Integrand size = 25, antiderivative size = 63 \[ \int \frac {(5-2 x)^2 \left (2+3 x+x^2\right )^2}{(4+3 x)^4} \, dx=\frac {61 x}{729}-\frac {26 x^2}{243}+\frac {4 x^3}{243}-\frac {2116}{6561 (4+3 x)^3}+\frac {46}{81 (4+3 x)^2}+\frac {401}{729 (4+3 x)}+\frac {1318 \log (4+3 x)}{2187} \] Output:
61/729*x-26/243*x^2+4/243*x^3-2116/6561/(4+3*x)^3+46/81/(4+3*x)^2+401/(291 6+2187*x)+1318/2187*ln(4+3*x)
Time = 0.04 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.89 \[ \int \frac {(5-2 x)^2 \left (2+3 x+x^2\right )^2}{(4+3 x)^4} \, dx=\frac {71212+151638 x+102699 x^2+8496 x^3-15147 x^4-2430 x^5+972 x^6+1318 (4+3 x)^3 \log (4+3 x)}{2187 (4+3 x)^3} \] Input:
Integrate[((5 - 2*x)^2*(2 + 3*x + x^2)^2)/(4 + 3*x)^4,x]
Output:
(71212 + 151638*x + 102699*x^2 + 8496*x^3 - 15147*x^4 - 2430*x^5 + 972*x^6 + 1318*(4 + 3*x)^3*Log[4 + 3*x])/(2187*(4 + 3*x)^3)
Time = 0.34 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1195, 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.
\(\displaystyle \int \frac {(5-2 x)^2 \left (x^2+3 x+2\right )^2}{(3 x+4)^4} \, dx\) |
\(\Big \downarrow \) 1195 |
\(\displaystyle \int \left (\frac {4 x^2}{81}-\frac {52 x}{243}+\frac {1318}{729 (3 x+4)}-\frac {401}{243 (3 x+4)^2}-\frac {92}{27 (3 x+4)^3}+\frac {2116}{729 (3 x+4)^4}+\frac {61}{729}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {4 x^3}{243}-\frac {26 x^2}{243}+\frac {61 x}{729}+\frac {401}{729 (3 x+4)}+\frac {46}{81 (3 x+4)^2}-\frac {2116}{6561 (3 x+4)^3}+\frac {1318 \log (3 x+4)}{2187}\) |
Input:
Int[((5 - 2*x)^2*(2 + 3*x + x^2)^2)/(4 + 3*x)^4,x]
Output:
(61*x)/729 - (26*x^2)/243 + (4*x^3)/243 - 2116/(6561*(4 + 3*x)^3) + 46/(81 *(4 + 3*x)^2) + 401/(729*(4 + 3*x)) + (1318*Log[4 + 3*x])/2187
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Time = 1.66 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.67
method | result | size |
risch | \(\frac {4 x^{3}}{243}-\frac {26 x^{2}}{243}+\frac {61 x}{729}+\frac {\frac {401}{81} x^{2}+\frac {3622}{243} x +\frac {70532}{6561}}{\left (3 x +4\right )^{3}}+\frac {1318 \ln \left (3 x +4\right )}{2187}\) | \(42\) |
norman | \(\frac {\frac {2545}{81} x^{2}+\frac {3938}{81} x -\frac {187}{27} x^{4}-\frac {10}{9} x^{5}+\frac {4}{9} x^{6}+\frac {153220}{6561}}{\left (3 x +4\right )^{3}}+\frac {1318 \ln \left (3 x +4\right )}{2187}\) | \(43\) |
default | \(\frac {4 x^{3}}{243}-\frac {26 x^{2}}{243}+\frac {61 x}{729}+\frac {46}{81 \left (3 x +4\right )^{2}}-\frac {2116}{6561 \left (3 x +4\right )^{3}}+\frac {401}{729 \left (3 x +4\right )}+\frac {1318 \ln \left (3 x +4\right )}{2187}\) | \(50\) |
