Integrand size = 27, antiderivative size = 137 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx=\frac {(2087+1528 x) \sqrt {2+5 x+3 x^2}}{3200 (3+2 x)^2}+\frac {(333+352 x) \left (2+5 x+3 x^2\right )^{3/2}}{240 (3+2 x)^4}-\frac {3}{32} \sqrt {3} \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )+\frac {2359 \text {arctanh}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{6400 \sqrt {5}} \]
1/240*(333+352*x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^4-3/32*arctanh(1/6*(5+6*x)*3 ^(1/2)/(3*x^2+5*x+2)^(1/2))*3^(1/2)+2359/32000*arctanh(1/10*(7+8*x)*5^(1/2 )/(3*x^2+5*x+2)^(1/2))*5^(1/2)+1/3200*(2087+1528*x)*(3*x^2+5*x+2)^(1/2)/(3 +2*x)^2
Time = 0.57 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.75 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx=\frac {\frac {5 \sqrt {2+5 x+3 x^2} \left (82989+211148 x+190412 x^2+60576 x^3\right )}{(3+2 x)^4}+7077 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )-9000 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{48000} \]
((5*Sqrt[2 + 5*x + 3*x^2]*(82989 + 211148*x + 190412*x^2 + 60576*x^3))/(3 + 2*x)^4 + 7077*Sqrt[5]*ArcTanh[Sqrt[2/5 + x + (3*x^2)/5]/(1 + x)] - 9000* Sqrt[3]*ArcTanh[Sqrt[2/3 + (5*x)/3 + x^2]/(1 + x)])/48000
Time = 0.33 (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, 1229, 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.
| \(\displaystyle \int \frac {(5-x) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^5} \, dx\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle \frac {(352 x+333) \left (3 x^2+5 x+2\right )^{3/2}}{240 (2 x+3)^4}-\frac {1}{160} \int \frac {(120 x+139) \sqrt {3 x^2+5 x+2}}{(2 x+3)^3}dx\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle \frac {1}{160} \left (\frac {1}{80} \int -\frac {2 (3600 x+3041)}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx+\frac {\sqrt {3 x^2+5 x+2} (1528 x+2087)}{20 (2 x+3)^2}\right )+\frac {(352 x+333) \left (3 x^2+5 x+2\right )^{3/2}}{240 (2 x+3)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{160} \left (\frac {(1528 x+2087) \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^2}-\frac {1}{40} \int \frac {3600 x+3041}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )+\frac {(352 x+333) \left (3 x^2+5 x+2\right )^{3/2}}{240 (2 x+3)^4}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {1}{160} \left (\frac {1}{40} \left (2359 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-1800 \int \frac {1}{\sqrt {3 x^2+5 x+2}}dx\right )+\frac {\sqrt {3 x^2+5 x+2} (1528 x+2087)}{20 (2 x+3)^2}\right )+\frac {(352 x+333) \left (3 x^2+5 x+2\right )^{3/2}}{240 (2 x+3)^4}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {1}{160} \left (\frac {1}{40} \left (2359 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-3600 \int \frac {1}{12-\frac {(6 x+5)^2}{3 x^2+5 x+2}}d\frac {6 x+5}{\sqrt {3 x^2+5 x+2}}\right )+\frac {\sqrt {3 x^2+5 x+2} (1528 x+2087)}{20 (2 x+3)^2}\right )+\frac {(352 x+333) \left (3 x^2+5 x+2\right )^{3/2}}{240 (2 x+3)^4}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{160} \left (\frac {1}{40} \left (2359 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-600 \sqrt {3} \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )\right )+\frac {\sqrt {3 x^2+5 x+2} (1528 x+2087)}{20 (2 x+3)^2}\right )+\frac {(352 x+333) \left (3 x^2+5 x+2\right )^{3/2}}{240 (2 x+3)^4}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {1}{160} \left (\frac {1}{40} \left (-4718 \int \frac {1}{20-\frac {(8 x+7)^2}{3 x^2+5 x+2}}d\left (-\frac {8 x+7}{\sqrt {3 x^2+5 x+2}}\right )-600 \sqrt {3} \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )\right )+\frac {\sqrt {3 x^2+5 x+2} (1528 x+2087)}{20 (2 x+3)^2}\right )+\frac {(352 x+333) \left (3 x^2+5 x+2\right )^{3/2}}{240 (2 x+3)^4}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{160} \left (\frac {1}{40} \left (\frac {2359 \text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {5}}-600 \sqrt {3} \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )\right )+\frac {\sqrt {3 x^2+5 x+2} (1528 x+2087)}{20 (2 x+3)^2}\right )+\frac {(352 x+333) \left (3 x^2+5 x+2\right )^{3/2}}{240 (2 x+3)^4}\) |
((333 + 352*x)*(2 + 5*x + 3*x^2)^(3/2))/(240*(3 + 2*x)^4) + (((2087 + 1528 *x)*Sqrt[2 + 5*x + 3*x^2])/(20*(3 + 2*x)^2) + (-600*Sqrt[3]*ArcTanh[(5 + 6 *x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])] + (2359*ArcTanh[(7 + 8*x)/(2*Sqrt[5 ]*Sqrt[2 + 5*x + 3*x^2])])/Sqrt[5])/40)/160
3.25.27.3.1 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[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 = 0.78
| method | result | size |
| risch | \(\frac {181728 x^{5}+874116 x^{4}+1706656 x^{3}+1685531 x^{2}+837241 x +165978}{9600 \left (3+2 x \right )^{4} \sqrt {3 x^{2}+5 x +2}}-\frac {3 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{32}-\frac {2359 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{32000}\) | \(107\) |
| trager | \(\frac {\left (60576 x^{3}+190412 x^{2}+211148 x +82989\right ) \sqrt {3 x^{2}+5 x +2}}{9600 \left (3+2 x \right )^{4}}+\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {3 x^{2}+5 x +2}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )\right )}{32}-\frac {2359 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}-7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{3+2 x}\right )}{32000}\) | \(128\) |
