Integrand size = 27, antiderivative size = 209 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{11}} \, dx=-\frac {13251 (7+8 x) \sqrt {2+5 x+3 x^2}}{1024000000 (3+2 x)^2}+\frac {4417 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{25600000 (3+2 x)^4}-\frac {4417 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{1600000 (3+2 x)^6}+\frac {1893 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{40000 (3+2 x)^8}-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{50 (3+2 x)^{10}}-\frac {29 \left (2+5 x+3 x^2\right )^{9/2}}{125 (3+2 x)^9}+\frac {13251 \text {arctanh}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{2048000000 \sqrt {5}} \]
4417/25600000*(7+8*x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^4-4417/1600000*(7+8*x)*( 3*x^2+5*x+2)^(5/2)/(3+2*x)^6+1893/40000*(7+8*x)*(3*x^2+5*x+2)^(7/2)/(3+2*x )^8-13/50*(3*x^2+5*x+2)^(9/2)/(3+2*x)^10-29/125*(3*x^2+5*x+2)^(9/2)/(3+2*x )^9+13251/10240000000*arctanh(1/10*(7+8*x)*5^(1/2)/(3*x^2+5*x+2)^(1/2))*5^ (1/2)-13251/1024000000*(7+8*x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^2
Time = 0.71 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.49 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{11}} \, dx=\frac {\frac {5 \sqrt {2+5 x+3 x^2} \left (3544392763+32786922608 x+128970753208 x^2+281702072128 x^3+372602220928 x^4+304078211712 x^5+148740043392 x^6+40186580992 x^7+5268182272 x^8+371791872 x^9\right )}{(3+2 x)^{10}}+13251 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )}{5120000000} \]
((5*Sqrt[2 + 5*x + 3*x^2]*(3544392763 + 32786922608*x + 128970753208*x^2 + 281702072128*x^3 + 372602220928*x^4 + 304078211712*x^5 + 148740043392*x^6 + 40186580992*x^7 + 5268182272*x^8 + 371791872*x^9))/(3 + 2*x)^10 + 13251 *Sqrt[5]*ArcTanh[Sqrt[2/5 + x + (3*x^2)/5]/(1 + x)])/5120000000
Time = 0.43 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.12, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1237, 27, 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.
| \(\displaystyle \int \frac {(5-x) \left (3 x^2+5 x+2\right )^{7/2}}{(2 x+3)^{11}} \, dx\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -\frac {1}{50} \int -\frac {3 (135-26 x) \left (3 x^2+5 x+2\right )^{7/2}}{2 (2 x+3)^{10}}dx-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{50 (2 x+3)^{10}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{100} \int \frac {(135-26 x) \left (3 x^2+5 x+2\right )^{7/2}}{(2 x+3)^{10}}dx-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{50 (2 x+3)^{10}}\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle \frac {3}{100} \left (\frac {631}{5} \int \frac {\left (3 x^2+5 x+2\right )^{7/2}}{(2 x+3)^9}dx-\frac {116 \left (3 x^2+5 x+2\right )^{9/2}}{15 (2 x+3)^9}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{50 (2 x+3)^{10}}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {3}{100} \left (\frac {631}{5} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{80 (2 x+3)^8}-\frac {7}{160} \int \frac {\left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^7}dx\right )-\frac {116 \left (3 x^2+5 x+2\right )^{9/2}}{15 (2 x+3)^9}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{50 (2 x+3)^{10}}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {3}{100} \left (\frac {631}{5} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{80 (2 x+3)^8}-\frac {7}{160} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}-\frac {1}{24} \int \frac {\left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^5}dx\right )\right )-\frac {116 \left (3 x^2+5 x+2\right )^{9/2}}{15 (2 x+3)^9}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{50 (2 x+3)^{10}}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {3}{100} \left (\frac {631}{5} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{80 (2 x+3)^8}-\frac {7}{160} \left (\frac {1}{24} \left (\frac {3}{80} \int \frac {\sqrt {3 x^2+5 x+2}}{(2 x+3)^3}dx-\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}\right )+\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}\right )\right )-\frac {116 \left (3 x^2+5 x+2\right )^{9/2}}{15 (2 x+3)^9}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{50 (2 x+3)^{10}}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {3}{100} \left (\frac {631}{5} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{80 (2 x+3)^8}-\frac {7}{160} \left (\frac {1}{24} \left (\frac {3}{80} \left (\frac {(8 x+7) \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^2}-\frac {1}{40} \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}\right )+\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}\right )\right )-\frac {116 \left (3 x^2+5 x+2\right )^{9/2}}{15 (2 x+3)^9}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{50 (2 x+3)^{10}}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {3}{100} \left (\frac {631}{5} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{80 (2 x+3)^8}-\frac {7}{160} \left (\frac {1}{24} \left (\frac {3}{80} \left (\frac {1}{20} \int \frac {1}{20-\frac {(8 x+7)^2}{3 x^2+5 x+2}}d\left (-\frac {8 x+7}{\sqrt {3 x^2+5 x+2}}\right )+\frac {\sqrt {3 x^2+5 x+2} (8 x+7)}{20 (2 x+3)^2}\right )-\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}\right )+\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}\right )\right )-\frac {116 \left (3 x^2+5 x+2\right )^{9/2}}{15 (2 x+3)^9}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{50 (2 x+3)^{10}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {3}{100} \left (\frac {631}{5} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{80 (2 x+3)^8}-\frac {7}{160} \left (\frac {1}{24} \left (\frac {3}{80} \left (\frac {(8 x+7) \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^2}-\frac {\text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{40 \sqrt {5}}\right )-\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}\right )+\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}\right )\right )-\frac {116 \left (3 x^2+5 x+2\right )^{9/2}}{15 (2 x+3)^9}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{50 (2 x+3)^{10}}\) |
(-13*(2 + 5*x + 3*x^2)^(9/2))/(50*(3 + 2*x)^10) + (3*((-116*(2 + 5*x + 3*x ^2)^(9/2))/(15*(3 + 2*x)^9) + (631*(((7 + 8*x)*(2 + 5*x + 3*x^2)^(7/2))/(8 0*(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]*Sqr t[2 + 5*x + 3*x^2])]/(40*Sqrt[5])))/80)/24))/160))/5))/100
3.25.62.3.1 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 = 0.50 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.52
| method | result | size |
| risch | \(\frac {1115375616 x^{11}+17663506176 x^{10}+147644238080 x^{9}+657689399680 x^{8}+1736308014080 x^{7}+2935677808128 x^{6}+3316273744448 x^{5}+2540627062120 x^{4}+1306618678120 x^{3}+432509297745 x^{2}+83295809031 x +7088785526}{1024000000 \left (3+2 x \right )^{10} \sqrt {3 x^{2}+5 x +2}}-\frac {13251 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{10240000000}\) | \(108\) |
| trager | \(\frac {\left (371791872 x^{9}+5268182272 x^{8}+40186580992 x^{7}+148740043392 x^{6}+304078211712 x^{5}+372602220928 x^{4}+281702072128 x^{3}+128970753208 x^{2}+32786922608 x +3544392763\right ) \sqrt {3 x^{2}+5 x +2}}{1024000000 \left (3+2 x \right )^{10}}-\frac {13251 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}-7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{3+2 x}\right )}{10240000000}\) | \(117\) |
