Integrand size = 27, antiderivative size = 167 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^6} \, dx=-\frac {(57845+26934 x) \sqrt {2+5 x+3 x^2}}{12800 (3+2 x)}+\frac {(17051+13074 x) \left (2+5 x+3 x^2\right )^{3/2}}{9600 (3+2 x)^3}+\frac {(119+114 x) \left (2+5 x+3 x^2\right )^{5/2}}{80 (3+2 x)^5}+\frac {177}{128} \sqrt {3} \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )-\frac {137111 \text {arctanh}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{25600 \sqrt {5}} \]
1/9600*(17051+13074*x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^3+1/80*(119+114*x)*(3*x ^2+5*x+2)^(5/2)/(3+2*x)^5+177/128*arctanh(1/6*(5+6*x)*3^(1/2)/(3*x^2+5*x+2 )^(1/2))*3^(1/2)-137111/128000*arctanh(1/10*(7+8*x)*5^(1/2)/(3*x^2+5*x+2)^ (1/2))*5^(1/2)-1/12800*(57845+26934*x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)
Time = 0.78 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.68 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^6} \, dx=\frac {-\frac {5 \sqrt {2+5 x+3 x^2} \left (12600183+37019838 x+41641148 x^2+21586808 x^3+4630848 x^4+172800 x^5\right )}{(3+2 x)^5}-411333 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )+531000 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{192000} \]
((-5*Sqrt[2 + 5*x + 3*x^2]*(12600183 + 37019838*x + 41641148*x^2 + 2158680 8*x^3 + 4630848*x^4 + 172800*x^5))/(3 + 2*x)^5 - 411333*Sqrt[5]*ArcTanh[Sq rt[2/5 + x + (3*x^2)/5]/(1 + x)] + 531000*Sqrt[3]*ArcTanh[Sqrt[2/3 + (5*x) /3 + x^2]/(1 + x)])/192000
Time = 0.39 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.07, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {1229, 1229, 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.
| \(\displaystyle \int \frac {(5-x) \left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^6} \, dx\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle \frac {(114 x+119) \left (3 x^2+5 x+2\right )^{5/2}}{80 (2 x+3)^5}-\frac {1}{160} \int \frac {(462 x+437) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^4}dx\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle \frac {1}{160} \left (\frac {1}{80} \int -\frac {2 (26934 x+22957) \sqrt {3 x^2+5 x+2}}{(2 x+3)^2}dx+\frac {(13074 x+17051) \left (3 x^2+5 x+2\right )^{3/2}}{60 (2 x+3)^3}\right )+\frac {(114 x+119) \left (3 x^2+5 x+2\right )^{5/2}}{80 (2 x+3)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{160} \left (\frac {(13074 x+17051) \left (3 x^2+5 x+2\right )^{3/2}}{60 (2 x+3)^3}-\frac {1}{40} \int \frac {(26934 x+22957) \sqrt {3 x^2+5 x+2}}{(2 x+3)^2}dx\right )+\frac {(114 x+119) \left (3 x^2+5 x+2\right )^{5/2}}{80 (2 x+3)^5}\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle \frac {1}{160} \left (\frac {1}{40} \left (\frac {1}{8} \int \frac {2 (212400 x+181489)}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-\frac {(26934 x+57845) \sqrt {3 x^2+5 x+2}}{2 (2 x+3)}\right )+\frac {(13074 x+17051) \left (3 x^2+5 x+2\right )^{3/2}}{60 (2 x+3)^3}\right )+\frac {(114 x+119) \left (3 x^2+5 x+2\right )^{5/2}}{80 (2 x+3)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{160} \left (\frac {1}{40} \left (\frac {1}{4} \int \frac {212400 x+181489}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-\frac {(26934 x+57845) \sqrt {3 x^2+5 x+2}}{2 (2 x+3)}\right )+\frac {(13074 x+17051) \left (3 x^2+5 x+2\right )^{3/2}}{60 (2 x+3)^3}\right )+\frac {(114 x+119) \left (3 x^2+5 x+2\right )^{5/2}}{80 (2 x+3)^5}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {1}{160} \left (\frac {1}{40} \left (\frac {1}{4} \left (106200 \int \frac {1}{\sqrt {3 x^2+5 x+2}}dx-137111 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {(26934 x+57845) \sqrt {3 x^2+5 x+2}}{2 (2 x+3)}\right )+\frac {(13074 x+17051) \left (3 x^2+5 x+2\right )^{3/2}}{60 (2 x+3)^3}\right )+\frac {(114 x+119) \left (3 x^2+5 x+2\right )^{5/2}}{80 (2 x+3)^5}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {1}{160} \left (\frac {1}{40} \left (\frac {1}{4} \left (212400 \int \frac {1}{12-\frac {(6 x+5)^2}{3 x^2+5 x+2}}d\frac {6 x+5}{\sqrt {3 x^2+5 x+2}}-137111 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {(26934 x+57845) \sqrt {3 x^2+5 x+2}}{2 (2 x+3)}\right )+\frac {(13074 x+17051) \left (3 x^2+5 x+2\right )^{3/2}}{60 (2 x+3)^3}\right )+\frac {(114 x+119) \left (3 x^2+5 x+2\right )^{5/2}}{80 (2 x+3)^5}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{160} \left (\frac {1}{40} \left (\frac {1}{4} \left (35400 \sqrt {3} \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-137111 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {(26934 x+57845) \sqrt {3 x^2+5 x+2}}{2 (2 x+3)}\right )+\frac {(13074 x+17051) \left (3 x^2+5 x+2\right )^{3/2}}{60 (2 x+3)^3}\right )+\frac {(114 x+119) \left (3 x^2+5 x+2\right )^{5/2}}{80 (2 x+3)^5}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {1}{160} \left (\frac {1}{40} \left (\frac {1}{4} \left (274222 \int \frac {1}{20-\frac {(8 x+7)^2}{3 x^2+5 x+2}}d\left (-\frac {8 x+7}{\sqrt {3 x^2+5 x+2}}\right )+35400 \sqrt {3} \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )\right )-\frac {(26934 x+57845) \sqrt {3 x^2+5 x+2}}{2 (2 x+3)}\right )+\frac {(13074 x+17051) \left (3 x^2+5 x+2\right )^{3/2}}{60 (2 x+3)^3}\right )+\frac {(114 x+119) \left (3 x^2+5 x+2\right )^{5/2}}{80 (2 x+3)^5}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{160} \left (\frac {1}{40} \left (\frac {1}{4} \left (35400 \sqrt {3} \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {137111 \text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {5}}\right )-\frac {(26934 x+57845) \sqrt {3 x^2+5 x+2}}{2 (2 x+3)}\right )+\frac {(13074 x+17051) \left (3 x^2+5 x+2\right )^{3/2}}{60 (2 x+3)^3}\right )+\frac {(114 x+119) \left (3 x^2+5 x+2\right )^{5/2}}{80 (2 x+3)^5}\) |
((119 + 114*x)*(2 + 5*x + 3*x^2)^(5/2))/(80*(3 + 2*x)^5) + (((17051 + 1307 4*x)*(2 + 5*x + 3*x^2)^(3/2))/(60*(3 + 2*x)^3) + (-1/2*((57845 + 26934*x)* Sqrt[2 + 5*x + 3*x^2])/(3 + 2*x) + (35400*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqr t[3]*Sqrt[2 + 5*x + 3*x^2])] - (137111*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/Sqrt[5])/4)/40)/160
3.25.42.3.1 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[(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 = 0.70
| method | result | size |
| risch | \(-\frac {518400 x^{7}+14756544 x^{6}+88260264 x^{5}+242119180 x^{4}+362438870 x^{3}+306182035 x^{2}+137040591 x +25200366}{38400 \left (3+2 x \right )^{5} \sqrt {3 x^{2}+5 x +2}}+\frac {177 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{128}+\frac {137111 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{128000}\) | \(117\) |
| trager | \(-\frac {\left (172800 x^{5}+4630848 x^{4}+21586808 x^{3}+41641148 x^{2}+37019838 x +12600183\right ) \sqrt {3 x^{2}+5 x +2}}{38400 \left (3+2 x \right )^{5}}-\frac {177 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {3 x^{2}+5 x +2}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )\right )}{128}-\frac {137111 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{3+2 x}\right )}{128000}\) | \(138\) |
| default | \(-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{800 \left (x +\frac {3}{2}\right )^{5}}-\frac {131 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{8000 \left (x +\frac {3}{2}\right )^{4}}-\frac {521 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{15000 \left (x +\frac {3}{2}\right )^{3}}-\frac {9349 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{300000 \left (x +\frac {3}{2}\right )^{2}}+\frac {11491 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{125000}-\frac {11491 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{62500 \left (x +\frac {3}{2}\right )}+\frac {6281 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{60000}+\frac {4361 \left (5+6 x \right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{16000}+\frac {177 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}\right ) \sqrt {3}}{128}+\frac {137111 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{128000}-\frac {137111 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{500000}-\frac {137111 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{240000}-\frac {137111 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}{128000}\) | \(279\) |
-1/38400*(518400*x^7+14756544*x^6+88260264*x^5+242119180*x^4+362438870*x^3 +306182035*x^2+137040591*x+25200366)/(3+2*x)^5/(3*x^2+5*x+2)^(1/2)+177/128 *ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)+137111/128000*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 = 209, normalized size of antiderivative = 1.25 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^6} \, dx=\frac {531000 \, \sqrt {3} {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 411333 \, \sqrt {5} {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )} \log \left (-\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} - 124 \, x^{2} - 212 \, x - 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 20 \, {\left (172800 \, x^{5} + 4630848 \, x^{4} + 21586808 \, x^{3} + 41641148 \, x^{2} + 37019838 \, x + 12600183\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{768000 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} \]
1/768000*(531000*sqrt(3)*(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)*log(4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49) + 411333*sqrt(5)*(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)*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*(172800*x^5 + 4630848*x^4 + 21586808*x^3 + 41641148*x^ 2 + 37019838*x + 12600183)*sqrt(3*x^2 + 5*x + 2))/(32*x^5 + 240*x^4 + 720* x^3 + 1080*x^2 + 810*x + 243)
\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^6} \, dx=- \int \left (- \frac {20 \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \left (- \frac {96 x \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \left (- \frac {165 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \left (- \frac {113 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \left (- \frac {15 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \frac {9 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\, dx \]
