\(\int \frac {(1-2 x)^3 (3+5 x)^3}{(2+3 x)^5} \, dx\) [1384]

Optimal result

Integrand size = 22, antiderivative size = 67 \[ \int \frac {(1-2 x)^3 (3+5 x)^3}{(2+3 x)^5} \, dx=\frac {9100 x}{729}-\frac {500 x^2}{243}+\frac {343}{8748 (2+3 x)^4}-\frac {1813}{2187 (2+3 x)^3}+\frac {10073}{1458 (2+3 x)^2}-\frac {66193}{2187 (2+3 x)}-\frac {14390}{729} \log (2+3 x) \]

[Out]

9100/729*x-500/243*x^2+343/8748/(2+3*x)^4-1813/2187/(2+3*x)^3+10073/1458/(2+3*x)^2-66193/2187/(2+3*x)-14390/72
9*ln(2+3*x)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 67, 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} \[ \int \frac {(1-2 x)^3 (3+5 x)^3}{(2+3 x)^5} \, dx=-\frac {500 x^2}{243}+\frac {9100 x}{729}-\frac {66193}{2187 (3 x+2)}+\frac {10073}{1458 (3 x+2)^2}-\frac {1813}{2187 (3 x+2)^3}+\frac {343}{8748 (3 x+2)^4}-\frac {14390}{729} \log (3 x+2) \]

[In]

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

[Out]

(9100*x)/729 - (500*x^2)/243 + 343/(8748*(2 + 3*x)^4) - 1813/(2187*(2 + 3*x)^3) + 10073/(1458*(2 + 3*x)^2) - 6
6193/(2187*(2 + 3*x)) - (14390*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 {9100}{729}-\frac {1000 x}{243}-\frac {343}{729 (2+3 x)^5}+\frac {1813}{243 (2+3 x)^4}-\frac {10073}{243 (2+3 x)^3}+\frac {66193}{729 (2+3 x)^2}-\frac {14390}{243 (2+3 x)}\right ) \, dx \\ & = \frac {9100 x}{729}-\frac {500 x^2}{243}+\frac {343}{8748 (2+3 x)^4}-\frac {1813}{2187 (2+3 x)^3}+\frac {10073}{1458 (2+3 x)^2}-\frac {66193}{2187 (2+3 x)}-\frac {14390}{729} \log (2+3 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.84 \[ \int \frac {(1-2 x)^3 (3+5 x)^3}{(2+3 x)^5} \, dx=\frac {-597785+675708 x+13894254 x^2+32163156 x^3+26244000 x^4+4957200 x^5-1458000 x^6-172680 (2+3 x)^4 \log (2+3 x)}{8748 (2+3 x)^4} \]

[In]

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

[Out]

(-597785 + 675708*x + 13894254*x^2 + 32163156*x^3 + 26244000*x^4 + 4957200*x^5 - 1458000*x^6 - 172680*(2 + 3*x
)^4*Log[2 + 3*x])/(8748*(2 + 3*x)^4)

Maple [A] (verified)

Time = 2.44 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.63

method	result	size
risch	\(-\frac {500 x^{2}}{243}+\frac {9100 x}{729}+\frac {-\frac {66193}{81} x^{3}-\frac {254699}{162} x^{2}-\frac {735691}{729} x -\frac {210065}{972}}{\left (2+3 x \right )^{4}}-\frac {14390 \ln \left (2+3 x \right )}{729}\)	\(42\)
norman	\(\frac {-\frac {2535691}{729} x -\frac {116731}{27} x^{3}-\frac {115411}{18} x^{2}+\frac {1700}{3} x^{5}-\frac {500}{3} x^{6}-\frac {5781785}{8748}}{\left (2+3 x \right )^{4}}-\frac {14390 \ln \left (2+3 x \right )}{729}\)	\(43\)
default	\(\frac {9100 x}{729}-\frac {500 x^{2}}{243}+\frac {343}{8748 \left (2+3 x \right )^{4}}-\frac {1813}{2187 \left (2+3 x \right )^{3}}+\frac {10073}{1458 \left (2+3 x \right )^{2}}-\frac {66193}{2187 \left (2+3 x \right )}-\frac {14390 \ln \left (2+3 x \right )}{729}\)	\(54\)
parallelrisch	\(-\frac {7776000 x^{6}[phone] \ln \left (\frac {2}{3}+x \right ) x^{4}-26438400 x^{5}[phone] \ln \left (\frac {2}{3}+x \right ) x^{3}-156108195 x^{4}[phone] \ln \left (\frac {2}{3}+x \right ) x^{2}-214577352 x^{3}[phone] \ln \left (\frac {2}{3}+x \right ) x -117143208 x^{2}[phone] \ln \left (\frac {2}{3}+x \right )-22732896 x}{46656 \left (2+3 x \right )^{4}}\)	\(79\)
meijerg	\(\frac {27 x \left (\frac {27}{8} x^{3}+9 x^{2}+9 x +4\right )}{128 \left (1+\frac {3 x}{2}\right )^{4}}-\frac {9 x^{2} \left (\frac {9}{4} x^{2}+6 x +6\right )}{128 \left (1+\frac {3 x}{2}\right )^{4}}-\frac {87 x^{3} \left (\frac {3 x}{2}+4\right )}{128 \left (1+\frac {3 x}{2}\right )^{4}}+\frac {179 x^{4}}{128 \left (1+\frac {3 x}{2}\right )^{4}}-\frac {29 x \left (\frac {3375}{8} x^{3}+585 x^{2}+315 x +60\right )}{324 \left (1+\frac {3 x}{2}\right )^{4}}-\frac {14390 \ln \left (1+\frac {3 x}{2}\right )}{729}-\frac {25 x \left (\frac {243}{4} x^{4}+\frac {3375}{8} x^{3}+585 x^{2}+315 x +60\right )}{243 \left (1+\frac {3 x}{2}\right )^{4}}+\frac {500 x \left (-\frac {1701}{16} x^{5}+\frac {1701}{4} x^{4}+\frac {23625}{8} x^{3}+4095 x^{2}+2205 x +420\right )}{5103 \left (1+\frac {3 x}{2}\right )^{4}}\)	\(176\)

