3.1501 \(\int \frac{(2+3 x)^8}{(1-2 x) (3+5 x)^2} \, dx\)

Optimal. Leaf size=72 \[ -\frac{2187 x^6}{100}-\frac{303993 x^5}{2500}-\frac{6194313 x^4}{20000}-\frac{24660207 x^3}{50000}-\frac{118543581 x^2}{200000}-\frac{3579885909 x}{5000000}-\frac{1}{4296875 (5 x+3)}-\frac{5764801 \log (1-2 x)}{15488}+\frac{266 \log (5 x+3)}{47265625} \]

[Out]

(-3579885909*x)/5000000 - (118543581*x^2)/200000 - (24660207*x^3)/50000 - (6194313*x^4)/20000 - (303993*x^5)/2
500 - (2187*x^6)/100 - 1/(4296875*(3 + 5*x)) - (5764801*Log[1 - 2*x])/15488 + (266*Log[3 + 5*x])/47265625

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Rubi [A]  time = 0.0338545, 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} \[ -\frac{2187 x^6}{100}-\frac{303993 x^5}{2500}-\frac{6194313 x^4}{20000}-\frac{24660207 x^3}{50000}-\frac{118543581 x^2}{200000}-\frac{3579885909 x}{5000000}-\frac{1}{4296875 (5 x+3)}-\frac{5764801 \log (1-2 x)}{15488}+\frac{266 \log (5 x+3)}{47265625} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3579885909*x)/5000000 - (118543581*x^2)/200000 - (24660207*x^3)/50000 - (6194313*x^4)/20000 - (303993*x^5)/2
500 - (2187*x^6)/100 - 1/(4296875*(3 + 5*x)) - (5764801*Log[1 - 2*x])/15488 + (266*Log[3 + 5*x])/47265625

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) (3+5 x)^2} \, dx &=\int \left (-\frac{3579885909}{5000000}-\frac{118543581 x}{100000}-\frac{73980621 x^2}{50000}-\frac{6194313 x^3}{5000}-\frac{303993 x^4}{500}-\frac{6561 x^5}{50}-\frac{5764801}{7744 (-1+2 x)}+\frac{1}{859375 (3+5 x)^2}+\frac{266}{9453125 (3+5 x)}\right ) \, dx\\ &=-\frac{3579885909 x}{5000000}-\frac{118543581 x^2}{200000}-\frac{24660207 x^3}{50000}-\frac{6194313 x^4}{20000}-\frac{303993 x^5}{2500}-\frac{2187 x^6}{100}-\frac{1}{4296875 (3+5 x)}-\frac{5764801 \log (1-2 x)}{15488}+\frac{266 \log (3+5 x)}{47265625}\\ \end{align*}

Mathematica [A]  time = 0.0589492, size = 64, normalized size = 0.89 \[ \frac{22 \left (-6014250000 x^6-33439230000 x^5-85171803750 x^4-135631138500 x^3-162997423875 x^2-196893724995 x-\frac{64}{5 x+3}-86057647830\right )-2251875390625 \log (3-6 x)+34048 \log (-3 (5 x+3))}{6050000000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(22*(-86057647830 - 196893724995*x - 162997423875*x^2 - 135631138500*x^3 - 85171803750*x^4 - 33439230000*x^5 -
6014250000*x^6 - 64/(3 + 5*x)) - 2251875390625*Log[3 - 6*x] + 34048*Log[-3*(3 + 5*x)])/6050000000

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Maple [A]  time = 0.006, size = 55, normalized size = 0.8 \begin{align*} -{\frac{2187\,{x}^{6}}{100}}-{\frac{303993\,{x}^{5}}{2500}}-{\frac{6194313\,{x}^{4}}{20000}}-{\frac{24660207\,{x}^{3}}{50000}}-{\frac{118543581\,{x}^{2}}{200000}}-{\frac{3579885909\,x}{5000000}}-{\frac{5764801\,\ln \left ( 2\,x-1 \right ) }{15488}}-{\frac{1}{12890625+21484375\,x}}+{\frac{266\,\ln \left ( 3+5\,x \right ) }{47265625}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2187/100*x^6-303993/2500*x^5-6194313/20000*x^4-24660207/50000*x^3-118543581/200000*x^2-3579885909/5000000*x-5
764801/15488*ln(2*x-1)-1/4296875/(3+5*x)+266/47265625*ln(3+5*x)

