\(\int \frac {(1-2 x)^3 (3+5 x)^3}{(2+3 x)^4} \, dx\) [208]

Optimal result

Integrand size = 22, antiderivative size = 63 \[ \int \frac {(1-2 x)^3 (3+5 x)^3}{(2+3 x)^4} \, dx=-\frac {24970 x}{729}+\frac {3550 x^2}{243}-\frac {1000 x^3}{243}+\frac {343}{6561 (2+3 x)^3}-\frac {1813}{1458 (2+3 x)^2}+\frac {10073}{729 (2+3 x)}+\frac {66193 \log (2+3 x)}{2187} \] Output:

-24970/729*x+3550/243*x^2-1000/243*x^3+343/6561/(2+3*x)^3-1813/1458/(2+3*x 
)^2+10073/(1458+2187*x)+66193/2187*ln(2+3*x)
 

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.83 \[ \int \frac {(1-2 x)^3 (3+5 x)^3}{(2+3 x)^4} \, dx=\frac {-\frac {3 \left (279268+1766567 x+3851166 x^2+3180480 x^3+414180 x^4-251100 x^5+162000 x^6\right )}{(2+3 x)^3}+132386 \log (2+3 x)}{4374} \] Input:

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

Output:

((-3*(279268 + 1766567*x + 3851166*x^2 + 3180480*x^3 + 414180*x^4 - 251100 
*x^5 + 162000*x^6))/(2 + 3*x)^3 + 132386*Log[2 + 3*x])/4374
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 63, 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.

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

\(\Big \downarrow \) 99

\(\displaystyle \int \left (-\frac {1000 x^2}{81}+\frac {7100 x}{243}+\frac {66193}{729 (3 x+2)}-\frac {10073}{243 (3 x+2)^2}+\frac {1813}{243 (3 x+2)^3}-\frac {343}{729 (3 x+2)^4}-\frac {24970}{729}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {1000 x^3}{243}+\frac {3550 x^2}{243}-\frac {24970 x}{729}+\frac {10073}{729 (3 x+2)}-\frac {1813}{1458 (3 x+2)^2}+\frac {343}{6561 (3 x+2)^3}+\frac {66193 \log (3 x+2)}{2187}\)

Input:

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

Output:

(-24970*x)/729 + (3550*x^2)/243 - (1000*x^3)/243 + 343/(6561*(2 + 3*x)^3) 
- 1813/(1458*(2 + 3*x)^2) + 10073/(729*(2 + 3*x)) + (66193*Log[2 + 3*x])/2 
187
 

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 = 42, normalized size of antiderivative = 0.67

method	result	size
risch	\(-\frac {1000 x^{3}}{243}+\frac {3550 x^{2}}{243}-\frac {24970 x}{729}+\frac {\frac {10073}{81} x^{2}+\frac {26257}{162} x +\frac {346654}{6561}}{\left (2+3 x \right )^{3}}+\frac {66193 \ln \left (2+3 x \right )}{2187}\)	\(42\)
norman	\(\frac {\frac {139433}{81} x^{2}+\frac {274897}{162} x -\frac {7670}{27} x^{4}+\frac {1550}{9} x^{5}-\frac {1000}{9} x^{6}+\frac {2983934}{6561}}{\left (2+3 x \right )^{3}}+\frac {66193 \ln \left (2+3 x \right )}{2187}\)	\(43\)
default	\(-\frac {1000 x^{3}}{243}+\frac {3550 x^{2}}{243}-\frac {24970 x}{729}+\frac {66193 \ln \left (2+3 x \right )}{2187}-\frac {1813}{1458 \left (2+3 x \right )^{2}}+\frac {10073}{729 \left (2+3 x \right )}+\frac {343}{6561 \left (2+3 x \right )^{3}}\)	\(50\)
parallelrisch	\(\frac {-1944000 x^{6}+3013200 x^{5}[phone] \ln \left (\frac {2}{3}+x \right ) x^{3}-4970160 x^{4}[phone] \ln \left (\frac {2}{3}+x \right ) x^{2}-26855406 x^{3}[phone] \ln \left (\frac {2}{3}+x \right ) x -23593284 x^{2}+4236352 \ln \left (\frac {2}{3}+x \right )-6118332 x}{17496 \left (2+3 x \right )^{3}}\)	\(70\)
meijerg	\(\frac {9 x \left (\frac {9}{4} x^{2}+\frac {9}{2} x +3\right )}{16 \left (1+\frac {3 x}{2}\right )^{3}}-\frac {9 x^{2} \left (3+\frac {3 x}{2}\right )}{32 \left (1+\frac {3 x}{2}\right )^{3}}-\frac {87 x^{3}}{16 \left (1+\frac {3 x}{2}\right )^{3}}-\frac {179 x \left (\frac {99}{2} x^{2}+45 x +12\right )}{648 \left (1+\frac {3 x}{2}\right )^{3}}+\frac {66193 \ln \left (1+\frac {3 x}{2}\right )}{2187}+\frac {58 x \left (\frac {405}{8} x^{3}+\frac {495}{2} x^{2}+225 x +60\right )}{81 \left (1+\frac {3 x}{2}\right )^{3}}+\frac {100 x \left (-\frac {243}{16} x^{4}+\frac {405}{8} x^{3}+\frac {495}{2} x^{2}+225 x +60\right )}{243 \left (1+\frac {3 x}{2}\right )^{3}}-\frac {4000 x \left (\frac {1701}{32} x^{5}-\frac {1701}{16} x^{4}+\frac {2835}{8} x^{3}+\frac {3465}{2} x^{2}+1575 x +420\right )}{15309 \left (1+\frac {3 x}{2}\right )^{3}}\)	\(169\)

