Integrand size = 27, antiderivative size = 128 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^2} \, dx=-\frac {1}{96} (361-726 x) \sqrt {2+5 x+3 x^2}-\frac {(21+x) \left (2+5 x+3 x^2\right )^{3/2}}{6 (3+2 x)}+\frac {3743 \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{192 \sqrt {3}}-\frac {161}{32} \sqrt {5} \text {arctanh}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right ) \]
-1/6*(21+x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)+3743/576*arctanh(1/6*(5+6*x)*3^(1/ 2)/(3*x^2+5*x+2)^(1/2))*3^(1/2)-161/32*arctanh(1/10*(7+8*x)*5^(1/2)/(3*x^2 +5*x+2)^(1/2))*5^(1/2)-1/96*(361-726*x)*(3*x^2+5*x+2)^(1/2)
Time = 0.39 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.80 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^2} \, dx=\frac {1}{288} \left (-\frac {3 \sqrt {2+5 x+3 x^2} \left (1755+256 x-364 x^2+48 x^3\right )}{3+2 x}-2898 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )+3743 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )\right ) \]
((-3*Sqrt[2 + 5*x + 3*x^2]*(1755 + 256*x - 364*x^2 + 48*x^3))/(3 + 2*x) - 2898*Sqrt[5]*ArcTanh[Sqrt[2/5 + x + (3*x^2)/5]/(1 + x)] + 3743*Sqrt[3]*Arc Tanh[Sqrt[2/3 + (5*x)/3 + x^2]/(1 + x)])/288
Time = 0.31 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1230, 27, 1231, 25, 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.
| \(\displaystyle \int \frac {(5-x) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^2} \, dx\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle -\frac {1}{8} \int -\frac {2 (121 x+101) \sqrt {3 x^2+5 x+2}}{2 x+3}dx-\frac {(x+21) \left (3 x^2+5 x+2\right )^{3/2}}{6 (2 x+3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \int \frac {(121 x+101) \sqrt {3 x^2+5 x+2}}{2 x+3}dx-\frac {(x+21) \left (3 x^2+5 x+2\right )^{3/2}}{6 (2 x+3)}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {1}{4} \left (-\frac {1}{48} \int -\frac {7486 x+6399}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-\frac {1}{24} \sqrt {3 x^2+5 x+2} (361-726 x)\right )-\frac {(x+21) \left (3 x^2+5 x+2\right )^{3/2}}{6 (2 x+3)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{48} \int \frac {7486 x+6399}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-\frac {1}{24} (361-726 x) \sqrt {3 x^2+5 x+2}\right )-\frac {(x+21) \left (3 x^2+5 x+2\right )^{3/2}}{6 (2 x+3)}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{48} \left (3743 \int \frac {1}{\sqrt {3 x^2+5 x+2}}dx-4830 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {1}{24} (361-726 x) \sqrt {3 x^2+5 x+2}\right )-\frac {(x+21) \left (3 x^2+5 x+2\right )^{3/2}}{6 (2 x+3)}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{48} \left (7486 \int \frac {1}{12-\frac {(6 x+5)^2}{3 x^2+5 x+2}}d\frac {6 x+5}{\sqrt {3 x^2+5 x+2}}-4830 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {1}{24} (361-726 x) \sqrt {3 x^2+5 x+2}\right )-\frac {(x+21) \left (3 x^2+5 x+2\right )^{3/2}}{6 (2 x+3)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{48} \left (\frac {3743 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {3}}-4830 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {1}{24} (361-726 x) \sqrt {3 x^2+5 x+2}\right )-\frac {(x+21) \left (3 x^2+5 x+2\right )^{3/2}}{6 (2 x+3)}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{48} \left (9660 \int \frac {1}{20-\frac {(8 x+7)^2}{3 x^2+5 x+2}}d\left (-\frac {8 x+7}{\sqrt {3 x^2+5 x+2}}\right )+\frac {3743 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {3}}\right )-\frac {1}{24} (361-726 x) \sqrt {3 x^2+5 x+2}\right )-\frac {(x+21) \left (3 x^2+5 x+2\right )^{3/2}}{6 (2 x+3)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{48} \left (\frac {3743 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {3}}-966 \sqrt {5} \text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )\right )-\frac {1}{24} (361-726 x) \sqrt {3 x^2+5 x+2}\right )-\frac {(x+21) \left (3 x^2+5 x+2\right )^{3/2}}{6 (2 x+3)}\) |
-1/6*((21 + x)*(2 + 5*x + 3*x^2)^(3/2))/(3 + 2*x) + (-1/24*((361 - 726*x)* Sqrt[2 + 5*x + 3*x^2]) + ((3743*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/Sqrt[3] - 966*Sqrt[5]*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5* x + 3*x^2])])/48)/4
3.25.24.3.1 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 = 0.33 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.84
| method | result | size |
| risch | \(-\frac {144 x^{5}-852 x^{4}-956 x^{3}+5817 x^{2}+9287 x +3510}{96 \left (3+2 x \right ) \sqrt {3 x^{2}+5 x +2}}+\frac {3743 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{576}+\frac {161 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{32}\) | \(107\) |
| trager | \(-\frac {\left (48 x^{3}-364 x^{2}+256 x +1755\right ) \sqrt {3 x^{2}+5 x +2}}{96 \left (3+2 x \right )}-\frac {3743 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {3 x^{2}+5 x +2}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )\right )}{576}+\frac {161 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}-7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{3+2 x}\right )}{32}\) | \(128\) |
| default | \(-\frac {161 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{60}+\frac {121 \left (5+6 x \right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{96}+\frac {3743 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}\right ) \sqrt {3}}{576}-\frac {161 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}{32}+\frac {161 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{32}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{10 \left (x +\frac {3}{2}\right )}+\frac {13 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{20}\) | \(158\) |
