3.1589 \(\int \frac{(2+3 x)^7}{(1-2 x)^2 (3+5 x)} \, dx\)

Optimal. Leaf size=65 \[ \frac{2187 x^5}{100}+\frac{13851 x^4}{100}+\frac{853659 x^3}{2000}+\frac{18237069 x^2}{20000}+\frac{370109547 x}{200000}+\frac{823543}{1408 (1-2 x)}+\frac{5764801 \log (1-2 x)}{3872}+\frac{\log (5 x+3)}{1890625} \]

[Out]

823543/(1408*(1 - 2*x)) + (370109547*x)/200000 + (18237069*x^2)/20000 + (853659*x^3)/2000 + (13851*x^4)/100 +
(2187*x^5)/100 + (5764801*Log[1 - 2*x])/3872 + Log[3 + 5*x]/1890625

________________________________________________________________________________________

Rubi [A]  time = 0.0312579, antiderivative size = 65, 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} \[ \frac{2187 x^5}{100}+\frac{13851 x^4}{100}+\frac{853659 x^3}{2000}+\frac{18237069 x^2}{20000}+\frac{370109547 x}{200000}+\frac{823543}{1408 (1-2 x)}+\frac{5764801 \log (1-2 x)}{3872}+\frac{\log (5 x+3)}{1890625} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^7/((1 - 2*x)^2*(3 + 5*x)),x]

[Out]

823543/(1408*(1 - 2*x)) + (370109547*x)/200000 + (18237069*x^2)/20000 + (853659*x^3)/2000 + (13851*x^4)/100 +
(2187*x^5)/100 + (5764801*Log[1 - 2*x])/3872 + Log[3 + 5*x]/1890625

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)^7}{(1-2 x)^2 (3+5 x)} \, dx &=\int \left (\frac{370109547}{200000}+\frac{18237069 x}{10000}+\frac{2560977 x^2}{2000}+\frac{13851 x^3}{25}+\frac{2187 x^4}{20}+\frac{823543}{704 (-1+2 x)^2}+\frac{5764801}{1936 (-1+2 x)}+\frac{1}{378125 (3+5 x)}\right ) \, dx\\ &=\frac{823543}{1408 (1-2 x)}+\frac{370109547 x}{200000}+\frac{18237069 x^2}{20000}+\frac{853659 x^3}{2000}+\frac{13851 x^4}{100}+\frac{2187 x^5}{100}+\frac{5764801 \log (1-2 x)}{3872}+\frac{\log (3+5 x)}{1890625}\\ \end{align*}

Mathematica [A]  time = 0.0324332, size = 60, normalized size = 0.92 \[ \frac{\frac{11 \left (4811400000 x^6+28066500000 x^5+78666390000 x^4+153656514000 x^3+306816622200 x^2-14798867886 x-158719988357\right )}{2 x-1}[phone] \log (5-10 x)+640 \log (5 x+3)}{1210000000} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^7/((1 - 2*x)^2*(3 + 5*x)),x]

[Out]

((11*(-158719988357 - 14798867886*x + 306816622200*x^2 + 153656514000*x^3 + 78666390000*x^4 + 28066500000*x^5
+ 4811400000*x^6))/(-1 + 2*x) + 1801500312500*Log[5 - 10*x] + 640*Log[3 + 5*x])/1210000000

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 50, normalized size = 0.8 \begin{align*}{\frac{2187\,{x}^{5}}{100}}+{\frac{13851\,{x}^{4}}{100}}+{\frac{853659\,{x}^{3}}{2000}}+{\frac{18237069\,{x}^{2}}{20000}}+{\frac{370109547\,x}{200000}}-{\frac{823543}{2816\,x-1408}}+{\frac{5764801\,\ln \left ( 2\,x-1 \right ) }{3872}}+{\frac{\ln \left ( 3+5\,x \right ) }{1890625}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^7/(1-2*x)^2/(3+5*x),x)

[Out]

2187/100*x^5+13851/100*x^4+853659/2000*x^3+18237069/20000*x^2+370109547/200000*x-823543/1408/(2*x-1)+5764801/3
872*ln(2*x-1)+1/1890625*ln(3+5*x)