Input:

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

Output:

-1000/243*x^3+3550/243*x^2-24970/729*x+27*(10073/2187*x^2+26257/4374*x+346 
654/177147)/(2+3*x)^3+66193/2187*ln(2+3*x)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.14 \[ \int \frac {(1-2 x)^3 (3+5 x)^3}{(2+3 x)^4} \, dx=-\frac {1458000 \, x^{6} - 2259900 \, x^{5} + 3727620 \, x^{4} + 17801640 \, x^{3} + 13015134 \, x^{2} - 397158 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (3 \, x + 2\right ) + 1468863 \, x - 693308}{13122 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \] Input:

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

Output:

-1/13122*(1458000*x^6 - 2259900*x^5 + 3727620*x^4 + 17801640*x^3 + 1301513 
4*x^2 - 397158*(27*x^3 + 54*x^2 + 36*x + 8)*log(3*x + 2) + 1468863*x - 693 
308)/(27*x^3 + 54*x^2 + 36*x + 8)
 

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.86 \[ \int \frac {(1-2 x)^3 (3+5 x)^3}{(2+3 x)^4} \, dx=- \frac {1000 x^{3}}{243} + \frac {3550 x^{2}}{243} - \frac {24970 x}{729} - \frac {- 1631826 x^{2} - 2126817 x - 693308}{354294 x^{3} + 708588 x^{2} + 472392 x + 104976} + \frac {66193 \log {\left (3 x + 2 \right )}}{2187} \] Input:

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

Output:

-1000*x**3/243 + 3550*x**2/243 - 24970*x/729 - (-1631826*x**2 - 2126817*x 
- 693308)/(354294*x**3 + 708588*x**2 + 472392*x + 104976) + 66193*log(3*x 
+ 2)/2187
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.81 \[ \int \frac {(1-2 x)^3 (3+5 x)^3}{(2+3 x)^4} \, dx=-\frac {1000}{243} \, x^{3} + \frac {3550}{243} \, x^{2} - \frac {24970}{729} \, x + \frac {7 \, {\left (233118 \, x^{2} + 303831 \, x + 99044\right )}}{13122 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {66193}{2187} \, \log \left (3 \, x + 2\right ) \] Input:

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

Output:

-1000/243*x^3 + 3550/243*x^2 - 24970/729*x + 7/13122*(233118*x^2 + 303831* 
x + 99044)/(27*x^3 + 54*x^2 + 36*x + 8) + 66193/2187*log(3*x + 2)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.67 \[ \int \frac {(1-2 x)^3 (3+5 x)^3}{(2+3 x)^4} \, dx=-\frac {1000}{243} \, x^{3} + \frac {3550}{243} \, x^{2} - \frac {24970}{729} \, x + \frac {7 \, {\left (233118 \, x^{2} + 303831 \, x + 99044\right )}}{13122 \, {\left (3 \, x + 2\right )}^{3}} + \frac {66193}{2187} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \] Input:

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

Output:

-1000/243*x^3 + 3550/243*x^2 - 24970/729*x + 7/13122*(233118*x^2 + 303831* 
x + 99044)/(3*x + 2)^3 + 66193/2187*log(abs(3*x + 2))
 

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.73 \[ \int \frac {(1-2 x)^3 (3+5 x)^3}{(2+3 x)^4} \, dx=\frac {66193\,\ln \left (x+\frac {2}{3}\right )}{2187}-\frac {24970\,x}{729}+\frac {\frac {10073\,x^2}{2187}+\frac {26257\,x}{4374}+\frac {346654}{177147}}{x^3+2\,x^2+\frac {4\,x}{3}+\frac {8}{27}}+\frac {3550\,x^2}{243}-\frac {1000\,x^3}{243} \] Input:

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

Output:

(66193*log(x + 2/3))/2187 - (24970*x)/729 + ((26257*x)/4374 + (10073*x^2)/ 
2187 + 346654/177147)/((4*x)/3 + 2*x^2 + x^3 + 8/27) + (3550*x^2)/243 - (1 
000*x^3)/243
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.30 \[ \int \frac {(1-2 x)^3 (3+5 x)^3}{(2+3 x)^4} \, dx=\frac {10723266 \,\mathrm {log}\left (3 x +2\right ) x^{3}[phone] \,\mathrm {log}\left (3 x +2\right ) x^{2}[phone] \,\mathrm {log}\left (3 x +2\right ) x +3177264 \,\mathrm {log}\left (3 x +2\right )-1458000 x^{6}+2259900 x^{5}-3727620 x^{4}-11294073 x^{3}+7207893 x +2621476}{354294 x^{3}+708588 x^{2}+472392 x +104976} \] Input:

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

Output:

(10723266*log(3*x + 2)*x**3 + 21446532*log(3*x + 2)*x**2 + 14297688*log(3* 
x + 2)*x + 3177264*log(3*x + 2) - 1458000*x**6 + 2259900*x**5 - 3727620*x* 
*4 - 11294073*x**3 + 7207893*x + 2621476)/(13122*(27*x**3 + 54*x**2 + 36*x 
 + 8))