Integrand size = 27, antiderivative size = 184 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{10}} \, dx=-\frac {329 (7+8 x) \sqrt {2+5 x+3 x^2}}{20480000 (3+2 x)^2}+\frac {329 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{1536000 (3+2 x)^4}-\frac {329 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{96000 (3+2 x)^6}+\frac {47 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{800 (3+2 x)^8}-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{45 (3+2 x)^9}+\frac {329 \text {arctanh}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{40960000 \sqrt {5}} \]
329/1536000*(7+8*x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^4-329/96000*(7+8*x)*(3*x^2 +5*x+2)^(5/2)/(3+2*x)^6+47/800*(7+8*x)*(3*x^2+5*x+2)^(7/2)/(3+2*x)^8-13/45 *(3*x^2+5*x+2)^(9/2)/(3+2*x)^9+329/204800000*arctanh(1/10*(7+8*x)*5^(1/2)/ (3*x^2+5*x+2)^(1/2))*5^(1/2)-329/20480000*(7+8*x)*(3*x^2+5*x+2)^(1/2)/(3+2 *x)^2
Time = 0.63 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.53 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{10}} \, dx=\frac {\frac {5 \sqrt {2+5 x+3 x^2} \left (578701331+5201574542 x+19810691268 x^2+41530110824 x^3+51825176720 x^4+38558367264 x^5+15895201728 x^6+2848109952 x^7+28394496 x^8\right )}{(3+2 x)^9}+2961 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )}{921600000} \]
((5*Sqrt[2 + 5*x + 3*x^2]*(578701331 + 5201574542*x + 19810691268*x^2 + 41 530110824*x^3 + 51825176720*x^4 + 38558367264*x^5 + 15895201728*x^6 + 2848 109952*x^7 + 28394496*x^8))/(3 + 2*x)^9 + 2961*Sqrt[5]*ArcTanh[Sqrt[2/5 + x + (3*x^2)/5]/(1 + x)])/921600000
Time = 0.38 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.11, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {1228, 1152, 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.
| \(\displaystyle \int \frac {(5-x) \left (3 x^2+5 x+2\right )^{7/2}}{(2 x+3)^{10}} \, dx\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle \frac {47}{10} \int \frac {\left (3 x^2+5 x+2\right )^{7/2}}{(2 x+3)^9}dx-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{45 (2 x+3)^9}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {47}{10} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{80 (2 x+3)^8}-\frac {7}{160} \int \frac {\left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^7}dx\right )-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{45 (2 x+3)^9}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {47}{10} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{80 (2 x+3)^8}-\frac {7}{160} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}-\frac {1}{24} \int \frac {\left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^5}dx\right )\right )-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{45 (2 x+3)^9}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {47}{10} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{80 (2 x+3)^8}-\frac {7}{160} \left (\frac {1}{24} \left (\frac {3}{80} \int \frac {\sqrt {3 x^2+5 x+2}}{(2 x+3)^3}dx-\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}\right )+\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}\right )\right )-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{45 (2 x+3)^9}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {47}{10} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{80 (2 x+3)^8}-\frac {7}{160} \left (\frac {1}{24} \left (\frac {3}{80} \left (\frac {(8 x+7) \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^2}-\frac {1}{40} \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}\right )+\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}\right )\right )-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{45 (2 x+3)^9}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {47}{10} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{80 (2 x+3)^8}-\frac {7}{160} \left (\frac {1}{24} \left (\frac {3}{80} \left (\frac {1}{20} \int \frac {1}{20-\frac {(8 x+7)^2}{3 x^2+5 x+2}}d\left (-\frac {8 x+7}{\sqrt {3 x^2+5 x+2}}\right )+\frac {\sqrt {3 x^2+5 x+2} (8 x+7)}{20 (2 x+3)^2}\right )-\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}\right )+\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}\right )\right )-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{45 (2 x+3)^9}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {47}{10} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{80 (2 x+3)^8}-\frac {7}{160} \left (\frac {1}{24} \left (\frac {3}{80} \left (\frac {(8 x+7) \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^2}-\frac {\text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{40 \sqrt {5}}\right )-\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}\right )+\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}\right )\right )-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{45 (2 x+3)^9}\) |
(-13*(2 + 5*x + 3*x^2)^(9/2))/(45*(3 + 2*x)^9) + (47*(((7 + 8*x)*(2 + 5*x + 3*x^2)^(7/2))/(80*(3 + 2*x)^8) - (7*(((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))/160))/10
3.25.61.3.1 Defintions of rubi rules used
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]
Time = 0.46 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.56
| method | result | size |
| risch | \(\frac {85183488 x^{10}+8686302336 x^{9}+61982943936 x^{8}+200847330336 x^{7}+380057769936 x^{6}+460832950600 x^{5}+370732981364 x^{4}+197718401614 x^{3}+67365359239 x^{2}+13296655739 x +1157402662}{184320000 \left (3+2 x \right )^{9} \sqrt {3 x^{2}+5 x +2}}-\frac {329 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{204800000}\) | \(103\) |
| trager | \(\frac {\left (28394496 x^{8}+2848109952 x^{7}+15895201728 x^{6}+38558367264 x^{5}+51825176720 x^{4}+41530110824 x^{3}+19810691268 x^{2}+5201574542 x +578701331\right ) \sqrt {3 x^{2}+5 x +2}}{184320000 \left (3+2 x \right )^{9}}-\frac {329 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}-7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{3+2 x}\right )}{204800000}\) | \(112\) |
| default | \(-\frac {893 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{384000 \left (x +\frac {3}{2}\right )^{6}}-\frac {1457 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{400000 \left (x +\frac {3}{2}\right )^{5}}-\frac {90287 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{16000000 \left (x +\frac {3}{2}\right )^{4}}-\frac {259393 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{30000000 \left (x +\frac {3}{2}\right )^{3}}-\frac {2621237 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{200000000 \left (x +\frac {3}{2}\right )^{2}}+\frac {491479 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{50000000}-\frac {491479 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{25000000 \left (x +\frac {3}{2}\right )}-\frac {191149 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{200000000}+\frac {9541 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{96000000}-\frac {329 \left (5+6 x \right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{25600000}-\frac {329 \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 )}{204800000}+\frac {329 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{800000000}+\frac {329 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{384000000}+\frac {47 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{200000000}-\frac {47 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{32000 \left (x +\frac {3}{2}\right )^{7}}+\frac {329 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}{204800000}-\frac {47 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{51200 \left (x +\frac {3}{2}\right )^{8}}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{23040 \left (x +\frac {3}{2}\right )^{9}}\) | \(369\) |
1/184320000*(85183488*x^10+8686302336*x^9+61982943936*x^8+200847330336*x^7 +380057769936*x^6+460832950600*x^5+370732981364*x^4+197718401614*x^3+67365 359239*x^2+13296655739*x+1157402662)/(3+2*x)^9/(3*x^2+5*x+2)^(1/2)-329/204 800000*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2) )
