\(\int \frac {(1-2 x)^2 (3+5 x)^3}{(2+3 x)^6} \, dx\) [1286]

Optimal result

Integrand size = 22, antiderivative size = 66 \[ \int \frac {(1-2 x)^2 (3+5 x)^3}{(2+3 x)^6} \, dx=\frac {49}{3645 (2+3 x)^5}-\frac {763}{2916 (2+3 x)^4}+\frac {4099}{2187 (2+3 x)^3}-\frac {8285}{1458 (2+3 x)^2}+\frac {3800}{729 (2+3 x)}+\frac {500}{729} \log (2+3 x) \]

[Out]

49/3645/(2+3*x)^5-763/2916/(2+3*x)^4+4099/2187/(2+3*x)^3-8285/1458/(2+3*x)^2+3800/729/(2+3*x)+500/729*ln(2+3*x
)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 66, 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} \[ \int \frac {(1-2 x)^2 (3+5 x)^3}{(2+3 x)^6} \, dx=\frac {3800}{729 (3 x+2)}-\frac {8285}{1458 (3 x+2)^2}+\frac {4099}{2187 (3 x+2)^3}-\frac {763}{2916 (3 x+2)^4}+\frac {49}{3645 (3 x+2)^5}+\frac {500}{729} \log (3 x+2) \]

[In]

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

[Out]

49/(3645*(2 + 3*x)^5) - 763/(2916*(2 + 3*x)^4) + 4099/(2187*(2 + 3*x)^3) - 8285/(1458*(2 + 3*x)^2) + 3800/(729
*(2 + 3*x)) + (500*Log[2 + 3*x])/729

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 {49}{243 (2+3 x)^6}+\frac {763}{243 (2+3 x)^5}-\frac {4099}{243 (2+3 x)^4}+\frac {8285}{243 (2+3 x)^3}-\frac {3800}{243 (2+3 x)^2}+\frac {500}{243 (2+3 x)}\right ) \, dx \\ & = \frac {49}{3645 (2+3 x)^5}-\frac {763}{2916 (2+3 x)^4}+\frac {4099}{2187 (2+3 x)^3}-\frac {8285}{1458 (2+3 x)^2}+\frac {3800}{729 (2+3 x)}+\frac {500}{729} \log (2+3 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.70 \[ \int \frac {(1-2 x)^2 (3+5 x)^3}{(2+3 x)^6} \, dx=\frac {1965218+13889625 x+36564120 x^2+42537150 x^3+18468000 x^4+30000 (2+3 x)^5 \log (20+30 x)}{43740 (2+3 x)^5} \]

[In]

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

[Out]

(1965218 + 13889625*x + 36564120*x^2 + 42537150*x^3 + 18468000*x^4 + 30000*(2 + 3*x)^5*Log[20 + 30*x])/(43740*
(2 + 3*x)^5)

Maple [A] (verified)

Time = 2.34 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.58

method	result	size
norman	\(\frac {\frac {1945}{2} x^{3}+\frac {3800}{9} x^{4}+\frac {203134}{243} x^{2}+\frac {925975}{2916} x +\frac {982609}{21870}}{\left (2+3 x \right )^{5}}+\frac {500 \ln \left (2+3 x \right )}{729}\)	\(38\)
risch	\(\frac {\frac {1945}{2} x^{3}+\frac {3800}{9} x^{4}+\frac {203134}{243} x^{2}+\frac {925975}{2916} x +\frac {982609}{21870}}{\left (2+3 x \right )^{5}}+\frac {500 \ln \left (2+3 x \right )}{729}\)	\(39\)
default	\(\frac {49}{3645 \left (2+3 x \right )^{5}}-\frac {763}{2916 \left (2+3 x \right )^{4}}+\frac {4099}{2187 \left (2+3 x \right )^{3}}-\frac {8285}{1458 \left (2+3 x \right )^{2}}+\frac {3800}{729 \left (2+3 x \right )}+\frac {500 \ln \left (2+3 x \right )}{729}\)	\(55\)
parallelrisch	\(\frac {38880000 \ln \left (\frac {2}{3}+x \right ) x^{5}[phone] \ln \left (\frac {2}{3}+x \right ) x^{4}-79591329 x^{5}[phone] \ln \left (\frac {2}{3}+x \right ) x^{3}-166808430 x^{4}[phone] \ln \left (\frac {2}{3}+x \right ) x^{2}-126874440 x^{3}[phone] \ln \left (\frac {2}{3}+x \right ) x -40817520 x^{2}+5120000 \ln \left (\frac {2}{3}+x \right )-4530720 x}{233280 \left (2+3 x \right )^{5}}\)	\(83\)
meijerg	\(\frac {27 x \left (\frac {81}{16} x^{4}+\frac {135}{8} x^{3}+\frac {45}{2} x^{2}+15 x +5\right )}{320 \left (1+\frac {3 x}{2}\right )^{5}}+\frac {27 x^{2} \left (\frac {27}{8} x^{3}+\frac {45}{4} x^{2}+15 x +10\right )}{1280 \left (1+\frac {3 x}{2}\right )^{5}}-\frac {69 x^{3} \left (\frac {9}{4} x^{2}+\frac {15}{2} x +10\right )}{640 \left (1+\frac {3 x}{2}\right )^{5}}-\frac {47 x^{4} \left (\frac {3 x}{2}+5\right )}{256 \left (1+\frac {3 x}{2}\right )^{5}}+\frac {5 x^{5}}{4 \left (1+\frac {3 x}{2}\right )^{5}}-\frac {25 x \left (\frac {11097}{16} x^{4}+\frac {10395}{8} x^{3}+\frac {2115}{2} x^{2}+405 x +60\right )}{1458 \left (1+\frac {3 x}{2}\right )^{5}}+\frac {500 \ln \left (1+\frac {3 x}{2}\right )}{729}\)	\(148\)

