3.1617 \(\int \frac {(2+3 x)^8}{(1-2 x)^2 (3+5 x)^3} \, dx\)

Optimal. Leaf size=80 \[ \frac {6561 x^4}{2000}+\frac {12393 x^3}{625}+\frac {6093711 x^2}{100000}+\frac {7680987 x}{50000}+\frac {5764801}{85184 (1-2 x)}-\frac {268}{103984375 (5 x+3)}-\frac {1}{18906250 (5 x+3)^2}+\frac {130943337 \log (1-2 x)}{937024}+\frac {6312 \log (5 x+3)}{228765625} \]

[Out]

5764801/85184/(1-2*x)+7680987/50000*x+6093711/100000*x^2+12393/625*x^3+6561/2000*x^4-1/18906250/(3+5*x)^2-268/
103984375/(3+5*x)[phone]/937024*ln(1-2*x)+6312/228765625*ln(3+5*x)

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Rubi [A]  time = 0.04, antiderivative size = 80, 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} \[ \frac {6561 x^4}{2000}+\frac {12393 x^3}{625}+\frac {6093711 x^2}{100000}+\frac {7680987 x}{50000}+\frac {5764801}{85184 (1-2 x)}-\frac {268}{103984375 (5 x+3)}-\frac {1}{18906250 (5 x+3)^2}+\frac {130943337 \log (1-2 x)}{937024}+\frac {6312 \log (5 x+3)}{228765625} \]

Antiderivative was successfully verified.

[In]

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

[Out]

5764801/(85184*(1 - 2*x)) + (7680987*x)/50000 + (6093711*x^2)/100000 + (12393*x^3)/625 + (6561*x^4)/2000 - 1/(
18906250*(3 + 5*x)^2) - 268/(103984375*(3 + 5*x)) + (130943337*Log[1 - 2*x])/937024 + (6312*Log[3 + 5*x])/2287
65625

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)^3} \, dx &=\int \left (\frac {7680987}{50000}+\frac {6093711 x}{50000}+\frac {37179 x^2}{625}+\frac {6561 x^3}{500}+\frac {5764801}{42592 (-1+2 x)^2}+\frac {130943337}{468512 (-1+2 x)}+\frac {1}{1890625 (3+5 x)^3}+\frac {268}{20796875 (3+5 x)^2}+\frac {6312}{45753125 (3+5 x)}\right ) \, dx\\ &=\frac {5764801}{85184 (1-2 x)}+\frac {7680987 x}{50000}+\frac {6093711 x^2}{100000}+\frac {12393 x^3}{625}+\frac {6561 x^4}{2000}-\frac {1}{18906250 (3+5 x)^2}-\frac {268}{103984375 (3+5 x)}+\frac {130943337 \log (1-2 x)}{937024}+\frac {6312 \log (3+5 x)}{228765625}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 74, normalized size = 0.92 \[ \frac {3 \left (\frac {11}{3} \left (21831727500 x^4+131960664000 x^3+405536467050 x^2+1022339369700 x+\frac {450375078125}{1-2 x}-\frac {17152}{5 x+3}-\frac {352}{(5 x+3)^2}[phone]\right )[phone] \log (3-6 x)+673280 \log (-3 (5 x+3))\right )}{73205000000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*((11*(536108166000 + 450375078125/(1 - 2*x) + 1022339369700*x + 405536467050*x^2 + 131960664000*x^3 + 21831
727500*x^4 - 352/(3 + 5*x)^2 - 17152/(3 + 5*x)))/3 + 3409982734375*Log[3 - 6*x] + 673280*Log[-3*(3 + 5*x)]))/7
3205000000

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fricas [A]  time = 0.62, size = 100, normalized size = 1.25 \[ \frac {12007450125000 \, x^{7} + 80983580287500 \, x^{6} + 270968124487500 \, x^{5} + 698838044478750 \, x^{4} + 327005737947900 \, x^{3} - 298950055409445 \, x^{2} + 2019840 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (5 \, x + 3\right ) + 10229948203125 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (2 \, x - 1\right ) - 249835373577966 \, x - 44587132164487}{73205000000 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/73205000000*(12007450125000*x^7 + 80983580287500*x^6 + 270968124487500*x^5 + 698838044478750*x^4 + 327005737
947900*x^3 - 298950055409445*x^2 + 2019840*(50*x^3 + 35*x^2 - 12*x - 9)*log(5*x + 3) + 10229948203125*(50*x^3
+ 35*x^2 - 12*x - 9)*log(2*x - 1) - 249835373577966*x - 44587132164487)/(50*x^3 + 35*x^2 - 12*x - 9)

