Integrand size = 27, antiderivative size = 160 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^3} \, dx=\frac {5 (3763-7854 x) \sqrt {2+5 x+3 x^2}}{1536}+\frac {5 (573+164 x) \left (2+5 x+3 x^2\right )^{3/2}}{192 (3+2 x)}-\frac {(29+2 x) \left (2+5 x+3 x^2\right )^{5/2}}{16 (3+2 x)^2}-\frac {199615 \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{3072 \sqrt {3}}+\frac {4295}{256} \sqrt {5} \text {arctanh}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right ) \]
5/192*(573+164*x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)-1/16*(29+2*x)*(3*x^2+5*x+2)^ (5/2)/(3+2*x)^2-199615/9216*arctanh(1/6*(5+6*x)*3^(1/2)/(3*x^2+5*x+2)^(1/2 ))*3^(1/2)+4295/256*arctanh(1/10*(7+8*x)*5^(1/2)/(3*x^2+5*x+2)^(1/2))*5^(1 /2)+5/1536*(3763-7854*x)*(3*x^2+5*x+2)^(1/2)
Time = 0.56 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.71 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^3} \, dx=\frac {-\frac {3 \sqrt {2+5 x+3 x^2} \left (-295719-290742 x-57292 x^2-14456 x^3-8544 x^4+1728 x^5\right )}{(3+2 x)^2}+154620 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )-199615 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{4608} \]
((-3*Sqrt[2 + 5*x + 3*x^2]*(-295719 - 290742*x - 57292*x^2 - 14456*x^3 - 8 544*x^4 + 1728*x^5))/(3 + 2*x)^2 + 154620*Sqrt[5]*ArcTanh[Sqrt[2/5 + x + ( 3*x^2)/5]/(1 + x)] - 199615*Sqrt[3]*ArcTanh[Sqrt[2/3 + (5*x)/3 + x^2]/(1 + x)])/4608
Time = 0.38 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.07, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {1230, 27, 1230, 27, 1231, 27, 1269, 1092, 219, 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.
| \(\displaystyle \int \frac {(5-x) \left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^3} \, dx\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle -\frac {5}{64} \int -\frac {2 (164 x+137) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^2}dx-\frac {(2 x+29) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{32} \int \frac {(164 x+137) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^2}dx-\frac {(2 x+29) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^2}\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle \frac {5}{32} \left (\frac {(164 x+573) \left (3 x^2+5 x+2\right )^{3/2}}{6 (2 x+3)}-\frac {1}{8} \int \frac {2 (2618 x+2209) \sqrt {3 x^2+5 x+2}}{2 x+3}dx\right )-\frac {(2 x+29) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{32} \left (\frac {(164 x+573) \left (3 x^2+5 x+2\right )^{3/2}}{6 (2 x+3)}-\frac {1}{4} \int \frac {(2618 x+2209) \sqrt {3 x^2+5 x+2}}{2 x+3}dx\right )-\frac {(2 x+29) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^2}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {5}{32} \left (\frac {1}{4} \left (\frac {1}{48} \int -\frac {2 (79846 x+68229)}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx+\frac {1}{12} \sqrt {3 x^2+5 x+2} (3763-7854 x)\right )+\frac {(164 x+573) \left (3 x^2+5 x+2\right )^{3/2}}{6 (2 x+3)}\right )-\frac {(2 x+29) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{32} \left (\frac {1}{4} \left (\frac {1}{12} (3763-7854 x) \sqrt {3 x^2+5 x+2}-\frac {1}{24} \int \frac {79846 x+68229}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )+\frac {(164 x+573) \left (3 x^2+5 x+2\right )^{3/2}}{6 (2 x+3)}\right )-\frac {(2 x+29) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^2}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {5}{32} \left (\frac {1}{4} \left (\frac {1}{24} \left (51540 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-39923 \int \frac {1}{\sqrt {3 x^2+5 x+2}}dx\right )+\frac {1}{12} \sqrt {3 x^2+5 x+2} (3763-7854 x)\right )+\frac {(164 x+573) \left (3 x^2+5 x+2\right )^{3/2}}{6 (2 x+3)}\right )-\frac {(2 x+29) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^2}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {5}{32} \left (\frac {1}{4} \left (\frac {1}{24} \left (51540 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-79846 \int \frac {1}{12-\frac {(6 x+5)^2}{3 x^2+5 x+2}}d\frac {6 x+5}{\sqrt {3 x^2+5 x+2}}\right )+\frac {1}{12} \sqrt {3 x^2+5 x+2} (3763-7854 x)\right )+\frac {(164 x+573) \left (3 x^2+5 x+2\right )^{3/2}}{6 (2 x+3)}\right )-\frac {(2 x+29) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {5}{32} \left (\frac {1}{4} \left (\frac {1}{24} \left (51540 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-\frac {39923 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {3}}\right )+\frac {1}{12} \sqrt {3 x^2+5 x+2} (3763-7854 x)\right )+\frac {(164 x+573) \left (3 x^2+5 x+2\right )^{3/2}}{6 (2 x+3)}\right )-\frac {(2 x+29) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^2}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {5}{32} \left (\frac {1}{4} \left (\frac {1}{24} \left (-103080 \int \frac {1}{20-\frac {(8 x+7)^2}{3 x^2+5 x+2}}d\left (-\frac {8 x+7}{\sqrt {3 x^2+5 x+2}}\right )-\frac {39923 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {3}}\right )+\frac {1}{12} \sqrt {3 x^2+5 x+2} (3763-7854 x)\right )+\frac {(164 x+573) \left (3 x^2+5 x+2\right )^{3/2}}{6 (2 x+3)}\right )-\frac {(2 x+29) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {5}{32} \left (\frac {1}{4} \left (\frac {1}{24} \left (10308 \sqrt {5} \text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )-\frac {39923 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {3}}\right )+\frac {1}{12} \sqrt {3 x^2+5 x+2} (3763-7854 x)\right )+\frac {(164 x+573) \left (3 x^2+5 x+2\right )^{3/2}}{6 (2 x+3)}\right )-\frac {(2 x+29) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^2}\) |
-1/16*((29 + 2*x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^2 + (5*(((573 + 164*x )*(2 + 5*x + 3*x^2)^(3/2))/(6*(3 + 2*x)) + (((3763 - 7854*x)*Sqrt[2 + 5*x + 3*x^2])/12 + ((-39923*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2] )])/Sqrt[3] + 10308*Sqrt[5]*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3* x^2])])/24)/4))/32
3.25.39.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
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[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Simp[p/(e^2*(m + 1)*(m + 2*p + 2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m + 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, - 1] || EqQ[p, 1] || (IntegerQ[p] && !RationalQ[m])) && NeQ[m, -1] && !ILtQ [m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ (c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)) Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c ^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x ] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || !R ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Integer Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Time = 0.36 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.73
| method | result | size |
| risch | \(-\frac {5184 x^{7}-16992 x^{6}-82632 x^{5}-261244 x^{4}-1187598 x^{3}-2455451 x^{2}-2060079 x -591438}{1536 \left (3+2 x \right )^{2} \sqrt {3 x^{2}+5 x +2}}-\frac {199615 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{9216}-\frac {4295 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{256}\) | \(117\) |
| trager | \(-\frac {\left (1728 x^{5}-8544 x^{4}-14456 x^{3}-57292 x^{2}-290742 x -295719\right ) \sqrt {3 x^{2}+5 x +2}}{1536 \left (3+2 x \right )^{2}}+\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3689405\right ) \ln \left (\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3689405\right ) x +8590 \sqrt {3 x^{2}+5 x +2}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3689405\right )}{3+2 x}\right )}{256}-\frac {199615 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+6 \sqrt {3 x^{2}+5 x +2}\right )}{9216}\) | \(138\) |
| default | \(\frac {83 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{50 \left (x +\frac {3}{2}\right )}+\frac {859 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{200}-\frac {109 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{64}-\frac {6545 \left (5+6 x \right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{1536}-\frac {199615 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}\right ) \sqrt {3}}{9216}+\frac {859 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{96}+\frac {4295 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}{256}-\frac {4295 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{256}-\frac {83 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{100}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{40 \left (x +\frac {3}{2}\right )^{2}}\) | \(216\) |
-1/1536*(5184*x^7-16992*x^6-82632*x^5-261244*x^4-1187598*x^3-2455451*x^2-2 060079*x-591438)/(3+2*x)^2/(3*x^2+5*x+2)^(1/2)-199615/9216*ln(1/3*(5/2+3*x )*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)-4295/256*5^(1/2)*arctanh(2/5*(-7/2- 4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))
