Integrand size = 32, antiderivative size = 143 \[ \int (1+2 x)^3 \sqrt {2-x+3 x^2} \left (1+3 x+4 x^2\right ) \, dx=\frac {5393 (1-6 x) \sqrt {2-x+3 x^2}}{15552}+\frac {17}{105} (1+2 x)^2 \left (2-x+3 x^2\right )^{3/2}+\frac {67}{378} (1+2 x)^3 \left (2-x+3 x^2\right )^{3/2}+\frac {2}{21} (1+2 x)^4 \left (2-x+3 x^2\right )^{3/2}-\frac {(75295+26982 x) \left (2-x+3 x^2\right )^{3/2}}{68040}+\frac {124039 \text {arcsinh}\left (\frac {1-6 x}{\sqrt {23}}\right )}{31104 \sqrt {3}} \]
17/105*(1+2*x)^2*(3*x^2-x+2)^(3/2)+67/378*(1+2*x)^3*(3*x^2-x+2)^(3/2)+2/21 *(1+2*x)^4*(3*x^2-x+2)^(3/2)-1/68040*(75295+26982*x)*(3*x^2-x+2)^(3/2)+124 039/93312*arcsinh(1/23*(1-6*x)*23^(1/2))*3^(1/2)+5393/15552*(1-6*x)*(3*x^2 -x+2)^(1/2)
Time = 0.42 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.56 \[ \int (1+2 x)^3 \sqrt {2-x+3 x^2} \left (1+3 x+4 x^2\right ) \, dx=\frac {6 \sqrt {2-x+3 x^2} \left (-543069+1493894 x+3280872 x^2+5497776 x^3+7491456 x^4+6462720 x^5+2488320 x^6\right )+4341365 \sqrt {3} \log \left (1-6 x+2 \sqrt {6-3 x+9 x^2}\right )}{3265920} \]
(6*Sqrt[2 - x + 3*x^2]*(-543069 + 1493894*x + 3280872*x^2 + 5497776*x^3 + 7491456*x^4 + 6462720*x^5 + 2488320*x^6) + 4341365*Sqrt[3]*Log[1 - 6*x + 2 *Sqrt[6 - 3*x + 9*x^2]])/3265920
Time = 0.38 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.14, 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, 1236, 27, 1225, 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.
| \(\displaystyle \int (2 x+1)^3 \sqrt {3 x^2-x+2} \left (4 x^2+3 x+1\right ) \, dx\) |
| \(\Big \downarrow \) 2184 |
| \(\displaystyle \frac {1}{84} \int -4 (8-67 x) (2 x+1)^3 \sqrt {3 x^2-x+2}dx+\frac {2}{21} \left (3 x^2-x+2\right )^{3/2} (2 x+1)^4\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{21} (2 x+1)^4 \left (3 x^2-x+2\right )^{3/2}-\frac {1}{21} \int (8-67 x) (2 x+1)^3 \sqrt {3 x^2-x+2}dx\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {1}{21} \left (\frac {67}{18} (2 x+1)^3 \left (3 x^2-x+2\right )^{3/2}-\frac {1}{18} \int \frac {3}{2} (565-612 x) (2 x+1)^2 \sqrt {3 x^2-x+2}dx\right )+\frac {2}{21} \left (3 x^2-x+2\right )^{3/2} (2 x+1)^4\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{21} \left (\frac {67}{18} (2 x+1)^3 \left (3 x^2-x+2\right )^{3/2}-\frac {1}{12} \int (565-612 x) (2 x+1)^2 \sqrt {3 x^2-x+2}dx\right )+\frac {2}{21} \left (3 x^2-x+2\right )^{3/2} (2 x+1)^4\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {1}{21} \left (\frac {1}{12} \left (\frac {204}{5} (2 x+1)^2 \left (3 x^2-x+2\right )^{3/2}-\frac {1}{15} \int 3 (2 x+1) (2998 x+4151) \sqrt {3 x^2-x+2}dx\right )+\frac {67}{18} \left (3 x^2-x+2\right )^{3/2} (2 x+1)^3\right )+\frac {2}{21} \left (3 x^2-x+2\right )^{3/2} (2 x+1)^4\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{21} \left (\frac {1}{12} \left (\frac {204}{5} (2 x+1)^2 \left (3 x^2-x+2\right )^{3/2}-\frac {1}{5} \int (2 x+1) (2998 x+4151) \sqrt {3 x^2-x+2}dx\right )+\frac {67}{18} \left (3 x^2-x+2\right )^{3/2} (2 x+1)^3\right )+\frac {2}{21} \left (3 x^2-x+2\right )^{3/2} (2 x+1)^4\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle \frac {1}{21} \left (\frac {1}{12} \left (\frac {1}{5} \left (-\frac {188755}{36} \int \sqrt {3 x^2-x+2}dx-\frac {1}{54} (26982 x+75295) \left (3 x^2-x+2\right )^{3/2}\right )+\frac {204}{5} \left (3 x^2-x+2\right )^{3/2} (2 x+1)^2\right )+\frac {67}{18} \left (3 x^2-x+2\right )^{3/2} (2 x+1)^3\right )+\frac {2}{21} \left (3 x^2-x+2\right )^{3/2} (2 x+1)^4\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{21} \left (\frac {1}{12} \left (\frac {1}{5} \left (-\frac {188755}{36} \left (\frac {23}{24} \int \frac {1}{\sqrt {3 x^2-x+2}}dx-\frac {1}{12} (1-6 x) \sqrt {3 x^2-x+2}\right )-\frac {1}{54} (26982 x+75295) \left (3 x^2-x+2\right )^{3/2}\right )+\frac {204}{5} \left (3 x^2-x+2\right )^{3/2} (2 x+1)^2\right )+\frac {67}{18} \left (3 x^2-x+2\right )^{3/2} (2 x+1)^3\right )+\frac {2}{21} \left (3 x^2-x+2\right )^{3/2} (2 x+1)^4\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle \frac {1}{21} \left (\frac {1}{12} \left (\frac {1}{5} \left (-\frac {188755}{36} \left (\frac {1}{24} \sqrt {\frac {23}{3}} \int \frac {1}{\sqrt {\frac {1}{23} (6 x-1)^2+1}}d(6 x-1)-\frac {1}{12} (1-6 x) \sqrt {3 x^2-x+2}\right )-\frac {1}{54} (26982 x+75295) \left (3 x^2-x+2\right )^{3/2}\right )+\frac {204}{5} \left (3 x^2-x+2\right )^{3/2} (2 x+1)^2\right )+\frac {67}{18} \left (3 x^2-x+2\right )^{3/2} (2 x+1)^3\right )+\frac {2}{21} \left (3 x^2-x+2\right )^{3/2} (2 x+1)^4\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {1}{21} \left (\frac {1}{12} \left (\frac {1}{5} \left (-\frac {188755}{36} \left (\frac {23 \text {arcsinh}\left (\frac {6 x-1}{\sqrt {23}}\right )}{24 \sqrt {3}}-\frac {1}{12} (1-6 x) \sqrt {3 x^2-x+2}\right )-\frac {1}{54} (26982 x+75295) \left (3 x^2-x+2\right )^{3/2}\right )+\frac {204}{5} \left (3 x^2-x+2\right )^{3/2} (2 x+1)^2\right )+\frac {67}{18} \left (3 x^2-x+2\right )^{3/2} (2 x+1)^3\right )+\frac {2}{21} \left (3 x^2-x+2\right )^{3/2} (2 x+1)^4\) |
(2*(1 + 2*x)^4*(2 - x + 3*x^2)^(3/2))/21 + ((67*(1 + 2*x)^3*(2 - x + 3*x^2 )^(3/2))/18 + ((204*(1 + 2*x)^2*(2 - x + 3*x^2)^(3/2))/5 + (-1/54*((75295 + 26982*x)*(2 - x + 3*x^2)^(3/2)) - (188755*(-1/12*((1 - 6*x)*Sqrt[2 - x + 3*x^2]) + (23*ArcSinh[(-1 + 6*x)/Sqrt[23]])/(24*Sqrt[3])))/36)/5)/12)/21
3.3.8.3.1 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 = 0.70 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.42
| method | result | size |
| risch | \(\frac {\left (2488320 x^{6}+6462720 x^{5}+7491456 x^{4}+5497776 x^{3}+3280872 x^{2}+1493894 x -543069\right ) \sqrt {3 x^{2}-x +2}}{544320}-\frac {124039 \sqrt {3}\, \operatorname {arcsinh}\left (\frac {6 \sqrt {23}\, \left (x -\frac {1}{6}\right )}{23}\right )}{93312}\) | \(60\) |
| trager | \(\left (\frac {32}{7} x^{6}+\frac {748}{63} x^{5}+\frac {1858}{135} x^{4}+\frac {38179}{3780} x^{3}+\frac {19529}{3240} x^{2}+\frac {746947}{272160} x -\frac {60341}{60480}\right ) \sqrt {3 x^{2}-x +2}+\frac {124039 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+6 \sqrt {3 x^{2}-x +2}\right )}{93312}\) | \(84\) |
