3.1671 \(\int \frac{(2+3 x)^8}{(1-2 x)^3 (3+5 x)} \, dx\)

Optimal. Leaf size=76 \[ -\frac{6561 x^5}{200}-\frac{408969 x^4}{1600}-\frac{124416 x^3}{125}-\frac{110180817 x^2}{40000}-\frac{2941619571 x}{400000}-\frac{188591347}{30976 (1-2 x)}+\frac{5764801}{5632 (1-2 x)^2}-\frac{2644396573 \log (1-2 x)}{340736}+\frac{\log (5 x+3)}{20796875} \]

[Out]

5764801/(5632*(1 - 2*x)^2) - 188591347/(30976*(1 - 2*x)) - (2941619571*x)/400000 - (110180817*x^2)/40000 - (12
4416*x^3)/125 - (408969*x^4)/1600 - (6561*x^5)/200 - (2644396573*Log[1 - 2*x])/340736 + Log[3 + 5*x]/20796875

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Rubi [A]  time = 0.0385941, 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} \[ -\frac{6561 x^5}{200}-\frac{408969 x^4}{1600}-\frac{124416 x^3}{125}-\frac{110180817 x^2}{40000}-\frac{2941619571 x}{400000}-\frac{188591347}{30976 (1-2 x)}+\frac{5764801}{5632 (1-2 x)^2}-\frac{2644396573 \log (1-2 x)}{340736}+\frac{\log (5 x+3)}{20796875} \]

Antiderivative was successfully verified.

[In]

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

[Out]

5764801/(5632*(1 - 2*x)^2) - 188591347/(30976*(1 - 2*x)) - (2941619571*x)/400000 - (110180817*x^2)/40000 - (12
4416*x^3)/125 - (408969*x^4)/1600 - (6561*x^5)/200 - (2644396573*Log[1 - 2*x])/340736 + Log[3 + 5*x]/20796875

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)} \, dx &=\int \left (-\frac{2941619571}{400000}-\frac{110180817 x}{20000}-\frac{373248 x^2}{125}-\frac{408969 x^3}{400}-\frac{6561 x^4}{40}-\frac{5764801}{1408 (-1+2 x)^3}-\frac{188591347}{15488 (-1+2 x)^2}-\frac{2644396573}{170368 (-1+2 x)}+\frac{1}{4159375 (3+5 x)}\right ) \, dx\\ &=\frac{5764801}{5632 (1-2 x)^2}-\frac{188591347}{30976 (1-2 x)}-\frac{2941619571 x}{400000}-\frac{110180817 x^2}{40000}-\frac{124416 x^3}{125}-\frac{408969 x^4}{1600}-\frac{6561 x^5}{200}-\frac{2644396573 \log (1-2 x)}{340736}+\frac{\log (3+5 x)}{20796875}\\ \end{align*}

Mathematica [A]  time = 0.0421093, size = 98, normalized size = 1.29 \[ -\frac{27}{200} (3 x+2)^5-\frac{2889 (3 x+2)^4}{1600}-\frac{17019 (3 x+2)^3}{1000}-\frac{5992353 (3 x+2)^2}{40000}-\frac{631722537 (3 x+2)}{400000}+\frac{188591347}{30976 (2 x-1)}+\frac{5764801}{5632 (1-2 x)^2}-\frac{2644396573 \log (3-6 x)}{340736}+\frac{\log (-3 (5 x+3))}{20796875} \]

Antiderivative was successfully verified.

[In]

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

[Out]

5764801/(5632*(1 - 2*x)^2) + 188591347/(30976*(-1 + 2*x)) - (631722537*(2 + 3*x))/400000 - (5992353*(2 + 3*x)^
2)/40000 - (17019*(2 + 3*x)^3)/1000 - (2889*(2 + 3*x)^4)/1600 - (27*(2 + 3*x)^5)/200 - (2644396573*Log[3 - 6*x
])/340736 + Log[-3*(3 + 5*x)]/20796875

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Maple [A]  time = 0.008, size = 59, normalized size = 0.8 \begin{align*} -{\frac{6561\,{x}^{5}}{200}}-{\frac{408969\,{x}^{4}}{1600}}-{\frac{124416\,{x}^{3}}{125}}-{\frac{110180817\,{x}^{2}}{40000}}-{\frac{2941619571\,x}{400000}}+{\frac{5764801}{5632\, \left ( 2\,x-1 \right ) ^{2}}}+{\frac{188591347}{61952\,x-30976}}-{\frac{2644396573\,\ln \left ( 2\,x-1 \right ) }{340736}}+{\frac{\ln \left ( 3+5\,x \right ) }{20796875}} \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),x)