[In]

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

[Out]

-500/243*x^2+9100/729*x+81*(-66193/6561*x^3-254699/13122*x^2-735691/59049*x-210065/78732)/(2+3*x)^4-14390/729*
ln(2+3*x)

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.22 \[ \int \frac {(1-2 x)^3 (3+5 x)^3}{(2+3 x)^5} \, dx=-\frac {486000 \, x^{6} - 1652400 \, x^{5} - 6566400 \, x^{4} - 4903452 \, x^{3} + 1186182 \, x^{2} + 57560 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (3 \, x + 2\right ) + 2360364 \, x + 630195}{2916 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]

[In]

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

[Out]

-1/2916*(486000*x^6 - 1652400*x^5 - 6566400*x^4 - 4903452*x^3 + 1186182*x^2 + 57560*(81*x^4 + 216*x^3 + 216*x^
2 + 96*x + 16)*log(3*x + 2) + 2360364*x + 630195)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.84 \[ \int \frac {(1-2 x)^3 (3+5 x)^3}{(2+3 x)^5} \, dx=- \frac {500 x^{2}}{243} + \frac {9100 x}{729} - \frac {2382948 x^{3} + 4584582 x^{2} + 2942764 x + 630195}{236196 x^{4} + 629856 x^{3} + 629856 x^{2} + 279936 x + 46656} - \frac {14390 \log {\left (3 x + 2 \right )}}{729} \]

[In]

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

[Out]

-500*x**2/243 + 9100*x/729 - (2382948*x**3 + 4584582*x**2 + 2942764*x + 630195)/(236196*x**4 + 629856*x**3 + 6
29856*x**2 + 279936*x + 46656) - 14390*log(3*x + 2)/729

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.84 \[ \int \frac {(1-2 x)^3 (3+5 x)^3}{(2+3 x)^5} \, dx=-\frac {500}{243} \, x^{2} + \frac {9100}{729} \, x - \frac {2382948 \, x^{3} + 4584582 \, x^{2} + 2942764 \, x + 630195}{2916 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} - \frac {14390}{729} \, \log \left (3 \, x + 2\right ) \]

[In]

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

[Out]

-500/243*x^2 + 9100/729*x - 1/2916*(2382948*x^3 + 4584582*x^2 + 2942764*x + 630195)/(81*x^4 + 216*x^3 + 216*x^
2 + 96*x + 16) - 14390/729*log(3*x + 2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.12 \[ \int \frac {(1-2 x)^3 (3+5 x)^3}{(2+3 x)^5} \, dx=\frac {100}{2187} \, {\left (3 \, x + 2\right )}^{2} {\left (\frac {111}{3 \, x + 2} - 5\right )} - \frac {66193}{2187 \, {\left (3 \, x + 2\right )}} + \frac {10073}{1458 \, {\left (3 \, x + 2\right )}^{2}} - \frac {1813}{2187 \, {\left (3 \, x + 2\right )}^{3}} + \frac {343}{8748 \, {\left (3 \, x + 2\right )}^{4}} + \frac {14390}{729} \, \log \left (\frac {{\left | 3 \, x + 2 \right |}}{3 \, {\left (3 \, x + 2\right )}^{2}}\right ) \]

[In]

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

[Out]

100/2187*(3*x + 2)^2*(111/(3*x + 2) - 5) - 66193/2187/(3*x + 2) + 10073/1458/(3*x + 2)^2 - 1813/2187/(3*x + 2)
^3 + 343/8748/(3*x + 2)^4 + 14390/729*log(1/3*abs(3*x + 2)/(3*x + 2)^2)

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.78 \[ \int \frac {(1-2 x)^3 (3+5 x)^3}{(2+3 x)^5} \, dx=\frac {9100\,x}{729}-\frac {14390\,\ln \left (x+\frac {2}{3}\right )}{729}-\frac {500\,x^2}{243}-\frac {\frac {66193\,x^3}{6561}+\frac {254699\,x^2}{13122}+\frac {735691\,x}{59049}+\frac {210065}{78732}}{x^4+\frac {8\,x^3}{3}+\frac {8\,x^2}{3}+\frac {32\,x}{27}+\frac {16}{81}} \]

[In]

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

[Out]

(9100*x)/729 - (14390*log(x + 2/3))/729 - (500*x^2)/243 - ((735691*x)/59049 + (254699*x^2)/13122 + (66193*x^3)
/6561 + 210065/78732)/((32*x)/27 + (8*x^2)/3 + (8*x^3)/3 + x^4 + 16/81)