Integrand size = 32, antiderivative size = 189 \[ \int (1+2 x)^3 \left (2-x+3 x^2\right )^{5/2} \left (1+3 x+4 x^2\right ) \, dx=\frac {2692081 (1-6 x) \sqrt {2-x+3 x^2}}{11943936}+\frac {117047 (1-6 x) \left (2-x+3 x^2\right )^{3/2}}{1492992}+\frac {5089 (1-6 x) \left (2-x+3 x^2\right )^{5/2}}{155520}-\frac {(26353-21350 x) \left (2-x+3 x^2\right )^{7/2}}{498960}+\frac {133 (1+2 x)^2 \left (2-x+3 x^2\right )^{7/2}}{1485}+\frac {29}{330} (1+2 x)^3 \left (2-x+3 x^2\right )^{7/2}+\frac {2}{33} (1+2 x)^4 \left (2-x+3 x^2\right )^{7/2}+\frac {61917863 \text {arcsinh}\left (\frac {1-6 x}{\sqrt {23}}\right )}{23887872 \sqrt {3}} \]
117047/1492992*(1-6*x)*(3*x^2-x+2)^(3/2)+5089/155520*(1-6*x)*(3*x^2-x+2)^( 5/2)-1/498960*(26353-21350*x)*(3*x^2-x+2)^(7/2)+133/1485*(1+2*x)^2*(3*x^2- x+2)^(7/2)+29/330*(1+2*x)^3*(3*x^2-x+2)^(7/2)+2/33*(1+2*x)^4*(3*x^2-x+2)^( 7/2)+61917863/71663616*arcsinh(1/23*(1-6*x)*23^(1/2))*3^(1/2)+2692081/1194 3936*(1-6*x)*(3*x^2-x+2)^(1/2)
Time = 0.84 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.53 \[ \int (1+2 x)^3 \left (2-x+3 x^2\right )^{5/2} \left (1+3 x+4 x^2\right ) \, dx=\frac {6 \sqrt {2-x+3 x^2} \left (9173509857+26646633218 x+72088585464 x^2+161269204752 x^3+263636134272 x^4+347247744768 x^5+415908006912 x^6+419978151936 x^7+308846297088 x^8+207681159168 x^9+120394874880 x^{10}\right )+23838377255 \sqrt {3} \log \left (1-6 x+2 \sqrt {6-3 x+9 x^2}\right )}{27590492160} \]
(6*Sqrt[2 - x + 3*x^2]*(9173509857 + 26646633218*x + 72088585464*x^2 + 161 269204752*x^3 + 263636134272*x^4 + 347247744768*x^5 + 415908006912*x^6 + 4 19978151936*x^7 + 308846297088*x^8 + 207681159168*x^9 + 120394874880*x^10) + 23838377255*Sqrt[3]*Log[1 - 6*x + 2*Sqrt[6 - 3*x + 9*x^2]])/27590492160
Time = 0.44 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.16, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.344, Rules used = {2184, 27, 1236, 27, 1236, 1225, 1087, 1087, 1087, 1090, 222}
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.