[Out]

-6561/200*x^5-408969/1600*x^4-124416/125*x^3-110180817/40000*x^2-2941619571/400000*x+5764801/5632/(2*x-1)^2+18
8591347/30976/(2*x-1)-2644396573/340736*ln(2*x-1)+1/20796875*ln(3+5*x)

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Maxima [A]  time = 1.06654, size = 80, normalized size = 1.05 \begin{align*} -\frac{6561}{200} \, x^{5} - \frac{408969}{1600} \, x^{4} - \frac{124416}{125} \, x^{3} - \frac{110180817}{40000} \, x^{2} - \frac{2941619571}{400000} \, x + \frac{823543 \,{\left (916 \, x - 381\right )}}{61952 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{1}{20796875} \, \log \left (5 \, x + 3\right ) - \frac{2644396573}{340736} \, \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),x, algorithm="maxima")

[Out]

-6561/200*x^5 - 408969/1600*x^4 - 124416/125*x^3 - 110180817/40000*x^2 - 2941619571/400000*x + 823543/61952*(9
16*x - 381)/(4*x^2 - 4*x + 1) + 1/20796875*log(5*x + 3) - 2644396573/340736*log(2*x - 1)

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Fricas [A]  time = 1.63589, size = 379, normalized size = 4.99 \begin{align*} -\frac{1397230560000 \, x^{7} + 9489524220000 \, x^{6} + 31855563036000 \, x^{5} + 77649212460600 \, x^{4} + 206501370522480 \, x^{3} - 283893518434680 \, x^{2} - 512 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (5 \, x + 3\right ) + 82637392906250 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (2 \, x - 1\right ) - 51350638082480 \, x + 53929198640625}{10648000000 \,{\left (4 \, x^{2} - 4 \, 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),x, algorithm="fricas")

[Out]

-1/10648000000*(1397230560000*x^7 + 9489524220000*x^6 + 31855563036000*x^5 + 77649212460600*x^4 + 206501370522
480*x^3 - 283893518434680*x^2 - 512*(4*x^2 - 4*x + 1)*log(5*x + 3) + 82637392906250*(4*x^2 - 4*x + 1)*log(2*x
- 1) - 51350638082480*x + 53929198640625)/(4*x^2 - 4*x + 1)

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Sympy [A]  time = 0.174255, size = 65, normalized size = 0.86 \begin{align*} - \frac{6561 x^{5}}{200} - \frac{408969 x^{4}}{1600} - \frac{124416 x^{3}}{125} - \frac{110180817 x^{2}}{40000} - \frac{2941619571 x}{400000} + \frac{754365388 x - 313769883}{247808 x^{2} - 247808 x + 61952} - \frac{2644396573 \log{\left (x - \frac{1}{2} \right )}}{340736} + \frac{\log{\left (x + \frac{3}{5} \right )}}{20796875} \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),x)

[Out]

-6561*x**5/200 - 408969*x**4/1600 - 124416*x**3/125 - 110180817*x**2/40000 - 2941619571*x/400000 + (754365388*
x - 313769883)/(247808*x**2 - 247808*x + 61952) - 2644396573*log(x - 1/2)/340736 + log(x + 3/5)/20796875

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Giac [A]  time = 1.80905, size = 76, normalized size = 1. \begin{align*} -\frac{6561}{200} \, x^{5} - \frac{408969}{1600} \, x^{4} - \frac{124416}{125} \, x^{3} - \frac{110180817}{40000} \, x^{2} - \frac{2941619571}{400000} \, x + \frac{823543 \,{\left (916 \, x - 381\right )}}{61952 \,{\left (2 \, x - 1\right )}^{2}} + \frac{1}{20796875} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac{2644396573}{340736} \, \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/(3+5*x),x, algorithm="giac")

[Out]

-6561/200*x^5 - 408969/1600*x^4 - 124416/125*x^3 - 110180817/40000*x^2 - 2941619571/400000*x + 823543/61952*(9
16*x - 381)/(2*x - 1)^2 + 1/20796875*log(abs(5*x + 3)) - 2644396573/340736*log(abs(2*x - 1))