Integrand size = 27, antiderivative size = 179 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^9} \, dx=\frac {1517 (7+8 x) \sqrt {2+5 x+3 x^2}}{5120000 (3+2 x)^2}-\frac {1517 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{384000 (3+2 x)^4}+\frac {1517 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{24000 (3+2 x)^6}-\frac {13 \left (2+5 x+3 x^2\right )^{7/2}}{40 (3+2 x)^8}-\frac {107 \left (2+5 x+3 x^2\right )^{7/2}}{350 (3+2 x)^7}-\frac {1517 \text {arctanh}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{10240000 \sqrt {5}} \]
-1517/384000*(7+8*x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^4+1517/24000*(7+8*x)*(3*x ^2+5*x+2)^(5/2)/(3+2*x)^6-13/40*(3*x^2+5*x+2)^(7/2)/(3+2*x)^8-107/350*(3*x ^2+5*x+2)^(7/2)/(3+2*x)^7-1517/51200000*arctanh(1/10*(7+8*x)*5^(1/2)/(3*x^ 2+5*x+2)^(1/2))*5^(1/2)+1517/5120000*(7+8*x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^2
Time = 0.70 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.52 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^9} \, dx=\frac {\frac {5 \sqrt {2+5 x+3 x^2} \left (325079151+2061624348 x+5395613996 x^2+7363989440 x^3+5486222160 x^4+2141523904 x^5+395685952 x^6+35495424 x^7\right )}{(3+2 x)^8}-31857 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )}{537600000} \]
((5*Sqrt[2 + 5*x + 3*x^2]*(325079151 + 2061624348*x + 5395613996*x^2 + 736 3989440*x^3 + 5486222160*x^4 + 2141523904*x^5 + 395685952*x^6 + 35495424*x ^7))/(3 + 2*x)^8 - 31857*Sqrt[5]*ArcTanh[Sqrt[2/5 + x + (3*x^2)/5]/(1 + x) ])/537600000
Time = 0.38 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.11, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {1237, 27, 1228, 1152, 1152, 1152, 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.
| \(\displaystyle \int \frac {(5-x) \left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^9} \, dx\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -\frac {1}{40} \int -\frac {(311-78 x) \left (3 x^2+5 x+2\right )^{5/2}}{2 (2 x+3)^8}dx-\frac {13 \left (3 x^2+5 x+2\right )^{7/2}}{40 (2 x+3)^8}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{80} \int \frac {(311-78 x) \left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^8}dx-\frac {13 \left (3 x^2+5 x+2\right )^{7/2}}{40 (2 x+3)^8}\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle \frac {1}{80} \left (\frac {1517}{5} \int \frac {\left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^7}dx-\frac {856 \left (3 x^2+5 x+2\right )^{7/2}}{35 (2 x+3)^7}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{7/2}}{40 (2 x+3)^8}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {1}{80} \left (\frac {1517}{5} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}-\frac {1}{24} \int \frac {\left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^5}dx\right )-\frac {856 \left (3 x^2+5 x+2\right )^{7/2}}{35 (2 x+3)^7}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{7/2}}{40 (2 x+3)^8}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {1}{80} \left (\frac {1517}{5} \left (\frac {1}{24} \left (\frac {3}{80} \int \frac {\sqrt {3 x^2+5 x+2}}{(2 x+3)^3}dx-\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}\right )+\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}\right )-\frac {856 \left (3 x^2+5 x+2\right )^{7/2}}{35 (2 x+3)^7}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{7/2}}{40 (2 x+3)^8}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {1}{80} \left (\frac {1517}{5} \left (\frac {1}{24} \left (\frac {3}{80} \left (\frac {(8 x+7) \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^2}-\frac {1}{40} \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}\right )+\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}\right )-\frac {856 \left (3 x^2+5 x+2\right )^{7/2}}{35 (2 x+3)^7}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{7/2}}{40 (2 x+3)^8}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {1}{80} \left (\frac {1517}{5} \left (\frac {1}{24} \left (\frac {3}{80} \left (\frac {1}{20} \int \frac {1}{20-\frac {(8 x+7)^2}{3 x^2+5 x+2}}d\left (-\frac {8 x+7}{\sqrt {3 x^2+5 x+2}}\right )+\frac {\sqrt {3 x^2+5 x+2} (8 