Integrand size = 27, antiderivative size = 197 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^9} \, dx=\frac {3 (559841+434104 x) \sqrt {2+5 x+3 x^2}}{4096000 (3+2 x)^2}+\frac {(20959+17096 x) \left (2+5 x+3 x^2\right )^{3/2}}{102400 (3+2 x)^4}+\frac {(881+664 x) \left (2+5 x+3 x^2\right )^{5/2}}{6400 (3+2 x)^6}+\frac {(757+808 x) \left (2+5 x+3 x^2\right )^{7/2}}{1120 (3+2 x)^8}-\frac {27}{512} \sqrt {3} \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )+\frac {1673211 \text {arctanh}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{8192000 \sqrt {5}} \]
1/102400*(20959+17096*x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^4+1/6400*(881+664*x)* (3*x^2+5*x+2)^(5/2)/(3+2*x)^6+1/1120*(757+808*x)*(3*x^2+5*x+2)^(7/2)/(3+2* x)^8-27/512*arctanh(1/6*(5+6*x)*3^(1/2)/(3*x^2+5*x+2)^(1/2))*3^(1/2)+16732 11/40960000*arctanh(1/10*(7+8*x)*5^(1/2)/(3*x^2+5*x+2)^(1/2))*5^(1/2)+3/40 96000*(559841+434104*x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^2
Time = 0.79 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.62 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^9} \, dx=\frac {\frac {5 \sqrt {2+5 x+3 x^2} \left (9818427389+48950756372 x+105874603844 x^2+129405924160 x^3+97176896240 x^4+45214440256 x^5+12182619328 x^6+1478785536 x^7\right )}{(3+2 x)^8}+11712477 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )-15120000 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{143360000} \]
((5*Sqrt[2 + 5*x + 3*x^2]*(9818427389 + 48950756372*x + 105874603844*x^2 + 129405924160*x^3 + 97176896240*x^4 + 45214440256*x^5 + 12182619328*x^6 + 1478785536*x^7))/(3 + 2*x)^8 + 11712477*Sqrt[5]*ArcTanh[Sqrt[2/5 + x + (3* x^2)/5]/(1 + x)] - 15120000*Sqrt[3]*ArcTanh[Sqrt[2/3 + (5*x)/3 + x^2]/(1 + x)])/143360000
Time = 0.44 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.08, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {1229, 27, 1229, 27, 1229, 27, 1229, 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.
| \(\displaystyle \int \frac {(5-x) \left (3 x^2+5 x+2\right )^{7/2}}{(2 x+3)^9} \, dx\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle \frac {(808 x+757) \left (3 x^2+5 x+2\right )^{7/2}}{1120 (2 x+3)^8}-\frac {1}{320} \int \frac {3 (80 x+97) \left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^7}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(808 x+757) \left (3 x^2+5 x+2\right )^{7/2}}{1120 (2 x+3)^8}-\frac {3}{320} \int \frac {(80 x+97) \left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^7}dx\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle \frac {(808 x+757) \left (3 x^2+5 x+2\right )^{7/2}}{1120 (2 x+3)^8}-\frac {3}{320} \left (-\frac {1}{240} \int -\frac {10 (1440 x+1223) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^5}dx-\frac {(664 x+881) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(808 x+757) \left (3 x^2+5 x+2\right )^{7/2}}{1120 (2 x+3)^8}-\frac {3}{320} \left (\frac {1}{24} \int \frac {(1440 x+1223) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^5}dx-\frac {(664 x+881) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}\right )\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle \frac {(808 x+757) \left (3 x^2+5 x+2\right )^{7/2}}{1120 (2 x+3)^8}-\frac {3}{320} \left (\frac {1}{24} \left (-\frac {1}{160} \int -\frac {6 (28800 x+24937) \sqrt {3 x^2+5 x+2}}{(2 x+3)^3}dx-\frac {(17096 x+20959) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}\right )-\frac {(664 x+881) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(808 x+757) \left (3 x^2+5 x+2\right )^{7/2}}{1120 (2 x+3)^8}-\frac {3}{320} \left (\frac {1}{24} \left (\frac {3}{80} \int \frac {(28800 x+24937) \sqrt {3 x^2+5 x+2}}{(2 x+3)^3}dx-\frac {(17096 x+20959) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}\right )-\frac {(664 x+881) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}\right )\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle \frac {(808 x+757) \left (3 x^2+5 x+2\right )^{7/2}}{1120 (2 x+3)^8}-\frac {3}{320} \left (\frac {1}{24} \left (\frac {3}{80} \left (-\frac {1}{80} \int -\frac {2 (864000 x+738263)}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-\frac {\sqrt {3 x^2+5 x+2} (434104 x+559841)}{20 (2 x+3)^2}\right )-\frac {(17096 x+20959) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}\right )-\frac {(664 x+881) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(808 x+757) \left (3 x^2+5 x+2\right )^{7/2}}{1120 (2 x+3)^8}-\frac {3}{320} \left (\frac {1}{24} \left (\frac {3}{80} \left (\frac {1}{40} \int \frac {864000 x+738263}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-\frac {(434104 x+559841) \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^2}\right )-\frac {(17096 x+20959) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}\right )-\frac {(664 x+881) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}\right )\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {(808 x+757) \left (3 x^2+5 x+2\right )^{7/2}}{1120 (2 x+3)^8}-\frac {3}{320} \left (\frac {1}{24} \left (\frac {3}{80} \left (\frac {1}{40} \left (432000 \int \frac {1}{\sqrt {3 x^2+5 x+2}}dx-557737 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {(434104 x+559841) \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^2}\right )-\frac {(17096 x+20959) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}\right )-\frac {(664 x+881) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}\right )\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {(808 x+757) \left (3 x^2+5 x+2\right )^{7/2}}{1120 (2 x+3)^8}-\frac {3}{320} \left (\frac {1}{24} \left (\frac {3}{80} \left (\frac {1}{40} \left (864000 \int \frac {1}{12-\frac {(6 x+5)^2}{3 x^2+5 x+2}}d\frac {6 x+5}{\sqrt {3 x^2+5 x+2}}-557737 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {(434104 x+559841) \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^2}\right )-\frac {(17096 x+20959) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}\right )-\frac {(664 x+881) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {(808 x+757) \left (3 x^2+5 x+2\right )^{7/2}}{1120 (2 x+3)^8}-\frac {3}{320} \left (\frac {1}{24} \left (\frac {3}{80} \left (\frac {1}{40} \left (144000 \sqrt {3} \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-557737 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {(434104 x+559841) \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^2}\right )-\frac {(17096 x+20959) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}\right )-\frac {(664 x+881) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}\right )\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {(808 x+757) \left (3 x^2+5 x+2\right )^{7/2}}{1120 (2 x+3)^8}-\frac {3}{320} \left (\frac {1}{24} \left (\frac {3}{80} \left (\frac {1}{40} \left (1115474 \int \frac {1}{20-\frac {(8 x+7)^2}{3 x^2+5 x+2}}d\left (-\frac {8 x+7}{\sqrt {3 x^2+5 x+2}}\right )+144000 \sqrt {3} \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )\right )-\frac {(434104 x+559841) \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^2}\right )-\frac {(17096 x+20959) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}\right )-\frac {(664 x+881) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {(808 x+757) \left (3 x^2+5 x+2\right )^{7/2}}{1120 (2 x+3)^8}-\frac {3}{320} \left (\frac {1}{24} \left (\frac {3}{80} \left (\frac {1}{40} \left (144000 \sqrt {3} \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {557737 \text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {5}}\right )-\frac {(434104 x+559841) \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^2}\right )-\frac {(17096 x+20959) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}\right )-\frac {(664 x+881) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}\right )\) |
((757 + 808*x)*(2 + 5*x + 3*x^2)^(7/2))/(1120*(3 + 2*x)^8) - (3*(-1/60*((8 81 + 664*x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^6 + (-1/40*((20959 + 17096* x)*(2 + 5*x + 3*x^2)^(3/2))/(3 + 2*x)^4 + (3*(-1/20*((559841 + 434104*x)*S qrt[2 + 5*x + 3*x^2])/(3 + 2*x)^2 + (144000*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*S qrt[3]*Sqrt[2 + 5*x + 3*x^2])] - (557737*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt [2 + 5*x + 3*x^2])])/Sqrt[5])/40))/80)/24))/320
3.25.60.3.1 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[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 = 0.48 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.64
| method | result | size |
| risch | \(\frac {4436356608 x^{9}+43941785664 x^{8}+199513988480 x^{7}+541968128656 x^{6}+964531134192 x^{5}+1159007224812 x^{4}+935037136656 x^{3}+485958271715 x^{2}+146993649689 x +19636854778}{28672000 \left (3+2 x \right )^{8} \sqrt {3 x^{2}+5 x +2}}-\frac {27 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{512}-\frac {1673211 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{40960000}\) | \(127\) |
| trager | \(\frac {\left (1478785536 x^{7}+12182619328 x^{6}+45214440256 x^{5}+97176896240 x^{4}+129405924160 x^{3}+105874603844 x^{2}+48950756372 x +9818427389\right ) \sqrt {3 x^{2}+5 x +2}}{28672000 \left (3+2 x \right )^{8}}+\frac {27 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {3 x^{2}+5 x +2}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )\right )}{512}+\frac {1673211 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{3+2 x}\right )}{40960000}\) | \(148\) |