Time = 0.28 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.09 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{10}} \, dx=\frac {2961 \, \sqrt {5} {\left (512 \, x^{9} + 6912 \, x^{8} + 41472 \, x^{7} + 145152 \, x^{6} + 326592 \, x^{5} + 489888 \, x^{4} + 489888 \, x^{3} + 314928 \, x^{2} + 118098 \, x + 19683\right )} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) + 20 \, {\left (28394496 \, x^{8} + 2848109952 \, x^{7} + 15895201728 \, x^{6} + 38558367264 \, x^{5} + 51825176720 \, x^{4} + 41530110824 \, x^{3} + 19810691268 \, x^{2} + 5201574542 \, x + 578701331\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{3686400000 \, {\left (512 \, x^{9} + 6912 \, x^{8} + 41472 \, x^{7} + 145152 \, x^{6} + 326592 \, x^{5} + 489888 \, x^{4} + 489888 \, x^{3} + 314928 \, x^{2} + 118098 \, x + 19683\right )}} \]
1/3686400000*(2961*sqrt(5)*(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)*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*(28394496*x^8 + 2848109952*x^7 + 15895201728*x^6 + 38558 367264*x^5 + 51825176720*x^4 + 41530110824*x^3 + 19810691268*x^2 + 5201574 542*x + 578701331)*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)
\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{10}} \, dx=- \int \left (- \frac {40 \sqrt {3 x^{2} + 5 x + 2}}{1024 x^{10} + 15360 x^{9} + 103680 x^{8} + 414720 x^{7} + 1088640 x^{6} + 1959552 x^{5} + 2449440 x^{4} + 2099520 x^{3} + 1180980 x^{2} + 393660 x + 59049}\right )\, dx - \int \left (- \frac {292 x \sqrt {3 x^{2} + 5 x + 2}}{1024 x^{10} + 15360 x^{9} + 103680 x^{8} + 414720 x^{7} + 1088640 x^{6} + 1959552 x^{5} + 2449440 x^{4} + 2099520 x^{3} + 1180980 x^{2} + 393660 x + 59049}\right )\, dx - \int \left (- \frac {870 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{1024 x^{10} + 15360 x^{9} + 103680 x^{8} + 414720 x^{7} + 1088640 x^{6} + 1959552 x^{5} + 2449440 x^{4} + 2099520 x^{3} + 1180980 x^{2} + 393660 x + 59049}\right )\, dx - \int \left (- \frac {1339 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{1024 x^{10} + 15360 x^{9} + 103680 x^{8} + 414720 x^{7} + 1088640 x^{6} + 1959552 x^{5} + 2449440 x^{4} + 2099520 x^{3} + 1180980 x^{2} + 393660 x + 59049}\right )\, dx - \int \left (- \frac {1090 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{1024 x^{10} + 15360 x^{9} + 103680 x^{8} + 414720 x^{7} + 1088640 x^{6} + 1959552 x^{5} + 2449440 x^{4} + 2099520 x^{3} + 1180980 x^{2} + 393660 x + 59049}\right )\, dx - \int \left (- \frac {396 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{1024 x^{10} + 15360 x^{9} + 103680 x^{8} + 414720 x^{7} + 1088640 x^{6} + 1959552 x^{5} + 2449440 x^{4} + 2099520 x^{3} + 1180980 x^{2} + 393660 x + 59049}\right )\, dx - \int \frac {27 x^{7} \sqrt {3 x^{2} + 5 x + 2}}{1024 x^{10} + 15360 x^{9} + 103680 x^{8} + 414720 x^{7} + 1088640 x^{6} + 1959552 x^{5} + 2449440 x^{4} + 2099520 x^{3} + 1180980 x^{2} + 393660 x + 59049}\, dx \]
-Integral(-40*sqrt(3*x**2 + 5*x + 2)/(1024*x**10 + 15360*x**9 + 103680*x** 8 + 414720*x**7 + 1088640*x**6 + 1959552*x**5 + 2449440*x**4 + 2099520*x** 3 + 1180980*x**2 + 393660*x + 59049), x) - Integral(-292*x*sqrt(3*x**2 + 5 *x + 2)/(1024*x**10 + 15360*x**9 + 103680*x**8 + 414720*x**7 + 1088640*x** 6 + 1959552*x**5 + 2449440*x**4 + 2099520*x**3 + 1180980*x**2 + 393660*x + 59049), x) - Integral(-870*x**2*sqrt(3*x**2 + 5*x + 2)/(1024*x**10 + 1536 0*x**9 + 103680*x**8 + 414720*x**7 + 1088640*x**6 + 1959552*x**5 + 2449440 *x**4 + 2099520*x**3 + 1180980*x**2 + 393660*x + 59049), x) - Integral(-13 39*x**3*sqrt(3*x**2 + 5*x + 2)/(1024*x**10 + 15360*x**9 + 103680*x**8 + 41 4720*x**7 + 1088640*x**6 + 1959552*x**5 + 2449440*x**4 + 2099520*x**3 + 11 80980*x**2 + 393660*x + 59049), x) - Integral(-1090*x**4*sqrt(3*x**2 + 5*x + 2)/(1024*x**10 + 15360*x**9 + 103680*x**8 + 414720*x**7 + 1088640*x**6 + 1959552*x**5 + 2449440*x**4 + 2099520*x**3 + 1180980*x**2 + 393660*x + 5 9049), x) - Integral(-396*x**5*sqrt(3*x**2 + 5*x + 2)/(1024*x**10 + 15360* x**9 + 103680*x**8 + 414720*x**7 + 1088640*x**6 + 1959552*x**5 + 2449440*x **4 + 2099520*x**3 + 1180980*x**2 + 393660*x + 59049), x) - Integral(27*x* *7*sqrt(3*x**2 + 5*x + 2)/(1024*x**10 + 15360*x**9 + 103680*x**8 + 414720* x**7 + 1088640*x**6 + 1959552*x**5 + 2449440*x**4 + 2099520*x**3 + 1180980 *x**2 + 393660*x + 59049), x)
Leaf count of result is larger than twice the leaf count of optimal. 513 vs. \(2 (154) = 308\).