| \(\displaystyle \int (2 x+1)^3 \left (3 x^2-x+2\right )^{5/2} \left (4 x^2+3 x+1\right ) \, dx\) |
| \(\Big \downarrow \) 2184 |
| \(\displaystyle \frac {1}{132} \int 4 (2 x+1)^3 (87 x+8) \left (3 x^2-x+2\right )^{5/2}dx+\frac {2}{33} \left (3 x^2-x+2\right )^{7/2} (2 x+1)^4\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{33} \int (2 x+1)^3 (87 x+8) \left (3 x^2-x+2\right )^{5/2}dx+\frac {2}{33} \left (3 x^2-x+2\right )^{7/2} (2 x+1)^4\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {1}{33} \left (\frac {1}{30} \int -\frac {9}{2} (111-532 x) (2 x+1)^2 \left (3 x^2-x+2\right )^{5/2}dx+\frac {29}{10} (2 x+1)^3 \left (3 x^2-x+2\right )^{7/2}\right )+\frac {2}{33} \left (3 x^2-x+2\right )^{7/2} (2 x+1)^4\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{33} \left (\frac {29}{10} (2 x+1)^3 \left (3 x^2-x+2\right )^{7/2}-\frac {3}{20} \int (111-532 x) (2 x+1)^2 \left (3 x^2-x+2\right )^{5/2}dx\right )+\frac {2}{33} \left (3 x^2-x+2\right )^{7/2} (2 x+1)^4\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {1}{33} \left (\frac {29}{10} (2 x+1)^3 \left (3 x^2-x+2\right )^{7/2}-\frac {3}{20} \left (\frac {1}{27} \int (5391-3050 x) (2 x+1) \left (3 x^2-x+2\right )^{5/2}dx-\frac {532}{27} (2 x+1)^2 \left (3 x^2-x+2\right )^{7/2}\right )\right )+\frac {2}{33} \left (3 x^2-x+2\right )^{7/2} (2 x+1)^4\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle \frac {1}{33} \left (\frac {29}{10} (2 x+1)^3 \left (3 x^2-x+2\right )^{7/2}-\frac {3}{20} \left (\frac {1}{27} \left (\frac {55979}{8} \int \left (3 x^2-x+2\right )^{5/2}dx+\frac {1}{84} (26353-21350 x) \left (3 x^2-x+2\right )^{7/2}\right )-\frac {532}{27} (2 x+1)^2 \left (3 x^2-x+2\right )^{7/2}\right )\right )+\frac {2}{33} \left (3 x^2-x+2\right )^{7/2} (2 x+1)^4\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{33} \left (\frac {29}{10} (2 x+1)^3 \left (3 x^2-x+2\right )^{7/2}-\frac {3}{20} \left (\frac {1}{27} \left (\frac {55979}{8} \left (\frac {115}{72} \int \left (3 x^2-x+2\right )^{3/2}dx-\frac {1}{36} (1-6 x) \left (3 x^2-x+2\right )^{5/2}\right )+\frac {1}{84} (26353-21350 x) \left (3 x^2-x+2\right )^{7/2}\right )-\frac {532}{27} (2 x+1)^2 \left (3 x^2-x+2\right )^{7/2}\right )\right )+\frac {2}{33} \left (3 x^2-x+2\right )^{7/2} (2 x+1)^4\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{33} \left (\frac {29}{10} (2 x+1)^3 \left (3 x^2-x+2\right )^{7/2}-\frac {3}{20} \left (\frac {1}{27} \left (\frac {55979}{8} \left (\frac {115}{72} \left (\frac {23}{16} \int \sqrt {3 x^2-x+2}dx-\frac {1}{24} (1-6 x) \left (3 x^2-x+2\right )^{3/2}\right )-\frac {1}{36} (1-6 x) \left (3 x^2-x+2\right )^{5/2}\right )+\frac {1}{84} (26353-21350 x) \left (3 x^2-x+2\right )^{7/2}\right )-\frac {532}{27} (2 x+1)^2 \left (3 x^2-x+2\right )^{7/2}\right )\right )+\frac {2}{33} \left (3 x^2-x+2\right )^{7/2} (2 x+1)^4\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{33} \left (\frac {29}{10} (2 x+1)^3 \left (3 x^2-x+2\right )^{7/2}-\frac {3}{20} \left (\frac {1}{27} \left (\frac {55979}{8} \left (\frac {115}{72} \left (\frac {23}{16} \left (\frac {23}{24} \int \frac {1}{\sqrt {3 x^2-x+2}}dx-\frac {1}{12} (1-6 x) \sqrt {3 x^2-x+2}\right )-\frac {1}{24} (1-6 x) \left (3 x^2-x+2\right )^{3/2}\right )-\frac {1}{36} (1-6 x) \left (3 x^2-x+2\right )^{5/2}\right )+\frac {1}{84} (26353-21350 x) \left (3 x^2-x+2\right )^{7/2}\right )-\frac {532}{27} (2 x+1)^2 \left (3 x^2-x+2\right )^{7/2}\right )\right )+\frac {2}{33} \left (3 x^2-x+2\right )^{7/2} (2 x+1)^4\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle \frac {1}{33} \left (\frac {29}{10} (2 x+1)^3 \left (3 x^2-x+2\right )^{7/2}-\frac {3}{20} \left (\frac {1}{27} \left (\frac {55979}{8} \left (\frac {115}{72} \left (\frac {23}{16} \left (\frac {1}{24} \sqrt {\frac {23}{3}} \int \frac {1}{\sqrt {\frac {1}{23} (6 x-1)^2+1}}d(6 x-1)-\frac {1}{12} (1-6 x) \sqrt {3 x^2-x+2}\right )-\frac {1}{24} (1-6 x) \left (3 x^2-x+2\right )^{3/2}\right )-\frac {1}{36} (1-6 x) \left (3 x^2-x+2\right )^{5/2}\right )+\frac {1}{84} (26353-21350 x) \left (3 x^2-x+2\right )^{7/2}\right )-\frac {532}{27} (2 x+1)^2 \left (3 x^2-x+2\right )^{7/2}\right )\right )+\frac {2}{33} \left (3 x^2-x+2\right )^{7/2} (2 x+1)^4\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {1}{33} \left (\frac {29}{10} (2 x+1)^3 \left (3 x^2-x+2\right )^{7/2}-\frac {3}{20} \left (\frac {1}{27} \left (\frac {55979}{8} \left (\frac {115}{72} \left (\frac {23}{16} \left (\frac {23 \text {arcsinh}\left (\frac {6 x-1}{\sqrt {23}}\right )}{24 \sqrt {3}}-\frac {1}{12} (1-6 x) \sqrt {3 x^2-x+2}\right )-\frac {1}{24} (1-6 x) \left (3 x^2-x+2\right )^{3/2}\right )-\frac {1}{36} (1-6 x) \left (3 x^2-x+2\right )^{5/2}\right )+\frac {1}{84} (26353-21350 x) \left (3 x^2-x+2\right )^{7/2}\right )-\frac {532}{27} (2 x+1)^2 \left (3 x^2-x+2\right )^{7/2}\right )\right )+\frac {2}{33} \left (3 x^2-x+2\right )^{7/2} (2 x+1)^4\) |
(2*(1 + 2*x)^4*(2 - x + 3*x^2)^(7/2))/33 + ((29*(1 + 2*x)^3*(2 - x + 3*x^2 )^(7/2))/10 - (3*((-532*(1 + 2*x)^2*(2 - x + 3*x^2)^(7/2))/27 + (((26353 - 21350*x)*(2 - x + 3*x^2)^(7/2))/84 + (55979*(-1/36*((1 - 6*x)*(2 - x + 3* x^2)^(5/2)) + (115*(-1/24*((1 - 6*x)*(2 - x + 3*x^2)^(3/2)) + (23*(-1/12*( (1 - 6*x)*Sqrt[2 - x + 3*x^2]) + (23*ArcSinh[(-1 + 6*x)/Sqrt[23]])/(24*Sqr t[3])))/16))/72))/8)/27))/20)/33
3.3.20.3.1 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[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
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[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/(2*c*(-4* (c/(b^2 - 4*a*c)))^p) Subst[Int[Simp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]
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])
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, S imp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(c*e^q*(m + q + 2*p + 1)) Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q - 1) - c *d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[ q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && Pol yQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && !(IGt Q[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Time = 0.68 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.42
| method | result | size |
| risch | \(\frac {\left (120394874880 x^{10}+207681159168 x^{9}+308846297088 x^{8}+419978151936 x^{7}+415908006912 x^{6}+347247744768 x^{5}+263636134272 x^{4}+161269204752 x^{3}+72088585464 x^{2}+26646633218 x +9173509857\right ) \sqrt {3 x^{2}-x +2}}{4598415360}-\frac {61917863 \sqrt {3}\, \operatorname {arcsinh}\left (\frac {6 \sqrt {23}\, \left (x -\frac {1}{6}\right )}{23}\right )}{71663616}\) | \(80\) |
