3.1686 \(\int \frac{(2+3 x)^6}{(1-2 x)^3 (3+5 x)^2} \, dx\)

Optimal. Leaf size=66 \[ -\frac{729 x^2}{400}-\frac{31347 x}{2000}-\frac{67228}{1331 (1-2 x)}-\frac{1}{831875 (5 x+3)}+\frac{117649}{7744 (1-2 x)^2}-\frac{7383075 \log (1-2 x)}{234256}+\frac{204 \log (5 x+3)}{9150625} \]

[Out]

117649/(7744*(1 - 2*x)^2) - 67228/(1331*(1 - 2*x)) - (31347*x)/2000 - (729*x^2)/400 - 1/(831875*(3 + 5*x)) - (
7383075*Log[1 - 2*x])/234256 + (204*Log[3 + 5*x])/9150625

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Rubi [A]  time = 0.0320095, antiderivative size = 66, 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} \[ -\frac{729 x^2}{400}-\frac{31347 x}{2000}-\frac{67228}{1331 (1-2 x)}-\frac{1}{831875 (5 x+3)}+\frac{117649}{7744 (1-2 x)^2}-\frac{7383075 \log (1-2 x)}{234256}+\frac{204 \log (5 x+3)}{9150625} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^6/((1 - 2*x)^3*(3 + 5*x)^2),x]

[Out]

117649/(7744*(1 - 2*x)^2) - 67228/(1331*(1 - 2*x)) - (31347*x)/2000 - (729*x^2)/400 - 1/(831875*(3 + 5*x)) - (
7383075*Log[1 - 2*x])/234256 + (204*Log[3 + 5*x])/9150625

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)^6}{(1-2 x)^3 (3+5 x)^2} \, dx &=\int \left (-\frac{31347}{2000}-\frac{729 x}{200}-\frac{117649}{1936 (-1+2 x)^3}-\frac{134456}{1331 (-1+2 x)^2}-\frac{7383075}{117128 (-1+2 x)}+\frac{1}{166375 (3+5 x)^2}+\frac{204}{1830125 (3+5 x)}\right ) \, dx\\ &=\frac{117649}{7744 (1-2 x)^2}-\frac{67228}{1331 (1-2 x)}-\frac{31347 x}{2000}-\frac{729 x^2}{400}-\frac{1}{831875 (3+5 x)}-\frac{7383075 \log (1-2 x)}{234256}+\frac{204 \log (3+5 x)}{9150625}\\ \end{align*}

Mathematica [A]  time = 0.0372138, size = 74, normalized size = 1.12 \[ -\frac{729 (1-2 x)^2}{1600}+\frac{2187}{250} (1-2 x)+\frac{67228}{1331 (2 x-1)}-\frac{1}{831875 (5 x+3)}+\frac{117649}{7744 (1-2 x)^2}-\frac{7383075 \log (1-2 x)}{234256}+\frac{204 \log (10 x+6)}{9150625} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^6/((1 - 2*x)^3*(3 + 5*x)^2),x]

[Out]

117649/(7744*(1 - 2*x)^2) + (2187*(1 - 2*x))/250 - (729*(1 - 2*x)^2)/1600 + 67228/(1331*(-1 + 2*x)) - 1/(83187
5*(3 + 5*x)) - (7383075*Log[1 - 2*x])/234256 + (204*Log[6 + 10*x])/9150625

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Maple [A]  time = 0.009, size = 53, normalized size = 0.8 \begin{align*} -{\frac{729\,{x}^{2}}{400}}-{\frac{31347\,x}{2000}}+{\frac{117649}{7744\, \left ( 2\,x-1 \right ) ^{2}}}+{\frac{67228}{2662\,x-1331}}-{\frac{7383075\,\ln \left ( 2\,x-1 \right ) }{234256}}-{\frac{1}{2495625+4159375\,x}}+{\frac{204\,\ln \left ( 3+5\,x \right ) }{9150625}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^6/(1-2*x)^3/(3+5*x)^2,x)

[Out]

