Integrand size = 27, antiderivative size = 229 \[ \int (5-x) (3+2 x)^4 \left (2+5 x+3 x^2\right )^{7/2} \, dx=-\frac {2595845 (5+6 x) \sqrt {2+5 x+3 x^2}}{5159780352}+\frac {2595845 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{644972544}-\frac {519169 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{13436928}+\frac {74167 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{186624}+\frac {205}{351} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}+\frac {439 (3+2 x)^3 \left (2+5 x+3 x^2\right )^{9/2}}{1404}-\frac {1}{39} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{9/2}+\frac {(852175+389394 x) \left (2+5 x+3 x^2\right )^{9/2}}{227448}+\frac {2595845 \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{10319560704 \sqrt {3}} \]
2595845/644972544*(5+6*x)*(3*x^2+5*x+2)^(3/2)-519169/13436928*(5+6*x)*(3*x ^2+5*x+2)^(5/2)+74167/186624*(5+6*x)*(3*x^2+5*x+2)^(7/2)+205/351*(3+2*x)^2 *(3*x^2+5*x+2)^(9/2)+439/1404*(3+2*x)^3*(3*x^2+5*x+2)^(9/2)-1/39*(3+2*x)^4 *(3*x^2+5*x+2)^(9/2)+1/227448*(852175+389394*x)*(3*x^2+5*x+2)^(9/2)+259584 5/30958682112*arctanh(1/6*(5+6*x)*3^(1/2)/(3*x^2+5*x+2)^(1/2))*3^(1/2)-259 5845/5159780352*(5+6*x)*(3*x^2+5*x+2)^(1/2)
Time = 1.03 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.48 \[ \int (5-x) (3+2 x)^4 \left (2+5 x+3 x^2\right )^{7/2} \, dx=\frac {-3 \sqrt {2+5 x+3 x^2} \left (-17510968283403-195441229635490 x-975104480077800 x^2-2865856228323984 x^3-5499074981552256 x^4-7203650864723712 x^5-6524509131334656 x^6-4022427759003648 x^7-1590604366381056 x^8-333952593887232 x^9-2110350163968 x^{10}+14643456638976 x^{11}+2229025112064 x^{12}\right )+33745985 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{201231433728} \]
(-3*Sqrt[2 + 5*x + 3*x^2]*(-17510968283403 - 195441229635490*x - 975104480 077800*x^2 - 2865856228323984*x^3 - 5499074981552256*x^4 - 720365086472371 2*x^5 - 6524509131334656*x^6 - 4022427759003648*x^7 - 1590604366381056*x^8 - 333952593887232*x^9 - 2110350163968*x^10 + 14643456638976*x^11 + 222902 5112064*x^12) + 33745985*Sqrt[3]*ArcTanh[Sqrt[2/3 + (5*x)/3 + x^2]/(1 + x) ])/201231433728
Time = 0.45 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.15, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {1236, 27, 1236, 27, 1236, 27, 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.
| \(\displaystyle \int (5-x) (2 x+3)^4 \left (3 x^2+5 x+2\right )^{7/2} \, dx\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {1}{39} \int \frac {1}{2} (2 x+3)^3 (878 x+1337) \left (3 x^2+5 x+2\right )^{7/2}dx-\frac {1}{39} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{9/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{78} \int (2 x+3)^3 (878 x+1337) \left (3 x^2+5 x+2\right )^{7/2}dx-\frac {1}{39} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{9/2}\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {1}{78} \left (\frac {1}{36} \int 15 (2 x+3)^2 (3608 x+4973) \left (3 x^2+5 x+2\right )^{7/2}dx+\frac {439}{18} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{9/2}\right )-\frac {1}{39} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{9/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{78} \left (\frac {5}{12} \int (2 x+3)^2 (3608 x+4973) \left (3 x^2+5 x+2\right )^{7/2}dx+\frac {439}{18} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{9/2}\right )-\frac {1}{39} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{9/2}\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {1}{78} \left (\frac {5}{12} \left (\frac {1}{33} \int 11 (2 x+3) (14422 x+19993) \left (3 x^2+5 x+2\right )^{7/2}dx+\frac {328}{3} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}\right )+\frac {439}{18} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{9/2}\right )-\frac {1}{39} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{9/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{78} \left (\frac {5}{12} \left (\frac {1}{3} \int (2 x+3) (14422 x+19993) \left (3 x^2+5 x+2\right )^{7/2}dx+\frac {328}{3} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}\right )+\frac {439}{18} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{9/2}\right )-\frac {1}{39} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{9/2}\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle \frac {1}{78} \left (\frac {5}{12} \left (\frac {1}{3} \left (\frac {964171}{90} \int \left (3 x^2+5 x+2\right )^{7/2}dx+\frac {1}{405} (389394 x+852175) \left (3 x^2+5 x+2\right )^{9/2}\right )+\frac {328}{3} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}\right )+\frac {439}{18} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{9/2}\right )-\frac {1}{39} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{9/2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{78} \left (\frac {5}{12} \left (\frac {1}{3} \left (\frac {964171}{90} \left (\frac {1}{48} (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}-\frac {7}{96} \int \left (3 x^2+5 x+2\right )^{5/2}dx\right )+\frac {1}{405} (389394 x+852175) \left (3 x^2+5 x+2\right )^{9/2}\right )+\frac {328}{3} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}\right )+\frac {439}{18} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{9/2}\right )-\frac {1}{39} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{9/2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{78} \left (\frac {5}{12} \left (\frac {1}{3} \left (\frac {964171}{90} \left (\frac {1}{48} (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}-\frac {7}{96} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \int \left (3 x^2+5 x+2\right )^{3/2}dx\right )\right )+\frac {1}{405} (389394 x+852175) \left (3 x^2+5 x+2\right )^{9/2}\right )+\frac {328}{3} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}\right )+\frac {439}{18} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{9/2}\right )-\frac {1}{39} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{9/2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{78} \left (\frac {5}{12} \left (\frac {1}{3} \left (\frac {964171}{90} \left (\frac {1}{48} (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}-\frac {7}{96} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \left (\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}-\frac {1}{16} \int \sqrt {3 x^2+5 x+2}dx\right )\right )\right )+\frac {1}{405} (389394 x+852175) \left (3 x^2+5 x+2\right )^{9/2}\right )+\frac {328}{3} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}\right )+\frac {439}{18} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{9/2}\right )-\frac {1}{39} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{9/2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{78} \left (\frac {5}{12} \left (\frac {1}{3} \left (\frac {964171}{90} \left (\frac {1}{48} (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}-\frac {7}{96} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \left (\frac {1}{16} \left (\frac {1}{24} \int \frac {1}{\sqrt {3 x^2+5 x+2}}dx-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )\right )+\frac {1}{405} (389394 x+852175) \left (3 x^2+5 x+2\right )^{9/2}\right )+\frac {328}{3} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}\right )+\frac {439}{18} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{9/2}\right )-\frac {1}{39} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{9/2}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {1}{78} \left (\frac {5}{12} \left (\frac {1}{3} \left (\frac {964171}{90} \left (\frac {1}{48} (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}-\frac {7}{96} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \left (\frac {1}{16} \left (\frac {1}{12} \int \frac {1}{12-\frac {(6 x+5)^2}{3 x^2+5 x+2}}d\frac {6 x+5}{\sqrt {3 x^2+5 x+2}}-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )\right )+\frac {1}{405} (389394 x+852175) \left (3 x^2+5 x+2\right )^{9/2}\right )+\frac {328}{3} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}\right )+\frac {439}{18} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{9/2}\right )-\frac {1}{39} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{9/2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{78} \left (\frac {5}{12} \left (\frac {1}{3} \left (\frac {964171}{90} \left (\frac {1}{48} (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}-\frac {7}{96} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \left (\frac {1}{16} \left (\frac {\text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{24 \sqrt {3}}-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )\right )+\frac {1}{405} (389394 x+852175) \left (3 x^2+5 x+2\right )^{9/2}\right )+\frac {328}{3} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}\right )+\frac {439}{18} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{9/2}\right )-\frac {1}{39} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{9/2}\) |