| trager | \(\left (\frac {288}{11} x^{10}+\frac {2484}{55} x^{9}+\frac {3694}{55} x^{8}+\frac {120557}{1320} x^{7}+\frac {557147}{6160} x^{6}+\frac {50238389}{665280} x^{5}+\frac {32692973}{570240} x^{4}+\frac {1119925033}{31933440} x^{3}+\frac {429098723}{27371520} x^{2}+\frac {13323316609}{2299207680} x +\frac {1019278873}{510935040}\right ) \sqrt {3 x^{2}-x +2}+\frac {61917863 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+6 \sqrt {3 x^{2}-x +2}\right )}{71663616}\) | \(104\) |
| default | \(-\frac {5089 \left (-1+6 x \right ) \left (3 x^{2}-x +2\right )^{\frac {5}{2}}}{155520}-\frac {117047 \left (-1+6 x \right ) \left (3 x^{2}-x +2\right )^{\frac {3}{2}}}{1492992}-\frac {2692081 \left (-1+6 x \right ) \sqrt {3 x^{2}-x +2}}{11943936}-\frac {61917863 \sqrt {3}\, \operatorname {arcsinh}\left (\frac {6 \sqrt {23}\, \left (x -\frac {1}{6}\right )}{23}\right )}{71663616}+\frac {92423 \left (3 x^{2}-x +2\right )^{\frac {7}{2}}}{498960}+\frac {32 x^{4} \left (3 x^{2}-x +2\right )^{\frac {7}{2}}}{33}+\frac {436 x^{3} \left (3 x^{2}-x +2\right )^{\frac {7}{2}}}{165}+\frac {4258 x^{2} \left (3 x^{2}-x +2\right )^{\frac {7}{2}}}{1485}+\frac {10073 x \left (3 x^{2}-x +2\right )^{\frac {7}{2}}}{7128}\) | \(153\) |
1/4598415360*(120394874880*x^10+207681159168*x^9+308846297088*x^8+41997815 1936*x^7+415908006912*x^6+347247744768*x^5+263636134272*x^4+161269204752*x ^3+72088585464*x^2+26646633218*x+9173509857)*(3*x^2-x+2)^(1/2)-61917863/71 663616*3^(1/2)*arcsinh(6/23*23^(1/2)*(x-1/6))
Time = 0.26 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.54 \[ \int (1+2 x)^3 \left (2-x+3 x^2\right )^{5/2} \left (1+3 x+4 x^2\right ) \, dx=\frac {1}{4598415360} \, {\left (120394874880 \, x^{10} + 207681159168 \, x^{9} + 308846297088 \, x^{8} + 419978151936 \, x^{7} + 415908006912 \, x^{6} + 347247744768 \, x^{5} + 263636134272 \, x^{4} + 161269204752 \, x^{3} + 72088585464 \, x^{2} + 26646633218 \, x + 9173509857\right )} \sqrt {3 \, x^{2} - x + 2} + \frac {61917863}{143327232} \, \sqrt {3} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} - x + 2} {\left (6 \, x - 1\right )} - 72 \, x^{2} + 24 \, x - 25\right ) \]
1/4598415360*(120394874880*x^10 + 207681159168*x^9 + 308846297088*x^8 + 41 9978151936*x^7 + 415908006912*x^6 + 347247744768*x^5 + 263636134272*x^4 + 161269204752*x^3 + 72088585464*x^2 + 26646633218*x + 9173509857)*sqrt(3*x^ 2 - x + 2) + 61917863/143327232*sqrt(3)*log(4*sqrt(3)*sqrt(3*x^2 - x + 2)* (6*x - 1) - 72*x^2 + 24*x - 25)
Time = 0.75 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.55 \[ \int (1+2 x)^3 \left (2-x+3 x^2\right )^{5/2} \left (1+3 x+4 x^2\right ) \, dx=\sqrt {3 x^{2} - x + 2} \cdot \left (\frac {288 x^{10}}{11} + \frac {2484 x^{9}}{55} + \frac {3694 x^{8}}{55} + \frac {120557 x^{7}}{1320} + \frac {557147 x^{6}}{6160} + \frac {50238389 x^{5}}{665280} + \frac {32692973 x^{4}}{570240} + \frac {1119925033 x^{3}}{31933440} + \frac {429098723 x^{2}}{27371520} + \frac {13323316609 x}{2299207680} + \frac {1019278873}{510935040}\right ) - \frac {61917863 \sqrt {3} \operatorname {asinh}{\left (\frac {6 \sqrt {23} \left (x - \frac {1}{6}\right )}{23} \right )}}{71663616} \]
