\(\int \frac {(2+3 x)^8}{(1-2 x) (3+5 x)} \, dx\) [1485]

Optimal result

Integrand size = 22, antiderivative size = 68 \[ \int \frac {(2+3 x)^8}{(1-2 x) (3+5 x)} \, dx=-\frac {40089855591 x}{10000000}-\frac {7136193339 x^2}{2000000}-\frac {345533877 x^3}{100000}-\frac {111146499 x^4}{40000}-\frac {8018271 x^5}{5000}-\frac {114453 x^6}{200}-\frac {6561 x^7}{70}-\frac {5764801 \log (1-2 x)}{2816}+\frac {\log (3+5 x)}{4296875} \]

[Out]

-40089855591/10000000*x-7136193339/2000000*x^2-345533877/100000*x^3-111146499/40000*x^4-8018271/5000*x^5-11445
3/200*x^6-6561/70*x^7-5764801/2816*ln(1-2*x)+1/4296875*ln(3+5*x)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 68, 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 = {84} \[ \int \frac {(2+3 x)^8}{(1-2 x) (3+5 x)} \, dx=-\frac {6561 x^7}{70}-\frac {114453 x^6}{200}-\frac {8018271 x^5}{5000}-\frac {111146499 x^4}{40000}-\frac {345533877 x^3}{100000}-\frac {7136193339 x^2}{2000000}-\frac {40089855591 x}{10000000}-\frac {5764801 \log (1-2 x)}{2816}+\frac {\log (5 x+3)}{4296875} \]

[In]

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

[Out]

(-40089855591*x)/10000000 - (7136193339*x^2)/2000000 - (345533877*x^3)/100000 - (111146499*x^4)/40000 - (80182
71*x^5)/5000 - (114453*x^6)/200 - (6561*x^7)/70 - (5764801*Log[1 - 2*x])/2816 + Log[3 + 5*x]/4296875

Rule 84

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {40089855591}{10000000}-\frac {7136193339 x}{1000000}-\frac {1036601631 x^2}{100000}-\frac {111146499 x^3}{10000}-\frac {8018271 x^4}{1000}-\frac {343359 x^5}{100}-\frac {6561 x^6}{10}-\frac {5764801}{1408 (-1+2 x)}+\frac {1}{859375 (3+5 x)}\right ) \, dx \\ & = -\frac {40089855591 x}{10000000}-\frac {7136193339 x^2}{2000000}-\frac {345533877 x^3}{100000}-\frac {111146499 x^4}{40000}-\frac {8018271 x^5}{5000}-\frac {114453 x^6}{200}-\frac {6561 x^7}{70}-\frac {5764801 \log (1-2 x)}{2816}+\frac {\log (3+5 x)}{4296875} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.91 \[ \int \frac {(2+3 x)^8}{(1-2 x) (3+5 x)} \, dx=-\frac {3 \left (40324556806+93542996379 x+83255588955 x^2+80624571300 x^3+64835457750 x^4+37418598000 x^5+13352850000 x^6+2187000000 x^7\right )}{70000000}-\frac {5764801 \log (3-6 x)}{2816}+\frac {\log (-3 (3+5 x))}{4296875} \]

[In]

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

[Out]

(-3*(40324556806 + 93542996379*x + 83255588955*x^2 + 80624571300*x^3 + 64835457750*x^4 + 37418598000*x^5 + 133
52850000*x^6 + 2187000000*x^7))/70000000 - (5764801*Log[3 - 6*x])/2816 + Log[-3*(3 + 5*x)]/4296875

Maple [A] (verified)

Time = 0.83 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.69

method	result	size
parallelrisch	\(-\frac {6561 x^{7}}{70}-\frac {114453 x^{6}}{200}-\frac {8018271 x^{5}}{5000}-\frac {111146499 x^{4}}{40000}-\frac {345533877 x^{3}}{100000}-\frac {7136193339 x^{2}}{2000000}-\frac {40089855591 x}{10000000}+\frac {\ln \left (x +\frac {3}{5}\right )}{4296875}-\frac {5764801 \ln \left (x -\frac {1}{2}\right )}{2816}\)	\(47\)
default	\(-\frac {6561 x^{7}}{70}-\frac {114453 x^{6}}{200}-\frac {8018271 x^{5}}{5000}-\frac {111146499 x^{4}}{40000}-\frac {345533877 x^{3}}{100000}-\frac {7136193339 x^{2}}{2000000}-\frac {40089855591 x}{10000000}+\frac {\ln \left (3+5 x \right )}{4296875}-\frac {5764801 \ln \left (-1+2 x \right )}{2816}\)	\(51\)
norman	\(-\frac {6561 x^{7}}{70}-\frac {114453 x^{6}}{200}-\frac {8018271 x^{5}}{5000}-\frac {111146499 x^{4}}{40000}-\frac {345533877 x^{3}}{100000}-\frac {7136193339 x^{2}}{2000000}-\frac {40089855591 x}{10000000}+\frac {\ln \left (3+5 x \right )}{4296875}-\frac {5764801 \ln \left (-1+2 x \right )}{2816}\)	\(51\)
risch	\(-\frac {6561 x^{7}}{70}-\frac {114453 x^{6}}{200}-\frac {8018271 x^{5}}{5000}-\frac {111146499 x^{4}}{40000}-\frac {345533877 x^{3}}{100000}-\frac {7136193339 x^{2}}{2000000}-\frac {40089855591 x}{10000000}+\frac {\ln \left (3+5 x \right )}{4296875}-\frac {5764801 \ln \left (-1+2 x \right )}{2816}\)	\(51\)

