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

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

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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} \begin {gather*} -\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} \end {gather*}

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*}

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Mathematica [A]  time = 0.06, size = 98, normalized size = 1.22 \begin {gather*} -\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} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(2+3 x)^8}{(1-2 x)^3 (3+5 x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.99, size = 100, normalized size = 1.25 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 1.24, size = 112, normalized size = 1.40 \begin {gather*} -\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 {gather*}

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))

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maple [A]  time = 0.01, size = 63, normalized size = 0.79 \begin {gather*} -\frac {6561 x^{4}}{800}-\frac {123201 x^{3}}{2000}-\frac {4863159 x^{2}}{20000}-\frac {81001863 x}{100000}-\frac {1845559863 \ln \left (2 x -1\right )}{1874048}+\frac {54 \ln \left (5 x +3\right )}{45753125}-\frac {1}{20796875 \left (5 x +3\right )}+\frac {5764801}{30976 \left (2 x -1\right )^{2}}+\frac {79883671}{85184 \left (2 x -1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6561/800*x^4-123201/2000*x^3-4863159/20000*x^2-81001863/100000*x-1/20796875/(5*x+3)+54/45753125*ln(5*x+3)+576
4801/30976/(2*x-1)^2+79883671/85184/(2*x-1)-1845559863/1874048*ln(2*x-1)

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maxima [A]  time = 0.50, size = 64, normalized size = 0.80 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 0.04, size = 60, normalized size = 0.75 \begin {gather*} \frac {54\,\ln \left (x+\frac {3}{5}\right )}{45753125}-\frac {1845559863\,\ln \left (x-\frac {1}{2}\right )}{1874048}-\frac {81001863\,x}{100000}-\frac {\frac {6240911796747\,x^2}{13310000000}+\frac {9946855297899\,x}{106480000000}-\frac {12005712797131}{106480000000}}{-x^3+\frac {2\,x^2}{5}+\frac {7\,x}{20}-\frac {3}{20}}-\frac {4863159\,x^2}{20000}-\frac {123201\,x^3}{2000}-\frac {6561\,x^4}{800} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(54*log(x + 3/5))/45753125 - (1845559863*log(x - 1/2))/1874048 - (81001863*x)/100000 - ((9946855297899*x)/1064
80000000 + (6240911796747*x^2)/13310000000 - 12005712797131/106480000000)/((7*x)/20 + (2*x^2)/5 - x^3 - 3/20)
- (4863159*x^2)/20000 - (123201*x^3)/2000 - (6561*x^4)/800

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sympy [A]  time = 0.22, size = 70, normalized size = 0.88 \begin {gather*} - \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 {gather*}

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 - 994685529
7899*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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