Integrand size = 27, antiderivative size = 197 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^6} \, dx=-\frac {21 (47145+21974 x) \sqrt {2+5 x+3 x^2}}{10240 (3+2 x)}+\frac {7 (42733+33142 x) \left (2+5 x+3 x^2\right )^{3/2}}{7680 (3+2 x)^3}+\frac {7 (1003+548 x) \left (2+5 x+3 x^2\right )^{5/2}}{960 (3+2 x)^4}-\frac {(27+5 x) \left (2+5 x+3 x^2\right )^{7/2}}{30 (3+2 x)^5}+\frac {30275 \sqrt {3} \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{1024}-\frac {2345091 \text {arctanh}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{20480 \sqrt {5}} \]
7/7680*(42733+33142*x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^3+7/960*(1003+548*x)*(3 *x^2+5*x+2)^(5/2)/(3+2*x)^4-1/30*(27+5*x)*(3*x^2+5*x+2)^(7/2)/(3+2*x)^5+30 275/1024*arctanh(1/6*(5+6*x)*3^(1/2)/(3*x^2+5*x+2)^(1/2))*3^(1/2)-2345091/ 102400*arctanh(1/10*(7+8*x)*5^(1/2)/(3*x^2+5*x+2)^(1/2))*5^(1/2)-21/10240* (47145+21974*x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)
Time = 0.72 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.62 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^6} \, dx=\frac {-\frac {5 \sqrt {2+5 x+3 x^2} \left (72189541+213122626 x+242016116 x^2+127665096 x^3+27897856 x^4+483840 x^5-257280 x^6+46080 x^7\right )}{(3+2 x)^5}-2345091 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )+3027500 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{51200} \]
((-5*Sqrt[2 + 5*x + 3*x^2]*(72189541 + 213122626*x + 242016116*x^2 + 12766 5096*x^3 + 27897856*x^4 + 483840*x^5 - 257280*x^6 + 46080*x^7))/(3 + 2*x)^ 5 - 2345091*Sqrt[5]*ArcTanh[Sqrt[2/5 + x + (3*x^2)/5]/(1 + x)] + 3027500*S qrt[3]*ArcTanh[Sqrt[2/3 + (5*x)/3 + x^2]/(1 + x)])/51200
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 = {1230, 27, 1230, 27, 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.
| \(\displaystyle \int \frac {(5-x) \left (3 x^2+5 x+2\right )^{7/2}}{(2 x+3)^6} \, dx\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle -\frac {7}{120} \int -\frac {2 (137 x+115) \left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^5}dx-\frac {(5 x+27) \left (3 x^2+5 x+2\right )^{7/2}}{30 (2 x+3)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7}{60} \int \frac {(137 x+115) \left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^5}dx-\frac {(5 x+27) \left (3 x^2+5 x+2\right )^{7/2}}{30 (2 x+3)^5}\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle \frac {7}{60} \left (\frac {(548 x+1003) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^4}-\frac {5}{64} \int \frac {2 (3278 x+2823) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^4}dx\right )-\frac {(5 x+27) \left (3 x^2+5 x+2\right )^{7/2}}{30 (2 x+3)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7}{60} \left (\frac {(548 x+1003) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^4}-\frac {5}{32} \int \frac {(3278 x+2823) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^4}dx\right )-\frac {(5 x+27) \left (3 x^2+5 x+2\right )^{7/2}}{30 (2 x+3)^5}\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle \frac {7}{60} \left (\frac {(548 x+1003) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^4}-\frac {5}{32} \left (-\frac {1}{80} \int -\frac {18 (21974 x+18777) \sqrt {3 x^2+5 x+2}}{(2 x+3)^2}dx-\frac {(33142 x+42733) \left (3 x^2+5 x+2\right )^{3/2}}{20 (2 x+3)^3}\right )\right )-\frac {(5 x+27) \left (3 x^2+5 x+2\right )^{7/2}}{30 (2 x+3)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7}{60} \left (\frac {(548 x+1003) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^4}-\frac {5}{32} \left (\frac {9}{40} \int \frac {(21974 x+18777) \sqrt {3 x^2+5 x+2}}{(2 x+3)^2}dx-\frac {(33142 x+42733) \left (3 x^2+5 x+2\right )^{3/2}}{20 (2 x+3)^3}\right )\right )-\frac {(5 x+27) \left (3 x^2+5 x+2\right )^{7/2}}{30 (2 x+3)^5}\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle \frac {7}{60} \left (\frac {(548 x+1003) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^4}-\frac {5}{32} \left (\frac {9}{40} \left (\frac {(21974 x+47145) \sqrt {3 x^2+5 x+2}}{2 (2 x+3)}-\frac {1}{8} \int \frac {2 (173000 x+147829)}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {(33142 x+42733) \left (3 x^2+5 x+2\right )^{3/2}}{20 (2 x+3)^3}\right )\right )-\frac {(5 x+27) \left (3 x^2+5 x+2\right )^{7/2}}{30 (2 x+3)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7}{60} \left (\frac {(548 x+1003) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^4}-\frac {5}{32} \left (\frac {9}{40} \left (\frac {(21974 x+47145) \sqrt {3 x^2+5 x+2}}{2 (2 x+3)}-\frac {1}{4} \int \frac {173000 x+147829}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {(33142 x+42733) \left (3 x^2+5 x+2\right )^{3/2}}{20 (2 x+3)^3}\right )\right )-\frac {(5 x+27) \left (3 x^2+5 x+2\right )^{7/2}}{30 (2 x+3)^5}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {7}{60} \left (\frac {(548 x+1003) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^4}-\frac {5}{32} \left (\frac {9}{40} \left (\frac {1}{4} \left (111671 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-86500 \int \frac {1}{\sqrt {3 x^2+5 x+2}}dx\right )+\frac {\sqrt {3 x^2+5 x+2} (21974 x+47145)}{2 (2 x+3)}\right )-\frac {(33142 x+42733) \left (3 x^2+5 x+2\right )^{3/2}}{20 (2 x+3)^3}\right )\right )-\frac {(5 x+27) \left (3 x^2+5 x+2\right )^{7/2}}{30 (2 x+3)^5}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {7}{60} \left (\frac {(548 x+1003) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^4}-\frac {5}{32} \left (\frac {9}{40} \left (\frac {1}{4} \left (111671 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-173000 \int \frac {1}{12-\frac {(6 x+5)^2}{3 x^2+5 x+2}}d\frac {6 x+5}{\sqrt {3 x^2+5 x+2}}\right )+\frac {\sqrt {3 x^2+5 x+2} (21974 x+47145)}{2 (2 x+3)}\right )-\frac {(33142 x+42733) \left (3 x^2+5 x+2\right )^{3/2}}{20 (2 x+3)^3}\right )\right )-\frac {(5 x+27) \left (3 x^2+5 x+2\right )^{7/2}}{30 (2 x+3)^5}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {7}{60} \left (\frac {(548 x+1003) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^4}-\frac {5}{32} \left (\frac {9}{40} \left (\frac {1}{4} \left (111671 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-\frac {86500 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {3}}\right )+\frac {\sqrt {3 x^2+5 x+2} (21974 x+47145)}{2 (2 x+3)}\right )-\frac {(33142 x+42733) \left (3 x^2+5 x+2\right )^{3/2}}{20 (2 x+3)^3}\right )\right )-\frac {(5 x+27) \left (3 x^2+5 x+2\right )^{7/2}}{30 (2 x+3)^5}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {7}{60} \left (\frac {(548 x+1003) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^4}-\frac {5}{32} \left (\frac {9}{40} \left (\frac {1}{4} \left (-223342 \int \frac {1}{20-\frac {(8 x+7)^2}{3 x^2+5 x+2}}d\left (-\frac {8 x+7}{\sqrt {3 x^2+5 x+2}}\right )-\frac {86500 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {3}}\right )+\frac {\sqrt {3 x^2+5 x+2} (21974 x+47145)}{2 (2 x+3)}\right )-\frac {(33142 x+42733) \left (3 x^2+5 x+2\right )^{3/2}}{20 (2 x+3)^3}\right )\right )-\frac {(5 x+27) \left (3 x^2+5 x+2\right )^{7/2}}{30 (2 x+3)^5}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {7}{60} \left (\frac {(548 x+1003) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^4}-\frac {5}{32} \left (\frac {9}{40} \left (\frac {1}{4} \left (\frac {111671 \text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {5}}-\frac {86500 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {3}}\right )+\frac {\sqrt {3 x^2+5 x+2} (21974 x+47145)}{2 (2 x+3)}\right )-\frac {(33142 x+42733) \left (3 x^2+5 x+2\right )^{3/2}}{20 (2 x+3)^3}\right )\right )-\frac {(5 x+27) \left (3 x^2+5 x+2\right )^{7/2}}{30 (2 x+3)^5}\) |
-1/30*((27 + 5*x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^5 + (7*(((1003 + 548* x)*(2 + 5*x + 3*x^2)^(5/2))/(16*(3 + 2*x)^4) - (5*(-1/20*((42733 + 33142*x )*(2 + 5*x + 3*x^2)^(3/2))/(3 + 2*x)^3 + (9*(((47145 + 21974*x)*Sqrt[2 + 5 *x + 3*x^2])/(2*(3 + 2*x)) + ((-86500*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/Sqrt[3] + (111671*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/Sqrt[5])/4))/40))/32))/60
3.25.57.3.1 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 = 0.40 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.64
| method | result | size |
| risch | \(-\frac {138240 x^{9}-541440 x^{8}+257280 x^{7}+85598208 x^{6}+523452248 x^{5}+1420169540 x^{4}+2104778650 x^{3}+1766213985 x^{2}+787192957 x +144379082}{10240 \left (3+2 x \right )^{5} \sqrt {3 x^{2}+5 x +2}}+\frac {30275 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{1024}+\frac {2345091 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{102400}\) | \(127\) |