-Integral(-20*sqrt(3*x**2 + 5*x + 2)/(64*x**6 + 576*x**5 + 2160*x**4 + 432 0*x**3 + 4860*x**2 + 2916*x + 729), x) - Integral(-96*x*sqrt(3*x**2 + 5*x + 2)/(64*x**6 + 576*x**5 + 2160*x**4 + 4320*x**3 + 4860*x**2 + 2916*x + 72 9), x) - Integral(-165*x**2*sqrt(3*x**2 + 5*x + 2)/(64*x**6 + 576*x**5 + 2 160*x**4 + 4320*x**3 + 4860*x**2 + 2916*x + 729), x) - Integral(-113*x**3* sqrt(3*x**2 + 5*x + 2)/(64*x**6 + 576*x**5 + 2160*x**4 + 4320*x**3 + 4860* x**2 + 2916*x + 729), x) - Integral(-15*x**4*sqrt(3*x**2 + 5*x + 2)/(64*x* *6 + 576*x**5 + 2160*x**4 + 4320*x**3 + 4860*x**2 + 2916*x + 729), x) - In tegral(9*x**5*sqrt(3*x**2 + 5*x + 2)/(64*x**6 + 576*x**5 + 2160*x**4 + 432 0*x**3 + 4860*x**2 + 2916*x + 729), x)
Leaf count of result is larger than twice the leaf count of optimal. 297 vs. \(2 (135) = 270\).
Time = 0.30 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.78 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^6} \, dx=\frac {9349}{100000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{25 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {131 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{500 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {521 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{1875 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {9349 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{75000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac {6281}{10000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {11491}{240000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {11491 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{25000 \, {\left (2 \, x + 3\right )}} + \frac {13083}{8000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {177}{128} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) + \frac {137111}{128000} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) - \frac {49891}{64000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]
9349/100000*(3*x^2 + 5*x + 2)^(5/2) - 13/25*(3*x^2 + 5*x + 2)^(7/2)/(32*x^ 5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 131/500*(3*x^2 + 5*x + 2 )^(7/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 521/1875*(3*x^2 + 5*x + 2)^(7/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 9349/75000*(3*x^2 + 5*x + 2)^(7/2 )/(4*x^2 + 12*x + 9) + 6281/10000*(3*x^2 + 5*x + 2)^(3/2)*x - 11491/240000 *(3*x^2 + 5*x + 2)^(3/2) - 11491/25000*(3*x^2 + 5*x + 2)^(5/2)/(2*x + 3) + 13083/8000*sqrt(3*x^2 + 5*x + 2)*x + 177/128*sqrt(3)*log(sqrt(3)*sqrt(3*x ^2 + 5*x + 2) + 3*x + 5/2) + 137111/128000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 49891/64000*sqrt(3*x^2 + 5*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 407 vs. \(2 (135) = 270\).
Time = 0.47 (sec) , antiderivative size = 407, normalized size of antiderivative = 2.44 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^6} \, dx=-\frac {137111}{128000} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) - \frac {177}{128} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) - \frac {9}{64} \, \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {27201072 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 316934472 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 4873277176 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} + 14374341276 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} + 80473660448 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 98380998102 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 236231795506 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 119385279741 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 103767800973 \, \sqrt {3} x + 13144069068 \, \sqrt {3} - 103767800973 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{38400 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{5}} \]
-137111/128000*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))) - 177/128*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x ^2 + 5*x + 2)) - 5)) - 9/64*sqrt(3*x^2 + 5*x + 2) - 1/38400*(27201072*(sqr t(3)*x - sqrt(3*x^2 + 5*x + 2))^9 + 316934472*sqrt(3)*(sqrt(3)*x - sqrt(3* x^2 + 5*x + 2))^8 + 4873277176*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^7 + 143 74341276*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^6 + 80473660448*(sqrt (3)*x - sqrt(3*x^2 + 5*x + 2))^5 + 98380998102*sqrt(3)*(sqrt(3)*x - sqrt(3 *x^2 + 5*x + 2))^4 + 236231795506*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 + 119385279741*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 103767800973* sqrt(3)*x + 13144069068*sqrt(3) - 103767800973*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)^5
Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^6} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2}}{{\left (2\,x+3\right )}^6} \,d x \]