\(\int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^3} \, dx\) [240]

Optimal result

Integrand size = 22, antiderivative size = 66 \[ \int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {4691 x}{15625}-\frac {7617 x^2}{6250}+\frac {2826 x^3}{3125}+\frac {513 x^4}{625}-\frac {648 x^5}{625}-\frac {1331}{781250 (3+5 x)^2}-\frac {15246}{390625 (3+5 x)}+\frac {63294 \log (3+5 x)}{390625} \] Output:

4691/15625*x-7617/6250*x^2+2826/3125*x^3+513/625*x^4-648/625*x^5-1331/7812 
50/(3+5*x)^2-15246/(1171875+1953125*x)+63294/390625*ln(3+5*x)
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.95 \[ \int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {21586298+83293560 x+53587800 x^2-81707500 x^3+15815625 x^4+148050000 x^5-41343750 x^6-101250000 x^7+632940 (3+5 x)^2 \log (6 (3+5 x))}{3906250 (3+5 x)^2} \] Input:

Integrate[((1 - 2*x)^3*(2 + 3*x)^4)/(3 + 5*x)^3,x]
 

Output:

(21586298 + 83293560*x + 53587800*x^2 - 81707500*x^3 + 15815625*x^4 + 1480 
50000*x^5 - 41343750*x^6 - 101250000*x^7 + 632940*(3 + 5*x)^2*Log[6*(3 + 5 
*x)])/(3906250*(3 + 5*x)^2)
 

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {99, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {(1-2 x)^3 (3 x+2)^4}{(5 x+3)^3} \, dx\)

\(\Big \downarrow \) 99

\(\displaystyle \int \left (-\frac {648 x^4}{125}+\frac {2052 x^3}{625}+\frac {8478 x^2}{3125}-\frac {7617 x}{3125}+\frac {63294}{78125 (5 x+3)}+\frac {15246}{78125 (5 x+3)^2}+\frac {1331}{78125 (5 x+3)^3}+\frac {4691}{15625}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {648 x^5}{625}+\frac {513 x^4}{625}+\frac {2826 x^3}{3125}-\frac {7617 x^2}{6250}+\frac {4691 x}{15625}-\frac {15246}{390625 (5 x+3)}-\frac {1331}{781250 (5 x+3)^2}+\frac {63294 \log (5 x+3)}{390625}\)

Input:

Int[((1 - 2*x)^3*(2 + 3*x)^4)/(3 + 5*x)^3,x]
 

Output:

(4691*x)/15625 - (7617*x^2)/6250 + (2826*x^3)/3125 + (513*x^4)/625 - (648* 
x^5)/625 - 1331/(781250*(3 + 5*x)^2) - 15246/(390625*(3 + 5*x)) + (63294*L 
og[3 + 5*x])/390625
 

Defintions of rubi rules used

rule 99

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Int[ExpandIntegrand[(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]))
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.71

method	result	size
risch	\(-\frac {648 x^{5}}{625}+\frac {513 x^{4}}{625}+\frac {2826 x^{3}}{3125}-\frac {7617 x^{2}}{6250}+\frac {4691 x}{15625}+\frac {-\frac {15246 x}{78125}-\frac {92807}{781250}}{\left (3+5 x \right )^{2}}+\frac {63294 \ln \left (3+5 x \right )}{390625}\)	\(47\)
default	\(-\frac {648 x^{5}}{625}+\frac {513 x^{4}}{625}+\frac {2826 x^{3}}{3125}-\frac {7617 x^{2}}{6250}+\frac {4691 x}{15625}-\frac {15246}{390625 \left (3+5 x \right )}+\frac {63294 \ln \left (3+5 x \right )}{390625}-\frac {1331}{781250 \left (3+5 x \right )^{2}}\)	\(51\)
norman	\(\frac {\frac {680354}{234375} x -\frac {229469}{140625} x^{2}-\frac {65366}{3125} x^{3}+\frac {5061}{1250} x^{4}+\frac {23688}{625} x^{5}-\frac {1323}{125} x^{6}-\frac {648}{25} x^{7}}{\left (3+5 x \right )^{2}}+\frac {63294 \ln \left (3+5 x \right )}{390625}\)	\(52\)
parallelrisch	\(\frac {-182250000 x^{7}-74418750 x^{6}[phone] x^{5}[phone] x^{4}[phone] \ln \left (x +\frac {3}{5}\right ) x^{2}-147073500 x^{3}[phone] \ln \left (x +\frac {3}{5}\right ) x -11473450 x^{2}[phone] \ln \left (x +\frac {3}{5}\right )[phone] x}{7031250 \left (3+5 x \right )^{2}}\)	\(66\)
meijerg	\(\frac {8 x \left (\frac {5 x}{3}+2\right )}{27 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {28 x \left (15 x +6\right )}{75 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {63294 \ln \left (1+\frac {5 x}{3}\right )}{390625}-\frac {14 x \left (\frac {100}{9} x^{2}+30 x +12\right )}{125 \left (1+\frac {5 x}{3}\right )^{2}}-\frac {1827 x \left (-\frac {625}{27} x^{3}+\frac {500}{9} x^{2}+150 x +60\right )}{6250 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {567 x \left (\frac {1250}{81} x^{4}-\frac {625}{27} x^{3}+\frac {500}{9} x^{2}+150 x +60\right )}{3125 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {729 x \left (-\frac {21875}{243} x^{5}+\frac {8750}{81} x^{4}-\frac {4375}{27} x^{3}+\frac {3500}{9} x^{2}+1050 x +420\right )}{15625 \left (1+\frac {5 x}{3}\right )^{2}}-\frac {6561 x \left (\frac {125000}{729} x^{6}-\frac {43750}{243} x^{5}+\frac {17500}{81} x^{4}-\frac {8750}{27} x^{3}+\frac {7000}{9} x^{2}+2100 x +840\right )}{390625 \left (1+\frac {5 x}{3}\right )^{2}}\)	\(190\)

