Integrand size = 32, antiderivative size = 164 \[ \int (1+2 x)^2 \left (2-x+3 x^2\right )^{5/2} \left (1+3 x+4 x^2\right ) \, dx=-\frac {154997 (1-6 x) \sqrt {2-x+3 x^2}}{4478976}-\frac {6739 (1-6 x) \left (2-x+3 x^2\right )^{3/2}}{559872}-\frac {293 (1-6 x) \left (2-x+3 x^2\right )^{5/2}}{58320}+\frac {37}{405} (1+2 x)^2 \left (2-x+3 x^2\right )^{7/2}+\frac {1}{15} (1+2 x)^3 \left (2-x+3 x^2\right )^{7/2}+\frac {(2731+3430 x) \left (2-x+3 x^2\right )^{7/2}}{17010}-\frac {3564931 \text {arcsinh}\left (\frac {1-6 x}{\sqrt {23}}\right )}{8957952 \sqrt {3}} \]
-6739/559872*(1-6*x)*(3*x^2-x+2)^(3/2)-293/58320*(1-6*x)*(3*x^2-x+2)^(5/2) +37/405*(1+2*x)^2*(3*x^2-x+2)^(7/2)+1/15*(1+2*x)^3*(3*x^2-x+2)^(7/2)+1/170 10*(2731+3430*x)*(3*x^2-x+2)^(7/2)-3564931/26873856*arcsinh(1/23*(1-6*x)*2 3^(1/2))*3^(1/2)-154997/4478976*(1-6*x)*(3*x^2-x+2)^(1/2)
Time = 0.76 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.58 \[ \int (1+2 x)^2 \left (2-x+3 x^2\right )^{5/2} \left (1+3 x+4 x^2\right ) \, dx=\frac {6 \sqrt {2-x+3 x^2} \left (387182961+692659234 x+1693765752 x^2+3096104976 x^3+4171579776 x^4+4996802304 x^5+5671627776 x^6+4427716608 x^7+2675441664 x^8+2257403904 x^9\right )-124772585 \sqrt {3} \log \left (1-6 x+2 \sqrt {6-3 x+9 x^2}\right )}{940584960} \]
(6*Sqrt[2 - x + 3*x^2]*(387182961 + 692659234*x + 1693765752*x^2 + 3096104 976*x^3 + 4171579776*x^4 + 4996802304*x^5 + 5671627776*x^6 + 4427716608*x^ 7 + 2675441664*x^8 + 2257403904*x^9) - 124772585*Sqrt[3]*Log[1 - 6*x + 2*S qrt[6 - 3*x + 9*x^2]])/940584960
Time = 0.40 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.15, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {2184, 27, 1236, 27, 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.
| \(\displaystyle \int (2 x+1)^2 \left (3 x^2-x+2\right )^{5/2} \left (4 x^2+3 x+1\right ) \, dx\) |
| \(\Big \downarrow \) 2184 |
| \(\displaystyle \frac {1}{120} \int 4 (2 x+1)^2 (74 x+13) \left (3 x^2-x+2\right )^{5/2}dx+\frac {1}{15} (2 x+1)^3 \left (3 x^2-x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{30} \int (2 x+1)^2 (74 x+13) \left (3 x^2-x+2\right )^{5/2}dx+\frac {1}{15} (2 x+1)^3 \left (3 x^2-x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {1}{30} \left (\frac {1}{27} \int 2 (2 x+1) (980 x+9) \left (3 x^2-x+2\right )^{5/2}dx+\frac {74}{27} (2 x+1)^2 \left (3 x^2-x+2\right )^{7/2}\right )+\frac {1}{15} (2 x+1)^3 \left (3 x^2-x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{30} \left (\frac {2}{27} \int (2 x+1) (980 x+9) \left (3 x^2-x+2\right )^{5/2}dx+\frac {74}{27} (2 x+1)^2 \left (3 x^2-x+2\right )^{7/2}\right )+\frac {1}{15} (2 x+1)^3 \left (3 x^2-x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle \frac {1}{30} \left (\frac {2}{27} \left (\frac {293}{4} \int \left (3 x^2-x+2\right )^{5/2}dx+\frac {1}{42} (3430 x+2731) \left (3 x^2-x+2\right )^{7/2}\right )+\frac {74}{27} (2 x+1)^2 \left (3 x^2-x+2\right )^{7/2}\right )+\frac {1}{15} (2 x+1)^3 \left (3 x^2-x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{30} \left (\frac {2}{27} \left (\frac {293}{4} \left (\frac {115}{72} \int \left (3 x^2-x+2\right )^{3/2}dx-\frac {1}{36} (1-6 x) \left (3 x^2-x+2\right )^{5/2}\right )+\frac {1}{42} (3430 x+2731) \left (3 x^2-x+2\right )^{7/2}\right )+\frac {74}{27} (2 x+1)^2 \left (3 x^2-x+2\right )^{7/2}\right )+\frac {1}{15} (2 x+1)^3 \left (3 x^2-x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{30} \left (\frac {2}{27} \left (\frac {293}{4} \left (\frac {115}{72} \left (\frac {23}{16} \int \sqrt {3 x^2-x+2}dx-\frac {1}{24} (1-6 x) \left (3 x^2-x+2\right )^{3/2}\right )-\frac {1}{36} (1-6 x) \left (3 x^2-x+2\right )^{5/2}\right )+\frac {1}{42} (3430 x+2731) \left (3 x^2-x+2\right )^{7/2}\right )+\frac {74}{27} (2 x+1)^2 \left (3 x^2-x+2\right )^{7/2}\right )+\frac {1}{15} (2 x+1)^3 \left (3 x^2-x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{30} \left (\frac {2}{27} \left (\frac {293}{4} \left (\frac {115}{72} \left (\frac {23}{16} \left (\frac {23}{24} \int \frac {1}{\sqrt {3 x^2-x+2}}dx-\frac {1}{12} (1-6 x) \sqrt {3 x^2-x+2}\right )-\frac {1}{24} (1-6 x) \left (3 x^2-x+2\right )^{3/2}\right )-\frac {1}{36} (1-6 x) \left (3 x^2-x+2\right )^{5/2}\right )+\frac {1}{42} (3430 x+2731) \left (3 x^2-x+2\right )^{7/2}\right )+\frac {74}{27} (2 x+1)^2 \left (3 x^2-x+2\right )^{7/2}\right )+\frac {1}{15} (2 x+1)^3 \left (3 x^2-x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle \frac {1}{30} \left (\frac {2}{27} \left (\frac {293}{4} \left (\frac {115}{72} \left (\frac {23}{16} \left (\frac {1}{24} \sqrt {\frac {23}{3}} \int \frac {1}{\sqrt {\frac {1}{23} (6 x-1)^2+1}}d(6 x-1)-\frac {1}{12} (1-6 x) \sqrt {3 x^2-x+2}\right )-\frac {1}{24} (1-6 x) \left (3 x^2-x+2\right )^{3/2}\right )-\frac {1}{36} (1-6 x) \left (3 x^2-x+2\right )^{5/2}\right )+\frac {1}{42} (3430 x+2731) \left (3 x^2-x+2\right )^{7/2}\right )+\frac {74}{27} (2 x+1)^2 \left (3 x^2-x+2\right )^{7/2}\right )+\frac {1}{15} (2 x+1)^3 \left (3 x^2-x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {1}{30} \left (\frac {2}{27} \left (\frac {293}{4} \left (\frac {115}{72} \left (\frac {23}{16} \left (\frac {23 \text {arcsinh}\left (\frac {6 x-1}{\sqrt {23}}\right )}{24 \sqrt {3}}-\frac {1}{12} (1-6 x) \sqrt {3 x^2-x+2}\right )-\frac {1}{24} (1-6 x) \left (3 x^2-x+2\right )^{3/2}\right )-\frac {1}{36} (1-6 x) \left (3 x^2-x+2\right )^{5/2}\right )+\frac {1}{42} (3430 x+2731) \left (3 x^2-x+2\right )^{7/2}\right )+\frac {74}{27} (2 x+1)^2 \left (3 x^2-x+2\right )^{7/2}\right )+\frac {1}{15} (2 x+1)^3 \left (3 x^2-x+2\right )^{7/2}\) |
((1 + 2*x)^3*(2 - x + 3*x^2)^(7/2))/15 + ((74*(1 + 2*x)^2*(2 - x + 3*x^2)^ (7/2))/27 + (2*(((2731 + 3430*x)*(2 - x + 3*x^2)^(7/2))/42 + (293*(-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*Sqrt[3])))/16))/72))/4))/27)/30
3.3.21.3.1 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 = 0.67 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.46
| method | result | size |
