3.1603 \(\int \frac {(2+3 x)^8}{(1-2 x)^2 (3+5 x)^2} \, dx\)

Optimal. Leaf size=76 \[ \frac {6561 x^5}{500}+\frac {168399 x^4}{2000}+\frac {2626101 x^3}{10000}+\frac {14171517 x^2}{25000}+\frac {231915717 x}{200000}+\frac {5764801}{15488 (1-2 x)}-\frac {1}{9453125 (5 x+3)}+\frac {79883671 \log (1-2 x)}{85184}+\frac {268 \log (5 x+3)}{103984375} \]

[Out]

5764801/15488/(1-2*x)[phone]/200000*x+14171517/25000*x^2+2626101/10000*x^3+168399/2000*x^4+6561/500*x^5-1/9
453125/(3+5*x)[phone]/85184*ln(1-2*x)+268/103984375*ln(3+5*x)

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Rubi [A]  time = 0.04, antiderivative size = 76, 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} \[ \frac {6561 x^5}{500}+\frac {168399 x^4}{2000}+\frac {2626101 x^3}{10000}+\frac {14171517 x^2}{25000}+\frac {231915717 x}{200000}+\frac {5764801}{15488 (1-2 x)}-\frac {1}{9453125 (5 x+3)}+\frac {79883671 \log (1-2 x)}{85184}+\frac {268 \log (5 x+3)}{103984375} \]

Antiderivative was successfully verified.

[In]

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

[Out]

5764801/(15488*(1 - 2*x)) + (231915717*x)/200000 + (14171517*x^2)/25000 + (2626101*x^3)/10000 + (168399*x^4)/2
000 + (6561*x^5)/500 - 1/(9453125*(3 + 5*x)) + (79883671*Log[1 - 2*x])/85184 + (268*Log[3 + 5*x])/103984375

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)^2} \, dx &=\int \left (\frac {231915717}{200000}+\frac {14171517 x}{12500}+\frac {7878303 x^2}{10000}+\frac {168399 x^3}{500}+\frac {6561 x^4}{100}+\frac {5764801}{7744 (-1+2 x)^2}+\frac {79883671}{42592 (-1+2 x)}+\frac {1}{1890625 (3+5 x)^2}+\frac {268}{20796875 (3+5 x)}\right ) \, dx\\ &=\frac {5764801}{15488 (1-2 x)}+\frac {231915717 x}{200000}+\frac {14171517 x^2}{25000}+\frac {2626101 x^3}{10000}+\frac {168399 x^4}{2000}+\frac {6561 x^5}{500}-\frac {1}{9453125 (3+5 x)}+\frac {79883671 \log (1-2 x)}{85184}+\frac {268 \log (3+5 x)}{103984375}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 95, normalized size = 1.25 \[ -\frac {2251875390881 x+1351125234247}{1210000000 \left (10 x^2+x-3\right )}+\frac {27}{500} (3 x+2)^5+\frac {999 (3 x+2)^4}{2000}+\frac {35703 (3 x+2)^3}{10000}+\frac {78921 (3 x+2)^2}{3125}+\frac {44471943 (3 x+2)}{200000}+\frac {79883671 \log (3-6 x)}{85184}+\frac {268 \log (-3 (5 x+3))}{103984375} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(44471943*(2 + 3*x))/200000 + (78921*(2 + 3*x)^2)/3125 + (35703*(2 + 3*x)^3)/10000 + (999*(2 + 3*x)^4)/2000 +
(27*(2 + 3*x)^5)/500 - (1351125234247 + 2251875390881*x)/(1210000000*(-3 + x + 10*x^2)) + (79883671*Log[3 - 6*
x])/85184 + (268*Log[-3*(3 + 5*x)])/103984375

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fricas [A]  time = 0.83, size = 79, normalized size = 1.04 \[ \frac {1746538200000 \, x^{7} + 11381607270000 \, x^{6} + 35550138195000 \, x^{5} + 75582410904000 \, x^{4} + 151398804021300 \, x^{3} - 7200755986050 \, x^{2} + 34304 \, {\left (10 \, x^{2} + x - 3\right )} \log \left (5 \, x + 3\right ) + 12481823593750 \, {\left (10 \, x^{2} + x - 3\right )} \log \left (2 \, x - 1\right ) - 71072602198741 \, x - 14862377576717}{13310000000 \, {\left (10 \, x^{2} + x - 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13310000000*(1746538200000*x^7 + 11381607270000*x^6 + 35550138195000*x^5 + 75582410904000*x^4 + 151398804021
300*x^3 - 7200755986050*x^2 + 34304*(10*x^2 + x - 3)*log(5*x + 3) + 12481823593750*(10*x^2 + x - 3)*log(2*x -
1) - 71072602198741*x - 14862377576717)/(10*x^2 + x - 3)