Time = 0.30 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.02 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^3} \, dx=\frac {199615 \, \sqrt {3} {\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 154620 \, \sqrt {5} {\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 12 \, {\left (1728 \, x^{5} - 8544 \, x^{4} - 14456 \, x^{3} - 57292 \, x^{2} - 290742 \, x - 295719\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{18432 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} \]
1/18432*(199615*sqrt(3)*(4*x^2 + 12*x + 9)*log(-4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49) + 154620*sqrt(5)*(4*x^2 + 12*x + 9) *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)) - 12*(1728*x^5 - 8544*x^4 - 14456*x^3 - 57292*x^2 - 2907 42*x - 295719)*sqrt(3*x^2 + 5*x + 2))/(4*x^2 + 12*x + 9)
\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^3} \, dx=- \int \left (- \frac {20 \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \left (- \frac {96 x \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \left (- \frac {165 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \left (- \frac {113 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \left (- \frac {15 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \frac {9 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\, dx \]
-Integral(-20*sqrt(3*x**2 + 5*x + 2)/(8*x**3 + 36*x**2 + 54*x + 27), x) - Integral(-96*x*sqrt(3*x**2 + 5*x + 2)/(8*x**3 + 36*x**2 + 54*x + 27), x) - Integral(-165*x**2*sqrt(3*x**2 + 5*x + 2)/(8*x**3 + 36*x**2 + 54*x + 27), x) - Integral(-113*x**3*sqrt(3*x**2 + 5*x + 2)/(8*x**3 + 36*x**2 + 54*x + 27), x) - Integral(-15*x**4*sqrt(3*x**2 + 5*x + 2)/(8*x**3 + 36*x**2 + 54 *x + 27), x) - Integral(9*x**5*sqrt(3*x**2 + 5*x + 2)/(8*x**3 + 36*x**2 + 54*x + 27), x)
Time = 0.27 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.18 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^3} \, dx=\frac {39}{40} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{10 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {327}{32} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {83}{192} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {83 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{20 \, {\left (2 \, x + 3\right )}} - \frac {6545}{256} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {199615}{9216} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) - \frac {4295}{256} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) + \frac {18815}{1536} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]
39/40*(3*x^2 + 5*x + 2)^(5/2) - 13/10*(3*x^2 + 5*x + 2)^(7/2)/(4*x^2 + 12* x + 9) - 327/32*(3*x^2 + 5*x + 2)^(3/2)*x + 83/192*(3*x^2 + 5*x + 2)^(3/2) + 83/20*(3*x^2 + 5*x + 2)^(5/2)/(2*x + 3) - 6545/256*sqrt(3*x^2 + 5*x + 2 )*x - 199615/9216*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) - 4295/256*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs (2*x + 3) - 2) + 18815/1536*sqrt(3*x^2 + 5*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (128) = 256\).
Time = 0.32 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.68 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^3} \, dx=-\frac {1}{1536} \, {\left (2 \, {\left (12 \, {\left (18 \, x - 143\right )} x + 2855\right )} x - 23731\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {4295}{256} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) + \frac {199615}{9216} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) + \frac {5 \, {\left (4214 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 15793 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 53551 \, \sqrt {3} x + 19053 \, \sqrt {3} - 53551 \, \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}}{128 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{2}} \]
-1/1536*(2*(12*(18*x - 143)*x + 2855)*x - 23731)*sqrt(3*x^2 + 5*x + 2) + 4 295/256*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))) + 199615/9216*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5)) + 5/128*(4214*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 + 15 793*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 53551*sqrt(3)*x + 1905 3*sqrt(3) - 53551*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)^2
Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^3} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2}}{{\left (2\,x+3\right )}^3} \,d x \]