\(\int \frac {(1-2 x)^3 (2+3 x)^5}{(3+5 x)^2} \, dx\) [1404]

Optimal result

Integrand size = 22, antiderivative size = 69 \[ \int \frac {(1-2 x)^3 (2+3 x)^5}{(3+5 x)^2} \, dx=\frac {1382328 x}{390625}-\frac {507023 x^2}{156250}-\frac {26594 x^3}{3125}+\frac {108387 x^4}{12500}+\frac {44982 x^5}{3125}-\frac {1026 x^6}{125}-\frac {1944 x^7}{175}-\frac {1331}{1953125 (3+5 x)}+\frac {19239 \log (3+5 x)}{1953125} \]

[Out]

1382328/390625*x-507023/156250*x^2-26594/3125*x^3+108387/12500*x^4+44982/3125*x^5-1026/125*x^6-1944/175*x^7-13
31/1953125/(3+5*x)+19239/1953125*ln(3+5*x)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 69, 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} \[ \int \frac {(1-2 x)^3 (2+3 x)^5}{(3+5 x)^2} \, dx=-\frac {1944 x^7}{175}-\frac {1026 x^6}{125}+\frac {44982 x^5}{3125}+\frac {108387 x^4}{12500}-\frac {26594 x^3}{3125}-\frac {507023 x^2}{156250}+\frac {1382328 x}{390625}-\frac {1331}{1953125 (5 x+3)}+\frac {19239 \log (5 x+3)}{1953125} \]

[In]

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

[Out]

(1382328*x)/390625 - (507023*x^2)/156250 - (26594*x^3)/3125 + (108387*x^4)/12500 + (44982*x^5)/3125 - (1026*x^
6)/125 - (1944*x^7)/175 - 1331/(1953125*(3 + 5*x)) + (19239*Log[3 + 5*x])/1953125

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 {1382328}{390625}-\frac {507023 x}{78125}-\frac {79782 x^2}{3125}+\frac {108387 x^3}{3125}+\frac {44982 x^4}{625}-\frac {6156 x^5}{125}-\frac {1944 x^6}{25}+\frac {1331}{390625 (3+5 x)^2}+\frac {19239}{390625 (3+5 x)}\right ) \, dx \\ & = \frac {1382328 x}{390625}-\frac {507023 x^2}{156250}-\frac {26594 x^3}{3125}+\frac {108387 x^4}{12500}+\frac {44982 x^5}{3125}-\frac {1026 x^6}{125}-\frac {1944 x^7}{175}-\frac {1331}{1953125 (3+5 x)}+\frac {19239 \log (3+5 x)}{1953125} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.96 \[ \int \frac {(1-2 x)^3 (2+3 x)^5}{(3+5 x)^2} \, dx=\frac {1247330759+4982083965 x+2176277250 x^2-11417376250 x^3-4521978125 x^4+23662603125 x^5+12946500000 x^6-20334375000 x^7-15187500000 x^8+2693460 (3+5 x) \log (6 (3+5 x))}{273437500 (3+5 x)} \]

[In]

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

[Out]

(1247330759 + 4982083965*x + 2176277250*x^2 - 11417376250*x^3 - 4521978125*x^4 + 23662603125*x^5 + 12946500000
*x^6 - 20334375000*x^7 - 15187500000*x^8 + 2693460*(3 + 5*x)*Log[6*(3 + 5*x)])/(273437500*(3 + 5*x))

Maple [A] (verified)