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giac [A]  time = 1.16, size = 112, normalized size = 1.47 \[ -\frac {{\left (5 \, x + 3\right )}^{5} {\left (\frac {1618458732}{5 \, x + 3} + \frac {15560361630}{{\left (5 \, x + 3\right )}^{2}} + \frac {171888467850}{{\left (5 \, x + 3\right )}^{3}} + \frac {2836763461125}{{\left (5 \, x + 3\right )}^{4}} - \frac {31204564033975}{{\left (5 \, x + 3\right )}^{5}} + 139723056\right )}}{16637500000 \, {\left (\frac {11}{5 \, x + 3} - 2\right )}} - \frac {1}{9453125 \, {\left (5 \, x + 3\right )}} - \frac {4688889417}{5000000} \, \log \left (\frac {{\left | 5 \, x + 3 \right |}}{5 \, {\left (5 \, x + 3\right )}^{2}}\right ) + \frac {79883671}{85184} \, \log \left ({\left | -\frac {11}{5 \, x + 3} + 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16637500000*(5*x + 3)^5*(1618458732/(5*x + 3) + 15560361630/(5*x + 3)^2 + 171888467850/(5*x + 3)^3 + 283676
3461125/(5*x + 3)^4 - 31204564033975/(5*x + 3)^5 + 139723056)/(11/(5*x + 3) - 2) - 1/9453125/(5*x + 3) - 46888
89417/5000000*log(1/5*abs(5*x + 3)/(5*x + 3)^2) + 79883671/85184*log(abs(-11/(5*x + 3) + 2))

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maple [A]  time = 0.01, size = 59, normalized size = 0.78 \[ \frac {6561 x^{5}}{500}+\frac {168399 x^{4}}{2000}+\frac {2626101 x^{3}}{10000}+\frac {14171517 x^{2}}{25000}+\frac {231915717 x}{200000}+\frac {79883671 \ln \left (2 x -1\right )}{85184}+\frac {268 \ln \left (5 x +3\right )}{103984375}-\frac {1}{9453125 \left (5 x +3\right )}-\frac {5764801}{15488 \left (2 x -1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6561/500*x^5+168399/2000*x^4+2626101/10000*x^3+14171517/25000*x^2+231915717/200000*x-1/9453125/(5*x+3)+268/103
984375*ln(5*x+3)-5764801/15488/(2*x-1)[phone]/85184*ln(2*x-1)

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maxima [A]  time = 0.55, size = 57, normalized size = 0.75 \[ \frac {6561}{500} \, x^{5} + \frac {168399}{2000} \, x^{4} + \frac {2626101}{10000} \, x^{3} + \frac {14171517}{25000} \, x^{2} + \frac {231915717}{200000} \, x - \frac {2251875390881 \, x + 1351125234247}{1210000000 \, {\left (10 \, x^{2} + x - 3\right )}} + \frac {268}{103984375} \, \log \left (5 \, x + 3\right ) + \frac {79883671}{85184} \, \log \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6561/500*x^5 + 168399/2000*x^4 + 2626101/10000*x^3 + 14171517/25000*x^2 + 231915717/200000*x - 1/1210000000*(2
251875390881*x + 1351125234247)/(10*x^2 + x - 3) + 268/103984375*log(5*x + 3) + 79883671/85184*log(2*x - 1)

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mupad [B]  time = 0.05, size = 53, normalized size = 0.70 \[ \frac {231915717\,x}{200000}+\frac {79883671\,\ln \left (x-\frac {1}{2}\right )}{85184}+\frac {268\,\ln \left (x+\frac {3}{5}\right )}{103984375}-\frac {\frac {2251875390881\,x}{12100000000}+\frac {1351125234247}{12100000000}}{x^2+\frac {x}{10}-\frac {3}{10}}+\frac {14171517\,x^2}{25000}+\frac {2626101\,x^3}{10000}+\frac {168399\,x^4}{2000}+\frac {6561\,x^5}{500} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(231915717*x)/200000 + (79883671*log(x - 1/2))/85184 + (268*log(x + 3/5))/103984375 - ((2251875390881*x)/12100
000000 + 1351125234247/12100000000)/(x/10 + x^2 - 3/10) + (14171517*x^2)/25000 + (2626101*x^3)/10000 + (168399
*x^4)/2000 + (6561*x^5)/500

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sympy [A]  time = 0.18, size = 68, normalized size = 0.89 \[ \frac {6561 x^{5}}{500} + \frac {168399 x^{4}}{2000} + \frac {2626101 x^{3}}{10000} + \frac {14171517 x^{2}}{25000} + \frac {231915717 x}{200000} + \frac {- 2251875390881 x - 1351125234247}{12100000000 x^{2} + 1210000000 x - 3630000000} + \frac {79883671 \log {\left (x - \frac {1}{2} \right )}}{85184} + \frac {268 \log {\left (x + \frac {3}{5} \right )}}{103984375} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6561*x**5/500 + 168399*x**4/2000 + 2626101*x**3/10000 + 14171517*x**2/25000 + 231915717*x/200000 + (-225187539
0881*x - 1351125234247)/(12100000000*x**2 + 1210000000*x - 3630000000) + 79883671*log(x - 1/2)/85184 + 268*log
(x + 3/5)/103984375

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