Integrand size = 27, antiderivative size = 154 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^8} \, dx=\frac {47 (7+8 x) \sqrt {2+5 x+3 x^2}}{128000 (3+2 x)^2}-\frac {47 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{9600 (3+2 x)^4}+\frac {47 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{600 (3+2 x)^6}-\frac {13 \left (2+5 x+3 x^2\right )^{7/2}}{35 (3+2 x)^7}-\frac {47 \text {arctanh}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{256000 \sqrt {5}} \]
-47/9600*(7+8*x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^4+47/600*(7+8*x)*(3*x^2+5*x+2 )^(5/2)/(3+2*x)^6-13/35*(3*x^2+5*x+2)^(7/2)/(3+2*x)^7-47/1280000*arctanh(1 /10*(7+8*x)*5^(1/2)/(3*x^2+5*x+2)^(1/2))*5^(1/2)+47/128000*(7+8*x)*(3*x^2+ 5*x+2)^(1/2)/(3+2*x)^2
Time = 0.57 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.57 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^8} \, dx=\frac {\frac {5 \sqrt {2+5 x+3 x^2} \left (6404247+39981058 x+100711840 x^2+127557120 x^3+81951440 x^4+22620128 x^5+1089792 x^6\right )}{(3+2 x)^7}-987 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )}{13440000} \]
((5*Sqrt[2 + 5*x + 3*x^2]*(6404247 + 39981058*x + 100711840*x^2 + 12755712 0*x^3 + 81951440*x^4 + 22620128*x^5 + 1089792*x^6))/(3 + 2*x)^7 - 987*Sqrt [5]*ArcTanh[Sqrt[2/5 + x + (3*x^2)/5]/(1 + x)])/13440000
Time = 0.32 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.10, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1228, 1152, 1152, 1152, 1154, 219}
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.
| \(\displaystyle \int \frac {(5-x) \left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^8} \, dx\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle \frac {47}{10} \int \frac {\left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^7}dx-\frac {13 \left (3 x^2+5 x+2\right )^{7/2}}{35 (2 x+3)^7}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {47}{10} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}-\frac {1}{24} \int \frac {\left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^5}dx\right )-\frac {13 \left (3 x^2+5 x+2\right )^{7/2}}{35 (2 x+3)^7}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {47}{10} \left (\frac {1}{24} \left (\frac {3}{80} \int \frac {\sqrt {3 x^2+5 x+2}}{(2 x+3)^3}dx-\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}\right )+\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{7/2}}{35 (2 x+3)^7}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {47}{10} \left (\frac {1}{24} \left (\frac {3}{80} \left (\frac {(8 x+7) \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^2}-\frac {1}{40} \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}\right )+\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{7/2}}{35 (2 x+3)^7}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {47}{10} \left (\frac {1}{24} \left (\frac {3}{80} \left (\frac {1}{20} \int \frac {1}{20-\frac {(8 x+7)^2}{3 x^2+5 x+2}}d\left (-\frac {8 x+7}{\sqrt {3 x^2+5 x+2}}\right )+\frac {\sqrt {3 x^2+5 x+2} (8 x+7)}{20 (2 x+3)^2}\right )-\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}\right )+\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{7/2}}{35 (2 x+3)^7}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {47}{10} \left (\frac {1}{24} \left (\frac {3}{80} \left (\frac {(8 x+7) \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^2}-\frac {\text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{40 \sqrt {5}}\right )-\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}\right )+\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{7/2}}{35 (2 x+3)^7}\) |
(-13*(2 + 5*x + 3*x^2)^(7/2))/(35*(3 + 2*x)^7) + (47*(((7 + 8*x)*(2 + 5*x + 3*x^2)^(5/2))/(60*(3 + 2*x)^6) + (-1/40*((7 + 8*x)*(2 + 5*x + 3*x^2)^(3/ 2))/(3 + 2*x)^4 + (3*(((7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(20*(3 + 2*x)^2) - ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])]/(40*Sqrt[5])))/80)/2 4))/10
3.25.44.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(-(d + e*x)^(m + 1))*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b *x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[p*((b^2 - 4*a *c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] && GtQ[p, 0]
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e *f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^ (m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x ] && EqQ[Simplify[m + 2*p + 3], 0]
Time = 0.40 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.60
| method | result | size |
| risch | \(\frac {3269376 x^{8}+73309344 x^{7}+361134544 x^{6}+837668816 x^{5}+1103824000 x^{4}+878616614 x^{3}+420541711 x^{2}+111983351 x +12808494}{2688000 \left (3+2 x \right )^{7} \sqrt {3 x^{2}+5 x +2}}+\frac {47 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{1280000}\) | \(93\) |
| trager | \(\frac {\left (1089792 x^{6}+22620128 x^{5}+81951440 x^{4}+127557120 x^{3}+100711840 x^{2}+39981058 x +6404247\right ) \sqrt {3 x^{2}+5 x +2}}{2688000 \left (3+2 x \right )^{7}}-\frac {47 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{3+2 x}\right )}{1280000}\) | \(102\) |