| default | \(-\frac {523 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{179200 \left (x +\frac {3}{2}\right )^{6}}-\frac {363 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{80000 \left (x +\frac {3}{2}\right )^{5}}-\frac {158331 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{22400000 \left (x +\frac {3}{2}\right )^{4}}-\frac {150503 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{14000000 \left (x +\frac {3}{2}\right )^{3}}-\frac {664383 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{40000000 \left (x +\frac {3}{2}\right )^{2}}+\frac {767427 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{70000000}-\frac {767427 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{35000000 \left (x +\frac {3}{2}\right )}-\frac {135591 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{40000000}-\frac {25627 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{6400000}-\frac {53211 \left (5+6 x \right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{5120000}-\frac {27 \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}}{512}-\frac {1673211 \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 )}{40960000}+\frac {1673211 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{160000000}+\frac {557737 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{25600000}+\frac {1673211 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{280000000}-\frac {81 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{44800 \left (x +\frac {3}{2}\right )^{7}}+\frac {1673211 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}{40960000}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{10240 \left (x +\frac {3}{2}\right )^{8}}\) | \(379\) |
1/28672000*(4436356608*x^9+43941785664*x^8+199513988480*x^7+541968128656*x ^6+964531134192*x^5+1159007224812*x^4+935037136656*x^3+485958271715*x^2+14 6993649689*x+19636854778)/(3+2*x)^8/(3*x^2+5*x+2)^(1/2)-27/512*ln(1/3*(5/2 +3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)-1673211/40960000*5^(1/2)*arctan h(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 = 263, normalized size of antiderivative = 1.34 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^9} \, dx=\frac {15120000 \, \sqrt {3} {\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 11712477 \, \sqrt {5} {\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) + 20 \, {\left (1478785536 \, x^{7} + 12182619328 \, x^{6} + 45214440256 \, x^{5} + 97176896240 \, x^{4} + 129405924160 \, x^{3} + 105874603844 \, x^{2} + 48950756372 \, x + 9818427389\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{573440000 \, {\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )}} \]
1/573440000*(15120000*sqrt(3)*(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(3)*sqrt (3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49) + 11712477*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)) + 20*(1478785536*x^7 + 12182619328*x ^6 + 45214440256*x^5 + 97176896240*x^4 + 129405924160*x^3 + 105874603844*x ^2 + 48950756372*x + 9818427389)*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)
\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^9} \, dx=- \int \left (- \frac {40 \sqrt {3 x^{2} + 5 x + 2}}{512 x^{9} + 6912 x^{8} + 41472 x^{7} + 145152 x^{6} + 326592 x^{5} + 489888 x^{4} + 489888 x^{3} + 314928 x^{2} + 118098 x + 19683}\right )\, dx - \int \left (- \frac {292 x \sqrt {3 x^{2} + 5 x + 2}}{512 x^{9} + 6912 x^{8} + 41472 x^{7} + 145152 x^{6} + 326592 x^{5} + 489888 x^{4} + 489888 x^{3} + 314928 x^{2} + 118098 x + 19683}\right )\, dx - \int \left (- \frac {870 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{512 x^{9} + 6912 x^{8} + 41472 x^{7} + 145152 x^{6} + 326592 x^{5} + 489888 x^{4} + 489888 x^{3} + 314928 x^{2} + 118098 x + 19683}\right )\, dx - \int \left (- \frac {1339 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{512 x^{9} + 6912 x^{8} + 41472 x^{7} + 145152 x^{6} + 326592 x^{5} + 489888 x^{4} + 489888 x^{3} + 314928 x^{2} + 118098 x + 19683}\right )\, dx - \int \left (- \frac {1090 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{512 x^{9} + 6912 x^{8} + 41472 x^{7} + 145152 x^{6} + 326592 x^{5} + 489888 x^{4} + 489888 x^{3} + 314928 x^{2} + 118098 x + 19683}\right )\, dx - \int \left (- \frac {396 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{512 x^{9} + 6912 x^{8} + 41472 x^{7} + 145152 x^{6} + 326592 x^{5} + 489888 x^{4} + 489888 x^{3} + 314928 x^{2} + 118098 x + 19683}\right )\, dx - \int \frac {27 x^{7} \sqrt {3 x^{2} + 5 x + 2}}{512 x^{9} + 6912 x^{8} + 41472 x^{7} + 145152 x^{6} + 326592 x^{5} + 489888 x^{4} + 489888 x^{3} + 314928 x^{2} + 118098 x + 19683}\, dx \]
-Integral(-40*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(-292*x*sqrt(3*x**2 + 5*x + 2)/(512*x**9 + 691 2*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(-870*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) - Integra l(-1339*x**3*sqrt(3*x**2 + 5*x + 2)/(512*x**9 + 6912*x**8 + 41472*x**7 + 1 45152*x**6 + 326592*x**5 + 489888*x**4 + 489888*x**3 + 314928*x**2 + 11809 8*x + 19683), x) - Integral(-1090*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(-396*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), x) - Inte gral(27*x**7*sqrt(3*x**2 + 5*x + 2)/(512*x**9 + 6912*x**8 + 41472*x**7 + 1 45152*x**6 + 326592*x**5 + 489888*x**4 + 489888*x**3 + 314928*x**2 + 11809 8*x + 19683), x)
Leaf count of result is larger than twice the leaf count of optimal. 479 vs. \(2 (161) = 322\).
