Integrand size = 27, antiderivative size = 204 \[ \int (5-x) (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2} \, dx=-\frac {637609 (5+6 x) \sqrt {2+5 x+3 x^2}}{1719926784}+\frac {637609 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{214990848}-\frac {637609 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{22394880}+\frac {91087 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{311040}+\frac {34}{99} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}-\frac {1}{36} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{9/2}+\frac {(863825+390798 x) \left (2+5 x+3 x^2\right )^{9/2}}{320760}+\frac {637609 \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{3439853568 \sqrt {3}} \]
637609/214990848*(5+6*x)*(3*x^2+5*x+2)^(3/2)-637609/22394880*(5+6*x)*(3*x^ 2+5*x+2)^(5/2)+91087/311040*(5+6*x)*(3*x^2+5*x+2)^(7/2)+34/99*(3+2*x)^2*(3 *x^2+5*x+2)^(9/2)-1/36*(3+2*x)^3*(3*x^2+5*x+2)^(9/2)+1/320760*(863825+3907 98*x)*(3*x^2+5*x+2)^(9/2)+637609/10319560704*arctanh(1/6*(5+6*x)*3^(1/2)/( 3*x^2+5*x+2)^(1/2))*3^(1/2)-637609/1719926784*(5+6*x)*(3*x^2+5*x+2)^(1/2)
Time = 0.94 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.52 \[ \int (5-x) (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2} \, dx=\frac {-3 \sqrt {2+5 x+3 x^2} \left (-8675936123685-91318722047870 x-425035984788120 x^2-1149328734822000 x^3-1992318117275520 x^4-2298912734198016 x^5-1766184385305600 x^6-866110416795648 x^7-235832896880640 x^8-15591566278656 x^9+8487838679040 x^{10}+1702727516160 x^{11}\right )+35068495 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{283787919360} \]
(-3*Sqrt[2 + 5*x + 3*x^2]*(-8675936123685 - 91318722047870*x - 42503598478 8120*x^2 - 1149328734822000*x^3 - 1992318117275520*x^4 - 2298912734198016* x^5 - 1766184385305600*x^6 - 866110416795648*x^7 - 235832896880640*x^8 - 1 5591566278656*x^9 + 8487838679040*x^10 + 1702727516160*x^11) + 35068495*Sq rt[3]*ArcTanh[Sqrt[2/3 + (5*x)/3 + x^2]/(1 + x)])/283787919360
Time = 0.41 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.15, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {1236, 27, 1236, 1225, 1087, 1087, 1087, 1087, 1092, 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.
| \(\displaystyle \int (5-x) (2 x+3)^3 \left (3 x^2+5 x+2\right )^{7/2} \, dx\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {1}{36} \int \frac {3}{2} (2 x+3)^2 (272 x+413) \left (3 x^2+5 x+2\right )^{7/2}dx-\frac {1}{36} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{9/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{24} \int (2 x+3)^2 (272 x+413) \left (3 x^2+5 x+2\right )^{7/2}dx-\frac {1}{36} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{9/2}\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {1}{24} \left (\frac {1}{33} \int (2 x+3) (14474 x+20351) \left (3 x^2+5 x+2\right )^{7/2}dx+\frac {272}{33} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}\right )-\frac {1}{36} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{9/2}\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle \frac {1}{24} \left (\frac {1}{33} \left (\frac {1001957}{90} \int \left (3 x^2+5 x+2\right )^{7/2}dx+\frac {1}{405} (390798 x+863825) \left (3 x^2+5 x+2\right )^{9/2}\right )+\frac {272}{33} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}\right )-\frac {1}{36} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{9/2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{24} \left (\frac {1}{33} \left (\frac {1001957}{90} \left (\frac {1}{48} (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}-\frac {7}{96} \int \left (3 x^2+5 x+2\right )^{5/2}dx\right )+\frac {1}{405} (390798 x+863825) \left (3 x^2+5 x+2\right )^{9/2}\right )+\frac {272}{33} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}\right )-\frac {1}{36} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{9/2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{24} \left (\frac {1}{33} \left (\frac {1001957}{90} \left (\frac {1}{48} (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}-\frac {7}{96} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \int \left (3 x^2+5 x+2\right )^{3/2}dx\right )\right )+\frac {1}{405} (390798 x+863825) \left (3 x^2+5 x+2\right )^{9/2}\right )+\frac {272}{33} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}\right )-\frac {1}{36} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{9/2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{24} \left (\frac {1}{33} \left (\frac {1001957}{90} \left (\frac {1}{48} (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}-\frac {7}{96} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \left (\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}-\frac {1}{16} \int \sqrt {3 x^2+5 x+2}dx\right )\right )\right )+\frac {1}{405} (390798 x+863825) \left (3 x^2+5 x+2\right )^{9/2}\right )+\frac {272}{33} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}\right )-\frac {1}{36} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{9/2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{24} \left (\frac {1}{33} \left (\frac {1001957}{90} \left (\frac {1}{48} (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}-\frac {7}{96} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \left (\frac {1}{16} \left (\frac {1}{24} \int \frac {1}{\sqrt {3 x^2+5 x+2}}dx-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )\right )+\frac {1}{405} (390798 x+863825) \left (3 x^2+5 x+2\right )^{9/2}\right )+\frac {272}{33} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}\right )-\frac {1}{36} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{9/2}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {1}{24} \left (\frac {1}{33} \left (\frac {1001957}{90} \left (\frac {1}{48} (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}-\frac {7}{96} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \left (\frac {1}{16} \left (\frac {1}{12} \int \frac {1}{12-\frac {(6 x+5)^2}{3 x^2+5 x+2}}d\frac {6 x+5}{\sqrt {3 x^2+5 x+2}}-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )\right )+\frac {1}{405} (390798 x+863825) \left (3 x^2+5 x+2\right )^{9/2}\right )+\frac {272}{33} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}\right )-\frac {1}{36} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{9/2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{24} \left (\frac {1}{33} \left (\frac {1001957}{90} \left (\frac {1}{48} (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}-\frac {7}{96} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \left (\frac {1}{16} \left (\frac {\text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{24 \sqrt {3}}-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )\right )+\frac {1}{405} (390798 x+863825) \left (3 x^2+5 x+2\right )^{9/2}\right )+\frac {272}{33} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}\right )-\frac {1}{36} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{9/2}\) |
-1/36*((3 + 2*x)^3*(2 + 5*x + 3*x^2)^(9/2)) + ((272*(3 + 2*x)^2*(2 + 5*x + 3*x^2)^(9/2))/33 + (((863825 + 390798*x)*(2 + 5*x + 3*x^2)^(9/2))/405 + ( 1001957*(((5 + 6*x)*(2 + 5*x + 3*x^2)^(7/2))/48 - (7*(((5 + 6*x)*(2 + 5*x + 3*x^2)^(5/2))/36 - (5*(((5 + 6*x)*(2 + 5*x + 3*x^2)^(3/2))/24 + (-1/12*( (5 + 6*x)*Sqrt[2 + 5*x + 3*x^2]) + ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5 *x + 3*x^2])]/(24*Sqrt[3]))/16))/72))/96))/90)/33)/24
3.25.48.3.1 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[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* p + 1))) Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
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[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c , d, e, f, g, p}, x] && !LeQ[p, -1]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1 )*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m *(c*e*f + c*d*g - b*e*g) + e*(p + 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[ {a, b, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && EqQ[f, 0])
Time = 0.36 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.49
| method | result | size |
| risch | \(-\frac {\left (1702727516160 x^{11}+8487838679040 x^{10}-15591566278656 x^{9}-235832896880640 x^{8}-866110416795648 x^{7}-1766184385305600 x^{6}-2298912734198016 x^{5}-1992318117275520 x^{4}-1149328734822000 x^{3}-425035984788120 x^{2}-91318722047870 x -8675936123685\right ) \sqrt {3 x^{2}+5 x +2}}{94595973120}+\frac {637609 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{10319560704}\) | \(100\) |
| trager | \(\left (-18 x^{11}-\frac {987}{11} x^{10}+\frac {36261}{220} x^{9}+\frac {1974499}{792} x^{8}+\frac {1740351757}{190080} x^{7}+\frac {1419579785}{76032} x^{6}+\frac {332597328443}{13685760} x^{5}+\frac {345888562027}{16422912} x^{4}+\frac {1596289909475}{131383296} x^{3}+\frac {3541966539901}{788299776} x^{2}+\frac {9131872204787}{9459597312} x +\frac {578395741579}{6306398208}\right ) \sqrt {3 x^{2}+5 x +2}-\frac {637609 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {3 x^{2}+5 x +2}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )\right )}{10319560704}\) | \(111\) |
| default | \(\frac {91087 \left (5+6 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{311040}-\frac {637609 \left (5+6 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{22394880}+\frac {637609 \left (5+6 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{214990848}-\frac {637609 \left (5+6 x \right ) \sqrt {3 x^{2}+5 x +2}}{1719926784}+\frac {637609 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{10319560704}+\frac {322939 \left (3 x^{2}+5 x +2\right )^{\frac {9}{2}}}{64152}-\frac {2 x^{3} \left (3 x^{2}+5 x +2\right )^{\frac {9}{2}}}{9}+\frac {37 x^{2} \left (3 x^{2}+5 x +2\right )^{\frac {9}{2}}}{99}+\frac {22807 x \left (3 x^{2}+5 x +2\right )^{\frac {9}{2}}}{5940}\) | \(170\) |
-1/94595973120*(1702727516160*x^11+8487838679040*x^10-15591566278656*x^9-2 35832896880640*x^8-866110416795648*x^7-1766184385305600*x^6-22989127341980 16*x^5-1992318117275520*x^4-1149328734822000*x^3-425035984788120*x^2-91318 722047870*x-8675936123685)*(3*x^2+5*x+2)^(1/2)+637609/10319560704*ln(1/3*( 5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)