x+7)}{20 (2 x+3)^2}\right )-\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}\right )+\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}\right )-\frac {856 \left (3 x^2+5 x+2\right )^{7/2}}{35 (2 x+3)^7}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{7/2}}{40 (2 x+3)^8}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{80} \left (\frac {1517}{5} \left (\frac {1}{24} \left (\frac {3}{80} \left (\frac {(8 x+7) \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^2}-\frac {\text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{40 \sqrt {5}}\right )-\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}\right )+\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}\right )-\frac {856 \left (3 x^2+5 x+2\right )^{7/2}}{35 (2 x+3)^7}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{7/2}}{40 (2 x+3)^8}\) |
(-13*(2 + 5*x + 3*x^2)^(7/2))/(40*(3 + 2*x)^8) + ((-856*(2 + 5*x + 3*x^2)^ (7/2))/(35*(3 + 2*x)^7) + (1517*(((7 + 8*x)*(2 + 5*x + 3*x^2)^(5/2))/(60*( 3 + 2*x)^6) + (-1/40*((7 + 8*x)*(2 + 5*x + 3*x^2)^(3/2))/(3 + 2*x)^4 + (3* (((7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(20*(3 + 2*x)^2) - ArcTanh[(7 + 8*x)/(2 *Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])]/(40*Sqrt[5])))/80)/24))/5)/80
3.25.45.3.1 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[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(-(d + e*x)^(m + 1))*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b *x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[p*((b^2 - 4*a *c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] && GtQ[p, 0]
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[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e *f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^ (m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x ] && EqQ[Simplify[m + 2*p + 3], 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b* x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1) *(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[ (c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 ] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Time = 0.42 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.55
| method | result | size |
| risch | \(\frac {106486272 x^{9}+1364534976 x^{8}+8473992320 x^{7}+27957657904 x^{6}+53806126928 x^{5}+63979233508 x^{4}+47890921904 x^{3}+22074587185 x^{2}+5748644451 x +650158302}{107520000 \left (3+2 x \right )^{8} \sqrt {3 x^{2}+5 x +2}}+\frac {1517 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{51200000}\) | \(98\) |
| trager | \(\frac {\left (35495424 x^{7}+395685952 x^{6}+2141523904 x^{5}+5486222160 x^{4}+7363989440 x^{3}+5395613996 x^{2}+2061624348 x +325079151\right ) \sqrt {3 x^{2}+5 x +2}}{107520000 \left (3+2 x \right )^{8}}+\frac {1517 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}-7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{3+2 x}\right )}{51200000}\) | \(107\) |
| default | \(-\frac {107 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{44800 \left (x +\frac {3}{2}\right )^{7}}-\frac {1517 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{384000 \left (x +\frac {3}{2}\right )^{6}}-\frac {1517 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{240000 \left (x +\frac {3}{2}\right )^{5}}-\frac {31857 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{3200000 \left (x +\frac {3}{2}\right )^{4}}-\frac {92537 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{6000000 \left (x +\frac {3}{2}\right )^{3}}-\frac {2820103 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{120000000 \left (x +\frac {3}{2}\right )^{2}}+\frac {881377 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{50000000}-\frac {881377 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{25000000 \left (x +\frac {3}{2}\right )}-\frac {43993 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{24000000}+\frac {1517 \left (5+6 x \right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{6400000}+\frac {1517 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{51200000}-\frac {1517 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{200000000}-\frac {1517 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{96000000}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{10240 \left (x +\frac {3}{2}\right )^{8}}-\frac {1517 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}{51200000}\) | \(311\) |