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giac [A]  time = 1.22, size = 112, normalized size = 1.40 \[ \frac {{\left (2 \, x - 1\right )}^{4} {\left (\frac {1230096557250}{2 \, x - 1} + \frac {11539159570125}{{\left (2 \, x - 1\right )}^{2}} + \frac {69299175042900}{{\left (2 \, x - 1\right )}^{3}} + \frac {182728002843460}{{\left (2 \, x - 1\right )}^{4}} + \frac {163740919200408}{{\left (2 \, x - 1\right )}^{5}} + 60037250625\right )}}{11712800000 \, {\left (\frac {11}{2 \, x - 1} + 5\right )}^{2}} - \frac {5764801}{85184 \, {\left (2 \, x - 1\right )}} - \frac {139743873}{1000000} \, \log \left (\frac {{\left | 2 \, x - 1 \right |}}{2 \, {\left (2 \, x - 1\right )}^{2}}\right ) + \frac {6312}{228765625} \, \log \left ({\left | -\frac {11}{2 \, x - 1} - 5 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11712800000*(2*x - 1)^4*(1230096557250/(2*x - 1) + 11539159570125/(2*x - 1)^2 + 69299175042900/(2*x - 1)^3 +
182728002843460/(2*x - 1)^4 + 163740919200408/(2*x - 1)^5 + 60037250625)/(11/(2*x - 1) + 5)^2 - 5764801/85184
/(2*x - 1) - 139743873/1000000*log(1/2*abs(2*x - 1)/(2*x - 1)^2) + 6312/228765625*log(abs(-11/(2*x - 1) - 5))

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maple [A]  time = 0.01, size = 63, normalized size = 0.79 \[ \frac {6561 x^{4}}{2000}+\frac {12393 x^{3}}{625}+\frac {6093711 x^{2}}{100000}+\frac {7680987 x}{50000}+\frac {130943337 \ln \left (2 x -1\right )}{937024}+\frac {6312 \ln \left (5 x +3\right )}{228765625}-\frac {1}{18906250 \left (5 x +3\right )^{2}}-\frac {268}{103984375 \left (5 x +3\right )}-\frac {5764801}{85184 \left (2 x -1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6561/2000*x^4+12393/625*x^3+6093711/100000*x^2+7680987/50000*x-1/18906250/(5*x+3)^2-268/103984375/(5*x+3)+6312
/228765625*ln(5*x+3)-5764801/85184/(2*x-1)[phone]/937024*ln(2*x-1)

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maxima [A]  time = 0.49, size = 64, normalized size = 0.80 \[ \frac {6561}{2000} \, x^{4} + \frac {12393}{625} \, x^{3} + \frac {6093711}{100000} \, x^{2} + \frac {7680987}{50000} \, x - \frac {11259377124645 \, x^{2} + 13511252361606 \, x + 4053375651317}{6655000000 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} + \frac {6312}{228765625} \, \log \left (5 \, x + 3\right ) + \frac {130943337}{937024} \, \log \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6561/2000*x^4 + 12393/625*x^3 + 6093711/100000*x^2 + 7680987/50000*x - 1/6655000000*(11259377124645*x^2 + 1351
1252361606*x + 4053375651317)/(50*x^3 + 35*x^2 - 12*x - 9) + 6312/228765625*log(5*x + 3) + 130943337/937024*lo
g(2*x - 1)

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mupad [B]  time = 1.06, size = 59, normalized size = 0.74 \[ \frac {7680987\,x}{50000}+\frac {130943337\,\ln \left (x-\frac {1}{2}\right )}{937024}+\frac {6312\,\ln \left (x+\frac {3}{5}\right )}{228765625}+\frac {\frac {2251875424929\,x^2}{66550000000}+\frac {6755626180803\,x}{166375000000}+\frac {4053375651317}{332750000000}}{-x^3-\frac {7\,x^2}{10}+\frac {6\,x}{25}+\frac {9}{50}}+\frac {6093711\,x^2}{100000}+\frac {12393\,x^3}{625}+\frac {6561\,x^4}{2000} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(7680987*x)/50000 + (130943337*log(x - 1/2))/937024 + (6312*log(x + 3/5))/228765625 + ((6755626180803*x)/16637
5000000 + (2251875424929*x^2)/66550000000 + 4053375651317/332750000000)/((6*x)/25 - (7*x^2)/10 - x^3 + 9/50) +
(6093711*x^2)/100000 + (12393*x^3)/625 + (6561*x^4)/2000

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sympy [A]  time = 0.20, size = 71, normalized size = 0.89 \[ \frac {6561 x^{4}}{2000} + \frac {12393 x^{3}}{625} + \frac {6093711 x^{2}}{100000} + \frac {7680987 x}{50000} + \frac {- 11259377124645 x^{2} - 13511252361606 x - 4053375651317}{332750000000 x^{3} + 232925000000 x^{2} - 79860000000 x - 59895000000} + \frac {130943337 \log {\left (x - \frac {1}{2} \right )}}{937024} + \frac {6312 \log {\left (x + \frac {3}{5} \right )}}{228765625} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6561*x**4/2000 + 12393*x**3/625 + 6093711*x**2/100000 + 7680987*x/50000 + (-11259377124645*x**2 - 135112523616
06*x - 4053375651317)/(332750000000*x**3 + 232925000000*x**2 - 79860000000*x - 59895000000) + 130943337*log(x
- 1/2)/937024 + 6312*log(x + 3/5)/228765625

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