Time = 0.29 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.53 \[ \int (5-x) (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2} \, dx=-\frac {1}{94595973120} \, {\left (1702727516160 \, x^{11} + 8487838679040 \, x^{10} - 15591566278656 \, x^{9} - 235832896880640 \, x^{8} - 866110416795648 \, x^{7} - 1766184385305600 \, x^{6} - 2298912734198016 \, x^{5} - 1992318117275520 \, x^{4} - 1149328734822000 \, x^{3} - 425035984788120 \, x^{2} - 91318722047870 \, x - 8675936123685\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {637609}{20639121408} \, \sqrt {3} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) \]
-1/94595973120*(1702727516160*x^11 + 8487838679040*x^10 - 15591566278656*x ^9 - 235832896880640*x^8 - 866110416795648*x^7 - 1766184385305600*x^6 - 22 98912734198016*x^5 - 1992318117275520*x^4 - 1149328734822000*x^3 - 4250359 84788120*x^2 - 91318722047870*x - 8675936123685)*sqrt(3*x^2 + 5*x + 2) + 6 37609/20639121408*sqrt(3)*log(4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49)
Time = 1.05 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.60 \[ \int (5-x) (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2} \, dx=\sqrt {3 x^{2} + 5 x + 2} \left (- 18 x^{11} - \frac {987 x^{10}}{11} + \frac {36261 x^{9}}{220} + \frac {1974499 x^{8}}{792} + \frac {1740351757 x^{7}}{190080} + \frac {1419579785 x^{6}}{76032} + \frac {332597328443 x^{5}}{13685760} + \frac {345888562027 x^{4}}{16422912} + \frac {1596289909475 x^{3}}{131383296} + \frac {3541966539901 x^{2}}{788299776} + \frac {9131872204787 x}{9459597312} + \frac {578395741579}{6306398208}\right ) + \frac {637609 \sqrt {3} \log {\left (6 x + 2 \sqrt {3} \sqrt {3 x^{2} + 5 x + 2} + 5 \right )}}{10319560704} \]
sqrt(3*x**2 + 5*x + 2)*(-18*x**11 - 987*x**10/11 + 36261*x**9/220 + 197449 9*x**8/792 + 1740351757*x**7/190080 + 1419579785*x**6/76032 + 332597328443 *x**5/13685760 + 345888562027*x**4/16422912 + 1596289909475*x**3/131383296 + 3541966539901*x**2/788299776 + 9131872204787*x/9459597312 + 57839574157 9/6306398208) + 637609*sqrt(3)*log(6*x + 2*sqrt(3)*sqrt(3*x**2 + 5*x + 2) + 5)/10319560704
Time = 0.27 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.02 \[ \int (5-x) (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2} \, dx=-\frac {2}{9} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}} x^{3} + \frac {37}{99} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}} x^{2} + \frac {22807}{5940} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}} x + \frac {322939}{64152} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}} + \frac {91087}{51840} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x + \frac {91087}{62208} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} - \frac {637609}{3732480} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x - \frac {637609}{4478976} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} + \frac {637609}{35831808} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {3188045}{214990848} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {637609}{286654464} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {637609}{10319560704} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac {3188045}{1719926784} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]
-2/9*(3*x^2 + 5*x + 2)^(9/2)*x^3 + 37/99*(3*x^2 + 5*x + 2)^(9/2)*x^2 + 228 07/5940*(3*x^2 + 5*x + 2)^(9/2)*x + 322939/64152*(3*x^2 + 5*x + 2)^(9/2) + 91087/51840*(3*x^2 + 5*x + 2)^(7/2)*x + 91087/62208*(3*x^2 + 5*x + 2)^(7/ 2) - 637609/3732480*(3*x^2 + 5*x + 2)^(5/2)*x - 637609/4478976*(3*x^2 + 5* x + 2)^(5/2) + 637609/35831808*(3*x^2 + 5*x + 2)^(3/2)*x + 3188045/2149908 48*(3*x^2 + 5*x + 2)^(3/2) - 637609/286654464*sqrt(3*x^2 + 5*x + 2)*x + 63 7609/10319560704*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) - 3188045/1719926784*sqrt(3*x^2 + 5*x + 2)
Time = 0.36 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.51 \[ \int (5-x) (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2} \, dx=-\frac {1}{94595973120} \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (8 \, {\left (6 \, {\left (36 \, {\left (2 \, {\left (48 \, {\left (54 \, {\left (20 \, {\left (66 \, x + 329\right )} x - 12087\right )} x - 9872495\right )} x - 1740351757\right )} x - 7097898925\right )} x - 332597328443\right )} x - 1729442810135\right )} x - 7981449547375\right )} x - 17709832699505\right )} x - 45659361023935\right )} x - 8675936123685\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {637609}{10319560704} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \]
-1/94595973120*(2*(12*(6*(8*(6*(36*(2*(48*(54*(20*(66*x + 329)*x - 12087)* x - 9872495)*x - 1740351757)*x - 7097898925)*x - 332597328443)*x - 1729442 810135)*x - 7981449547375)*x - 17709832699505)*x - 45659361023935)*x - 867 5936123685)*sqrt(3*x^2 + 5*x + 2) - 637609/10319560704*sqrt(3)*log(abs(-2* sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5))
Timed out. \[ \int (5-x) (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2} \, dx=-\int {\left (2\,x+3\right )}^3\,\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{7/2} \,d x \]