| risch | \(\frac {\left (2257403904 x^{9}+2675441664 x^{8}+4427716608 x^{7}+5671627776 x^{6}+4996802304 x^{5}+4171579776 x^{4}+3096104976 x^{3}+1693765752 x^{2}+692659234 x +387182961\right ) \sqrt {3 x^{2}-x +2}}{156764160}+\frac {3564931 \sqrt {3}\, \operatorname {arcsinh}\left (\frac {6 \sqrt {23}\, \left (x -\frac {1}{6}\right )}{23}\right )}{26873856}\) | \(75\) |
| trager | \(\left (\frac {72}{5} x^{9}+\frac {256}{15} x^{8}+\frac {1271}{45} x^{7}+\frac {22793}{630} x^{6}+\frac {722917}{22680} x^{5}+\frac {517309}{19440} x^{4}+\frac {21500729}{1088640} x^{3}+\frac {10081939}{933120} x^{2}+\frac {346329617}{78382080} x +\frac {43020329}{17418240}\right ) \sqrt {3 x^{2}-x +2}-\frac {3564931 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+6 \sqrt {3 x^{2}-x +2}\right )}{26873856}\) | \(99\) |
| default | \(\frac {293 \left (-1+6 x \right ) \left (3 x^{2}-x +2\right )^{\frac {5}{2}}}{58320}+\frac {6739 \left (-1+6 x \right ) \left (3 x^{2}-x +2\right )^{\frac {3}{2}}}{559872}+\frac {154997 \left (-1+6 x \right ) \sqrt {3 x^{2}-x +2}}{4478976}+\frac {3564931 \sqrt {3}\, \operatorname {arcsinh}\left (\frac {6 \sqrt {23}\, \left (x -\frac {1}{6}\right )}{23}\right )}{26873856}+\frac {5419 \left (3 x^{2}-x +2\right )^{\frac {7}{2}}}{17010}+\frac {8 x^{3} \left (3 x^{2}-x +2\right )^{\frac {7}{2}}}{15}+\frac {472 x^{2} \left (3 x^{2}-x +2\right )^{\frac {7}{2}}}{405}+\frac {235 x \left (3 x^{2}-x +2\right )^{\frac {7}{2}}}{243}\) | \(136\) |
1/156764160*(2257403904*x^9+2675441664*x^8+4427716608*x^7+5671627776*x^6+4 996802304*x^5+4171579776*x^4+3096104976*x^3+1693765752*x^2+692659234*x+387 182961)*(3*x^2-x+2)^(1/2)+3564931/26873856*3^(1/2)*arcsinh(6/23*23^(1/2)*( x-1/6))
Time = 0.26 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.60 \[ \int (1+2 x)^2 \left (2-x+3 x^2\right )^{5/2} \left (1+3 x+4 x^2\right ) \, dx=\frac {1}{156764160} \, {\left (2257403904 \, x^{9} + 2675441664 \, x^{8} + 4427716608 \, x^{7} + 5671627776 \, x^{6} + 4996802304 \, x^{5} + 4171579776 \, x^{4} + 3096104976 \, x^{3} + 1693765752 \, x^{2} + 692659234 \, x + 387182961\right )} \sqrt {3 \, x^{2} - x + 2} + \frac {3564931}{53747712} \, \sqrt {3} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} - x + 2} {\left (6 \, x - 1\right )} - 72 \, x^{2} + 24 \, x - 25\right ) \]
1/156764160*(2257403904*x^9 + 2675441664*x^8 + 4427716608*x^7 + 5671627776 *x^6 + 4996802304*x^5 + 4171579776*x^4 + 3096104976*x^3 + 1693765752*x^2 + 692659234*x + 387182961)*sqrt(3*x^2 - x + 2) + 3564931/53747712*sqrt(3)*l og(-4*sqrt(3)*sqrt(3*x^2 - x + 2)*(6*x - 1) - 72*x^2 + 24*x - 25)
Time = 0.66 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.59 \[ \int (1+2 x)^2 \left (2-x+3 x^2\right )^{5/2} \left (1+3 x+4 x^2\right ) \, dx=\sqrt {3 x^{2} - x + 2} \cdot \left (\frac {72 x^{9}}{5} + \frac {256 x^{8}}{15} + \frac {1271 x^{7}}{45} + \frac {22793 x^{6}}{630} + \frac {722917 x^{5}}{22680} + \frac {517309 x^{4}}{19440} + \frac {21500729 x^{3}}{1088640} + \frac {10081939 x^{2}}{933120} + \frac {346329617 x}{78382080} + \frac {43020329}{17418240}\right ) + \frac {3564931 \sqrt {3} \operatorname {asinh}{\left (\frac {6 \sqrt {23} \left (x - \frac {1}{6}\right )}{23} \right )}}{26873856} \]