parallelrisch | \(\frac {2916 x^{6}-7290 x^{5}+106758 \ln \left (x +\frac {4}{3}\right ) x^{3}-45441 x^{4}+153220+427032 \ln \left (x +\frac {4}{3}\right ) x^{2}+569376 \ln \left (x +\frac {4}{3}\right ) x +206145 x^{2}+253056 \ln \left (x +\frac {4}{3}\right )+318978 x}{6561 \left (3 x +4\right )^{3}}\) | \(66\) |
meijerg | \(\frac {25 x \left (\frac {9}{16} x^{2}+\frac {9}{4} x +3\right )}{192 \left (1+\frac {3 x}{4}\right )^{3}}+\frac {55 x^{2} \left (\frac {3 x}{4}+3\right )}{384 \left (1+\frac {3 x}{4}\right )^{3}}+\frac {101 x^{3}}{768 \left (1+\frac {3 x}{4}\right )^{3}}+\frac {31 x \left (\frac {99}{8} x^{2}+\frac {45}{2} x +12\right )}{648 \left (1+\frac {3 x}{4}\right )^{3}}+\frac {1318 \ln \left (1+\frac {3 x}{4}\right )}{2187}-\frac {43 x \left (\frac {405}{64} x^{3}+\frac {495}{8} x^{2}+\frac {225}{2} x +60\right )}{1215 \left (1+\frac {3 x}{4}\right )^{3}}-\frac {8 x \left (-\frac {243}{256} x^{4}+\frac {405}{64} x^{3}+\frac {495}{8} x^{2}+\frac {225}{2} x +60\right )}{729 \left (1+\frac {3 x}{4}\right )^{3}}+\frac {64 x \left (\frac {1701}{1024} x^{5}-\frac {1701}{256} x^{4}+\frac {2835}{64} x^{3}+\frac {3465}{8} x^{2}+\frac {1575}{2} x +420\right )}{15309 \left (1+\frac {3 x}{4}\right )^{3}}\) | \(169\) |
Input:
int((5-2*x)^2*(x^2+3*x+2)^2/(3*x+4)^4,x,method=_RETURNVERBOSE)
Output:
4/243*x^3-26/243*x^2+61/729*x+27*(401/2187*x^2+3622/6561*x+70532/177147)/( 3*x+4)^3+1318/2187*ln(3*x+4)
Time = 0.07 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.14 \[ \int \frac {(5-2 x)^2 \left (2+3 x+x^2\right )^2}{(4+3 x)^4} \, dx=\frac {2916 \, x^{6} - 7290 \, x^{5} - 45441 \, x^{4} - 34884 \, x^{3} + 66609 \, x^{2} + 3954 \, {\left (27 \, x^{3} + 108 \, x^{2} + 144 \, x + 64\right )} \log \left (3 \, x + 4\right ) + 132930 \, x + 70532}{6561 \, {\left (27 \, x^{3} + 108 \, x^{2} + 144 \, x + 64\right )}} \] Input:
integrate((5-2*x)^2*(x^2+3*x+2)^2/(4+3*x)^4,x, algorithm="fricas")
Output:
1/6561*(2916*x^6 - 7290*x^5 - 45441*x^4 - 34884*x^3 + 66609*x^2 + 3954*(27 *x^3 + 108*x^2 + 144*x + 64)*log(3*x + 4) + 132930*x + 70532)/(27*x^3 + 10 8*x^2 + 144*x + 64)
Time = 0.07 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.84 \[ \int \frac {(5-2 x)^2 \left (2+3 x+x^2\right )^2}{(4+3 x)^4} \, dx=\frac {4 x^{3}}{243} - \frac {26 x^{2}}{243} + \frac {61 x}{729} + \frac {32481 x^{2} + 97794 x + 70532}{177147 x^{3} + 708588 x^{2} + 944784 x + 419904} + \frac {1318 \log {\left (3 x + 4 \right )}}{2187} \] Input:
integrate((5-2*x)**2*(x**2+3*x+2)**2/(4+3*x)**4,x)
Output:
4*x**3/243 - 26*x**2/243 + 61*x/729 + (32481*x**2 + 97794*x + 70532)/(1771 47*x**3 + 708588*x**2 + 944784*x + 419904) + 1318*log(3*x + 4)/2187
Time = 0.03 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.81 \[ \int \frac {(5-2 x)^2 \left (2+3 x+x^2\right )^2}{(4+3 x)^4} \, dx=\frac {4}{243} \, x^{3} - \frac {26}{243} \, x^{2} + \frac {61}{729} \, x + \frac {32481 \, x^{2} + 97794 \, x + 70532}{6561 \, {\left (27 \, x^{3} + 108 \, x^{2} + 144 \, x + 64\right )}} + \frac {1318}{2187} \, \log \left (3 \, x + 4\right ) \] Input:
integrate((5-2*x)^2*(x^2+3*x+2)^2/(4+3*x)^4,x, algorithm="maxima")
Output:
4/243*x^3 - 26/243*x^2 + 61/729*x + 1/6561*(32481*x^2 + 97794*x + 70532)/( 27*x^3 + 108*x^2 + 144*x + 64) + 1318/2187*log(3*x + 4)
Time = 0.29 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.67 \[ \int \frac {(5-2 x)^2 \left (2+3 x+x^2\right )^2}{(4+3 x)^4} \, dx=\frac {4}{243} \, x^{3} - \frac {26}{243} \, x^{2} + \frac {61}{729} \, x + \frac {32481 \, x^{2} + 97794 \, x + 70532}{6561 \, {\left (3 \, x + 4\right )}^{3}} + \frac {1318}{2187} \, \log \left ({\left | 3 \, x + 4 \right |}\right ) \] Input:
integrate((5-2*x)^2*(x^2+3*x+2)^2/(4+3*x)^4,x, algorithm="giac")
Output:
4/243*x^3 - 26/243*x^2 + 61/729*x + 1/6561*(32481*x^2 + 97794*x + 70532)/( 3*x + 4)^3 + 1318/2187*log(abs(3*x + 4))
Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.73 \[ \int \frac {(5-2 x)^2 \left (2+3 x+x^2\right )^2}{(4+3 x)^4} \, dx=\frac {61\,x}{729}+\frac {1318\,\ln \left (x+\frac {4}{3}\right )}{2187}+\frac {\frac {401\,x^2}{2187}+\frac {3622\,x}{6561}+\frac {70532}{177147}}{x^3+4\,x^2+\frac {16\,x}{3}+\frac {64}{27}}-\frac {26\,x^2}{243}+\frac {4\,x^3}{243} \] Input:
int(((2*x - 5)^2*(3*x + x^2 + 2)^2)/(3*x + 4)^4,x)
Output:
(61*x)/729 + (1318*log(x + 4/3))/2187 + ((3622*x)/6561 + (401*x^2)/2187 + 70532/177147)/((16*x)/3 + 4*x^2 + x^3 + 64/27) - (26*x^2)/243 + (4*x^3)/24 3
Time = 0.16 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.30 \[ \int \frac {(5-2 x)^2 \left (2+3 x+x^2\right )^2}{(4+3 x)^4} \, dx=\frac {427032 \,\mathrm {log}\left (3 x +4\right ) x^{3}+1708128 \,\mathrm {log}\left (3 x +4\right ) x^{2}+2277504 \,\mathrm {log}\left (3 x +4\right ) x +1012224 \,\mathrm {log}\left (3 x +4\right )+11664 x^{6}-29160 x^{5}-181764 x^{4}-206145 x^{3}+176472 x +124240}{708588 x^{3}+2834352 x^{2}+3779136 x +1679616} \] Input:
int((5-2*x)^2*(x^2+3*x+2)^2/(4+3*x)^4,x)
Output:
(427032*log(3*x + 4)*x**3 + 1708128*log(3*x + 4)*x**2 + 2277504*log(3*x + 4)*x + 1012224*log(3*x + 4) + 11664*x**6 - 29160*x**5 - 181764*x**4 - 2061 45*x**3 + 176472*x + 124240)/(26244*(27*x**3 + 108*x**2 + 144*x + 64))