| trager | \(-\frac {\left (46080 x^{7}-257280 x^{6}+483840 x^{5}+27897856 x^{4}+127665096 x^{3}+242016116 x^{2}+213122626 x +72189541\right ) \sqrt {3 x^{2}+5 x +2}}{10240 \left (3+2 x \right )^{5}}+\frac {30275 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+6 \sqrt {3 x^{2}+5 x +2}\right )}{1024}-\frac {2345091 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{3+2 x}\right )}{102400}\) | \(148\) |
| default | \(-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{800 \left (x +\frac {3}{2}\right )^{5}}-\frac {27 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{8000 \left (x +\frac {3}{2}\right )^{4}}-\frac {251 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{5000 \left (x +\frac {3}{2}\right )^{3}}+\frac {10023 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{100000 \left (x +\frac {3}{2}\right )^{2}}+\frac {19059 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{25000}-\frac {19059 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{12500 \left (x +\frac {3}{2}\right )}+\frac {122871 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{100000}+\frac {37037 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{16000}+\frac {37233 \left (5+6 x \right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{6400}+\frac {30275 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}\right ) \sqrt {3}}{1024}+\frac {2345091 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{102400}-\frac {2345091 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{400000}-\frac {781697 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{64000}-\frac {335013 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{100000}-\frac {2345091 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}{102400}\) | \(316\) |
-1/10240*(138240*x^9-541440*x^8+257280*x^7+85598208*x^6+523452248*x^5+1420 169540*x^4+2104778650*x^3+1766213985*x^2+787192957*x+144379082)/(3+2*x)^5/ (3*x^2+5*x+2)^(1/2)+30275/1024*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2 ))*3^(1/2)+2345091/102400*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/ 2)^2-16*x-19)^(1/2))
Time = 0.28 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.11 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^6} \, dx=\frac {3027500 \, \sqrt {3} {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 2345091 \, \sqrt {5} {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )} \log \left (-\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} - 124 \, x^{2} - 212 \, x - 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 20 \, {\left (46080 \, x^{7} - 257280 \, x^{6} + 483840 \, x^{5} + 27897856 \, x^{4} + 127665096 \, x^{3} + 242016116 \, x^{2} + 213122626 \, x + 72189541\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{204800 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} \]
1/204800*(3027500*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) + 2345091*sqrt(5)*(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)*l og(-(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*(46080*x^7 - 257280*x^6 + 483840*x^5 + 27897856*x^4 + 127665096*x^3 + 242016116*x^2 + 213122626*x + 72189541)*sqrt(3*x^2 + 5*x + 2))/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)
\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^6} \, dx=- \int \left (- \frac {40 \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \left (- \frac {292 x \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \left (- \frac {870 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \left (- \frac {1339 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \left (- \frac {1090 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \left (- \frac {396 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \frac {27 x^{7} \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\, dx \]
-Integral(-40*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(-292*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 + 7 29), x) - Integral(-870*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), x) - Integral(-1339*x** 3*sqrt(3*x**2 + 5*x + 2)/(64*x**6 + 576*x**5 + 2160*x**4 + 4320*x**3 + 486 0*x**2 + 2916*x + 729), x) - Integral(-1090*x**4*sqrt(3*x**2 + 5*x + 2)/(6 4*x**6 + 576*x**5 + 2160*x**4 + 4320*x**3 + 4860*x**2 + 2916*x + 729), x) - Integral(-396*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), x) - Integral(27*x**7*sqrt(3*x* *2 + 5*x + 2)/(64*x**6 + 576*x**5 + 2160*x**4 + 4320*x**3 + 4860*x**2 + 29 16*x + 729), x)
Leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (161) = 322\).