| default | \(-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{4480 \left (x +\frac {3}{2}\right )^{7}}-\frac {47 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{9600 \left (x +\frac {3}{2}\right )^{6}}-\frac {47 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{6000 \left (x +\frac {3}{2}\right )^{5}}-\frac {987 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{80000 \left (x +\frac {3}{2}\right )^{4}}-\frac {2867 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{150000 \left (x +\frac {3}{2}\right )^{3}}-\frac {87373 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{3000000 \left (x +\frac {3}{2}\right )^{2}}+\frac {27307 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{1250000}-\frac {27307 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{625000 \left (x +\frac {3}{2}\right )}-\frac {1363 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{600000}+\frac {47 \left (5+6 x \right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{160000}+\frac {47 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{1280000}-\frac {47 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{5000000}-\frac {47 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{2400000}-\frac {47 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}{1280000}\) | \(290\) |
1/2688000*(3269376*x^8+73309344*x^7+361134544*x^6+837668816*x^5+1103824000 *x^4+878616614*x^3+420541711*x^2+111983351*x+12808494)/(3+2*x)^7/(3*x^2+5* x+2)^(1/2)+47/1280000*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2 -16*x-19)^(1/2))
Time = 0.28 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.11 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^8} \, dx=\frac {987 \, \sqrt {5} {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )} \log \left (-\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} - 124 \, x^{2} - 212 \, x - 89}{4 \, x^{2} + 12 \, x + 9}\right ) + 20 \, {\left (1089792 \, x^{6} + 22620128 \, x^{5} + 81951440 \, x^{4} + 127557120 \, x^{3} + 100711840 \, x^{2} + 39981058 \, x + 6404247\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{53760000 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} \]
1/53760000*(987*sqrt(5)*(128*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 22680 *x^3 + 20412*x^2 + 10206*x + 2187)*log(-(4*sqrt(5)*sqrt(3*x^2 + 5*x + 2)*( 8*x + 7) - 124*x^2 - 212*x - 89)/(4*x^2 + 12*x + 9)) + 20*(1089792*x^6 + 2 2620128*x^5 + 81951440*x^4 + 127557120*x^3 + 100711840*x^2 + 39981058*x + 6404247)*sqrt(3*x^2 + 5*x + 2))/(128*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187)
\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^8} \, dx=- \int \left (- \frac {20 \sqrt {3 x^{2} + 5 x + 2}}{256 x^{8} + 3072 x^{7} + 16128 x^{6} + 48384 x^{5} + 90720 x^{4} + 108864 x^{3} + 81648 x^{2} + 34992 x + 6561}\right )\, dx - \int \left (- \frac {96 x \sqrt {3 x^{2} + 5 x + 2}}{256 x^{8} + 3072 x^{7} + 16128 x^{6} + 48384 x^{5} + 90720 x^{4} + 108864 x^{3} + 81648 x^{2} + 34992 x + 6561}\right )\, dx - \int \left (- \frac {165 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{256 x^{8} + 3072 x^{7} + 16128 x^{6} + 48384 x^{5} + 90720 x^{4} + 108864 x^{3} + 81648 x^{2} + 34992 x + 6561}\right )\, dx - \int \left (- \frac {113 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{256 x^{8} + 3072 x^{7} + 16128 x^{6} + 48384 x^{5} + 90720 x^{4} + 108864 x^{3} + 81648 x^{2} + 34992 x + 6561}\right )\, dx - \int \left (- \frac {15 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{256 x^{8} + 3072 x^{7} + 16128 x^{6} + 48384 x^{5} + 90720 x^{4} + 108864 x^{3} + 81648 x^{2} + 34992 x + 6561}\right )\, dx - \int \frac {9 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{256 x^{8} + 3072 x^{7} + 16128 x^{6} + 48384 x^{5} + 90720 x^{4} + 108864 x^{3} + 81648 x^{2} + 34992 x + 6561}\, dx \]
-Integral(-20*sqrt(3*x**2 + 5*x + 2)/(256*x**8 + 3072*x**7 + 16128*x**6 + 48384*x**5 + 90720*x**4 + 108864*x**3 + 81648*x**2 + 34992*x + 6561), x) - Integral(-96*x*sqrt(3*x**2 + 5*x + 2)/(256*x**8 + 3072*x**7 + 16128*x**6 + 48384*x**5 + 90720*x**4 + 108864*x**3 + 81648*x**2 + 34992*x + 6561), x) - Integral(-165*x**2*sqrt(3*x**2 + 5*x + 2)/(256*x**8 + 3072*x**7 + 16128 *x**6 + 48384*x**5 + 90720*x**4 + 108864*x**3 + 81648*x**2 + 34992*x + 656 1), x) - Integral(-113*x**3*sqrt(3*x**2 + 5*x + 2)/(256*x**8 + 3072*x**7 + 16128*x**6 + 48384*x**5 + 90720*x**4 + 108864*x**3 + 81648*x**2 + 34992*x + 6561), x) - Integral(-15*x**4*sqrt(3*x**2 + 5*x + 2)/(256*x**8 + 3072*x **7 + 16128*x**6 + 48384*x**5 + 90720*x**4 + 108864*x**3 + 81648*x**2 + 34 992*x + 6561), x) - Integral(9*x**5*sqrt(3*x**2 + 5*x + 2)/(256*x**8 + 307 2*x**7 + 16128*x**6 + 48384*x**5 + 90720*x**4 + 108864*x**3 + 81648*x**2 + 34992*x + 6561), x)
Leaf count of result is larger than twice the leaf count of optimal. 367 vs. \(2 (128) = 256\).
