Integrand size = 27, antiderivative size = 167 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx=\frac {(14083+10952 x) \sqrt {2+5 x+3 x^2}}{25600 (3+2 x)^2}+\frac {(437+328 x) \left (2+5 x+3 x^2\right )^{3/2}}{1920 (3+2 x)^4}+\frac {(109+116 x) \left (2+5 x+3 x^2\right )^{5/2}}{120 (3+2 x)^6}-\frac {9}{128} \sqrt {3} \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )+\frac {13931 \text {arctanh}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{51200 \sqrt {5}} \]
1/1920*(437+328*x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^4+1/120*(109+116*x)*(3*x^2+ 5*x+2)^(5/2)/(3+2*x)^6-9/128*arctanh(1/6*(5+6*x)*3^(1/2)/(3*x^2+5*x+2)^(1/ 2))*3^(1/2)+13931/256000*arctanh(1/10*(7+8*x)*5^(1/2)/(3*x^2+5*x+2)^(1/2)) *5^(1/2)+1/25600*(14083+10952*x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^2
Time = 0.67 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.68 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx=\frac {\frac {5 \sqrt {2+5 x+3 x^2} \left (4015849+14921560 x+22854480 x^2+18217760 x^3+7629680 x^4+1351296 x^5\right )}{(3+2 x)^6}+41793 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )-54000 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{384000} \]
((5*Sqrt[2 + 5*x + 3*x^2]*(4015849 + 14921560*x + 22854480*x^2 + 18217760* x^3 + 7629680*x^4 + 1351296*x^5))/(3 + 2*x)^6 + 41793*Sqrt[5]*ArcTanh[Sqrt [2/5 + x + (3*x^2)/5]/(1 + x)] - 54000*Sqrt[3]*ArcTanh[Sqrt[2/3 + (5*x)/3 + x^2]/(1 + x)])/384000
Time = 0.40 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.07, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {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.
| \(\displaystyle \int \frac {(5-x) \left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^7} \, dx\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle \frac {(116 x+109) \left (3 x^2+5 x+2\right )^{5/2}}{120 (2 x+3)^6}-\frac {1}{240} \int \frac {5 (36 x+43) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^5}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(116 x+109) \left (3 x^2+5 x+2\right )^{5/2}}{120 (2 x+3)^6}-\frac {1}{48} \int \frac {(36 x+43) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^5}dx\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle \frac {1}{48} \left (\frac {1}{160} \int -\frac {6 (720 x+611) \sqrt {3 x^2+5 x+2}}{(2 x+3)^3}dx+\frac {(328 x+437) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}\right )+\frac {(116 x+109) \left (3 x^2+5 x+2\right )^{5/2}}{120 (2 x+3)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{48} \left (\frac {(328 x+437) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}-\frac {3}{80} \int \frac {(720 x+611) \sqrt {3 x^2+5 x+2}}{(2 x+3)^3}dx\right )+\frac {(116 x+109) \left (3 x^2+5 x+2\right )^{5/2}}{120 (2 x+3)^6}\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle \frac {1}{48} \left (\frac {(328 x+437) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}-\frac {3}{80} \left (-\frac {1}{80} \int -\frac {2 (21600 x+18469)}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-\frac {\sqrt {3 x^2+5 x+2} (10952 x+14083)}{20 (2 x+3)^2}\right )\right )+\frac {(116 x+109) \left (3 x^2+5 x+2\right )^{5/2}}{120 (2 x+3)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{48} \left (\frac {(328 x+437) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}-\frac {3}{80} \left (\frac {1}{40} \int \frac {21600 x+18469}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-\frac {(10952 x+14083) \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^2}\right )\right )+\frac {(116 x+109) \left (3 x^2+5 x+2\right )^{5/2}}{120 (2 x+3)^6}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {1}{48} \left (\frac {(328 x+437) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}-\frac {3}{80} \left (\frac {1}{40} \left (10800 \int \frac {1}{\sqrt {3 x^2+5 x+2}}dx-13931 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {(10952 x+14083) \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^2}\right )\right )+\frac {(116 x+109) \left (3 x^2+5 x+2\right )^{5/2}}{120 (2 x+3)^6}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {1}{48} \left (\frac {(328 x+437) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}-\frac {3}{80} \left (\frac {1}{40} \left (21600 \int \frac {1}{12-\frac {(6 x+5)^2}{3 x^2+5 x+2}}d\frac {6 x+5}{\sqrt {3 x^2+5 x+2}}-13931 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {(10952 x+14083) \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^2}\right )\right )+\frac {(116 x+109) \left (3 x^2+5 x+2\right )^{5/2}}{120 (2 x+3)^6}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{48} \left (\frac {(328 x+437) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}-\frac {3}{80} \left (\frac {1}{40} \left (3600 \sqrt {3} \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-13931 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {(10952 x+14083) \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^2}\right )\right )+\frac {(116 x+109) \left (3 x^2+5 x+2\right )^{5/2}}{120 (2 x+3)^6}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {1}{48} \left (\frac {(328 x+437) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}-\frac {3}{80} \left (\frac {1}{40} \left (27862 \int \frac {1}{20-\frac {(8 x+7)^2}{3 x^2+5 x+2}}d\left (-\frac {8 x+7}{\sqrt {3 x^2+5 x+2}}\right )+3600 \sqrt {3} \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )\right )-\frac {(10952 x+14083) \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^2}\right )\right )+\frac {(116 x+109) \left (3 x^2+5 x+2\right )^{5/2}}{120 (2 x+3)^6}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{48} \left (\frac {(328 x+437) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}-\frac {3}{80} \left (\frac {1}{40} \left (3600 \sqrt {3} \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {13931 \text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {5}}\right )-\frac {(10952 x+14083) \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^2}\right )\right )+\frac {(116 x+109) \left (3 x^2+5 x+2\right )^{5/2}}{120 (2 x+3)^6}\) |
