3.1684 \(\int \frac{(2+3 x)^8}{(1-2 x)^3 (3+5 x)^2} \, dx\)

Optimal. Leaf size=80 \[ -\frac{6561 x^4}{800}-\frac{123201 x^3}{2000}-\frac{4863159 x^2}{20000}-\frac{81001863 x}{100000}-\frac{79883671}{85184 (1-2 x)}-\frac{1}{20796875 (5 x+3)}+\frac{5764801}{30976 (1-2 x)^2}-\frac{1845559863 \log (1-2 x)}{1874048}+\frac{54 \log (5 x+3)}{45753125} \]

[Out]

5764801/(30976*(1 - 2*x)^2) - 79883671/(85184*(1 - 2*x)) - (81001863*x)/100000 - (4863159*x^2)/20000 - (123201
*x^3)/2000 - (6561*x^4)/800 - 1/(20796875*(3 + 5*x)) - (1845559863*Log[1 - 2*x])/1874048 + (54*Log[3 + 5*x])/4
5753125

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Rubi [A]  time = 0.0437822, 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} \[ -\frac{6561 x^4}{800}-\frac{123201 x^3}{2000}-\frac{4863159 x^2}{20000}-\frac{81001863 x}{100000}-\frac{79883671}{85184 (1-2 x)}-\frac{1}{20796875 (5 x+3)}+\frac{5764801}{30976 (1-2 x)^2}-\frac{1845559863 \log (1-2 x)}{1874048}+\frac{54 \log (5 x+3)}{45753125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

5764801/(30976*(1 - 2*x)^2) - 79883671/(85184*(1 - 2*x)) - (81001863*x)/100000 - (4863159*x^2)/20000 - (123201
*x^3)/2000 - (6561*x^4)/800 - 1/(20796875*(3 + 5*x)) - (1845559863*Log[1 - 2*x])/1874048 + (54*Log[3 + 5*x])/4
5753125

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 (3+5 x)^2} \, dx &=\int \left (-\frac{81001863}{100000}-\frac{4863159 x}{10000}-\frac{369603 x^2}{2000}-\frac{6561 x^3}{200}-\frac{5764801}{7744 (-1+2 x)^3}-\frac{79883671}{42592 (-1+2 x)^2}-\frac{1845559863}{937024 (-1+2 x)}+\frac{1}{4159375 (3+5 x)^2}+\frac{54}{9150625 (3+5 x)}\right ) \, dx\\ &=\frac{5764801}{30976 (1-2 x)^2}-\frac{79883671}{85184 (1-2 x)}-\frac{81001863 x}{100000}-\frac{4863159 x^2}{20000}-\frac{123201 x^3}{2000}-\frac{6561 x^4}{800}-\frac{1}{20796875 (3+5 x)}-\frac{1845559863 \log (1-2 x)}{1874048}+\frac{54 \log (3+5 x)}{45753125}\\ \end{align*}

Mathematica [A]  time = 0.0471255, size = 98, normalized size = 1.22 \[ -\frac{81}{800} (3 x+2)^4-\frac{2943 (3 x+2)^3}{2000}-\frac{315171 (3 x+2)^2}{20000}-\frac{18607401 (3 x+2)}{100000}+\frac{79883671}{85184 (2 x-1)}-\frac{1}{20796875 (5 x+3)}+\frac{5764801}{30976 (1-2 x)^2}-\frac{1845559863 \log (3-6 x)}{1874048}+\frac{54 \log (-3 (5 x+3))}{45753125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

5764801/(30976*(1 - 2*x)^2) + 79883671/(85184*(-1 + 2*x)) - (18607401*(2 + 3*x))/100000 - (315171*(2 + 3*x)^2)
/20000 - (2943*(2 + 3*x)^3)/2000 - (81*(2 + 3*x)^4)/800 - 1/(20796875*(3 + 5*x)) - (1845559863*Log[3 - 6*x])/1
874048 + (54*Log[-3*(3 + 5*x)])/45753125

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Maple [A]  time = 0.009, size = 63, normalized size = 0.8 \begin{align*} -{\frac{6561\,{x}^{4}}{800}}-{\frac{123201\,{x}^{3}}{2000}}-{\frac{4863159\,{x}^{2}}{20000}}-{\frac{81001863\,x}{100000}}+{\frac{5764801}{30976\, \left ( 2\,x-1 \right ) ^{2}}}+{\frac{79883671}{170368\,x-85184}}-{\frac{1845559863\,\ln \left ( 2\,x-1 \right ) }{1874048}}-{\frac{1}{62390625+103984375\,x}}+{\frac{54\,\ln \left ( 3+5\,x \right ) }{45753125}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6561/800*x^4-123201/2000*x^3-4863159/20000*x^2-81001863/100000*x+5764801/30976/(2*x-1)^2+79883671/85184/(2*x-
1)-1845559863/1874048*ln(2*x-1)-1/20796875/(3+5*x)+54/45753125*ln(3+5*x)