| default | \(-\frac {11989 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{6400000 \left (x +\frac {3}{2}\right )^{6}}-\frac {58683 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{20000000 \left (x +\frac {3}{2}\right )^{5}}-\frac {3636453 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{800000000 \left (x +\frac {3}{2}\right )^{4}}-\frac {3482489 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{500000000 \left (x +\frac {3}{2}\right )^{3}}-\frac {105574503 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{10000000000 \left (x +\frac {3}{2}\right )^{2}}+\frac {19795101 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{2500000000}-\frac {19795101 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{1250000000 \left (x +\frac {3}{2}\right )}-\frac {7698831 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{10000000000}+\frac {128093 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{1600000000}-\frac {13251 \left (5+6 x \right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{1280000000}-\frac {13251 \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 )}{10240000000}-\frac {13 \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 )^{10}}+\frac {13251 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{40000000000}+\frac {4417 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{6400000000}+\frac {1893 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{10000000000}-\frac {1893 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{1600000 \left (x +\frac {3}{2}\right )^{7}}+\frac {13251 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}{10240000000}-\frac {1893 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{2560000 \left (x +\frac {3}{2}\right )^{8}}-\frac {29 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{64000 \left (x +\frac {3}{2}\right )^{9}}\) | \(390\) |
1/1024000000*(1115375616*x^11+17663506176*x^10+147644238080*x^9+6576893996 80*x^8+1736308014080*x^7+2935677808128*x^6+3316273744448*x^5+2540627062120 *x^4+1306618678120*x^3+432509297745*x^2+83295809031*x+7088785526)/(3+2*x)^ 10/(3*x^2+5*x+2)^(1/2)-13251/10240000000*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 = 215, normalized size of antiderivative = 1.03 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{11}} \, dx=\frac {13251 \, \sqrt {5} {\left (1024 \, x^{10} + 15360 \, x^{9} + 103680 \, x^{8} + 414720 \, x^{7} + 1088640 \, x^{6} + 1959552 \, x^{5} + 2449440 \, x^{4} + 2099520 \, x^{3} + 1180980 \, x^{2} + 393660 \, x + 59049\right )} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) + 20 \, {\left (371791872 \, x^{9} + 5268182272 \, x^{8} + 40186580992 \, x^{7} + 148740043392 \, x^{6} + 304078211712 \, x^{5} + 372602220928 \, x^{4} + 281702072128 \, x^{3} + 128970753208 \, x^{2} + 32786922608 \, x + 3544392763\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{20480000000 \, {\left (1024 \, x^{10} + 15360 \, x^{9} + 103680 \, x^{8} + 414720 \, x^{7} + 1088640 \, x^{6} + 1959552 \, x^{5} + 2449440 \, x^{4} + 2099520 \, x^{3} + 1180980 \, x^{2} + 393660 \, x + 59049\right )}} \]
1/20480000000*(13251*sqrt(5)*(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)*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*(371791872*x^9 + 5268182272*x^8 + 40186580992*x^7 + 148740043392*x^6 + 304078211712*x^5 + 372602220928*x^4 + 281702072128*x^3 + 128970753208*x^2 + 32786922608*x + 3544392763)*sqrt( 3*x^2 + 5*x + 2))/(1024*x^10 + 15360*x^9 + 103680*x^8 + 414720*x^7 + 10886 40*x^6 + 1959552*x^5 + 2449440*x^4 + 2099520*x^3 + 1180980*x^2 + 393660*x + 59049)
