Integrand size = 27, antiderivative size = 165 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^4} \, dx=-\frac {5 (736+343 x) \sqrt {2+5 x+3 x^2}}{64 (3+2 x)}+\frac {5 (93+43 x) \left (2+5 x+3 x^2\right )^{3/2}}{48 (3+2 x)^2}-\frac {(8+x) \left (2+5 x+3 x^2\right )^{5/2}}{6 (3+2 x)^3}+\frac {13505 \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{256 \sqrt {3}}-\frac {3487}{256} \sqrt {5} \text {arctanh}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right ) \]
5/48*(93+43*x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^2-1/6*(8+x)*(3*x^2+5*x+2)^(5/2) /(3+2*x)^3+13505/768*arctanh(1/6*(5+6*x)*3^(1/2)/(3*x^2+5*x+2)^(1/2))*3^(1 /2)-3487/256*arctanh(1/10*(7+8*x)*5^(1/2)/(3*x^2+5*x+2)^(1/2))*5^(1/2)-5/6 4*(736+343*x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)
Time = 0.58 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.68 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^4} \, dx=\frac {1}{384} \left (-\frac {2 \sqrt {2+5 x+3 x^2} \left (89224+143533 x+64332 x^2+1944 x^3-1896 x^4+288 x^5\right )}{(3+2 x)^3}-10461 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )+13505 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )\right ) \]
((-2*Sqrt[2 + 5*x + 3*x^2]*(89224 + 143533*x + 64332*x^2 + 1944*x^3 - 1896 *x^4 + 288*x^5))/(3 + 2*x)^3 - 10461*Sqrt[5]*ArcTanh[Sqrt[2/5 + x + (3*x^2 )/5]/(1 + x)] + 13505*Sqrt[3]*ArcTanh[Sqrt[2/3 + (5*x)/3 + x^2]/(1 + x)])/ 384
Time = 0.39 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.09, 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, 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.
| \(\displaystyle \int \frac {(5-x) \left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^4} \, dx\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle -\frac {5}{72} \int -\frac {6 (43 x+36) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^3}dx-\frac {(x+8) \left (3 x^2+5 x+2\right )^{5/2}}{6 (2 x+3)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{12} \int \frac {(43 x+36) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^3}dx-\frac {(x+8) \left (3 x^2+5 x+2\right )^{5/2}}{6 (2 x+3)^3}\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle \frac {5}{12} \left (\frac {(43 x+93) \left (3 x^2+5 x+2\right )^{3/2}}{4 (2 x+3)^2}-\frac {3}{32} \int \frac {4 (343 x+293) \sqrt {3 x^2+5 x+2}}{(2 x+3)^2}dx\right )-\frac {(x+8) \left (3 x^2+5 x+2\right )^{5/2}}{6 (2 x+3)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{12} \left (\frac {(43 x+93) \left (3 x^2+5 x+2\right )^{3/2}}{4 (2 x+3)^2}-\frac {3}{8} \int \frac {(343 x+293) \sqrt {3 x^2+5 x+2}}{(2 x+3)^2}dx\right )-\frac {(x+8) \left (3 x^2+5 x+2\right )^{5/2}}{6 (2 x+3)^3}\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle \frac {5}{12} \left (\frac {(43 x+93) \left (3 x^2+5 x+2\right )^{3/2}}{4 (2 x+3)^2}-\frac {3}{8} \left (\frac {(343 x+736) \sqrt {3 x^2+5 x+2}}{2 (2 x+3)}-\frac {1}{8} \int \frac {2 (2701 x+2308)}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )\right )-\frac {(x+8) \left (3 x^2+5 x+2\right )^{5/2}}{6 (2 x+3)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{12} \left (\frac {(43 x+93) \left (3 x^2+5 x+2\right )^{3/2}}{4 (2 x+3)^2}-\frac {3}{8} \left (\frac {(343 x+736) \sqrt {3 x^2+5 x+2}}{2 (2 x+3)}-\frac {1}{4} \int \frac {2701 x+2308}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )\right )-\frac {(x+8) \left (3 x^2+5 x+2\right )^{5/2}}{6 (2 x+3)^3}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {5}{12} \left (\frac {(43 x+93) \left (3 x^2+5 x+2\right )^{3/2}}{4 (2 x+3)^2}-\frac {3}{8} \left (\frac {1}{4} \left (\frac {3487}{2} \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-\frac {2701}{2} \int \frac {1}{\sqrt {3 x^2+5 x+2}}dx\right )+\frac {\sqrt {3 x^2+5 x+2} (343 x+736)}{2 (2 x+3)}\right )\right )-\frac {(x+8) \left (3 x^2+5 x+2\right )^{5/2}}{6 (2 x+3)^3}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {5}{12} \left (\frac {(43 x+93) \left (3 x^2+5 x+2\right )^{3/2}}{4 (2 x+3)^2}-\frac {3}{8} \left (\frac {1}{4} \left (\frac {3487}{2} \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-2701 \int \frac {1}{12-\frac {(6 x+5)^2}{3 x^2+5 x+2}}d\frac {6 x+5}{\sqrt {3 x^2+5 x+2}}\right )+\frac {\sqrt {3 x^2+5 x+2} (343 x+736)}{2 (2 x+3)}\right )\right )-\frac {(x+8) \left (3 x^2+5 x+2\right )^{5/2}}{6 (2 x+3)^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {5}{12} \left (\frac {(43 x+93) \left (3 x^2+5 x+2\right )^{3/2}}{4 (2 x+3)^2}-\frac {3}{8} \left (\frac {1}{4} \left (\frac {3487}{2} \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-\frac {2701 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{2 \sqrt {3}}\right )+\frac {\sqrt {3 x^2+5 x+2} (343 x+736)}{2 (2 x+3)}\right )\right )-\frac {(x+8) \left (3 x^2+5 x+2\right )^{5/2}}{6 (2 x+3)^3}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {5}{12} \left (\frac {(43 x+93) \left (3 x^2+5 x+2\right )^{3/2}}{4 (2 x+3)^2}-\frac {3}{8} \left (\frac {1}{4} \left (-3487 \int \frac {1}{20-\frac {(8 x+7)^2}{3 x^2+5 x+2}}d\left (-\frac {8 x+7}{\sqrt {3 x^2+5 x+2}}\right )-\frac {2701 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{2 \sqrt {3}}\right )+\frac {\sqrt {3 x^2+5 x+2} (343 x+736)}{2 (2 x+3)}\right )\right )-\frac {(x+8) \left (3 x^2+5 x+2\right )^{5/2}}{6 (2 x+3)^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {5}{12} \left (\frac {(43 x+93) \left (3 x^2+5 x+2\right )^{3/2}}{4 (2 x+3)^2}-\frac {3}{8} \left (\frac {1}{4} \left (\frac {3487 \text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{2 \sqrt {5}}-\frac {2701 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{2 \sqrt {3}}\right )+\frac {\sqrt {3 x^2+5 x+2} (343 x+736)}{2 (2 x+3)}\right )\right )-\frac {(x+8) \left (3 x^2+5 x+2\right )^{5/2}}{6 (2 x+3)^3}\) |
-1/6*((8 + x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^3 + (5*(((93 + 43*x)*(2 + 5*x + 3*x^2)^(3/2))/(4*(3 + 2*x)^2) - (3*(((736 + 343*x)*Sqrt[2 + 5*x + 3 *x^2])/(2*(3 + 2*x)) + ((-2701*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(2*Sqrt[3]) + (3487*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(2*Sqrt[5]))/4))/8))/12
3.25.40.3.1 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[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 = 0.38 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.71
| method | result | size |
| risch | \(-\frac {864 x^{7}-4248 x^{6}-3072 x^{5}+198924 x^{4}+756147 x^{3}+1114001 x^{2}+733186 x +178448}{192 \left (3+2 x \right )^{3} \sqrt {3 x^{2}+5 x +2}}+\frac {13505 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{768}+\frac {3487 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{256}\) | \(117\) |
| trager | \(-\frac {\left (288 x^{5}-1896 x^{4}+1944 x^{3}+64332 x^{2}+143533 x +89224\right ) \sqrt {3 x^{2}+5 x +2}}{192 \left (3+2 x \right )^{3}}-\frac {3487 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{3+2 x}\right )}{256}-\frac {13505 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {3 x^{2}+5 x +2}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )\right )}{768}\) | \(138\) |
| default | \(\frac {67 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{600 \left (x +\frac {3}{2}\right )^{2}}-\frac {197 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{125 \left (x +\frac {3}{2}\right )}-\frac {3487 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{1000}+\frac {329 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{240}+\frac {443 \left (5+6 x \right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{128}+\frac {13505 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}\right ) \sqrt {3}}{768}-\frac {3487 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{480}-\frac {3487 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}{256}+\frac {3487 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{256}+\frac {197 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{250}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{120 \left (x +\frac {3}{2}\right )^{3}}\) | \(237\) |
-1/192*(864*x^7-4248*x^6-3072*x^5+198924*x^4+756147*x^3+1114001*x^2+733186 *x+178448)/(3+2*x)^3/(3*x^2+5*x+2)^(1/2)+13505/768*ln(1/3*(5/2+3*x)*3^(1/2 )+(3*x^2+5*x+2)^(1/2))*3^(1/2)+3487/256*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^( 1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))
Time = 0.29 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.08 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^4} \, dx=\frac {13505 \, \sqrt {3} {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 10461 \, \sqrt {5} {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} \log \left (-\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} - 124 \, x^{2} - 212 \, x - 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 8 \, {\left (288 \, x^{5} - 1896 \, x^{4} + 1944 \, x^{3} + 64332 \, x^{2} + 143533 \, x + 89224\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{1536 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} \]