-1/39*((3 + 2*x)^4*(2 + 5*x + 3*x^2)^(9/2)) + ((439*(3 + 2*x)^3*(2 + 5*x + 3*x^2)^(9/2))/18 + (5*((328*(3 + 2*x)^2*(2 + 5*x + 3*x^2)^(9/2))/3 + (((8 52175 + 389394*x)*(2 + 5*x + 3*x^2)^(9/2))/405 + (964171*(((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)/3))/12)/78
3.25.47.3.1 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 = 0.34 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.46
| method | result | size |
| risch | \(-\frac {\left (2229025112064 x^{12}+14643456638976 x^{11}-2110350163968 x^{10}-333952593887232 x^{9}-1590604366381056 x^{8}-4022427759003648 x^{7}-6524509131334656 x^{6}-7203650864723712 x^{5}-5499074981552256 x^{4}-2865856228323984 x^{3}-975104480077800 x^{2}-195441229635490 x -17510968283403\right ) \sqrt {3 x^{2}+5 x +2}}{67077144576}+\frac {2595845 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{30958682112}\) | \(105\) |
| trager | \(\left (-\frac {432}{13} x^{12}-\frac {2838}{13} x^{11}+\frac {409}{13} x^{10}+\frac {258889}{52} x^{9}+\frac {5122021}{216} x^{8}+\frac {8082617507}{134784} x^{7}+\frac {26220538883}{269568} x^{6}+\frac {1042194858901}{9704448} x^{5}+\frac {367192506781}{4478976} x^{4}+\frac {19901779363361}{465813504} x^{3}+\frac {40629353336575}{2794881024} x^{2}+\frac {97720614817745}{33538572288} x +\frac {5836989427801}{22359048192}\right ) \sqrt {3 x^{2}+5 x +2}-\frac {2595845 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {3 x^{2}+5 x +2}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )\right )}{30958682112}\) | \(116\) |
| default | \(\frac {74167 \left (5+6 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{186624}-\frac {519169 \left (5+6 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{13436928}+\frac {2595845 \left (5+6 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{644972544}-\frac {2595845 \left (5+6 x \right ) \sqrt {3 x^{2}+5 x +2}}{5159780352}+\frac {2595845 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{30958682112}+\frac {3495529 \left (3 x^{2}+5 x +2\right )^{\frac {9}{2}}}{227448}-\frac {16 x^{4} \left (3 x^{2}+5 x +2\right )^{\frac {9}{2}}}{39}+\frac {14 x^{3} \left (3 x^{2}+5 x +2\right )^{\frac {9}{2}}}{351}+\frac {2827 x^{2} \left (3 x^{2}+5 x +2\right )^{\frac {9}{2}}}{351}+\frac {84521 x \left (3 x^{2}+5 x +2\right )^{\frac {9}{2}}}{4212}\) | \(187\) |
-1/67077144576*(2229025112064*x^12+14643456638976*x^11-2110350163968*x^10- 333952593887232*x^9-1590604366381056*x^8-4022427759003648*x^7-652450913133 4656*x^6-7203650864723712*x^5-5499074981552256*x^4-2865856228323984*x^3-97 5104480077800*x^2-195441229635490*x-17510968283403)*(3*x^2+5*x+2)^(1/2)+25 95845/30958682112*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)
Time = 0.28 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.49 \[ \int (5-x) (3+2 x)^4 \left (2+5 x+3 x^2\right )^{7/2} \, dx=-\frac {1}{67077144576} \, {\left (2229025112064 \, x^{12} + 14643456638976 \, x^{11} - 2110350163968 \, x^{10} - 333952593887232 \, x^{9} - 1590604366381056 \, x^{8} - 4022427759003648 \, x^{7} - 6524509131334656 \, x^{6} - 7203650864723712 \, x^{5} - 5499074981552256 \, x^{4} - 2865856228323984 \, x^{3} - 975104480077800 \, x^{2} - 195441229635490 \, x - 17510968283403\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {2595845}{61917364224} \, \sqrt {3} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) \]
-1/67077144576*(2229025112064*x^12 + 14643456638976*x^11 - 2110350163968*x ^10 - 333952593887232*x^9 - 1590604366381056*x^8 - 4022427759003648*x^7 - 6524509131334656*x^6 - 7203650864723712*x^5 - 5499074981552256*x^4 - 28658 56228323984*x^3 - 975104480077800*x^2 - 195441229635490*x - 17510968283403 )*sqrt(3*x^2 + 5*x + 2) + 2595845/61917364224*sqrt(3)*log(4*sqrt(3)*sqrt(3 *x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49)