\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{11}} \, dx=- \int \left (- \frac {40 \sqrt {3 x^{2} + 5 x + 2}}{2048 x^{11} + 33792 x^{10} + 253440 x^{9} + 1140480 x^{8} + 3421440 x^{7} + 7185024 x^{6} + 10777536 x^{5} + 11547360 x^{4} + 8660520 x^{3} + 4330260 x^{2} + 1299078 x + 177147}\right )\, dx - \int \left (- \frac {292 x \sqrt {3 x^{2} + 5 x + 2}}{2048 x^{11} + 33792 x^{10} + 253440 x^{9} + 1140480 x^{8} + 3421440 x^{7} + 7185024 x^{6} + 10777536 x^{5} + 11547360 x^{4} + 8660520 x^{3} + 4330260 x^{2} + 1299078 x + 177147}\right )\, dx - \int \left (- \frac {870 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{2048 x^{11} + 33792 x^{10} + 253440 x^{9} + 1140480 x^{8} + 3421440 x^{7} + 7185024 x^{6} + 10777536 x^{5} + 11547360 x^{4} + 8660520 x^{3} + 4330260 x^{2} + 1299078 x + 177147}\right )\, dx - \int \left (- \frac {1339 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{2048 x^{11} + 33792 x^{10} + 253440 x^{9} + 1140480 x^{8} + 3421440 x^{7} + 7185024 x^{6} + 10777536 x^{5} + 11547360 x^{4} + 8660520 x^{3} + 4330260 x^{2} + 1299078 x + 177147}\right )\, dx - \int \left (- \frac {1090 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{2048 x^{11} + 33792 x^{10} + 253440 x^{9} + 1140480 x^{8} + 3421440 x^{7} + 7185024 x^{6} + 10777536 x^{5} + 11547360 x^{4} + 8660520 x^{3} + 4330260 x^{2} + 1299078 x + 177147}\right )\, dx - \int \left (- \frac {396 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{2048 x^{11} + 33792 x^{10} + 253440 x^{9} + 1140480 x^{8} + 3421440 x^{7} + 7185024 x^{6} + 10777536 x^{5} + 11547360 x^{4} + 8660520 x^{3} + 4330260 x^{2} + 1299078 x + 177147}\right )\, dx - \int \frac {27 x^{7} \sqrt {3 x^{2} + 5 x + 2}}{2048 x^{11} + 33792 x^{10} + 253440 x^{9} + 1140480 x^{8} + 3421440 x^{7} + 7185024 x^{6} + 10777536 x^{5} + 11547360 x^{4} + 8660520 x^{3} + 4330260 x^{2} + 1299078 x + 177147}\, dx \]
-Integral(-40*sqrt(3*x**2 + 5*x + 2)/(2048*x**11 + 33792*x**10 + 253440*x* *9 + 1140480*x**8 + 3421440*x**7 + 7185024*x**6 + 10777536*x**5 + 11547360 *x**4 + 8660520*x**3 + 4330260*x**2 + 1299078*x + 177147), x) - Integral(- 292*x*sqrt(3*x**2 + 5*x + 2)/(2048*x**11 + 33792*x**10 + 253440*x**9 + 114 0480*x**8 + 3421440*x**7 + 7185024*x**6 + 10777536*x**5 + 11547360*x**4 + 8660520*x**3 + 4330260*x**2 + 1299078*x + 177147), x) - Integral(-870*x**2 *sqrt(3*x**2 + 5*x + 2)/(2048*x**11 + 33792*x**10 + 253440*x**9 + 1140480* x**8 + 3421440*x**7 + 7185024*x**6 + 10777536*x**5 + 11547360*x**4 + 86605 20*x**3 + 4330260*x**2 + 1299078*x + 177147), x) - Integral(-1339*x**3*sqr t(3*x**2 + 5*x + 2)/(2048*x**11 + 33792*x**10 + 253440*x**9 + 1140480*x**8 + 3421440*x**7 + 7185024*x**6 + 10777536*x**5 + 11547360*x**4 + 8660520*x **3 + 4330260*x**2 + 1299078*x + 177147), x) - Integral(-1090*x**4*sqrt(3* x**2 + 5*x + 2)/(2048*x**11 + 33792*x**10 + 253440*x**9 + 1140480*x**8 + 3 421440*x**7 + 7185024*x**6 + 10777536*x**5 + 11547360*x**4 + 8660520*x**3 + 4330260*x**2 + 1299078*x + 177147), x) - Integral(-396*x**5*sqrt(3*x**2 + 5*x + 2)/(2048*x**11 + 33792*x**10 + 253440*x**9 + 1140480*x**8 + 342144 0*x**7 + 7185024*x**6 + 10777536*x**5 + 11547360*x**4 + 8660520*x**3 + 433 0260*x**2 + 1299078*x + 177147), x) - Integral(27*x**7*sqrt(3*x**2 + 5*x + 2)/(2048*x**11 + 33792*x**10 + 253440*x**9 + 1140480*x**8 + 3421440*x**7 + 7185024*x**6 + 10777536*x**5 + 11547360*x**4 + 8660520*x**3 + 4330260...
Leaf count of result is larger than twice the leaf count of optimal. 579 vs. \(2 (175) = 350\).
