Integrand size = 32, antiderivative size = 141 \[ \int (1+2 x)^2 \left (2-x+3 x^2\right )^{3/2} \left (1+3 x+4 x^2\right ) \, dx=\frac {2093 (1-6 x) \sqrt {2-x+3 x^2}}{27648}+\frac {91 (1-6 x) \left (2-x+3 x^2\right )^{3/2}}{3456}+\frac {8}{63} (1+2 x)^2 \left (2-x+3 x^2\right )^{5/2}+\frac {1}{12} (1+2 x)^3 \left (2-x+3 x^2\right )^{5/2}+\frac {13 (29+50 x) \left (2-x+3 x^2\right )^{5/2}}{2520}+\frac {48139 \text {arcsinh}\left (\frac {1-6 x}{\sqrt {23}}\right )}{55296 \sqrt {3}} \]
91/3456*(1-6*x)*(3*x^2-x+2)^(3/2)+8/63*(1+2*x)^2*(3*x^2-x+2)^(5/2)+1/12*(1 +2*x)^3*(3*x^2-x+2)^(5/2)+13/2520*(29+50*x)*(3*x^2-x+2)^(5/2)+48139/165888 *arcsinh(1/23*(1-6*x)*23^(1/2))*3^(1/2)+2093/27648*(1-6*x)*(3*x^2-x+2)^(1/ 2)
Time = 0.54 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.60 \[ \int (1+2 x)^2 \left (2-x+3 x^2\right )^{3/2} \left (1+3 x+4 x^2\right ) \, dx=\frac {6 \sqrt {2-x+3 x^2} \left (1517367+2735918 x+5694024 x^2+10119792 x^3+12173952 x^4+10656000 x^5+9262080 x^6+5806080 x^7\right )+1684865 \sqrt {3} \log \left (1-6 x+2 \sqrt {6-3 x+9 x^2}\right )}{5806080} \]
(6*Sqrt[2 - x + 3*x^2]*(1517367 + 2735918*x + 5694024*x^2 + 10119792*x^3 + 12173952*x^4 + 10656000*x^5 + 9262080*x^6 + 5806080*x^7) + 1684865*Sqrt[3 ]*Log[1 - 6*x + 2*Sqrt[6 - 3*x + 9*x^2]])/5806080
Time = 0.38 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.14, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {2184, 27, 1236, 27, 1225, 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.
| \(\displaystyle \int (2 x+1)^2 \left (3 x^2-x+2\right )^{3/2} \left (4 x^2+3 x+1\right ) \, dx\) |
| \(\Big \downarrow \) 2184 |
| \(\displaystyle \frac {1}{96} \int 4 (2 x+1)^2 (64 x+5) \left (3 x^2-x+2\right )^{3/2}dx+\frac {1}{12} \left (3 x^2-x+2\right )^{5/2} (2 x+1)^3\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{24} \int (2 x+1)^2 (64 x+5) \left (3 x^2-x+2\right )^{3/2}dx+\frac {1}{12} \left (3 x^2-x+2\right )^{5/2} (2 x+1)^3\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {1}{24} \left (\frac {1}{21} \int -13 (19-90 x) (2 x+1) \left (3 x^2-x+2\right )^{3/2}dx+\frac {64}{21} (2 x+1)^2 \left (3 x^2-x+2\right )^{5/2}\right )+\frac {1}{12} \left (3 x^2-x+2\right )^{5/2} (2 x+1)^3\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{24} \left (\frac {64}{21} (2 x+1)^2 \left (3 x^2-x+2\right )^{5/2}-\frac {13}{21} \int (19-90 x) (2 x+1) \left (3 x^2-x+2\right )^{3/2}dx\right )+\frac {1}{12} \left (3 x^2-x+2\right )^{5/2} (2 x+1)^3\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle \frac {1}{24} \left (\frac {64}{21} (2 x+1)^2 \left (3 x^2-x+2\right )^{5/2}-\frac {13}{21} \left (\frac {49}{2} \int \left (3 x^2-x+2\right )^{3/2}dx-\frac {1}{5} (50 x+29) \left (3 x^2-x+2\right )^{5/2}\right )\right )+\frac {1}{12} \left (3 x^2-x+2\right )^{5/2} (2 x+1)^3\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{24} \left (\frac {64}{21} (2 x+1)^2 \left (3 x^2-x+2\right )^{5/2}-\frac {13}{21} \left (\frac {49}{2} \left (\frac {23}{16} \int \sqrt {3 x^2-x+2}dx-\frac {1}{24} (1-6 x) \left (3 x^2-x+2\right )^{3/2}\right )-\frac {1}{5} (50 x+29) \left (3 x^2-x+2\right )^{5/2}\right )\right )+\frac {1}{12} \left (3 x^2-x+2\right )^{5/2} (2 x+1)^3\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{24} \left (\frac {64}{21} (2 x+1)^2 \left (3 x^2-x+2\right )^{5/2}-\frac {13}{21} \left (\frac {49}{2} \left (\frac {23}{16} \left (\frac {23}{24} \int \frac {1}{\sqrt {3 x^2-x+2}}dx-\frac {1}{12} (1-6 x) \sqrt {3 x^2-x+2}\right )-\frac {1}{24} (1-6 x) \left (3 x^2-x+2\right )^{3/2}\right )-\frac {1}{5} (50 x+29) \left (3 x^2-x+2\right )^{5/2}\right )\right )+\frac {1}{12} \left (3 x^2-x+2\right )^{5/2} (2 x+1)^3\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle \frac {1}{24} \left (\frac {64}{21} (2 x+1)^2 \left (3 x^2-x+2\right )^{5/2}-\frac {13}{21} \left (\frac {49}{2} \left (\frac {23}{16} \left (\frac {1}{24} \sqrt {\frac {23}{3}} \int \frac {1}{\sqrt {\frac {1}{23} (6 x-1)^2+1}}d(6 x-1)-\frac {1}{12} (1-6 x) \sqrt {3 x^2-x+2}\right )-\frac {1}{24} (1-6 x) \left (3 x^2-x+2\right )^{3/2}\right )-\frac {1}{5} (50 x+29) \left (3 x^2-x+2\right )^{5/2}\right )\right )+\frac {1}{12} \left (3 x^2-x+2\right )^{5/2} (2 x+1)^3\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {1}{24} \left (\frac {64}{21} (2 x+1)^2 \left (3 x^2-x+2\right )^{5/2}-\frac {13}{21} \left (\frac {49}{2} \left (\frac {23}{16} \left (\frac {23 \text {arcsinh}\left (\frac {6 x-1}{\sqrt {23}}\right )}{24 \sqrt {3}}-\frac {1}{12} (1-6 x) \sqrt {3 x^2-x+2}\right )-\frac {1}{24} (1-6 x) \left (3 x^2-x+2\right )^{3/2}\right )-\frac {1}{5} (50 x+29) \left (3 x^2-x+2\right )^{5/2}\right )\right )+\frac {1}{12} \left (3 x^2-x+2\right )^{5/2} (2 x+1)^3\) |
((1 + 2*x)^3*(2 - x + 3*x^2)^(5/2))/12 + ((64*(1 + 2*x)^2*(2 - x + 3*x^2)^ (5/2))/21 - (13*(-1/5*((29 + 50*x)*(2 - x + 3*x^2)^(5/2)) + (49*(-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))/2))/21)/24
3.3.15.3.1 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 = 0.79 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.46
| method | result | size |
| risch | \(\frac {\left (5806080 x^{7}+9262080 x^{6}+10656000 x^{5}+12173952 x^{4}+10119792 x^{3}+5694024 x^{2}+2735918 x +1517367\right ) \sqrt {3 x^{2}-x +2}}{967680}-\frac {48139 \sqrt {3}\, \operatorname {arcsinh}\left (\frac {6 \sqrt {23}\, \left (x -\frac {1}{6}\right )}{23}\right )}{165888}\) | \(65\) |
| trager | \(\left (6 x^{7}+\frac {67}{7} x^{6}+\frac {925}{84} x^{5}+\frac {4529}{360} x^{4}+\frac {210829}{20160} x^{3}+\frac {33893}{5760} x^{2}+\frac {1367959}{483840} x +\frac {505789}{322560}\right ) \sqrt {3 x^{2}-x +2}-\frac {48139 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {3 x^{2}-x +2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )\right )}{165888}\) | \(91\) |
| default | \(-\frac {91 \left (-1+6 x \right ) \left (3 x^{2}-x +2\right )^{\frac {3}{2}}}{3456}-\frac {2093 \left (-1+6 x \right ) \sqrt {3 x^{2}-x +2}}{27648}-\frac {48139 \sqrt {3}\, \operatorname {arcsinh}\left (\frac {6 \sqrt {23}\, \left (x -\frac {1}{6}\right )}{23}\right )}{165888}+\frac {907 \left (3 x^{2}-x +2\right )^{\frac {5}{2}}}{2520}+\frac {2 x^{3} \left (3 x^{2}-x +2\right )^{\frac {5}{2}}}{3}+\frac {95 x^{2} \left (3 x^{2}-x +2\right )^{\frac {5}{2}}}{63}+\frac {319 x \left (3 x^{2}-x +2\right )^{\frac {5}{2}}}{252}\) | \(117\) |
1/967680*(5806080*x^7+9262080*x^6+10656000*x^5+12173952*x^4+10119792*x^3+5 694024*x^2+2735918*x+1517367)*(3*x^2-x+2)^(1/2)-48139/165888*3^(1/2)*arcsi nh(6/23*23^(1/2)*(x-1/6))
