3.1516 \(\int \frac{(2+3 x)^8}{(1-2 x) (3+5 x)^3} \, dx\)

Optimal. Leaf size=76 \[ -\frac{6561 x^5}{1250}-\frac{264627 x^4}{10000}-\frac{1535517 x^3}{25000}-\frac{9268263 x^2}{100000}-\frac{62934003 x}{500000}-\frac{266}{47265625 (5 x+3)}-\frac{1}{8593750 (5 x+3)^2}-\frac{5764801 \log (1-2 x)}{85184}+\frac{31024 \log (5 x+3)}{519921875} \]

[Out]

(-62934003*x)/500000 - (9268263*x^2)/100000 - (1535517*x^3)/25000 - (264627*x^4)/10000 - (6561*x^5)/1250 - 1/(
8593750*(3 + 5*x)^2) - 266/(47265625*(3 + 5*x)) - (5764801*Log[1 - 2*x])/85184 + (31024*Log[3 + 5*x])/51992187
5

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Rubi [A]  time = 0.0383633, antiderivative size = 76, 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} \[ -\frac{6561 x^5}{1250}-\frac{264627 x^4}{10000}-\frac{1535517 x^3}{25000}-\frac{9268263 x^2}{100000}-\frac{62934003 x}{500000}-\frac{266}{47265625 (5 x+3)}-\frac{1}{8593750 (5 x+3)^2}-\frac{5764801 \log (1-2 x)}{85184}+\frac{31024 \log (5 x+3)}{519921875} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-62934003*x)/500000 - (9268263*x^2)/100000 - (1535517*x^3)/25000 - (264627*x^4)/10000 - (6561*x^5)/1250 - 1/(
8593750*(3 + 5*x)^2) - 266/(47265625*(3 + 5*x)) - (5764801*Log[1 - 2*x])/85184 + (31024*Log[3 + 5*x])/51992187
5

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)^3} \, dx &=\int \left (-\frac{62934003}{500000}-\frac{9268263 x}{50000}-\frac{4606551 x^2}{25000}-\frac{264627 x^3}{2500}-\frac{6561 x^4}{250}-\frac{5764801}{42592 (-1+2 x)}+\frac{1}{859375 (3+5 x)^3}+\frac{266}{9453125 (3+5 x)^2}+\frac{31024}{103984375 (3+5 x)}\right ) \, dx\\ &=-\frac{62934003 x}{500000}-\frac{9268263 x^2}{100000}-\frac{1535517 x^3}{25000}-\frac{264627 x^4}{10000}-\frac{6561 x^5}{1250}-\frac{1}{8593750 (3+5 x)^2}-\frac{266}{47265625 (3+5 x)}-\frac{5764801 \log (1-2 x)}{85184}+\frac{31024 \log (3+5 x)}{519921875}\\ \end{align*}

Mathematica [A]  time = 0.0709512, size = 68, normalized size = 0.89 \[ \frac{22 \left (-7938810000 x^5-40024833750 x^4-92898778500 x^3-140182477875 x^2-190375359075 x-\frac{8512}{5 x+3}-\frac{176}{(5 x+3)^2}-85278446550\right )-2251875390625 \log (3-6 x)+1985536 \log (-3 (5 x+3))}{33275000000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(22*(-85278446550 - 190375359075*x - 140182477875*x^2 - 92898778500*x^3 - 40024833750*x^4 - 7938810000*x^5 - 1
76/(3 + 5*x)^2 - 8512/(3 + 5*x)) - 2251875390625*Log[3 - 6*x] + 1985536*Log[-3*(3 + 5*x)])/33275000000

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Maple [A]  time = 0.009, size = 59, normalized size = 0.8 \begin{align*} -{\frac{6561\,{x}^{5}}{1250}}-{\frac{264627\,{x}^{4}}{10000}}-{\frac{1535517\,{x}^{3}}{25000}}-{\frac{9268263\,{x}^{2}}{100000}}-{\frac{62934003\,x}{500000}}-{\frac{5764801\,\ln \left ( 2\,x-1 \right ) }{85184}}-{\frac{1}{8593750\, \left ( 3+5\,x \right ) ^{2}}}-{\frac{266}{141796875+236328125\,x}}+{\frac{31024\,\ln \left ( 3+5\,x \right ) }{519921875}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6561/1250*x^5-264627/10000*x^4-1535517/25000*x^3-9268263/100000*x^2-62934003/500000*x-5764801/85184*ln(2*x-1)
-1/8593750/(3+5*x)^2-266/47265625/(3+5*x)+31024/519921875*ln(3+5*x)