Time = 0.30 (sec) , antiderivative size = 579, normalized size of antiderivative = 2.77 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{11}} \, dx=\frac {316723509}{10000000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{50 \, {\left (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 )}} - \frac {29 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{125 \, {\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 {1893 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{10000 \, {\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 {1893 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{12500 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac {11989 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{100000 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {58683 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{625000 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {3636453 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{50000000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {3482489 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{62500000 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {105574503 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{2500000000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {23096493}{5000000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x - \frac {153963369}{40000000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {19795101 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{500000000 \, {\left (2 \, x + 3\right )}} + \frac {384279}{800000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {2566277}{6400000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {39753}{640000000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {13251}{10240000000} \, \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 {251769}{5120000000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]
316723509/10000000000*(3*x^2 + 5*x + 2)^(7/2) - 13/50*(3*x^2 + 5*x + 2)^(9 /2)/(1024*x^10 + 15360*x^9 + 103680*x^8 + 414720*x^7 + 1088640*x^6 + 19595 52*x^5 + 2449440*x^4 + 2099520*x^3 + 1180980*x^2 + 393660*x + 59049) - 29/ 125*(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 + 489888*x^3 + 314928*x^2 + 118098*x + 19683) - 1 893/10000*(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 + 81648*x^2 + 34992*x + 6561) - 1893/12500*(3 *x^2 + 5*x + 2)^(9/2)/(128*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 22680*x ^3 + 20412*x^2 + 10206*x + 2187) - 11989/100000*(3*x^2 + 5*x + 2)^(9/2)/(6 4*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729) - 58683/6 25000*(3*x^2 + 5*x + 2)^(9/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810 *x + 243) - 3636453/50000000*(3*x^2 + 5*x + 2)^(9/2)/(16*x^4 + 96*x^3 + 21 6*x^2 + 216*x + 81) - 3482489/62500000*(3*x^2 + 5*x + 2)^(9/2)/(8*x^3 + 36 *x^2 + 54*x + 27) - 105574503/2500000000*(3*x^2 + 5*x + 2)^(9/2)/(4*x^2 + 12*x + 9) - 23096493/5000000000*(3*x^2 + 5*x + 2)^(5/2)*x - 153963369/4000 0000000*(3*x^2 + 5*x + 2)^(5/2) - 19795101/500000000*(3*x^2 + 5*x + 2)^(7/ 2)/(2*x + 3) + 384279/800000000*(3*x^2 + 5*x + 2)^(3/2)*x + 2566277/640000 0000*(3*x^2 + 5*x + 2)^(3/2) - 39753/640000000*sqrt(3*x^2 + 5*x + 2)*x - 1 3251/10240000000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 251769/5120000000*sqrt(3*x^2 + 5*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 614 vs. \(2 (175) = 350\).
Time = 0.34 (sec) , antiderivative size = 614, normalized size of antiderivative = 2.94 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{11}} \, dx=\frac {13251}{10240000000} \, \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 {6784512 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{19} + 83137358592 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{18} + 2689605043456 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{17} + 9174489217536 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{16} - 53080570863872 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{15} - 898783135722624 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{14} - 13174687008250752 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{13} - 40507172795248512 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{12} - 270169596727110016 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{11} - 458790099197766656 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{10} - 1833183533173743552 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} - 1939024456450048032 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} - 4903074367120921776 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} - 3280073192617110456 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} - 5164856211259534888 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} - 2082844158764403144 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} - 1869656136275991262 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} - 391066159205340747 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} - 153124376229353121 \, \sqrt {3} x - 9387541838830536 \, \sqrt {3} + 153124376229353121 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{1024000000 \, {\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 )}^{10}} \]
13251/10240000000*sqrt(5)*log(abs(-4*sqrt(3)*x - 2*sqrt(5) - 6*sqrt(3) + 4 *sqrt(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))) - 1/1024000000*(6784512*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^19 + 83137358592*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^18 + 268 9605043456*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^17 + 9174489217536*sqrt(3)* (sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^16 - 53080570863872*(sqrt(3)*x - sqrt( 3*x^2 + 5*x + 2))^15 - 898783135722624*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5 *x + 2))^14 - 13174687008250752*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^13 - 4 0507172795248512*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^12 - 27016959 6727110016*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^11 - 458790099197766656*sqr t(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^10 - 1833183533173743552*(sqrt(3) *x - sqrt(3*x^2 + 5*x + 2))^9 - 1939024456450048032*sqrt(3)*(sqrt(3)*x - s qrt(3*x^2 + 5*x + 2))^8 - 4903074367120921776*(sqrt(3)*x - sqrt(3*x^2 + 5* x + 2))^7 - 3280073192617110456*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2) )^6 - 5164856211259534888*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 - 20828441 58764403144*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 - 18696561362759 91262*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 - 391066159205340747*sqrt(3)*( sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 - 153124376229353121*sqrt(3)*x - 9387 541838830536*sqrt(3) + 153124376229353121*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...
Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{11}} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{7/2}}{{\left (2\,x+3\right )}^{11}} \,d x \]