[In]

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

[Out]

-6561/70*x^7-114453/200*x^6-8018271/5000*x^5-111146499/40000*x^4-345533877/100000*x^3-7136193339/2000000*x^2-4
0089855591/10000000*x+1/4296875*ln(x+3/5)-5764801/2816*ln(x-1/2)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.74 \[ \int \frac {(2+3 x)^8}{(1-2 x) (3+5 x)} \, dx=-\frac {6561}{70} \, x^{7} - \frac {114453}{200} \, x^{6} - \frac {8018271}{5000} \, x^{5} - \frac {111146499}{40000} \, x^{4} - \frac {345533877}{100000} \, x^{3} - \frac {7136193339}{2000000} \, x^{2} - \frac {40089855591}{10000000} \, x + \frac {1}{4296875} \, \log \left (5 \, x + 3\right ) - \frac {5764801}{2816} \, \log \left (2 \, x - 1\right ) \]

[In]

integrate((2+3*x)^8/(1-2*x)/(3+5*x),x, algorithm="fricas")

[Out]

-6561/70*x^7 - 114453/200*x^6 - 8018271/5000*x^5 - 111146499/40000*x^4 - 345533877/100000*x^3 - 7136193339/200
0000*x^2 - 40089855591/10000000*x + 1/4296875*log(5*x + 3) - 5764801/2816*log(2*x - 1)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.93 \[ \int \frac {(2+3 x)^8}{(1-2 x) (3+5 x)} \, dx=- \frac {6561 x^{7}}{70} - \frac {114453 x^{6}}{200} - \frac {8018271 x^{5}}{5000} - \frac {111146499 x^{4}}{40000} - \frac {345533877 x^{3}}{100000} - \frac {7136193339 x^{2}}{2000000} - \frac {40089855591 x}{10000000} - \frac {5764801 \log {\left (x - \frac {1}{2} \right )}}{2816} + \frac {\log {\left (x + \frac {3}{5} \right )}}{4296875} \]

[In]

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

[Out]

-6561*x**7/70 - 114453*x**6/200 - 8018271*x**5/5000 - 111146499*x**4/40000 - 345533877*x**3/100000 - 713619333
9*x**2/2000000 - 40089855591*x/10000000 - 5764801*log(x - 1/2)/2816 + log(x + 3/5)/4296875

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.74 \[ \int \frac {(2+3 x)^8}{(1-2 x) (3+5 x)} \, dx=-\frac {6561}{70} \, x^{7} - \frac {114453}{200} \, x^{6} - \frac {8018271}{5000} \, x^{5} - \frac {111146499}{40000} \, x^{4} - \frac {345533877}{100000} \, x^{3} - \frac {7136193339}{2000000} \, x^{2} - \frac {40089855591}{10000000} \, x + \frac {1}{4296875} \, \log \left (5 \, x + 3\right ) - \frac {5764801}{2816} \, \log \left (2 \, x - 1\right ) \]

[In]

integrate((2+3*x)^8/(1-2*x)/(3+5*x),x, algorithm="maxima")

[Out]

-6561/70*x^7 - 114453/200*x^6 - 8018271/5000*x^5 - 111146499/40000*x^4 - 345533877/100000*x^3 - 7136193339/200
0000*x^2 - 40089855591/10000000*x + 1/4296875*log(5*x + 3) - 5764801/2816*log(2*x - 1)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.76 \[ \int \frac {(2+3 x)^8}{(1-2 x) (3+5 x)} \, dx=-\frac {6561}{70} \, x^{7} - \frac {114453}{200} \, x^{6} - \frac {8018271}{5000} \, x^{5} - \frac {111146499}{40000} \, x^{4} - \frac {345533877}{100000} \, x^{3} - \frac {7136193339}{2000000} \, x^{2} - \frac {40089855591}{10000000} \, x + \frac {1}{4296875} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {5764801}{2816} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]

[In]

integrate((2+3*x)^8/(1-2*x)/(3+5*x),x, algorithm="giac")

[Out]

-6561/70*x^7 - 114453/200*x^6 - 8018271/5000*x^5 - 111146499/40000*x^4 - 345533877/100000*x^3 - 7136193339/200
0000*x^2 - 40089855591/10000000*x + 1/4296875*log(abs(5*x + 3)) - 5764801/2816*log(abs(2*x - 1))

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.68 \[ \int \frac {(2+3 x)^8}{(1-2 x) (3+5 x)} \, dx=\frac {\ln \left (x+\frac {3}{5}\right )}{4296875}-\frac {5764801\,\ln \left (x-\frac {1}{2}\right )}{2816}-\frac {40089855591\,x}{10000000}-\frac {7136193339\,x^2}{2000000}-\frac {345533877\,x^3}{100000}-\frac {111146499\,x^4}{40000}-\frac {8018271\,x^5}{5000}-\frac {114453\,x^6}{200}-\frac {6561\,x^7}{70} \]

[In]

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

[Out]

log(x + 3/5)/4296875 - (5764801*log(x - 1/2))/2816 - (40089855591*x)/10000000 - (7136193339*x^2)/2000000 - (34
5533877*x^3)/100000 - (111146499*x^4)/40000 - (8018271*x^5)/5000 - (114453*x^6)/200 - (6561*x^7)/70