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Maxima [A]  time = 2.71945, size = 80, normalized size = 1.05 \begin{align*} -\frac{6561}{1250} \, x^{5} - \frac{264627}{10000} \, x^{4} - \frac{1535517}{25000} \, x^{3} - \frac{9268263}{100000} \, x^{2} - \frac{62934003}{500000} \, x - \frac{2660 \, x + 1607}{94531250 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{31024}{519921875} \, \log \left (5 \, x + 3\right ) - \frac{5764801}{85184} \, \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)^3,x, algorithm="maxima")

[Out]

-6561/1250*x^5 - 264627/10000*x^4 - 1535517/25000*x^3 - 9268263/100000*x^2 - 62934003/500000*x - 1/94531250*(2
660*x + 1607)/(25*x^2 + 30*x + 9) + 31024/519921875*log(5*x + 3) - 5764801/85184*log(2*x - 1)

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Fricas [A]  time = 1.34631, size = 383, normalized size = 5.04 \begin{align*} -\frac{4366345500000 \, x^{7} + 27253273162500 \, x^{6} + 79082602830000 \, x^{5} + 146338473723750 \, x^{4} + 215620841031750 \, x^{3} + 153403867608750 \, x^{2} - 1985536 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) + 2251875390625 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (2 \, x - 1\right ) + 37694322033170 \, x + 565664}{33275000000 \,{\left (25 \, x^{2} + 30 \, x + 9\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)^3,x, algorithm="fricas")

[Out]

-1/33275000000*(4366345500000*x^7 + 27253273162500*x^6 + 79082602830000*x^5 + 146338473723750*x^4 + 2156208410
31750*x^3 + 153403867608750*x^2 - 1985536*(25*x^2 + 30*x + 9)*log(5*x + 3) + 2251875390625*(25*x^2 + 30*x + 9)
*log(2*x - 1) + 37694322033170*x + 565664)/(25*x^2 + 30*x + 9)

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Sympy [A]  time = 0.176517, size = 66, normalized size = 0.87 \begin{align*} - \frac{6561 x^{5}}{1250} - \frac{264627 x^{4}}{10000} - \frac{1535517 x^{3}}{25000} - \frac{9268263 x^{2}}{100000} - \frac{62934003 x}{500000} - \frac{2660 x + 1607}{2363281250 x^{2} + 2835937500 x + 850781250} - \frac{5764801 \log{\left (x - \frac{1}{2} \right )}}{85184} + \frac{31024 \log{\left (x + \frac{3}{5} \right )}}{519921875} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6561*x**5/1250 - 264627*x**4/10000 - 1535517*x**3/25000 - 9268263*x**2/100000 - 62934003*x/500000 - (2660*x +
1607)/(2363281250*x**2 + 2835937500*x + 850781250) - 5764801*log(x - 1/2)/85184 + 31024*log(x + 3/5)/51992187
5

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Giac [A]  time = 2.59279, size = 76, normalized size = 1. \begin{align*} -\frac{6561}{1250} \, x^{5} - \frac{264627}{10000} \, x^{4} - \frac{1535517}{25000} \, x^{3} - \frac{9268263}{100000} \, x^{2} - \frac{62934003}{500000} \, x - \frac{2660 \, x + 1607}{94531250 \,{\left (5 \, x + 3\right )}^{2}} + \frac{31024}{519921875} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac{5764801}{85184} \, \log \left ({\left | 2 \, x - 1 \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)^3,x, algorithm="giac")

[Out]

-6561/1250*x^5 - 264627/10000*x^4 - 1535517/25000*x^3 - 9268263/100000*x^2 - 62934003/500000*x - 1/94531250*(2
660*x + 1607)/(5*x + 3)^2 + 31024/519921875*log(abs(5*x + 3)) - 5764801/85184*log(abs(2*x - 1))