Integrand size = 27, antiderivative size = 137 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^4} \, dx=-\frac {(845+402 x) \sqrt {2+5 x+3 x^2}}{160 (3+2 x)}+\frac {(383+342 x) \left (2+5 x+3 x^2\right )^{3/2}}{120 (3+2 x)^3}+\frac {51}{32} \sqrt {3} \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )-\frac {1973 \text {arctanh}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{320 \sqrt {5}} \]
1/120*(383+342*x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^3+51/32*arctanh(1/6*(5+6*x)* 3^(1/2)/(3*x^2+5*x+2)^(1/2))*3^(1/2)-1973/1600*arctanh(1/10*(7+8*x)*5^(1/2 )/(3*x^2+5*x+2)^(1/2))*5^(1/2)-1/160*(845+402*x)*(3*x^2+5*x+2)^(1/2)/(3+2* x)
Time = 0.43 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.75 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^4} \, dx=\frac {-\frac {5 \sqrt {2+5 x+3 x^2} \left (19751+30878 x+13176 x^2+720 x^3\right )}{(3+2 x)^3}-5919 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )+7650 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{2400} \]
((-5*Sqrt[2 + 5*x + 3*x^2]*(19751 + 30878*x + 13176*x^2 + 720*x^3))/(3 + 2 *x)^3 - 5919*Sqrt[5]*ArcTanh[Sqrt[2/5 + x + (3*x^2)/5]/(1 + x)] + 7650*Sqr t[3]*ArcTanh[Sqrt[2/3 + (5*x)/3 + x^2]/(1 + x)])/2400
Time = 0.32 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {1229, 1230, 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.
| \(\displaystyle \int \frac {(5-x) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^4} \, dx\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle \frac {(342 x+383) \left (3 x^2+5 x+2\right )^{3/2}}{120 (2 x+3)^3}-\frac {1}{80} \int \frac {(402 x+361) \sqrt {3 x^2+5 x+2}}{(2 x+3)^2}dx\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle \frac {1}{80} \left (\frac {1}{8} \int \frac {2 (3060 x+2617)}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-\frac {(402 x+845) \sqrt {3 x^2+5 x+2}}{2 (2 x+3)}\right )+\frac {(342 x+383) \left (3 x^2+5 x+2\right )^{3/2}}{120 (2 x+3)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{80} \left (\frac {1}{4} \int \frac {3060 x+2617}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-\frac {(402 x+845) \sqrt {3 x^2+5 x+2}}{2 (2 x+3)}\right )+\frac {(342 x+383) \left (3 x^2+5 x+2\right )^{3/2}}{120 (2 x+3)^3}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {1}{80} \left (\frac {1}{4} \left (1530 \int \frac {1}{\sqrt {3 x^2+5 x+2}}dx-1973 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {(402 x+845) \sqrt {3 x^2+5 x+2}}{2 (2 x+3)}\right )+\frac {(342 x+383) \left (3 x^2+5 x+2\right )^{3/2}}{120 (2 x+3)^3}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {1}{80} \left (\frac {1}{4} \left (3060 \int \frac {1}{12-\frac {(6 x+5)^2}{3 x^2+5 x+2}}d\frac {6 x+5}{\sqrt {3 x^2+5 x+2}}-1973 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {(402 x+845) \sqrt {3 x^2+5 x+2}}{2 (2 x+3)}\right )+\frac {(342 x+383) \left (3 x^2+5 x+2\right )^{3/2}}{120 (2 x+3)^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{80} \left (\frac {1}{4} \left (510 \sqrt {3} \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-1973 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {(402 x+845) \sqrt {3 x^2+5 x+2}}{2 (2 x+3)}\right )+\frac {(342 x+383) \left (3 x^2+5 x+2\right )^{3/2}}{120 (2 x+3)^3}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {1}{80} \left (\frac {1}{4} \left (3946 \int \frac {1}{20-\frac {(8 x+7)^2}{3 x^2+5 x+2}}d\left (-\frac {8 x+7}{\sqrt {3 x^2+5 x+2}}\right )+510 \sqrt {3} \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )\right )-\frac {(402 x+845) \sqrt {3 x^2+5 x+2}}{2 (2 x+3)}\right )+\frac {(342 x+383) \left (3 x^2+5 x+2\right )^{3/2}}{120 (2 x+3)^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{80} \left (\frac {1}{4} \left (510 \sqrt {3} \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {1973 \text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {5}}\right )-\frac {(402 x+845) \sqrt {3 x^2+5 x+2}}{2 (2 x+3)}\right )+\frac {(342 x+383) \left (3 x^2+5 x+2\right )^{3/2}}{120 (2 x+3)^3}\) |
((383 + 342*x)*(2 + 5*x + 3*x^2)^(3/2))/(120*(3 + 2*x)^3) + (-1/2*((845 + 402*x)*Sqrt[2 + 5*x + 3*x^2])/(3 + 2*x) + (510*Sqrt[3]*ArcTanh[(5 + 6*x)/( 2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])] - (1973*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqr t[2 + 5*x + 3*x^2])])/Sqrt[5])/4)/80
3.25.26.3.1 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))*((a + b*x + c*x^2 )^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2))*(c* d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x), x] - Simp[p/(e^2*(m + 1 )*(m + 2)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2 )^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m + 1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c *(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1) - b*(d*g*( m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g }, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] && !ILtQ[m + 2*p + 3, 0]
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[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 = 0.34 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.78
| method | result | size |
| risch | \(-\frac {2160 x^{5}+43128 x^{4}+159954 x^{3}+239995 x^{2}+160511 x +39502}{480 \left (3+2 x \right )^{3} \sqrt {3 x^{2}+5 x +2}}+\frac {51 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{32}+\frac {1973 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{1600}\) | \(107\) |
| trager | \(-\frac {\left (720 x^{3}+13176 x^{2}+30878 x +19751\right ) \sqrt {3 x^{2}+5 x +2}}{480 \left (3+2 x \right )^{3}}-\frac {51 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {3 x^{2}+5 x +2}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )\right )}{32}-\frac {1973 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{3+2 x}\right )}{1600}\) | \(128\) |