((109 + 116*x)*(2 + 5*x + 3*x^2)^(5/2))/(120*(3 + 2*x)^6) + (((437 + 328*x )*(2 + 5*x + 3*x^2)^(3/2))/(40*(3 + 2*x)^4) - (3*(-1/20*((14083 + 10952*x) *Sqrt[2 + 5*x + 3*x^2])/(3 + 2*x)^2 + (3600*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*S qrt[3]*Sqrt[2 + 5*x + 3*x^2])] - (13931*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[ 2 + 5*x + 3*x^2])])/Sqrt[5])/40))/80)/48
3.25.43.3.1 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 = 0.39 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.70
| method | result | size |
| risch | \(\frac {4053888 x^{7}+29645520 x^{6}+95504272 x^{5}+174911600 x^{4}+195472600 x^{3}+132364307 x^{2}+49922365 x +8031698}{76800 \left (3+2 x \right )^{6} \sqrt {3 x^{2}+5 x +2}}-\frac {9 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{128}-\frac {13931 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{256000}\) | \(117\) |
| trager | \(\frac {\left (1351296 x^{5}+7629680 x^{4}+18217760 x^{3}+22854480 x^{2}+14921560 x +4015849\right ) \sqrt {3 x^{2}+5 x +2}}{76800 \left (3+2 x \right )^{6}}+\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {3 x^{2}+5 x +2}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )\right )}{128}-\frac {13931 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}-7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{3+2 x}\right )}{256000}\) | \(138\) |
| default | \(-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{1920 \left (x +\frac {3}{2}\right )^{6}}-\frac {23 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{2400 \left (x +\frac {3}{2}\right )^{5}}-\frac {249 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{16000 \left (x +\frac {3}{2}\right )^{4}}-\frac {709 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{30000 \left (x +\frac {3}{2}\right )^{3}}-\frac {22271 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{600000 \left (x +\frac {3}{2}\right )^{2}}+\frac {6089 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{250000}-\frac {6089 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{125000 \left (x +\frac {3}{2}\right )}-\frac {1001 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{120000}-\frac {431 \left (5+6 x \right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{32000}-\frac {9 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}\right ) \sqrt {3}}{128}-\frac {13931 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{256000}+\frac {13931 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{1000000}+\frac {13931 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{480000}+\frac {13931 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}{256000}\) | \(300\) |
1/76800*(4053888*x^7+29645520*x^6+95504272*x^5+174911600*x^4+195472600*x^3 +132364307*x^2+49922365*x+8031698)/(3+2*x)^6/(3*x^2+5*x+2)^(1/2)-9/128*ln( 1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)-13931/256000*5^(1/2)*ar ctanh(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 = 223, normalized size of antiderivative = 1.34 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx=\frac {54000 \, \sqrt {3} {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 41793 \, \sqrt {5} {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) + 20 \, {\left (1351296 \, x^{5} + 7629680 \, x^{4} + 18217760 \, x^{3} + 22854480 \, x^{2} + 14921560 \, x + 4015849\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{1536000 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} \]
1/1536000*(54000*sqrt(3)*(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^ 2 + 2916*x + 729)*log(-4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49) + 41793*sqrt(5)*(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 48 60*x^2 + 2916*x + 729)*log((4*sqrt(5)*sqrt(3*x^2 + 5*x + 2)*(8*x + 7) + 12 4*x^2 + 212*x + 89)/(4*x^2 + 12*x + 9)) + 20*(1351296*x^5 + 7629680*x^4 + 18217760*x^3 + 22854480*x^2 + 14921560*x + 4015849)*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)
\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx=- \int \left (- \frac {20 \sqrt {3 x^{2} + 5 x + 2}}{128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187}\right )\, dx - \int \left (- \frac {96 x \sqrt {3 x^{2} + 5 x + 2}}{128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187}\right )\, dx - \int \left (- \frac {165 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187}\right )\, dx - \int \left (- \frac {113 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187}\right )\, dx - \int \left (- \frac {15 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187}\right )\, dx - \int \frac {9 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187}\, dx \]