Time = 0.27 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.62 \[ \int (1+2 x)^2 \left (2-x+3 x^2\right )^{3/2} \left (1+3 x+4 x^2\right ) \, dx=\frac {1}{967680} \, {\left (5806080 \, x^{7} + 9262080 \, x^{6} + 10656000 \, x^{5} + 12173952 \, x^{4} + 10119792 \, x^{3} + 5694024 \, x^{2} + 2735918 \, x + 1517367\right )} \sqrt {3 \, x^{2} - x + 2} + \frac {48139}{331776} \, \sqrt {3} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} - x + 2} {\left (6 \, x - 1\right )} - 72 \, x^{2} + 24 \, x - 25\right ) \]
1/967680*(5806080*x^7 + 9262080*x^6 + 10656000*x^5 + 12173952*x^4 + 101197 92*x^3 + 5694024*x^2 + 2735918*x + 1517367)*sqrt(3*x^2 - x + 2) + 48139/33 1776*sqrt(3)*log(4*sqrt(3)*sqrt(3*x^2 - x + 2)*(6*x - 1) - 72*x^2 + 24*x - 25)
Time = 0.54 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.58 \[ \int (1+2 x)^2 \left (2-x+3 x^2\right )^{3/2} \left (1+3 x+4 x^2\right ) \, dx=\sqrt {3 x^{2} - x + 2} \cdot \left (6 x^{7} + \frac {67 x^{6}}{7} + \frac {925 x^{5}}{84} + \frac {4529 x^{4}}{360} + \frac {210829 x^{3}}{20160} + \frac {33893 x^{2}}{5760} + \frac {1367959 x}{483840} + \frac {505789}{322560}\right ) - \frac {48139 \sqrt {3} \operatorname {asinh}{\left (\frac {6 \sqrt {23} \left (x - \frac {1}{6}\right )}{23} \right )}}{165888} \]
sqrt(3*x**2 - x + 2)*(6*x**7 + 67*x**6/7 + 925*x**5/84 + 4529*x**4/360 + 2 10829*x**3/20160 + 33893*x**2/5760 + 1367959*x/483840 + 505789/322560) - 4 8139*sqrt(3)*asinh(6*sqrt(23)*(x - 1/6)/23)/165888
Time = 0.27 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.98 \[ \int (1+2 x)^2 \left (2-x+3 x^2\right )^{3/2} \left (1+3 x+4 x^2\right ) \, dx=\frac {2}{3} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {5}{2}} x^{3} + \frac {95}{63} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {5}{2}} x^{2} + \frac {319}{252} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {5}{2}} x + \frac {907}{2520} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {5}{2}} - \frac {91}{576} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}} x + \frac {91}{3456} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}} - \frac {2093}{4608} \, \sqrt {3 \, x^{2} - x + 2} x - \frac {48139}{165888} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (6 \, x - 1\right )}\right ) + \frac {2093}{27648} \, \sqrt {3 \, x^{2} - x + 2} \]
2/3*(3*x^2 - x + 2)^(5/2)*x^3 + 95/63*(3*x^2 - x + 2)^(5/2)*x^2 + 319/252* (3*x^2 - x + 2)^(5/2)*x + 907/2520*(3*x^2 - x + 2)^(5/2) - 91/576*(3*x^2 - x + 2)^(3/2)*x + 91/3456*(3*x^2 - x + 2)^(3/2) - 2093/4608*sqrt(3*x^2 - x + 2)*x - 48139/165888*sqrt(3)*arcsinh(1/23*sqrt(23)*(6*x - 1)) + 2093/276 48*sqrt(3*x^2 - x + 2)
Time = 0.29 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.59 \[ \int (1+2 x)^2 \left (2-x+3 x^2\right )^{3/2} \left (1+3 x+4 x^2\right ) \, dx=\frac {1}{967680} \, {\left (2 \, {\left (12 \, {\left (2 \, {\left (8 \, {\left (30 \, {\left (12 \, {\left (42 \, x + 67\right )} x + 925\right )} x + 31703\right )} x + 210829\right )} x + 237251\right )} x + 1367959\right )} x + 1517367\right )} \sqrt {3 \, x^{2} - x + 2} + \frac {48139}{165888} \, \sqrt {3} \log \left (-2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - x + 2}\right )} + 1\right ) \]
1/967680*(2*(12*(2*(8*(30*(12*(42*x + 67)*x + 925)*x + 31703)*x + 210829)* x + 237251)*x + 1367959)*x + 1517367)*sqrt(3*x^2 - x + 2) + 48139/165888*s qrt(3)*log(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 - x + 2)) + 1)
Timed out. \[ \int (1+2 x)^2 \left (2-x+3 x^2\right )^{3/2} \left (1+3 x+4 x^2\right ) \, dx=\int {\left (2\,x+1\right )}^2\,{\left (3\,x^2-x+2\right )}^{3/2}\,\left (4\,x^2+3\,x+1\right ) \,d x \]