| default | \(-\frac {17 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{300 \left (x +\frac {3}{2}\right )^{3}}-\frac {1129 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{12000 \left (x +\frac {3}{2}\right )^{2}}-\frac {911 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{7500 \left (x +\frac {3}{2}\right )}+\frac {2359 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{60000}-\frac {109 \left (5+6 x \right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{4000}-\frac {3 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}\right ) \sqrt {3}}{32}+\frac {2359 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}{32000}-\frac {2359 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{32000}+\frac {911 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{15000}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{320 \left (x +\frac {3}{2}\right )^{4}}\) | \(221\) |
1/9600*(181728*x^5+874116*x^4+1706656*x^3+1685531*x^2+837241*x+165978)/(3+ 2*x)^4/(3*x^2+5*x+2)^(1/2)-3/32*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/ 2))*3^(1/2)-2359/32000*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 = 183, normalized size of antiderivative = 1.34 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx=\frac {9000 \, \sqrt {3} {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 7077 \, \sqrt {5} {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) + 20 \, {\left (60576 \, x^{3} + 190412 \, x^{2} + 211148 \, x + 82989\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{192000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} \]
1/192000*(9000*sqrt(3)*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(-4*sqr t(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49) + 7077*sqrt(5) *(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*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*(60576*x^3 + 190412*x^2 + 211148*x + 82989)*sqrt(3*x^2 + 5*x + 2))/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)
\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx=- \int \left (- \frac {10 \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \left (- \frac {23 x \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \left (- \frac {10 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \frac {3 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\, dx \]
-Integral(-10*sqrt(3*x**2 + 5*x + 2)/(32*x**5 + 240*x**4 + 720*x**3 + 1080 *x**2 + 810*x + 243), x) - Integral(-23*x*sqrt(3*x**2 + 5*x + 2)/(32*x**5 + 240*x**4 + 720*x**3 + 1080*x**2 + 810*x + 243), x) - Integral(-10*x**2*s qrt(3*x**2 + 5*x + 2)/(32*x**5 + 240*x**4 + 720*x**3 + 1080*x**2 + 810*x + 243), x) - Integral(3*x**3*sqrt(3*x**2 + 5*x + 2)/(32*x**5 + 240*x**4 + 7 20*x**3 + 1080*x**2 + 810*x + 243), x)
Leaf count of result is larger than twice the leaf count of optimal. 227 vs. \(2 (109) = 218\).
Time = 0.28 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.66 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx=\frac {1129}{4000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{20 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {34 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{75 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {1129 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{3000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {327}{2000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {3}{32} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) - \frac {2359}{32000} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) + \frac {179}{16000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {911 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{3000 \, {\left (2 \, x + 3\right )}} \]
1129/4000*(3*x^2 + 5*x + 2)^(3/2) - 13/20*(3*x^2 + 5*x + 2)^(5/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 34/75*(3*x^2 + 5*x + 2)^(5/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 1129/3000*(3*x^2 + 5*x + 2)^(5/2)/(4*x^2 + 12*x + 9) - 327/2000*sqrt(3*x^2 + 5*x + 2)*x - 3/32*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) - 2359/32000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) + 179/16000*sqrt(3*x^2 + 5*x + 2 ) - 911/3000*(3*x^2 + 5*x + 2)^(3/2)/(2*x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 251 vs. \(2 (109) = 218\).
Time = 0.46 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.83 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx=\frac {3}{32} \, \sqrt {3} \log \left (\frac {{\left | -2 \, \sqrt {3} + 2 \, \sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {2 \, \sqrt {5}}{2 \, x + 3} \right |}}{{\left | 2 \, \sqrt {3} + 2 \, \sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {2 \, \sqrt {5}}{2 \, x + 3} \right |}}\right ) \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {2359}{32000} \, \sqrt {5} \log \left ({\left | \sqrt {5} {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )} - 4 \right |}\right ) \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {1}{19200} \, {\left (\frac {5 \, {\left (\frac {10 \, {\left (\frac {195 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}{2 \, x + 3} - 488 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} + 4109 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} - 7572 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )} \sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} \]
3/32*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)) - 2359/32000*sqrt(5)*log(a bs(sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3)) - 4))*sgn(1/(2*x + 3)) - 1/19200*(5*(10*(195*sgn(1/(2*x + 3))/(2*x + 3) - 48 8*sgn(1/(2*x + 3)))/(2*x + 3) + 4109*sgn(1/(2*x + 3)))/(2*x + 3) - 7572*sg n(1/(2*x + 3)))*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3)
Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{{\left (2\,x+3\right )}^5} \,d x \]