Time = 0.29 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.65 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^6} \, dx=-\frac {30069}{100000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{25 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {27 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{500 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {251 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{625 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} + \frac {10023 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{25000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac {368613}{50000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x + \frac {112329}{400000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {19059 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{5000 \, {\left (2 \, x + 3\right )}} + \frac {111111}{8000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {40957}{64000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {111699}{3200} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {30275}{1024} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) + \frac {2345091}{102400} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) - \frac {855771}{51200} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]
-30069/100000*(3*x^2 + 5*x + 2)^(7/2) - 13/25*(3*x^2 + 5*x + 2)^(9/2)/(32* x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 27/500*(3*x^2 + 5*x + 2)^(9/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 251/625*(3*x^2 + 5*x + 2)^(9/2)/(8*x^3 + 36*x^2 + 54*x + 27) + 10023/25000*(3*x^2 + 5*x + 2)^(9/ 2)/(4*x^2 + 12*x + 9) + 368613/50000*(3*x^2 + 5*x + 2)^(5/2)*x + 112329/40 0000*(3*x^2 + 5*x + 2)^(5/2) - 19059/5000*(3*x^2 + 5*x + 2)^(7/2)/(2*x + 3 ) + 111111/8000*(3*x^2 + 5*x + 2)^(3/2)*x - 40957/64000*(3*x^2 + 5*x + 2)^ (3/2) + 111699/3200*sqrt(3*x^2 + 5*x + 2)*x + 30275/1024*sqrt(3)*log(sqrt( 3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) + 2345091/102400*sqrt(5)*log(sqrt(5) *sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 855771/51200 *sqrt(3*x^2 + 5*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 417 vs. \(2 (161) = 322\).
Time = 0.34 (sec) , antiderivative size = 417, normalized size of antiderivative = 2.12 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^6} \, dx=-\frac {3}{512} \, {\left (2 \, {\left (12 \, x - 157\right )} x + 2067\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {2345091}{102400} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) - \frac {30275}{1024} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) - \frac {60397264 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 739203704 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 11836231432 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} + 36096211012 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} + 207702455456 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 259725515674 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 635418284542 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 326158305587 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 287216072451 \, \sqrt {3} x + 36785380096 \, \sqrt {3} - 287216072451 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{10240 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{5}} \]
-3/512*(2*(12*x - 157)*x + 2067)*sqrt(3*x^2 + 5*x + 2) - 2345091/102400*sq rt(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))) - 30275/1024*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5)) - 1/10240*(60397264*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^9 + 7392037 04*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^8 + 11836231432*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^7 + 36096211012*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^6 + 207702455456*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 + 259725 515674*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 + 635418284542*(sqrt( 3)*x - sqrt(3*x^2 + 5*x + 2))^3 + 326158305587*sqrt(3)*(sqrt(3)*x - sqrt(3 *x^2 + 5*x + 2))^2 + 287216072451*sqrt(3)*x + 36785380096*sqrt(3) - 287216 072451*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. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^6} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{7/2}}{{\left (2\,x+3\right )}^6} \,d x \]