[In]

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

[Out]

(1945/2*x^3+3800/9*x^4+203134/243*x^2+925975/2916*x+982609/21870)/(2+3*x)^5+500/729*ln(2+3*x)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.24 \[ \int \frac {(1-2 x)^2 (3+5 x)^3}{(2+3 x)^6} \, dx=\frac {18468000 \, x^{4} + 42537150 \, x^{3} + 36564120 \, x^{2} + 30000 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (3 \, x + 2\right ) + 13889625 \, x + 1965218}{43740 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

[In]

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

[Out]

1/43740*(18468000*x^4 + 42537150*x^3 + 36564120*x^2 + 30000*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x +
32)*log(3*x + 2) + 13889625*x + 1965218)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.82 \[ \int \frac {(1-2 x)^2 (3+5 x)^3}{(2+3 x)^6} \, dx=\frac {18468000 x^{4} + 42537150 x^{3} + 36564120 x^{2} + 13889625 x + 1965218}{10628820 x^{5} + 35429400 x^{4} + 47239200 x^{3} + 31492800 x^{2} + 10497600 x + 1399680} + \frac {500 \log {\left (3 x + 2 \right )}}{729} \]

[In]

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

[Out]

(18468000*x**4 + 42537150*x**3 + 36564120*x**2 + 13889625*x + 1965218)/(10628820*x**5 + 35429400*x**4 + 472392
00*x**3 + 31492800*x**2 + 10497600*x + 1399680) + 500*log(3*x + 2)/729

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.88 \[ \int \frac {(1-2 x)^2 (3+5 x)^3}{(2+3 x)^6} \, dx=\frac {18468000 \, x^{4} + 42537150 \, x^{3} + 36564120 \, x^{2} + 13889625 \, x + 1965218}{43740 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {500}{729} \, \log \left (3 \, x + 2\right ) \]

[In]

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

[Out]

1/43740*(18468000*x^4 + 42537150*x^3 + 36564120*x^2 + 13889625*x + 1965218)/(243*x^5 + 810*x^4 + 1080*x^3 + 72
0*x^2 + 240*x + 32) + 500/729*log(3*x + 2)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.59 \[ \int \frac {(1-2 x)^2 (3+5 x)^3}{(2+3 x)^6} \, dx=\frac {18468000 \, x^{4} + 42537150 \, x^{3} + 36564120 \, x^{2} + 13889625 \, x + 1965218}{43740 \, {\left (3 \, x + 2\right )}^{5}} + \frac {500}{729} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \]

[In]

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

[Out]

1/43740*(18468000*x^4 + 42537150*x^3 + 36564120*x^2 + 13889625*x + 1965218)/(3*x + 2)^5 + 500/729*log(abs(3*x
+ 2))

Mupad [B] (verification not implemented)

Time = 1.36 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.80 \[ \int \frac {(1-2 x)^2 (3+5 x)^3}{(2+3 x)^6} \, dx=\frac {500\,\ln \left (x+\frac {2}{3}\right )}{729}+\frac {\frac {3800\,x^4}{2187}+\frac {1945\,x^3}{486}+\frac {203134\,x^2}{59049}+\frac {925975\,x}{708588}+\frac {982609}{5314410}}{x^5+\frac {10\,x^4}{3}+\frac {40\,x^3}{9}+\frac {80\,x^2}{27}+\frac {80\,x}{81}+\frac {32}{243}} \]

[In]

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

[Out]

(500*log(x + 2/3))/729 + ((925975*x)/708588 + (203134*x^2)/59049 + (1945*x^3)/486 + (3800*x^4)/2187 + 982609/5
314410)/((80*x)/81 + (80*x^2)/27 + (40*x^3)/9 + (10*x^4)/3 + x^5 + 32/243)