Time = 0.29 (sec) , antiderivative size = 367, normalized size of antiderivative = 2.38 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^8} \, dx=\frac {87373}{1000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{35 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac {47 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{150 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {94 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{375 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {987 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{5000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {2867 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{18750 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {87373 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{750000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {1363}{100000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {27307}{2400000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {27307 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{250000 \, {\left (2 \, x + 3\right )}} + \frac {141}{80000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {47}{1280000} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) + \frac {893}{640000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]
87373/1000000*(3*x^2 + 5*x + 2)^(5/2) - 13/35*(3*x^2 + 5*x + 2)^(7/2)/(128 *x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187) - 47/150*(3*x^2 + 5*x + 2)^(7/2)/(64*x^6 + 576*x^5 + 2160*x^4 + 432 0*x^3 + 4860*x^2 + 2916*x + 729) - 94/375*(3*x^2 + 5*x + 2)^(7/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 987/5000*(3*x^2 + 5*x + 2) ^(7/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 2867/18750*(3*x^2 + 5*x + 2)^(7/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 87373/750000*(3*x^2 + 5*x + 2)^( 7/2)/(4*x^2 + 12*x + 9) - 1363/100000*(3*x^2 + 5*x + 2)^(3/2)*x - 27307/24 00000*(3*x^2 + 5*x + 2)^(3/2) - 27307/250000*(3*x^2 + 5*x + 2)^(5/2)/(2*x + 3) + 141/80000*sqrt(3*x^2 + 5*x + 2)*x + 47/1280000*sqrt(5)*log(sqrt(5)* sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) + 893/640000*sq rt(3*x^2 + 5*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 461 vs. \(2 (128) = 256\).
Time = 0.42 (sec) , antiderivative size = 461, normalized size of antiderivative = 2.99 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^8} \, dx=-\frac {47}{1280000} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) - \frac {72512832 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{13} + 651952224 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{12} + 6898276448 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 8494566864 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{10} - 58878767920 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} - 326450774496 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} - 2207907445056 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} - 3147944405424 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} - 9314774279636 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} - 6492162811470 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} - 9472821206534 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} - 3070624865553 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} - 1792565462541 \, \sqrt {3} x - 158637115728 \, \sqrt {3} + 1792565462541 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{2688000 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{7}} \]
-47/1280000*sqrt(5)*log(abs(-4*sqrt(3)*x - 2*sqrt(5) - 6*sqrt(3) + 4*sqrt( 3*x^2 + 5*x + 2))/abs(-4*sqrt(3)*x + 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))) - 1/2688000*(72512832*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^13 + 651952224*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^12 + 6898276448*(s qrt(3)*x - sqrt(3*x^2 + 5*x + 2))^11 + 8494566864*sqrt(3)*(sqrt(3)*x - sqr t(3*x^2 + 5*x + 2))^10 - 58878767920*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^9 - 326450774496*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^8 - 2207907445 056*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^7 - 3147944405424*sqrt(3)*(sqrt(3) *x - sqrt(3*x^2 + 5*x + 2))^6 - 9314774279636*(sqrt(3)*x - sqrt(3*x^2 + 5* x + 2))^5 - 6492162811470*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 - 9472821206534*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 - 3070624865553*sqrt(3 )*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 - 1792565462541*sqrt(3)*x - 158637 115728*sqrt(3) + 1792565462541*sqrt(3*x^2 + 5*x + 2))/(2*(sqrt(3)*x - sqrt (3*x^2 + 5*x + 2))^2 + 6*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) + 11) ^7
Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^8} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2}}{{\left (2\,x+3\right )}^8} \,d x \]