Time = 0.29 (sec) , antiderivative size = 513, normalized size of antiderivative = 2.79 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{10}} \, dx=\frac {7863711}{200000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{45 \, {\left (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 )}} - \frac {47 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{200 \, {\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 {47 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{250 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac {893 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{6000 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {1457 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{12500 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {90287 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{1000000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {259393 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{3750000 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {2621237 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{50000000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {573447}{100000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x - \frac {3822651}{800000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {491479 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{10000000 \, {\left (2 \, x + 3\right )}} + \frac {9541}{16000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {191149}{384000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {987}{12800000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {329}{204800000} \, \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 {6251}{102400000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]
7863711/200000000*(3*x^2 + 5*x + 2)^(7/2) - 13/45*(3*x^2 + 5*x + 2)^(9/2)/ (512*x^9 + 6912*x^8 + 41472*x^7 + 145152*x^6 + 326592*x^5 + 489888*x^4 + 4 89888*x^3 + 314928*x^2 + 118098*x + 19683) - 47/200*(3*x^2 + 5*x + 2)^(9/2 )/(256*x^8 + 3072*x^7 + 16128*x^6 + 48384*x^5 + 90720*x^4 + 108864*x^3 + 8 1648*x^2 + 34992*x + 6561) - 47/250*(3*x^2 + 5*x + 2)^(9/2)/(128*x^7 + 134 4*x^6 + 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187) - 8 93/6000*(3*x^2 + 5*x + 2)^(9/2)/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729) - 1457/12500*(3*x^2 + 5*x + 2)^(9/2)/(32*x^5 + 24 0*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 90287/1000000*(3*x^2 + 5*x + 2 )^(9/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 259393/3750000*(3*x^2 + 5*x + 2)^(9/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 2621237/50000000*(3*x^2 + 5 *x + 2)^(9/2)/(4*x^2 + 12*x + 9) - 573447/100000000*(3*x^2 + 5*x + 2)^(5/2 )*x - 3822651/800000000*(3*x^2 + 5*x + 2)^(5/2) - 491479/10000000*(3*x^2 + 5*x + 2)^(7/2)/(2*x + 3) + 9541/16000000*(3*x^2 + 5*x + 2)^(3/2)*x + 1911 49/384000000*(3*x^2 + 5*x + 2)^(3/2) - 987/12800000*sqrt(3*x^2 + 5*x + 2)* x - 329/204800000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 6251/102400000*sqrt(3*x^2 + 5*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 563 vs. \(2 (154) = 308\).
Time = 0.35 (sec) , antiderivative size = 563, normalized size of antiderivative = 3.06 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{10}} \, dx=\frac {329}{204800000} \, \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 {14930678016 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{17} + 204061569408 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{16} + 3866707486848 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{15} + 14840812733760 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{14} + 114102022608000 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{13} + 198779998219488 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{12} + 649357338634272 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 207317438979984 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{10} - 2217334591351040 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} - 5247913396815000 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} - 20151247122371016 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} - 17924557725783828 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} - 35125577732048328 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} - 16953161853593070 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} - 17752204726475250 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} - 4253745315948057 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} - 1882391465118753 \, \sqrt {3} x - 129047626217736 \, \sqrt {3} + 1882391465118753 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{184320000 \, {\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 )}^{9}} \]
329/204800000*sqrt(5)*log(abs(-4*sqrt(3)*x - 2*sqrt(5) - 6*sqrt(3) + 4*sqr t(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))) - 1/184320000*(14930678016*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^17 + 204061569408*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^16 + 386 6707486848*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^15 + 14840812733760*sqrt(3) *(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^14 + 114102022608000*(sqrt(3)*x - sqr t(3*x^2 + 5*x + 2))^13 + 198779998219488*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^12 + 649357338634272*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^11 + 2 07317438979984*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^10 - 2217334591 351040*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^9 - 5247913396815000*sqrt(3)*(s qrt(3)*x - sqrt(3*x^2 + 5*x + 2))^8 - 20151247122371016*(sqrt(3)*x - sqrt( 3*x^2 + 5*x + 2))^7 - 17924557725783828*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^6 - 35125577732048328*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 - 16 953161853593070*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 - 1775220472 6475250*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 - 4253745315948057*sqrt(3)*( sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 - 1882391465118753*sqrt(3)*x - 129047 626217736*sqrt(3) + 1882391465118753*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)^9
Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{10}} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{7/2}}{{\left (2\,x+3\right )}^{10}} \,d x \]