Time = 1.11 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.57 \[ \int (5-x) (3+2 x)^4 \left (2+5 x+3 x^2\right )^{7/2} \, dx=\sqrt {3 x^{2} + 5 x + 2} \left (- \frac {432 x^{12}}{13} - \frac {2838 x^{11}}{13} + \frac {409 x^{10}}{13} + \frac {258889 x^{9}}{52} + \frac {5122021 x^{8}}{216} + \frac {8082617507 x^{7}}{134784} + \frac {26220538883 x^{6}}{269568} + \frac {1042194858901 x^{5}}{9704448} + \frac {367192506781 x^{4}}{4478976} + \frac {19901779363361 x^{3}}{465813504} + \frac {40629353336575 x^{2}}{2794881024} + \frac {97720614817745 x}{33538572288} + \frac {5836989427801}{22359048192}\right ) + \frac {2595845 \sqrt {3} \log {\left (6 x + 2 \sqrt {3} \sqrt {3 x^{2} + 5 x + 2} + 5 \right )}}{30958682112} \]
sqrt(3*x**2 + 5*x + 2)*(-432*x**12/13 - 2838*x**11/13 + 409*x**10/13 + 258 889*x**9/52 + 5122021*x**8/216 + 8082617507*x**7/134784 + 26220538883*x**6 /269568 + 1042194858901*x**5/9704448 + 367192506781*x**4/4478976 + 1990177 9363361*x**3/465813504 + 40629353336575*x**2/2794881024 + 97720614817745*x /33538572288 + 5836989427801/22359048192) + 2595845*sqrt(3)*log(6*x + 2*sq rt(3)*sqrt(3*x**2 + 5*x + 2) + 5)/30958682112
Time = 0.27 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.98 \[ \int (5-x) (3+2 x)^4 \left (2+5 x+3 x^2\right )^{7/2} \, dx=-\frac {16}{39} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}} x^{4} + \frac {14}{351} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}} x^{3} + \frac {2827}{351} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}} x^{2} + \frac {84521}{4212} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}} x + \frac {3495529}{227448} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}} + \frac {74167}{31104} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x + \frac {370835}{186624} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} - \frac {519169}{2239488} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x - \frac {2595845}{13436928} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} + \frac {2595845}{107495424} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {12979225}{644972544} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {2595845}{859963392} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {2595845}{30958682112} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac {12979225}{5159780352} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]
-16/39*(3*x^2 + 5*x + 2)^(9/2)*x^4 + 14/351*(3*x^2 + 5*x + 2)^(9/2)*x^3 + 2827/351*(3*x^2 + 5*x + 2)^(9/2)*x^2 + 84521/4212*(3*x^2 + 5*x + 2)^(9/2)* x + 3495529/227448*(3*x^2 + 5*x + 2)^(9/2) + 74167/31104*(3*x^2 + 5*x + 2) ^(7/2)*x + 370835/186624*(3*x^2 + 5*x + 2)^(7/2) - 519169/2239488*(3*x^2 + 5*x + 2)^(5/2)*x - 2595845/13436928*(3*x^2 + 5*x + 2)^(5/2) + 2595845/107 495424*(3*x^2 + 5*x + 2)^(3/2)*x + 12979225/644972544*(3*x^2 + 5*x + 2)^(3 /2) - 2595845/859963392*sqrt(3*x^2 + 5*x + 2)*x + 2595845/30958682112*sqrt (3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) - 12979225/5159780352*s qrt(3*x^2 + 5*x + 2)
Time = 0.35 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.48 \[ \int (5-x) (3+2 x)^4 \left (2+5 x+3 x^2\right )^{7/2} \, dx=-\frac {1}{67077144576} \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (8 \, {\left (6 \, {\left (36 \, {\left (2 \, {\left (48 \, {\left (54 \, {\left (4 \, {\left (6 \, {\left (72 \, x + 473\right )} x - 409\right )} x - 258889\right )} x - 66586273\right )} x - 8082617507\right )} x - 26220538883\right )} x - 1042194858901\right )} x - 4773502588153\right )} x - 19901779363361\right )} x - 40629353336575\right )} x - 97720614817745\right )} x - 17510968283403\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {2595845}{30958682112} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \]
-1/67077144576*(2*(12*(6*(8*(6*(36*(2*(48*(54*(4*(6*(72*x + 473)*x - 409)* x - 258889)*x - 66586273)*x - 8082617507)*x - 26220538883)*x - 10421948589 01)*x - 4773502588153)*x - 19901779363361)*x - 40629353336575)*x - 9772061 4817745)*x - 17510968283403)*sqrt(3*x^2 + 5*x + 2) - 2595845/30958682112*s qrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5))
Timed out. \[ \int (5-x) (3+2 x)^4 \left (2+5 x+3 x^2\right )^{7/2} \, dx=-\int {\left (2\,x+3\right )}^4\,\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{7/2} \,d x \]