sqrt(3*x**2 - x + 2)*(288*x**10/11 + 2484*x**9/55 + 3694*x**8/55 + 120557* x**7/1320 + 557147*x**6/6160 + 50238389*x**5/665280 + 32692973*x**4/570240 + 1119925033*x**3/31933440 + 429098723*x**2/27371520 + 13323316609*x/2299 207680 + 1019278873/510935040) - 61917863*sqrt(3)*asinh(6*sqrt(23)*(x - 1/ 6)/23)/71663616
Time = 0.30 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.97 \[ \int (1+2 x)^3 \left (2-x+3 x^2\right )^{5/2} \left (1+3 x+4 x^2\right ) \, dx=\frac {32}{33} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {7}{2}} x^{4} + \frac {436}{165} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {7}{2}} x^{3} + \frac {4258}{1485} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {7}{2}} x^{2} + \frac {10073}{7128} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {7}{2}} x + \frac {92423}{498960} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {7}{2}} - \frac {5089}{25920} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {5}{2}} x + \frac {5089}{155520} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {5}{2}} - \frac {117047}{248832} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}} x + \frac {117047}{1492992} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}} - \frac {2692081}{1990656} \, \sqrt {3 \, x^{2} - x + 2} x - \frac {61917863}{71663616} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (6 \, x - 1\right )}\right ) + \frac {2692081}{11943936} \, \sqrt {3 \, x^{2} - x + 2} \]
32/33*(3*x^2 - x + 2)^(7/2)*x^4 + 436/165*(3*x^2 - x + 2)^(7/2)*x^3 + 4258 /1485*(3*x^2 - x + 2)^(7/2)*x^2 + 10073/7128*(3*x^2 - x + 2)^(7/2)*x + 924 23/498960*(3*x^2 - x + 2)^(7/2) - 5089/25920*(3*x^2 - x + 2)^(5/2)*x + 508 9/155520*(3*x^2 - x + 2)^(5/2) - 117047/248832*(3*x^2 - x + 2)^(3/2)*x + 1 17047/1492992*(3*x^2 - x + 2)^(3/2) - 2692081/1990656*sqrt(3*x^2 - x + 2)* x - 61917863/71663616*sqrt(3)*arcsinh(1/23*sqrt(23)*(6*x - 1)) + 2692081/1 1943936*sqrt(3*x^2 - x + 2)
Time = 0.28 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.52 \[ \int (1+2 x)^3 \left (2-x+3 x^2\right )^{5/2} \left (1+3 x+4 x^2\right ) \, dx=\frac {1}{4598415360} \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (8 \, {\left (6 \, {\left (36 \, {\left (14 \, {\left (48 \, {\left (18 \, {\left (40 \, x + 69\right )} x + 1847\right )} x + 120557\right )} x + 1671441\right )} x + 50238389\right )} x + 228850811\right )} x + 1119925033\right )} x + 3003691061\right )} x + 13323316609\right )} x + 9173509857\right )} \sqrt {3 \, x^{2} - x + 2} + \frac {61917863}{71663616} \, \sqrt {3} \log \left (-2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - x + 2}\right )} + 1\right ) \]
1/4598415360*(2*(12*(6*(8*(6*(36*(14*(48*(18*(40*x + 69)*x + 1847)*x + 120 557)*x + 1671441)*x + 50238389)*x + 228850811)*x + 1119925033)*x + 3003691 061)*x + 13323316609)*x + 9173509857)*sqrt(3*x^2 - x + 2) + 61917863/71663 616*sqrt(3)*log(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 - x + 2)) + 1)
Timed out. \[ \int (1+2 x)^3 \left (2-x+3 x^2\right )^{5/2} \left (1+3 x+4 x^2\right ) \, dx=\int {\left (2\,x+1\right )}^3\,{\left (3\,x^2-x+2\right )}^{5/2}\,\left (4\,x^2+3\,x+1\right ) \,d x \]