Input:

int((1-2*x)^3*(2+3*x)^4/(3+5*x)^3,x,method=_RETURNVERBOSE)
 

Output:

-648/625*x^5+513/625*x^4+2826/3125*x^3-7617/6250*x^2+4691/15625*x+25*(-152 
46/1953125*x-92807/19531250)/(3+5*x)^2+63294/390625*ln(3+5*x)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.02 \[ \int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^3} \, dx=-\frac {20250000 \, x^{7} + 8268750 \, x^{6} - 29610000 \, x^{5} - 3163125 \, x^{4} + 16341500 \, x^{3} + 1532625 \, x^{2} - 126588 \, {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) - 1958490 \, x + 92807}{781250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \] Input:

integrate((1-2*x)^3*(2+3*x)^4/(3+5*x)^3,x, algorithm="fricas")
 

Output:

-1/781250*(20250000*x^7 + 8268750*x^6 - 29610000*x^5 - 3163125*x^4 + 16341 
500*x^3 + 1532625*x^2 - 126588*(25*x^2 + 30*x + 9)*log(5*x + 3) - 1958490* 
x + 92807)/(25*x^2 + 30*x + 9)
 

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.85 \[ \int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^3} \, dx=- \frac {648 x^{5}}{625} + \frac {513 x^{4}}{625} + \frac {2826 x^{3}}{3125} - \frac {7617 x^{2}}{6250} + \frac {4691 x}{15625} - \frac {152460 x + 92807}{19531250 x^{2} + 23437500 x + 7031250} + \frac {63294 \log {\left (5 x + 3 \right )}}{390625} \] Input:

integrate((1-2*x)**3*(2+3*x)**4/(3+5*x)**3,x)
                                                                                    
                                                                                    
 

Output:

-648*x**5/625 + 513*x**4/625 + 2826*x**3/3125 - 7617*x**2/6250 + 4691*x/15 
625 - (152460*x + 92807)/(19531250*x**2 + 23437500*x + 7031250) + 63294*lo 
g(5*x + 3)/390625
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.77 \[ \int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^3} \, dx=-\frac {648}{625} \, x^{5} + \frac {513}{625} \, x^{4} + \frac {2826}{3125} \, x^{3} - \frac {7617}{6250} \, x^{2} + \frac {4691}{15625} \, x - \frac {121 \, {\left (1260 \, x + 767\right )}}{781250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {63294}{390625} \, \log \left (5 \, x + 3\right ) \] Input:

integrate((1-2*x)^3*(2+3*x)^4/(3+5*x)^3,x, algorithm="maxima")
 

Output:

-648/625*x^5 + 513/625*x^4 + 2826/3125*x^3 - 7617/6250*x^2 + 4691/15625*x 
- 121/781250*(1260*x + 767)/(25*x^2 + 30*x + 9) + 63294/390625*log(5*x + 3 
)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.71 \[ \int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^3} \, dx=-\frac {648}{625} \, x^{5} + \frac {513}{625} \, x^{4} + \frac {2826}{3125} \, x^{3} - \frac {7617}{6250} \, x^{2} + \frac {4691}{15625} \, x - \frac {121 \, {\left (1260 \, x + 767\right )}}{781250 \, {\left (5 \, x + 3\right )}^{2}} + \frac {63294}{390625} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) \] Input:

integrate((1-2*x)^3*(2+3*x)^4/(3+5*x)^3,x, algorithm="giac")
 

Output:

-648/625*x^5 + 513/625*x^4 + 2826/3125*x^3 - 7617/6250*x^2 + 4691/15625*x 
- 121/781250*(1260*x + 767)/(5*x + 3)^2 + 63294/390625*log(abs(5*x + 3))
 

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.71 \[ \int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {4691\,x}{15625}+\frac {63294\,\ln \left (x+\frac {3}{5}\right )}{390625}-\frac {\frac {15246\,x}{1953125}+\frac {92807}{19531250}}{x^2+\frac {6\,x}{5}+\frac {9}{25}}-\frac {7617\,x^2}{6250}+\frac {2826\,x^3}{3125}+\frac {513\,x^4}{625}-\frac {648\,x^5}{625} \] Input:

int(-((2*x - 1)^3*(3*x + 2)^4)/(5*x + 3)^3,x)
 

Output:

(4691*x)/15625 + (63294*log(x + 3/5))/390625 - ((15246*x)/1953125 + 92807/ 
19531250)/((6*x)/5 + x^2 + 9/25) - (7617*x^2)/6250 + (2826*x^3)/3125 + (51 
3*x^4)/625 - (648*x^5)/625
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.11 \[ \int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {3164700 \,\mathrm {log}\left (5 x +3\right ) x^{2}+3797640 \,\mathrm {log}\left (5 x +3\right ) x +1139292 \,\mathrm {log}\left (5 x +3\right )-20250000 x^{7}-8268750 x^{6}[phone] x^{5}+3163125 x^{4}-16341500 x^{3}-3164700 x^{2}-680354}{19531250 x^{2}[phone] x +7031250} \] Input:

int((1-2*x)^3*(2+3*x)^4/(3+5*x)^3,x)
 

Output:

(3164700*log(5*x + 3)*x**2 + 3797640*log(5*x + 3)*x + 1139292*log(5*x + 3) 
 - 20250000*x**7 - 8268750*x**6 + 29610000*x**5 + 3163125*x**4 - 16341500* 
x**3 - 3164700*x**2 - 680354)/(781250*(25*x**2 + 30*x + 9))