-Integral(-20*sqrt(3*x**2 + 5*x + 2)/(128*x**7 + 1344*x**6 + 6048*x**5 + 1 5120*x**4 + 22680*x**3 + 20412*x**2 + 10206*x + 2187), x) - Integral(-96*x *sqrt(3*x**2 + 5*x + 2)/(128*x**7 + 1344*x**6 + 6048*x**5 + 15120*x**4 + 2 2680*x**3 + 20412*x**2 + 10206*x + 2187), x) - Integral(-165*x**2*sqrt(3*x **2 + 5*x + 2)/(128*x**7 + 1344*x**6 + 6048*x**5 + 15120*x**4 + 22680*x**3 + 20412*x**2 + 10206*x + 2187), x) - Integral(-113*x**3*sqrt(3*x**2 + 5*x + 2)/(128*x**7 + 1344*x**6 + 6048*x**5 + 15120*x**4 + 22680*x**3 + 20412* x**2 + 10206*x + 2187), x) - Integral(-15*x**4*sqrt(3*x**2 + 5*x + 2)/(128 *x**7 + 1344*x**6 + 6048*x**5 + 15120*x**4 + 22680*x**3 + 20412*x**2 + 102 06*x + 2187), x) - Integral(9*x**5*sqrt(3*x**2 + 5*x + 2)/(128*x**7 + 1344 *x**6 + 6048*x**5 + 15120*x**4 + 22680*x**3 + 20412*x**2 + 10206*x + 2187) , x)
Leaf count of result is larger than twice the leaf count of optimal. 343 vs. \(2 (135) = 270\).
Time = 0.28 (sec) , antiderivative size = 343, normalized size of antiderivative = 2.05 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx=\frac {22271}{200000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{30 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {23 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{75 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {249 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{1000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {709 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{3750 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {22271 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{150000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {1001}{20000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {6089}{480000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {6089 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{50000 \, {\left (2 \, x + 3\right )}} - \frac {1293}{16000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {9}{128} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) - \frac {13931}{256000} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) + \frac {5311}{128000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]
22271/200000*(3*x^2 + 5*x + 2)^(5/2) - 13/30*(3*x^2 + 5*x + 2)^(7/2)/(64*x ^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729) - 23/75*(3*x ^2 + 5*x + 2)^(7/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 249/1000*(3*x^2 + 5*x + 2)^(7/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81 ) - 709/3750*(3*x^2 + 5*x + 2)^(7/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 22271/ 150000*(3*x^2 + 5*x + 2)^(7/2)/(4*x^2 + 12*x + 9) - 1001/20000*(3*x^2 + 5* x + 2)^(3/2)*x - 6089/480000*(3*x^2 + 5*x + 2)^(3/2) - 6089/50000*(3*x^2 + 5*x + 2)^(5/2)/(2*x + 3) - 1293/16000*sqrt(3*x^2 + 5*x + 2)*x - 9/128*sqr t(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) - 13931/256000*sqrt(5) *log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) + 5311/128000*sqrt(3*x^2 + 5*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 444 vs. \(2 (135) = 270\).
Time = 0.48 (sec) , antiderivative size = 444, normalized size of antiderivative = 2.66 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx=\frac {13931}{256000} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) + \frac {9}{128} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) + \frac {20435424 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 269619696 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{10} + 4893810640 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 17834042400 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 129909086880 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} + 219870810528 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} + 791797675536 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 672745449240 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 1187868124850 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 460902113505 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 318216938187 \, \sqrt {3} x + 32907940848 \, \sqrt {3} - 318216938187 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{76800 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{6}} \]
13931/256000*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))) + 9/128*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5)) + 1/76800*(20435424*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^1 1 + 269619696*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^10 + 4893810640* (sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^9 + 17834042400*sqrt(3)*(sqrt(3)*x - s qrt(3*x^2 + 5*x + 2))^8 + 129909086880*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) ^7 + 219870810528*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^6 + 79179767 5536*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 + 672745449240*sqrt(3)*(sqrt(3) *x - sqrt(3*x^2 + 5*x + 2))^4 + 1187868124850*(sqrt(3)*x - sqrt(3*x^2 + 5* x + 2))^3 + 460902113505*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 3 18216938187*sqrt(3)*x + 32907940848*sqrt(3) - 318216938187*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)^6
Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2}}{{\left (2\,x+3\right )}^7} \,d x \]