-729/400*x^2-31347/2000*x+117649/7744/(2*x-1)^2+67228/1331/(2*x-1)-7383075/234256*ln(2*x-1)-1/831875/(3+5*x)+2
04/9150625*ln(3+5*x)

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Maxima [A]  time = 2.62146, size = 73, normalized size = 1.11 \begin{align*} -\frac{729}{400} \, x^{2} - \frac{31347}{2000} \, x + \frac{26891199744 \, x^{2} + 6733304631 \, x - 5640849439}{53240000 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} + \frac{204}{9150625} \, \log \left (5 \, x + 3\right ) - \frac{7383075}{234256} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^6/(1-2*x)^3/(3+5*x)^2,x, algorithm="maxima")

[Out]

-729/400*x^2 - 31347/2000*x + 1/53240000*(26891199744*x^2 + 6733304631*x - 5640849439)/(20*x^3 - 8*x^2 - 7*x +
3) + 204/9150625*log(5*x + 3) - 7383075/234256*log(2*x - 1)

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Fricas [A]  time = 1.52363, size = 333, normalized size = 5.05 \begin{align*} -\frac{21346578000 \, x^{5} + 175041939600 \, x^{4} - 80903530620 \, x^{3} - 356854410264 \, x^{2} - 13056 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (5 \, x + 3\right ) + 18457687500 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (2 \, x - 1\right ) - 46529265321 \, x + 62049343829}{585640000 \,{\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)^6/(1-2*x)^3/(3+5*x)^2,x, algorithm="fricas")

[Out]

-1/585640000*(21346578000*x^5 + 175041939600*x^4 - 80903530620*x^3 - 356854410264*x^2 - 13056*(20*x^3 - 8*x^2
- 7*x + 3)*log(5*x + 3) + 18457687500*(20*x^3 - 8*x^2 - 7*x + 3)*log(2*x - 1) - 46529265321*x + 62049343829)/(
20*x^3 - 8*x^2 - 7*x + 3)

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Sympy [A]  time = 0.190832, size = 56, normalized size = 0.85 \begin{align*} - \frac{729 x^{2}}{400} - \frac{31347 x}{2000} + \frac{26891199744 x^{2} + 6733304631 x - 5640849439}{1064800000 x^{3} - 425920000 x^{2} - 372680000 x + 159720000} - \frac{7383075 \log{\left (x - \frac{1}{2} \right )}}{234256} + \frac{204 \log{\left (x + \frac{3}{5} \right )}}{9150625} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**6/(1-2*x)**3/(3+5*x)**2,x)

[Out]

-729*x**2/400 - 31347*x/2000 + (26891199744*x**2 + 6733304631*x - 5640849439)/(1064800000*x**3 - 425920000*x**
2 - 372680000*x + 159720000) - 7383075*log(x - 1/2)/234256 + 204*log(x + 3/5)/9150625

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Giac [A]  time = 2.36413, size = 127, normalized size = 1.92 \begin{align*} -\frac{{\left (5 \, x + 3\right )}^{2}{\left (\frac{555011028}{5 \, x + 3} - \frac{13845990449}{{\left (5 \, x + 3\right )}^{2}} + \frac{50757096489}{{\left (5 \, x + 3\right )}^{3}} + 21346578\right )}}{73205000 \,{\left (\frac{11}{5 \, x + 3} - 2\right )}^{2}} - \frac{1}{831875 \,{\left (5 \, x + 3\right )}} + \frac{315171}{10000} \, \log \left (\frac{{\left | 5 \, x + 3 \right |}}{5 \,{\left (5 \, x + 3\right )}^{2}}\right ) - \frac{7383075}{234256} \, \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)^6/(1-2*x)^3/(3+5*x)^2,x, algorithm="giac")

[Out]

-1/73205000*(5*x + 3)^2*(555011028/(5*x + 3) - 13845990449/(5*x + 3)^2 + 50757096489/(5*x + 3)^3 + 21346578)/(
11/(5*x + 3) - 2)^2 - 1/831875/(5*x + 3) + 315171/10000*log(1/5*abs(5*x + 3)/(5*x + 3)^2) - 7383075/234256*log
(abs(-11/(5*x + 3) + 2))