________________________________________________________________________________________

Maxima [A]  time = 1.06816, size = 66, normalized size = 1.02 \begin{align*} \frac{2187}{100} \, x^{5} + \frac{13851}{100} \, x^{4} + \frac{853659}{2000} \, x^{3} + \frac{18237069}{20000} \, x^{2} + \frac{370109547}{200000} \, x - \frac{823543}{1408 \,{\left (2 \, x - 1\right )}} + \frac{1}{1890625} \, \log \left (5 \, x + 3\right ) + \frac{5764801}{3872} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^7/(1-2*x)^2/(3+5*x),x, algorithm="maxima")

[Out]

2187/100*x^5 + 13851/100*x^4 + 853659/2000*x^3 + 18237069/20000*x^2 + 370109547/200000*x - 823543/1408/(2*x -
1) + 1/1890625*log(5*x + 3) + 5764801/3872*log(2*x - 1)

________________________________________________________________________________________

Fricas [A]  time = 1.3075, size = 290, normalized size = 4.46 \begin{align*} \frac{10585080000 \, x^{6} + 61746300000 \, x^{5} + 173066058000 \, x^{4} + 338044330800 \, x^{3} + 674996568840 \, x^{2} + 128 \,{\left (2 \, x - 1\right )} \log \left (5 \, x + 3\right ) + 360300062500 \,{\left (2 \, x - 1\right )} \log \left (2 \, x - 1\right ) - 447832551870 \, x - 141546453125}{242000000 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^7/(1-2*x)^2/(3+5*x),x, algorithm="fricas")

[Out]

1/242000000*(10585080000*x^6 + 61746300000*x^5 + 173066058000*x^4 + 338044330800*x^3 + 674996568840*x^2 + 128*
(2*x - 1)*log(5*x + 3) + 360300062500*(2*x - 1)*log(2*x - 1) - 447832551870*x - 141546453125)/(2*x - 1)

________________________________________________________________________________________

Sympy [A]  time = 0.139685, size = 56, normalized size = 0.86 \begin{align*} \frac{2187 x^{5}}{100} + \frac{13851 x^{4}}{100} + \frac{853659 x^{3}}{2000} + \frac{18237069 x^{2}}{20000} + \frac{370109547 x}{200000} + \frac{5764801 \log{\left (x - \frac{1}{2} \right )}}{3872} + \frac{\log{\left (x + \frac{3}{5} \right )}}{1890625} - \frac{823543}{2816 x - 1408} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**7/(1-2*x)**2/(3+5*x),x)

[Out]

2187*x**5/100 + 13851*x**4/100 + 853659*x**3/2000 + 18237069*x**2/20000 + 370109547*x/200000 + 5764801*log(x -
1/2)/3872 + log(x + 3/5)/1890625 - 823543/(2816*x - 1408)

________________________________________________________________________________________

Giac [A]  time = 3.09818, size = 122, normalized size = 1.88 \begin{align*} \frac{27}{400000} \,{\left (2 \, x - 1\right )}^{5}{\left (\frac{178875}{2 \, x - 1} + \frac{1404675}{{\left (2 \, x - 1\right )}^{2}} + \frac{6619260}{{\left (2 \, x - 1\right )}^{3}} + \frac{23397131}{{\left (2 \, x - 1\right )}^{4}} + 10125\right )} - \frac{823543}{1408 \,{\left (2 \, x - 1\right )}} - \frac{744421617}{500000} \, \log \left (\frac{{\left | 2 \, x - 1 \right |}}{2 \,{\left (2 \, x - 1\right )}^{2}}\right ) + \frac{1}{1890625} \, \log \left ({\left | -\frac{11}{2 \, x - 1} - 5 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^7/(1-2*x)^2/(3+5*x),x, algorithm="giac")

[Out]

27/400000*(2*x - 1)^5*(178875/(2*x - 1) + 1404675/(2*x - 1)^2 + 6619260/(2*x - 1)^3 + 23397131/(2*x - 1)^4 + 1
0125) - 823543/1408/(2*x - 1) - 744421617/500000*log(1/2*abs(2*x - 1)/(2*x - 1)^2) + 1/1890625*log(abs(-11/(2*
x - 1) - 5))