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*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

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Rubi [A]  time = 0.0420433, antiderivative size = 80, 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} \[ \frac{6561 x^4}{2000}+\frac{12393 x^3}{625}+\frac{6093711 x^2}{100000}+\frac{7680987 x}{50000}+\frac{5764801}{85184 (1-2 x)}-\frac{268}{103984375 (5 x+3)}-\frac{1}{18906250 (5 x+3)^2}+\frac{130943337 \log (1-2 x)}{937024}+\frac{6312 \log (5 x+3)}{228765625} \]

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*}

Mathematica [A]  time = 0.0763912, 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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Maple [A]  time = 0.008, size = 63, normalized size = 0.8 \begin{align*}{\frac{6561\,{x}^{4}}{2000}}+{\frac{12393\,{x}^{3}}{625}}+{\frac{6093711\,{x}^{2}}{100000}}+{\frac{7680987\,x}{50000}}-{\frac{5764801}{170368\,x-85184}}+{\frac{130943337\,\ln \left ( 2\,x-1 \right ) }{937024}}-{\frac{1}{18906250\, \left ( 3+5\,x \right ) ^{2}}}-{\frac{268}{311953125+519921875\,x}}+{\frac{6312\,\ln \left ( 3+5\,x \right ) }{228765625}} \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)^3,x)

[Out]

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

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Maxima [A]  time = 1.09374, size = 86, normalized size = 1.08 \begin{align*} \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 ) \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)^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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Fricas [A]  time = 1.585, size = 435, normalized size = 5.44 \begin{align*} \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 )}} \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)^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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Sympy [A]  time = 0.183499, size = 70, normalized size = 0.88 \begin{align*} \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} \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)**3,x)

[Out]

6561*x**4/2000 + 12393*x**3/625 + 6093711*x**2/100000 + 7680987*x/50000 - (11259377124645*x**2 + 1351125236160
6*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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Giac [A]  time = 2.51926, size = 151, normalized size = 1.89 \begin{align*} \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 ) \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)^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))