| default | \(-\frac {37 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{600 \left (x +\frac {3}{2}\right )^{2}}-\frac {158 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{375 \left (x +\frac {3}{2}\right )}-\frac {1973 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{3000}+\frac {121 \left (5+6 x \right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{400}+\frac {51 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}\right ) \sqrt {3}}{32}-\frac {1973 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}{1600}+\frac {1973 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{1600}+\frac {79 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{375}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{120 \left (x +\frac {3}{2}\right )^{3}}\) | \(200\) |
-1/480*(2160*x^5+43128*x^4+159954*x^3+239995*x^2+160511*x+39502)/(3+2*x)^3 /(3*x^2+5*x+2)^(1/2)+51/32*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3 ^(1/2)+1973/1600*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x -19)^(1/2))
Time = 0.29 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.23 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^4} \, dx=\frac {7650 \, \sqrt {3} {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 5919 \, \sqrt {5} {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} \log \left (-\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} - 124 \, x^{2} - 212 \, x - 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 20 \, {\left (720 \, x^{3} + 13176 \, x^{2} + 30878 \, x + 19751\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{9600 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} \]
1/9600*(7650*sqrt(3)*(8*x^3 + 36*x^2 + 54*x + 27)*log(4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49) + 5919*sqrt(5)*(8*x^3 + 36*x^ 2 + 54*x + 27)*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*(720*x^3 + 13176*x^2 + 30878*x + 197 51)*sqrt(3*x^2 + 5*x + 2))/(8*x^3 + 36*x^2 + 54*x + 27)
\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^4} \, dx=- \int \left (- \frac {10 \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \left (- \frac {23 x \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \left (- \frac {10 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \frac {3 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\, dx \]
-Integral(-10*sqrt(3*x**2 + 5*x + 2)/(16*x**4 + 96*x**3 + 216*x**2 + 216*x + 81), x) - Integral(-23*x*sqrt(3*x**2 + 5*x + 2)/(16*x**4 + 96*x**3 + 21 6*x**2 + 216*x + 81), x) - Integral(-10*x**2*sqrt(3*x**2 + 5*x + 2)/(16*x* *4 + 96*x**3 + 216*x**2 + 216*x + 81), x) - Integral(3*x**3*sqrt(3*x**2 + 5*x + 2)/(16*x**4 + 96*x**3 + 216*x**2 + 216*x + 81), x)
Time = 0.28 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.39 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^4} \, dx=\frac {37}{200} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{15 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {37 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{150 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac {363}{200} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {51}{32} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) + \frac {1973}{1600} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) - \frac {763}{800} \, \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {79 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{75 \, {\left (2 \, x + 3\right )}} \]
37/200*(3*x^2 + 5*x + 2)^(3/2) - 13/15*(3*x^2 + 5*x + 2)^(5/2)/(8*x^3 + 36 *x^2 + 54*x + 27) - 37/150*(3*x^2 + 5*x + 2)^(5/2)/(4*x^2 + 12*x + 9) + 36 3/200*sqrt(3*x^2 + 5*x + 2)*x + 51/32*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) + 1973/1600*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/a bs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 763/800*sqrt(3*x^2 + 5*x + 2) - 79/7 5*(3*x^2 + 5*x + 2)^(3/2)/(2*x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 305 vs. \(2 (109) = 218\).
Time = 0.34 (sec) , antiderivative size = 305, normalized size of antiderivative = 2.23 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^4} \, dx=-\frac {1973}{1600} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) - \frac {51}{32} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) - \frac {3}{16} \, \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {62484 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 390510 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 2835190 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 3307455 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 5598195 \, \sqrt {3} x + 1227924 \, \sqrt {3} - 5598195 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{480 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{3}} \]
-1973/1600*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))) - 51/32*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5 *x + 2)) - 5)) - 3/16*sqrt(3*x^2 + 5*x + 2) - 1/480*(62484*(sqrt(3)*x - sq rt(3*x^2 + 5*x + 2))^5 + 390510*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2) )^4 + 2835190*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 + 3307455*sqrt(3)*(sqr t(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 5598195*sqrt(3)*x + 1227924*sqrt(3) - 5598195*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)^3
Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^4} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{{\left (2\,x+3\right )}^4} \,d x \]