1/1536*(13505*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) + 10461*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)) - 8*(288*x^5 - 1896*x^4 + 1944*x^3 + 64 332*x^2 + 143533*x + 89224)*sqrt(3*x^2 + 5*x + 2))/(8*x^3 + 36*x^2 + 54*x + 27)
\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^4} \, dx=- \int \left (- \frac {20 \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \left (- \frac {96 x \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \left (- \frac {165 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \left (- \frac {113 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \left (- \frac {15 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \frac {9 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\, dx \]
-Integral(-20*sqrt(3*x**2 + 5*x + 2)/(16*x**4 + 96*x**3 + 216*x**2 + 216*x + 81), x) - Integral(-96*x*sqrt(3*x**2 + 5*x + 2)/(16*x**4 + 96*x**3 + 21 6*x**2 + 216*x + 81), x) - Integral(-165*x**2*sqrt(3*x**2 + 5*x + 2)/(16*x **4 + 96*x**3 + 216*x**2 + 216*x + 81), x) - Integral(-113*x**3*sqrt(3*x** 2 + 5*x + 2)/(16*x**4 + 96*x**3 + 216*x**2 + 216*x + 81), x) - Integral(-1 5*x**4*sqrt(3*x**2 + 5*x + 2)/(16*x**4 + 96*x**3 + 216*x**2 + 216*x + 81), x) - Integral(9*x**5*sqrt(3*x**2 + 5*x + 2)/(16*x**4 + 96*x**3 + 216*x**2 + 216*x + 81), x)
Time = 0.29 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.33 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^4} \, dx=-\frac {67}{200} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{15 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} + \frac {67 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{150 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac {329}{40} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {197}{480} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {197 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{50 \, {\left (2 \, x + 3\right )}} + \frac {1329}{64} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {13505}{768} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) + \frac {3487}{256} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) - \frac {159}{16} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]
-67/200*(3*x^2 + 5*x + 2)^(5/2) - 13/15*(3*x^2 + 5*x + 2)^(7/2)/(8*x^3 + 3 6*x^2 + 54*x + 27) + 67/150*(3*x^2 + 5*x + 2)^(7/2)/(4*x^2 + 12*x + 9) + 3 29/40*(3*x^2 + 5*x + 2)^(3/2)*x - 197/480*(3*x^2 + 5*x + 2)^(3/2) - 197/50 *(3*x^2 + 5*x + 2)^(5/2)/(2*x + 3) + 1329/64*sqrt(3*x^2 + 5*x + 2)*x + 135 05/768*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) + 3487/256*s qrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 159/16*sqrt(3*x^2 + 5*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 315 vs. \(2 (133) = 266\).
Time = 0.35 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.91 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^4} \, dx=-\frac {1}{128} \, {\left (2 \, {\left (12 \, x - 133\right )} x + 1197\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {3487}{256} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) - \frac {13505}{768} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) - \frac {203604 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 1334970 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 10053790 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 12051375 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 20819415 \, \sqrt {3} x + 4639299 \, \sqrt {3} - 20819415 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{384 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{3}} \]
-1/128*(2*(12*x - 133)*x + 1197)*sqrt(3*x^2 + 5*x + 2) - 3487/256*sqrt(5)* log(abs(-4*sqrt(3)*x - 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))/ab s(-4*sqrt(3)*x + 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))) - 13505 /768*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5)) - 1/384*(203604*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 + 1334970*sqrt(3)*(s qrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 + 10053790*(sqrt(3)*x - sqrt(3*x^2 + 5 *x + 2))^3 + 12051375*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 2081 9415*sqrt(3)*x + 4639299*sqrt(3) - 20819415*sqrt(3*x^2 + 5*x + 2))/(2*(sqr t(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. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^4} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2}}{{\left (2\,x+3\right )}^4} \,d x \]