3}{5} \right )}}{9453125} - \frac {5764801}{5632 x - 2816} \]

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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