sqrt(3*x**2 - x + 2)*(72*x**9/5 + 256*x**8/15 + 1271*x**7/45 + 22793*x**6/ 630 + 722917*x**5/22680 + 517309*x**4/19440 + 21500729*x**3/1088640 + 1008 1939*x**2/933120 + 346329617*x/78382080 + 43020329/17418240) + 3564931*sqr t(3)*asinh(6*sqrt(23)*(x - 1/6)/23)/26873856
Time = 0.28 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.02 \[ \int (1+2 x)^2 \left (2-x+3 x^2\right )^{5/2} \left (1+3 x+4 x^2\right ) \, dx=\frac {8}{15} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {7}{2}} x^{3} + \frac {472}{405} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {7}{2}} x^{2} + \frac {235}{243} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {7}{2}} x + \frac {5419}{17010} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {7}{2}} + \frac {293}{9720} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {5}{2}} x - \frac {293}{58320} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {5}{2}} + \frac {6739}{93312} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}} x - \frac {6739}{559872} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}} + \frac {154997}{746496} \, \sqrt {3 \, x^{2} - x + 2} x + \frac {3564931}{26873856} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (6 \, x - 1\right )}\right ) - \frac {154997}{4478976} \, \sqrt {3 \, x^{2} - x + 2} \]
8/15*(3*x^2 - x + 2)^(7/2)*x^3 + 472/405*(3*x^2 - x + 2)^(7/2)*x^2 + 235/2 43*(3*x^2 - x + 2)^(7/2)*x + 5419/17010*(3*x^2 - x + 2)^(7/2) + 293/9720*( 3*x^2 - x + 2)^(5/2)*x - 293/58320*(3*x^2 - x + 2)^(5/2) + 6739/93312*(3*x ^2 - x + 2)^(3/2)*x - 6739/559872*(3*x^2 - x + 2)^(3/2) + 154997/746496*sq rt(3*x^2 - x + 2)*x + 3564931/26873856*sqrt(3)*arcsinh(1/23*sqrt(23)*(6*x - 1)) - 154997/4478976*sqrt(3*x^2 - x + 2)
Time = 0.28 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.57 \[ \int (1+2 x)^2 \left (2-x+3 x^2\right )^{5/2} \left (1+3 x+4 x^2\right ) \, dx=\frac {1}{156764160} \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (8 \, {\left (6 \, {\left (36 \, {\left (14 \, {\left (24 \, {\left (27 \, x + 32\right )} x + 1271\right )} x + 22793\right )} x + 722917\right )} x + 3621163\right )} x + 21500729\right )} x + 70573573\right )} x + 346329617\right )} x + 387182961\right )} \sqrt {3 \, x^{2} - x + 2} - \frac {3564931}{26873856} \, \sqrt {3} \log \left (-2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - x + 2}\right )} + 1\right ) \]
1/156764160*(2*(12*(6*(8*(6*(36*(14*(24*(27*x + 32)*x + 1271)*x + 22793)*x + 722917)*x + 3621163)*x + 21500729)*x + 70573573)*x + 346329617)*x + 387 182961)*sqrt(3*x^2 - x + 2) - 3564931/26873856*sqrt(3)*log(-2*sqrt(3)*(sqr t(3)*x - sqrt(3*x^2 - x + 2)) + 1)
Timed out. \[ \int (1+2 x)^2 \left (2-x+3 x^2\right )^{5/2} \left (1+3 x+4 x^2\right ) \, dx=\int {\left (2\,x+1\right )}^2\,{\left (3\,x^2-x+2\right )}^{5/2}\,\left (4\,x^2+3\,x+1\right ) \,d x \]