Time = 0.30 (sec) , antiderivative size = 479, normalized size of antiderivative = 2.43 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^9} \, dx=\frac {1993149}{40000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{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 {81 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{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 {523 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{2800 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {363 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{2500 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {158331 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{1400000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {150503 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{1750000 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {664383 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{10000000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {406773}{20000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x - \frac {1038609}{160000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {767427 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{14000000 \, {\left (2 \, x + 3\right )}} - \frac {76881}{3200000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {45197}{25600000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {159633}{2560000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {27}{512} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) - \frac {1673211}{40960000} \, \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 {608991}{20480000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]
1993149/40000000*(3*x^2 + 5*x + 2)^(7/2) - 13/40*(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 + 8164 8*x^2 + 34992*x + 6561) - 81/350*(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) - 523/ 2800*(3*x^2 + 5*x + 2)^(9/2)/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 486 0*x^2 + 2916*x + 729) - 363/2500*(3*x^2 + 5*x + 2)^(9/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 158331/1400000*(3*x^2 + 5*x + 2)^(9 /2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 150503/1750000*(3*x^2 + 5*x + 2)^(9/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 664383/10000000*(3*x^2 + 5*x + 2)^(9/2)/(4*x^2 + 12*x + 9) - 406773/20000000*(3*x^2 + 5*x + 2)^(5/2)*x - 1038609/160000000*(3*x^2 + 5*x + 2)^(5/2) - 767427/14000000*(3*x^2 + 5*x + 2)^(7/2)/(2*x + 3) - 76881/3200000*(3*x^2 + 5*x + 2)^(3/2)*x + 45197/2560 0000*(3*x^2 + 5*x + 2)^(3/2) - 159633/2560000*sqrt(3*x^2 + 5*x + 2)*x - 27 /512*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) - 1673211/4096 0000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) + 608991/20480000*sqrt(3*x^2 + 5*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 546 vs. \(2 (161) = 322\).
Time = 0.35 (sec) , antiderivative size = 546, normalized size of antiderivative = 2.77 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^9} \, dx=\frac {1673211}{40960000} \, \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 {27}{512} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) + \frac {25982914944 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{15} + 475461282240 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{14} + 12329944383680 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{13} + 66497191380480 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{12} + 747738478510240 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 2056338758898032 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{10} + 12823219634258640 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 20470141041874560 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 75774797457107080 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} + 72179382871515780 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} + 157788604924552196 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 86325470670757920 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 102935771527447390 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 28057073003987265 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 14067886443441495 \, \sqrt {3} x + 1086949713645432 \, \sqrt {3} - 14067886443441495 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{28672000 \, {\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}} \]
1673211/40960000*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))) + 27/512*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3* x^2 + 5*x + 2)) - 5)) + 1/28672000*(25982914944*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^15 + 475461282240*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^14 + 12329944383680*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^13 + 66497191380480* sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^12 + 747738478510240*(sqrt(3)* x - sqrt(3*x^2 + 5*x + 2))^11 + 2056338758898032*sqrt(3)*(sqrt(3)*x - sqrt (3*x^2 + 5*x + 2))^10 + 12823219634258640*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^9 + 20470141041874560*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^8 + 75774797457107080*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^7 + 7217938287151578 0*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^6 + 157788604924552196*(sqrt (3)*x - sqrt(3*x^2 + 5*x + 2))^5 + 86325470670757920*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 + 102935771527447390*(sqrt(3)*x - sqrt(3*x^2 + 5* x + 2))^3 + 28057073003987265*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^ 2 + 14067886443441495*sqrt(3)*x + 1086949713645432*sqrt(3) - 1406788644344 1495*sqrt(3*x^2 + 5*x + 2))/(2*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 6*s qrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) + 11)^8
Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^9} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{7/2}}{{\left (2\,x+3\right )}^9} \,d x \]