Time = 0.81 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.72

method	result	size
risch	\(-\frac {1944 x^{7}}{175}-\frac {1026 x^{6}}{125}+\frac {44982 x^{5}}{3125}+\frac {108387 x^{4}}{12500}-\frac {26594 x^{3}}{3125}-\frac {507023 x^{2}}{156250}+\frac {1382328 x}{390625}-\frac {1331}{9765625 \left (x +\frac {3}{5}\right )}+\frac {19239 \ln \left (3+5 x \right )}{1953125}\)	\(50\)
default	\(\frac {1382328 x}{390625}-\frac {507023 x^{2}}{156250}-\frac {26594 x^{3}}{3125}+\frac {108387 x^{4}}{12500}+\frac {44982 x^{5}}{3125}-\frac {1026 x^{6}}{125}-\frac {1944 x^{7}}{175}-\frac {1331}{1953125 \left (3+5 x \right )}+\frac {19239 \ln \left (3+5 x \right )}{1953125}\)	\(52\)
norman	\(\frac {\frac {12442283}{1171875} x +\frac {1243587}{156250} x^{2}-\frac {1304843}{31250} x^{3}-\frac {206719}{12500} x^{4}+\frac {1081719}{12500} x^{5}+\frac {29592}{625} x^{6}-\frac {13014}{175} x^{7}-\frac {1944}{35} x^{8}}{3+5 x}+\frac {19239 \ln \left (3+5 x \right )}{1953125}\)	\(57\)
parallelrisch	\(\frac {-9112500000 x^{8}-12200625000 x^{7}[phone] x^{6}[phone] x^{5}-2713186875 x^{4}-6850425750 x^{3}+8080380 \ln \left (x +\frac {3}{5}\right ) x [phone] x^{2}+4848228 \ln \left (x +\frac {3}{5}\right )[phone] x}{492187500+820312500 x}\)	\(62\)
meijerg	\(\frac {16 x}{45 \left (1+\frac {5 x}{3}\right )}+\frac {19239 \ln \left (1+\frac {5 x}{3}\right )}{1953125}-\frac {112 x \left (5 x +6\right )}{25 \left (1+\frac {5 x}{3}\right )}+\frac {462 x \left (-\frac {50}{9} x^{2}+10 x +12\right )}{125 \left (1+\frac {5 x}{3}\right )}+\frac {126 x \left (\frac {625}{27} x^{3}-\frac {250}{9} x^{2}+50 x +60\right )}{125 \left (1+\frac {5 x}{3}\right )}-\frac {23247 x \left (-\frac {625}{27} x^{4}+\frac {625}{27} x^{3}-\frac {250}{9} x^{2}+50 x +60\right )}{12500 \left (1+\frac {5 x}{3}\right )}-\frac {2187 x \left (\frac {43750}{243} x^{5}-\frac {4375}{27} x^{4}+\frac {4375}{27} x^{3}-\frac {1750}{9} x^{2}+350 x +420\right )}{78125 \left (1+\frac {5 x}{3}\right )}+\frac {72171 x \left (-\frac {312500}{729} x^{6}+\frac {87500}{243} x^{5}-\frac {8750}{27} x^{4}+\frac {8750}{27} x^{3}-\frac {3500}{9} x^{2}+700 x +840\right )}{781250 \left (1+\frac {5 x}{3}\right )}-\frac {157464 x \left (\frac {390625}{243} x^{7}-\frac {312500}{243} x^{6}+\frac {87500}{81} x^{5}-\frac {8750}{9} x^{4}+\frac {8750}{9} x^{3}-\frac {3500}{3} x^{2}+2100 x +2520\right )}{13671875 \left (1+\frac {5 x}{3}\right )}\)	\(230\)

[In]

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

[Out]

-1944/175*x^7-1026/125*x^6+44982/3125*x^5+108387/12500*x^4-26594/3125*x^3-507023/156250*x^2+1382328/390625*x-1
331/9765625/(x+3/5)+19239/1953125*ln(3+5*x)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.90 \[ \int \frac {(1-2 x)^3 (2+3 x)^5}{(3+5 x)^2} \, dx=-\frac {3037500000 \, x^{8} + 4066875000 \, x^{7} - 2589300000 \, x^{6} - 4732520625 \, x^{5} + 904395625 \, x^{4} + 2283475250 \, x^{3} - 435255450 \, x^{2} - 538692 \, {\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) - 580577760 \, x + 37268}{54687500 \, {\left (5 \, x + 3\right )}} \]

[In]

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

[Out]