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Maxima [A]  time = 1.07025, size = 73, normalized size = 1.01 \begin{align*} -\frac{2187}{100} \, x^{6} - \frac{303993}{2500} \, x^{5} - \frac{6194313}{20000} \, x^{4} - \frac{24660207}{50000} \, x^{3} - \frac{118543581}{200000} \, x^{2} - \frac{3579885909}{5000000} \, x - \frac{1}{4296875 \,{\left (5 \, x + 3\right )}} + \frac{266}{47265625} \, \log \left (5 \, x + 3\right ) - \frac{5764801}{15488} \, \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)/(3+5*x)^2,x, algorithm="maxima")

[Out]

-2187/100*x^6 - 303993/2500*x^5 - 6194313/20000*x^4 - 24660207/50000*x^3 - 118543581/200000*x^2 - 3579885909/5
000000*x - 1/4296875/(5*x + 3) + 266/47265625*log(5*x + 3) - 5764801/15488*log(2*x - 1)

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Fricas [A]  time = 1.33161, size = 329, normalized size = 4.57 \begin{align*} -\frac{661567500000 \, x^{7} + 4075255800000 \, x^{6} + 11575887592500 \, x^{5} + 20540764282500 \, x^{4} + 26881371767250 \, x^{3} + 32416139725200 \, x^{2} - 34048 \,{\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) + 2251875390625 \,{\left (5 \, x + 3\right )} \log \left (2 \, x - 1\right ) + 12994985849670 \, x + 1408}{6050000000 \,{\left (5 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6050000000*(661567500000*x^7 + 4075255800000*x^6 + 11575887592500*x^5 + 20540764282500*x^4 + 26881371767250
*x^3 + 32416139725200*x^2 - 34048*(5*x + 3)*log(5*x + 3) + 2251875390625*(5*x + 3)*log(2*x - 1) + 129949858496
70*x + 1408)/(5*x + 3)

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Sympy [A]  time = 0.157419, size = 65, normalized size = 0.9 \begin{align*} - \frac{2187 x^{6}}{100} - \frac{303993 x^{5}}{2500} - \frac{6194313 x^{4}}{20000} - \frac{24660207 x^{3}}{50000} - \frac{118543581 x^{2}}{200000} - \frac{3579885909 x}{5000000} - \frac{5764801 \log{\left (x - \frac{1}{2} \right )}}{15488} + \frac{266 \log{\left (x + \frac{3}{5} \right )}}{47265625} - \frac{1}{21484375 x + 12890625} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2187*x**6/100 - 303993*x**5/2500 - 6194313*x**4/20000 - 24660207*x**3/50000 - 118543581*x**2/200000 - 3579885
909*x/5000000 - 5764801*log(x - 1/2)/15488 + 266*log(x + 3/5)/47265625 - 1/(21484375*x + 12890625)

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Giac [A]  time = 1.53222, size = 134, normalized size = 1.86 \begin{align*} -\frac{27}{125000000} \,{\left (5 \, x + 3\right )}^{6}{\left (\frac{63504}{5 \, x + 3} + \frac{466830}{{\left (5 \, x + 3\right )}^{2}} + \frac{3450300}{{\left (5 \, x + 3\right )}^{3}} + \frac{28481775}{{\left (5 \, x + 3\right )}^{4}} + \frac{313308485}{{\left (5 \, x + 3\right )}^{5}} + 6480\right )} - \frac{1}{4296875 \,{\left (5 \, x + 3\right )}} + \frac{18610540137}{50000000} \, \log \left (\frac{{\left | 5 \, x + 3 \right |}}{5 \,{\left (5 \, x + 3\right )}^{2}}\right ) - \frac{5764801}{15488} \, \log \left ({\left | -\frac{11}{5 \, x + 3} + 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27/125000000*(5*x + 3)^6*(63504/(5*x + 3) + 466830/(5*x + 3)^2 + 3450300/(5*x + 3)^3 + 28481775/(5*x + 3)^4 +
313308485/(5*x + 3)^5 + 6480) - 1/4296875/(5*x + 3) + 18610540137/50000000*log(1/5*abs(5*x + 3)/(5*x + 3)^2)
- 5764801/15488*log(abs(-11/(5*x + 3) + 2))