-1/96*(144*x^5-852*x^4-956*x^3+5817*x^2+9287*x+3510)/(3+2*x)/(3*x^2+5*x+2) ^(1/2)+3743/576*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)+161/ 32*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 = 139, normalized size of antiderivative = 1.09 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^2} \, dx=\frac {3743 \, \sqrt {3} {\left (2 \, x + 3\right )} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 2898 \, \sqrt {5} {\left (2 \, x + 3\right )} \log \left (-\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} - 124 \, x^{2} - 212 \, x - 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 12 \, {\left (48 \, x^{3} - 364 \, x^{2} + 256 \, x + 1755\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{1152 \, {\left (2 \, x + 3\right )}} \]
1/1152*(3743*sqrt(3)*(2*x + 3)*log(4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49) + 2898*sqrt(5)*(2*x + 3)*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* (48*x^3 - 364*x^2 + 256*x + 1755)*sqrt(3*x^2 + 5*x + 2))/(2*x + 3)
\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^2} \, dx=- \int \left (- \frac {10 \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \left (- \frac {23 x \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \left (- \frac {10 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \frac {3 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\, dx \]
-Integral(-10*sqrt(3*x**2 + 5*x + 2)/(4*x**2 + 12*x + 9), x) - Integral(-2 3*x*sqrt(3*x**2 + 5*x + 2)/(4*x**2 + 12*x + 9), x) - Integral(-10*x**2*sqr t(3*x**2 + 5*x + 2)/(4*x**2 + 12*x + 9), x) - Integral(3*x**3*sqrt(3*x**2 + 5*x + 2)/(4*x**2 + 12*x + 9), x)
Time = 0.29 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.05 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^2} \, dx=-\frac {1}{12} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {121}{16} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {3743}{576} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) + \frac {161}{32} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) - \frac {361}{96} \, \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{4 \, {\left (2 \, x + 3\right )}} \]
-1/12*(3*x^2 + 5*x + 2)^(3/2) + 121/16*sqrt(3*x^2 + 5*x + 2)*x + 3743/576* sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) + 161/32*sqrt(5)*lo g(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 361 /96*sqrt(3*x^2 + 5*x + 2) - 13/4*(3*x^2 + 5*x + 2)^(3/2)/(2*x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 481 vs. \(2 (100) = 200\).
Time = 0.52 (sec) , antiderivative size = 481, normalized size of antiderivative = 3.76 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^2} \, dx=-\frac {3743}{576} \, \sqrt {3} \log \left (\frac {{\left | -2 \, \sqrt {3} + 2 \, \sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {2 \, \sqrt {5}}{2 \, x + 3} \right |}}{{\left | 2 \, \sqrt {3} + 2 \, \sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {2 \, \sqrt {5}}{2 \, x + 3} \right |}}\right ) \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + \frac {161}{32} \, \sqrt {5} \log \left ({\left | \sqrt {5} {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )} - 4 \right |}\right ) \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {65}{32} \, \sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + \frac {4069 \, {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{5} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 4308 \, \sqrt {5} {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{4} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 14464 \, {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{3} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + 17388 \, \sqrt {5} {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{2} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + 12627 \, {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 17928 \, \sqrt {5} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}{96 \, {\left ({\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{2} - 3\right )}^{3}} \]
-3743/576*sqrt(3)*log(abs(-2*sqrt(3) + 2*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + 2*sqrt(5)/(2*x + 3))/abs(2*sqrt(3) + 2*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + 2*sqrt(5)/(2*x + 3)))*sgn(1/(2*x + 3)) + 161/32*sqrt(5)*log( abs(sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3)) - 4))*sgn(1/(2*x + 3)) - 65/32*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3)*sgn(1 /(2*x + 3)) + 1/96*(4069*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5) /(2*x + 3))^5*sgn(1/(2*x + 3)) - 4308*sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^4*sgn(1/(2*x + 3)) - 14464*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^3*sgn(1/(2*x + 3)) + 17388* sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^2*sgn (1/(2*x + 3)) + 12627*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2 *x + 3))*sgn(1/(2*x + 3)) - 17928*sqrt(5)*sgn(1/(2*x + 3)))/((sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^2 - 3)^3
Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^2} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{{\left (2\,x+3\right )}^2} \,d x \]