1/107520000*(106486272*x^9+1364534976*x^8+8473992320*x^7+27957657904*x^6+5 3806126928*x^5+63979233508*x^4+47890921904*x^3+22074587185*x^2+5748644451* x+650158302)/(3+2*x)^8/(3*x^2+5*x+2)^(1/2)+1517/51200000*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 = 186, normalized size of antiderivative = 1.04 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^9} \, dx=\frac {31857 \, \sqrt {5} {\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )} \log \left (-\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} - 124 \, x^{2} - 212 \, x - 89}{4 \, x^{2} + 12 \, x + 9}\right ) + 20 \, {\left (35495424 \, x^{7} + 395685952 \, x^{6} + 2141523904 \, x^{5} + 5486222160 \, x^{4} + 7363989440 \, x^{3} + 5395613996 \, x^{2} + 2061624348 \, x + 325079151\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{2150400000 \, {\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )}} \]
1/2150400000*(31857*sqrt(5)*(256*x^8 + 3072*x^7 + 16128*x^6 + 48384*x^5 + 90720*x^4 + 108864*x^3 + 81648*x^2 + 34992*x + 6561)*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)) + 2 0*(35495424*x^7 + 395685952*x^6 + 2141523904*x^5 + 5486222160*x^4 + 736398 9440*x^3 + 5395613996*x^2 + 2061624348*x + 325079151)*sqrt(3*x^2 + 5*x + 2 ))/(256*x^8 + 3072*x^7 + 16128*x^6 + 48384*x^5 + 90720*x^4 + 108864*x^3 + 81648*x^2 + 34992*x + 6561)
\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^9} \, dx=- \int \left (- \frac {20 \sqrt {3 x^{2} + 5 x + 2}}{512 x^{9} + 6912 x^{8} + 41472 x^{7} + 145152 x^{6} + 326592 x^{5} + 489888 x^{4} + 489888 x^{3} + 314928 x^{2} + 118098 x + 19683}\right )\, dx - \int \left (- \frac {96 x \sqrt {3 x^{2} + 5 x + 2}}{512 x^{9} + 6912 x^{8} + 41472 x^{7} + 145152 x^{6} + 326592 x^{5} + 489888 x^{4} + 489888 x^{3} + 314928 x^{2} + 118098 x + 19683}\right )\, dx - \int \left (- \frac {165 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{512 x^{9} + 6912 x^{8} + 41472 x^{7} + 145152 x^{6} + 326592 x^{5} + 489888 x^{4} + 489888 x^{3} + 314928 x^{2} + 118098 x + 19683}\right )\, dx - \int \left (- \frac {113 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{512 x^{9} + 6912 x^{8} + 41472 x^{7} + 145152 x^{6} + 326592 x^{5} + 489888 x^{4} + 489888 x^{3} + 314928 x^{2} + 118098 x + 19683}\right )\, dx - \int \left (- \frac {15 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{512 x^{9} + 6912 x^{8} + 41472 x^{7} + 145152 x^{6} + 326592 x^{5} + 489888 x^{4} + 489888 x^{3} + 314928 x^{2} + 118098 x + 19683}\right )\, dx - \int \frac {9 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{512 x^{9} + 6912 x^{8} + 41472 x^{7} + 145152 x^{6} + 326592 x^{5} + 489888 x^{4} + 489888 x^{3} + 314928 x^{2} + 118098 x + 19683}\, dx \]
-Integral(-20*sqrt(3*x**2 + 5*x + 2)/(512*x**9 + 6912*x**8 + 41472*x**7 + 145152*x**6 + 326592*x**5 + 489888*x**4 + 489888*x**3 + 314928*x**2 + 1180 98*x + 19683), x) - Integral(-96*x*sqrt(3*x**2 + 5*x + 2)/(512*x**9 + 6912 *x**8 + 41472*x**7 + 145152*x**6 + 326592*x**5 + 489888*x**4 + 489888*x**3 + 314928*x**2 + 118098*x + 19683), x) - Integral(-165*x**2*sqrt(3*x**2 + 5*x + 2)/(512*x**9 + 6912*x**8 + 41472*x**7 + 145152*x**6 + 326592*x**5 + 489888*x**4 + 489888*x**3 + 314928*x**2 + 118098*x + 19683), x) - Integral (-113*x**3*sqrt(3*x**2 + 5*x + 2)/(512*x**9 + 6912*x**8 + 41472*x**7 + 145 152*x**6 + 326592*x**5 + 489888*x**4 + 489888*x**3 + 314928*x**2 + 118098* x + 19683), x) - Integral(-15*x**4*sqrt(3*x**2 + 5*x + 2)/(512*x**9 + 6912 *x**8 + 41472*x**7 + 145152*x**6 + 326592*x**5 + 489888*x**4 + 489888*x**3 + 314928*x**2 + 118098*x + 19683), x) - Integral(9*x**5*sqrt(3*x**2 + 5*x + 2)/(512*x**9 + 6912*x**8 + 41472*x**7 + 145152*x**6 + 326592*x**5 + 489 888*x**4 + 489888*x**3 + 314928*x**2 + 118098*x + 19683), x)
Leaf count of result is larger than twice the leaf count of optimal. 423 vs. \(2 (149) = 298\).
