\(\int \frac {(2+3 x)^7}{(1-2 x)^2 (3+5 x)} \, dx\) [1589]

Optimal result

Integrand size = 22, antiderivative size = 65 \[ \int \frac {(2+3 x)^7}{(1-2 x)^2 (3+5 x)} \, dx=\frac {823543}{1408 (1-2 x)}+\frac {370109547 x}{200000}+\frac {18237069 x^2}{20000}+\frac {853659 x^3}{2000}+\frac {13851 x^4}{100}+\frac {2187 x^5}{100}+\frac {5764801 \log (1-2 x)}{3872}+\frac {\log (3+5 x)}{1890625} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {90} \[ \int \frac {(2+3 x)^7}{(1-2 x)^2 (3+5 x)} \, dx=\frac {2187 x^5}{100}+\frac {13851 x^4}{100}+\frac {853659 x^3}{2000}+\frac {18237069 x^2}{20000}+\frac {370109547 x}{200000}+\frac {823543}{1408 (1-2 x)}+\frac {5764801 \log (1-2 x)}{3872}+\frac {\log (5 x+3)}{1890625} \]

[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 90

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*} \text {integral}& = \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] (verified)

Time = 0.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.92 \[ \int \frac {(2+3 x)^7}{(1-2 x)^2 (3+5 x)} \, dx=\frac {\frac {11 \left (-158719988357-14798867886 x+306816622200 x^2+153656514000 x^3+78666390000 x^4+28066500000 x^5+4811400000 x^6\right )}{-1+2 x}[phone] \log (5-10 x)+640 \log (3+5 x)}{1210000000} \]

[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] (verified)

Time = 0.87 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.74

method	result	size
risch	\(\frac {2187 x^{5}}{100}+\frac {13851 x^{4}}{100}+\frac {853659 x^{3}}{2000}+\frac {18237069 x^{2}}{20000}+\frac {370109547 x}{200000}-\frac {823543}{2816 \left (x -\frac {1}{2}\right )}+\frac {5764801 \ln \left (-1+2 x \right )}{3872}+\frac {\ln \left (3+5 x \right )}{1890625}\)	\(48\)
default	\(\frac {2187 x^{5}}{100}+\frac {13851 x^{4}}{100}+\frac {853659 x^{3}}{2000}+\frac {18237069 x^{2}}{20000}+\frac {370109547 x}{200000}+\frac {\ln \left (3+5 x \right )}{1890625}-\frac {823543}{1408 \left (-1+2 x \right )}+\frac {5764801 \ln \left (-1+2 x \right )}{3872}\)	\(50\)
norman	\(\frac {-\frac {1661194223}{550000} x +\frac {139462101}{50000} x^{2}+\frac {6984387}{5000} x^{3}+\frac {715149}{1000} x^{4}+\frac {5103}{20} x^{5}+\frac {2187}{50} x^{6}}{-1+2 x}+\frac {5764801 \ln \left (-1+2 x \right )}{3872}+\frac {\ln \left (3+5 x \right )}{1890625}\)	\(55\)
parallelrisch	\(\frac {2646270000 x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}+64 \ln \left (x +\frac {3}{5}\right ) x [phone] \ln \left (x -\frac {1}{2}\right ) x [phone] x^{2}-32 \ln \left (x +\frac {3}{5}\right )-90075015625 \ln \left (x -\frac {1}{2}\right )-182731364530 x}{-60500000+121000000 x}\)	\(65\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00 \[ \int \frac {(2+3 x)^7}{(1-2 x)^2 (3+5 x)} \, dx=\frac {10585080000 \, x^{6} + 61746300000 \, x^{5} + 173066058000 \, x^{4} + 338044330800 \, x^{3} + 674996568840 \, x^{2} + 128 \, {\left (2 \, x - 1\right )} \log \left (5 \, x + 3\right ) + 360300062500 \, {\left (2 \, x - 1\right )} \log \left (2 \, x - 1\right ) - 447832551870 \, x - 141546453125}{242000000 \, {\left (2 \, x - 1\right )}} \]

[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] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.86 \[ \int \frac {(2+3 x)^7}{(1-2 x)^2 (3+5 x)} \, dx=\frac {2187 x^{5}}{100} + \frac {13851 x^{4}}{100} + \frac {853659 x^{3}}{2000} + \frac {18237069 x^{2}}{20000} + \frac {370109547 x}{200000} + \frac {5764801 \log {\left (x - \frac {1}{2} \right )}}{3872} + \frac {\log {\left (x + \frac {3}{5} \right )}}{1890625} - \frac {823543}{2816 x - 1408} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.75 \[ \int \frac {(2+3 x)^7}{(1-2 x)^2 (3+5 x)} \, dx=\frac {2187}{100} \, x^{5} + \frac {13851}{100} \, x^{4} + \frac {853659}{2000} \, x^{3} + \frac {18237069}{20000} \, x^{2} + \frac {370109547}{200000} \, x - \frac {823543}{1408 \, {\left (2 \, x - 1\right )}} + \frac {1}{1890625} \, \log \left (5 \, x + 3\right ) + \frac {5764801}{3872} \, \log \left (2 \, x - 1\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.38 \[ \int \frac {(2+3 x)^7}{(1-2 x)^2 (3+5 x)} \, dx=\frac {27}{400000} \, {\left (2 \, x - 1\right )}^{5} {\left (\frac {178875}{2 \, x - 1} + \frac {1404675}{{\left (2 \, x - 1\right )}^{2}} + \frac {6619260}{{\left (2 \, x - 1\right )}^{3}} + \frac {23397131}{{\left (2 \, x - 1\right )}^{4}} + 10125\right )} - \frac {823543}{1408 \, {\left (2 \, x - 1\right )}} - \frac {744421617}{500000} \, \log \left (\frac {{\left | 2 \, x - 1 \right |}}{2 \, {\left (2 \, x - 1\right )}^{2}}\right ) + \frac {1}{1890625} \, \log \left ({\left | -\frac {11}{2 \, x - 1} - 5 \right |}\right ) \]

[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))

Mupad [B] (verification not implemented)

Time = 1.30 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.69 \[ \int \frac {(2+3 x)^7}{(1-2 x)^2 (3+5 x)} \, dx=\frac {370109547\,x}{200000}+\frac {5764801\,\ln \left (x-\frac {1}{2}\right )}{3872}+\frac {\ln \left (x+\frac {3}{5}\right )}{1890625}-\frac {823543}{2816\,\left (x-\frac {1}{2}\right )}+\frac {18237069\,x^2}{20000}+\frac {853659\,x^3}{2000}+\frac {13851\,x^4}{100}+\frac {2187\,x^5}{100} \]

[In]

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

[Out]

(370109547*x)/200000 + (5764801*log(x - 1/2))/3872 + log(x + 3/5)/1890625 - 823543/(2816*(x - 1/2)) + (1823706
9*x^2)/20000 + (853659*x^3)/2000 + (13851*x^4)/100 + (2187*x^5)/100