-1/54687500*(3037500000*x^8 + 4066875000*x^7 - 2589300000*x^6 - 4732520625*x^5 + 904395625*x^4 + 2283475250*x^
3 - 435255450*x^2 - 538692*(5*x + 3)*log(5*x + 3) - 580577760*x + 37268)/(5*x + 3)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.88 \[ \int \frac {(1-2 x)^3 (2+3 x)^5}{(3+5 x)^2} \, dx=- \frac {1944 x^{7}}{175} - \frac {1026 x^{6}}{125} + \frac {44982 x^{5}}{3125} + \frac {108387 x^{4}}{12500} - \frac {26594 x^{3}}{3125} - \frac {507023 x^{2}}{156250} + \frac {1382328 x}{390625} + \frac {19239 \log {\left (5 x + 3 \right )}}{1953125} - \frac {1331}{9765625 x + 5859375} \]

[In]

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

[Out]

-1944*x**7/175 - 1026*x**6/125 + 44982*x**5/3125 + 108387*x**4/12500 - 26594*x**3/3125 - 507023*x**2/156250 +
1382328*x/390625 + 19239*log(5*x + 3)/1953125 - 1331/(9765625*x + 5859375)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.74 \[ \int \frac {(1-2 x)^3 (2+3 x)^5}{(3+5 x)^2} \, dx=-\frac {1944}{175} \, x^{7} - \frac {1026}{125} \, x^{6} + \frac {44982}{3125} \, x^{5} + \frac {108387}{12500} \, x^{4} - \frac {26594}{3125} \, x^{3} - \frac {507023}{156250} \, x^{2} + \frac {1382328}{390625} \, x - \frac {1331}{1953125 \, {\left (5 \, x + 3\right )}} + \frac {19239}{1953125} \, \log \left (5 \, x + 3\right ) \]

[In]

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

[Out]

-1944/175*x^7 - 1026/125*x^6 + 44982/3125*x^5 + 108387/12500*x^4 - 26594/3125*x^3 - 507023/156250*x^2 + 138232
8/390625*x - 1331/1953125/(5*x + 3) + 19239/1953125*log(5*x + 3)

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.35 \[ \int \frac {(1-2 x)^3 (2+3 x)^5}{(3+5 x)^2} \, dx=\frac {1}{273437500} \, {\left (5 \, x + 3\right )}^{7} {\left (\frac {672840}{5 \, x + 3} - \frac {3503304}{{\left (5 \, x + 3\right )}^{2}} + \frac {2251305}{{\left (5 \, x + 3\right )}^{3}} + \frac {16557100}{{\left (5 \, x + 3\right )}^{4}} + \frac {20720140}{{\left (5 \, x + 3\right )}^{5}} + \frac {15264480}{{\left (5 \, x + 3\right )}^{6}} - 38880\right )} - \frac {1331}{1953125 \, {\left (5 \, x + 3\right )}} - \frac {19239}{1953125} \, \log \left (\frac {{\left | 5 \, x + 3 \right |}}{5 \, {\left (5 \, x + 3\right )}^{2}}\right ) \]

[In]

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

[Out]

1/273437500*(5*x + 3)^7*(672840/(5*x + 3) - 3503304/(5*x + 3)^2 + 2251305/(5*x + 3)^3 + 16557100/(5*x + 3)^4 +
20720140/(5*x + 3)^5 + 15264480/(5*x + 3)^6 - 38880) - 1331/1953125/(5*x + 3) - 19239/1953125*log(1/5*abs(5*x
+ 3)/(5*x + 3)^2)

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.71 \[ \int \frac {(1-2 x)^3 (2+3 x)^5}{(3+5 x)^2} \, dx=\frac {1382328\,x}{390625}+\frac {19239\,\ln \left (x+\frac {3}{5}\right )}{1953125}-\frac {1331}{9765625\,\left (x+\frac {3}{5}\right )}-\frac {507023\,x^2}{156250}-\frac {26594\,x^3}{3125}+\frac {108387\,x^4}{12500}+\frac {44982\,x^5}{3125}-\frac {1026\,x^6}{125}-\frac {1944\,x^7}{175} \]

[In]

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

[Out]

(1382328*x)/390625 + (19239*log(x + 3/5))/1953125 - 1331/(9765625*(x + 3/5)) - (507023*x^2)/156250 - (26594*x^
3)/3125 + (108387*x^4)/12500 + (44982*x^5)/3125 - (1026*x^6)/125 - (1944*x^7)/175