Time = 0.29 (sec) , antiderivative size = 423, normalized size of antiderivative = 2.36 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^9} \, dx=\frac {2820103}{40000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{40 \, {\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )}} - \frac {107 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{350 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac {1517 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{6000 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {1517 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{7500 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {31857 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{200000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {92537 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{750000 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {2820103 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{30000000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {43993}{4000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {881377}{96000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {881377 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{10000000 \, {\left (2 \, x + 3\right )}} + \frac {4551}{3200000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {1517}{51200000} \, \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 {28823}{25600000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]
2820103/40000000*(3*x^2 + 5*x + 2)^(5/2) - 13/40*(3*x^2 + 5*x + 2)^(7/2)/( 256*x^8 + 3072*x^7 + 16128*x^6 + 48384*x^5 + 90720*x^4 + 108864*x^3 + 8164 8*x^2 + 34992*x + 6561) - 107/350*(3*x^2 + 5*x + 2)^(7/2)/(128*x^7 + 1344* x^6 + 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187) - 151 7/6000*(3*x^2 + 5*x + 2)^(7/2)/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4 860*x^2 + 2916*x + 729) - 1517/7500*(3*x^2 + 5*x + 2)^(7/2)/(32*x^5 + 240* x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 31857/200000*(3*x^2 + 5*x + 2)^( 7/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 92537/750000*(3*x^2 + 5*x + 2)^(7/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 2820103/30000000*(3*x^2 + 5*x + 2)^(7/2)/(4*x^2 + 12*x + 9) - 43993/4000000*(3*x^2 + 5*x + 2)^(3/2)*x - 88 1377/96000000*(3*x^2 + 5*x + 2)^(3/2) - 881377/10000000*(3*x^2 + 5*x + 2)^ (5/2)/(2*x + 3) + 4551/3200000*sqrt(3*x^2 + 5*x + 2)*x + 1517/51200000*sqr t(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2 ) + 28823/25600000*sqrt(3*x^2 + 5*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 512 vs. \(2 (149) = 298\).
Time = 0.40 (sec) , antiderivative size = 512, normalized size of antiderivative = 2.86 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^9} \, dx=-\frac {1517}{51200000} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) + \frac {4077696 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{15} - 2811291840 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{14} - 54242130880 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{13} + 23829496320 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{12} + 4407279220960 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 22617729467088 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{10} + 195051199819760 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 377875254407040 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 1580087388997720 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} + 1627784736400620 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} + 3742975645158764 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 2115026806109280 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 2573382759804010 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 709918795444635 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 358308332266605 \, \sqrt {3} x + 27766562618088 \, \sqrt {3} - 358308332266605 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{107520000 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{8}} \]
-1517/51200000*sqrt(5)*log(abs(-4*sqrt(3)*x - 2*sqrt(5) - 6*sqrt(3) + 4*sq rt(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))) + 1/107520000*(4077696*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) ^15 - 2811291840*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^14 - 54242130 880*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^13 + 23829496320*sqrt(3)*(sqrt(3)* x - sqrt(3*x^2 + 5*x + 2))^12 + 4407279220960*(sqrt(3)*x - sqrt(3*x^2 + 5* x + 2))^11 + 22617729467088*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^10 + 195051199819760*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^9 + 377875254407040 *sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^8 + 1580087388997720*(sqrt(3) *x - sqrt(3*x^2 + 5*x + 2))^7 + 1627784736400620*sqrt(3)*(sqrt(3)*x - sqrt (3*x^2 + 5*x + 2))^6 + 3742975645158764*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2) )^5 + 2115026806109280*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 + 257 3382759804010*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 + 709918795444635*sqrt (3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 358308332266605*sqrt(3)*x + 27 766562618088*sqrt(3) - 358308332266605*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)^8
Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^9} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2}}{{\left (2\,x+3\right )}^9} \,d x \]