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Maxima [A]  time = 1.0434, size = 86, normalized size = 1.08 \begin{align*} -\frac{6561}{800} \, x^{4} - \frac{123201}{2000} \, x^{3} - \frac{4863159}{20000} \, x^{2} - \frac{81001863}{100000} \, x + \frac{49927294373976 \, x^{2} + 9946855297899 \, x - 12005712797131}{5324000000 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} + \frac{54}{45753125} \, \log \left (5 \, x + 3\right ) - \frac{1845559863}{1874048} \, \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/(3+5*x)^2,x, algorithm="maxima")

[Out]

-6561/800*x^4 - 123201/2000*x^3 - 4863159/20000*x^2 - 81001863/100000*x + 1/5324000000*(49927294373976*x^2 + 9
946855297899*x - 12005712797131)/(20*x^3 - 8*x^2 - 7*x + 3) + 54/45753125*log(5*x + 3) - 1845559863/1874048*lo
g(2*x - 1)

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Fricas [A]  time = 1.45391, size = 424, normalized size = 5.3 \begin{align*} -\frac{9605960100000 \, x^{7} + 68309049600000 \, x^{6} + 252583384185000 \, x^{5} + 811024095717000 \, x^{4} - 468362848619160 \, x^{3} - 838544848893576 \, x^{2} - 69120 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (5 \, x + 3\right ) + 57673745718750 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (2 \, x - 1\right ) + 32898384865071 \, x + 132062840768441}{58564000000 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, 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/(3+5*x)^2,x, algorithm="fricas")

[Out]

-1/58564000000*(9605960100000*x^7 + 68309049600000*x^6 + 252583384185000*x^5 + 811024095717000*x^4 - 468362848
619160*x^3 - 838544848893576*x^2 - 69120*(20*x^3 - 8*x^2 - 7*x + 3)*log(5*x + 3) + 57673745718750*(20*x^3 - 8*
x^2 - 7*x + 3)*log(2*x - 1) + 32898384865071*x + 132062840768441)/(20*x^3 - 8*x^2 - 7*x + 3)

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Sympy [A]  time = 0.194518, size = 70, normalized size = 0.88 \begin{align*} - \frac{6561 x^{4}}{800} - \frac{123201 x^{3}}{2000} - \frac{4863159 x^{2}}{20000} - \frac{81001863 x}{100000} + \frac{49927294373976 x^{2} + 9946855297899 x - 12005712797131}{106480000000 x^{3} - 42592000000 x^{2} - 37268000000 x + 15972000000} - \frac{1845559863 \log{\left (x - \frac{1}{2} \right )}}{1874048} + \frac{54 \log{\left (x + \frac{3}{5} \right )}}{45753125} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6561*x**4/800 - 123201*x**3/2000 - 4863159*x**2/20000 - 81001863*x/100000 + (49927294373976*x**2 + 9946855297
899*x - 12005712797131)/(106480000000*x**3 - 42592000000*x**2 - 37268000000*x + 15972000000) - 1845559863*log(
x - 1/2)/1874048 + 54*log(x + 3/5)/45753125

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Giac [A]  time = 1.85674, size = 151, normalized size = 1.89 \begin{align*} -\frac{{\left (5 \, x + 3\right )}^{4}{\left (\frac{11185606872}{5 \, x + 3} + \frac{158583727962}{{\left (5 \, x + 3\right )}^{2}} + \frac{3495217526460}{{\left (5 \, x + 3\right )}^{3}} - \frac{86510680819405}{{\left (5 \, x + 3\right )}^{4}} + \frac{317205578854725}{{\left (5 \, x + 3\right )}^{5}} + 768476808\right )}}{14641000000 \,{\left (\frac{11}{5 \, x + 3} - 2\right )}^{2}} - \frac{1}{20796875 \,{\left (5 \, x + 3\right )}} + \frac{393919443}{400000} \, \log \left (\frac{{\left | 5 \, x + 3 \right |}}{5 \,{\left (5 \, x + 3\right )}^{2}}\right ) - \frac{1845559863}{1874048} \, \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/(3+5*x)^2,x, algorithm="giac")

[Out]

-1/14641000000*(5*x + 3)^4*(11185606872/(5*x + 3) + 158583727962/(5*x + 3)^2 + 3495217526460/(5*x + 3)^3 - 865
10680819405/(5*x + 3)^4 + 317205578854725/(5*x + 3)^5 + 768476808)/(11/(5*x + 3) - 2)^2 - 1/20796875/(5*x + 3)
+ 393919443/400000*log(1/5*abs(5*x + 3)/(5*x + 3)^2) - 1845559863/1874048*log(abs(-11/(5*x + 3) + 2))