| default | \(-\frac {5393 \left (-1+6 x \right ) \sqrt {3 x^{2}-x +2}}{15552}-\frac {124039 \sqrt {3}\, \operatorname {arcsinh}\left (\frac {6 \sqrt {23}\, \left (x -\frac {1}{6}\right )}{23}\right )}{93312}-\frac {45739 \left (3 x^{2}-x +2\right )^{\frac {3}{2}}}{68040}+\frac {32 x^{4} \left (3 x^{2}-x +2\right )^{\frac {3}{2}}}{21}+\frac {844 x^{3} \left (3 x^{2}-x +2\right )^{\frac {3}{2}}}{189}+\frac {1594 x^{2} \left (3 x^{2}-x +2\right )^{\frac {3}{2}}}{315}+\frac {7849 x \left (3 x^{2}-x +2\right )^{\frac {3}{2}}}{3780}\) | \(115\) |
1/544320*(2488320*x^6+6462720*x^5+7491456*x^4+5497776*x^3+3280872*x^2+1493 894*x-543069)*(3*x^2-x+2)^(1/2)-124039/93312*3^(1/2)*arcsinh(6/23*23^(1/2) *(x-1/6))
Time = 0.34 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.58 \[ \int (1+2 x)^3 \sqrt {2-x+3 x^2} \left (1+3 x+4 x^2\right ) \, dx=\frac {1}{544320} \, {\left (2488320 \, x^{6} + 6462720 \, x^{5} + 7491456 \, x^{4} + 5497776 \, x^{3} + 3280872 \, x^{2} + 1493894 \, x - 543069\right )} \sqrt {3 \, x^{2} - x + 2} + \frac {124039}{186624} \, \sqrt {3} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} - x + 2} {\left (6 \, x - 1\right )} - 72 \, x^{2} + 24 \, x - 25\right ) \]
1/544320*(2488320*x^6 + 6462720*x^5 + 7491456*x^4 + 5497776*x^3 + 3280872* x^2 + 1493894*x - 543069)*sqrt(3*x^2 - x + 2) + 124039/186624*sqrt(3)*log( 4*sqrt(3)*sqrt(3*x^2 - x + 2)*(6*x - 1) - 72*x^2 + 24*x - 25)
Time = 0.48 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.53 \[ \int (1+2 x)^3 \sqrt {2-x+3 x^2} \left (1+3 x+4 x^2\right ) \, dx=\sqrt {3 x^{2} - x + 2} \cdot \left (\frac {32 x^{6}}{7} + \frac {748 x^{5}}{63} + \frac {1858 x^{4}}{135} + \frac {38179 x^{3}}{3780} + \frac {19529 x^{2}}{3240} + \frac {746947 x}{272160} - \frac {60341}{60480}\right ) - \frac {124039 \sqrt {3} \operatorname {asinh}{\left (\frac {6 \sqrt {23} \left (x - \frac {1}{6}\right )}{23} \right )}}{93312} \]
sqrt(3*x**2 - x + 2)*(32*x**6/7 + 748*x**5/63 + 1858*x**4/135 + 38179*x**3 /3780 + 19529*x**2/3240 + 746947*x/272160 - 60341/60480) - 124039*sqrt(3)* asinh(6*sqrt(23)*(x - 1/6)/23)/93312
Time = 0.27 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.88 \[ \int (1+2 x)^3 \sqrt {2-x+3 x^2} \left (1+3 x+4 x^2\right ) \, dx=\frac {32}{21} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}} x^{4} + \frac {844}{189} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}} x^{3} + \frac {1594}{315} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}} x^{2} + \frac {7849}{3780} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}} x - \frac {45739}{68040} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}} - \frac {5393}{2592} \, \sqrt {3 \, x^{2} - x + 2} x - \frac {124039}{93312} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (6 \, x - 1\right )}\right ) + \frac {5393}{15552} \, \sqrt {3 \, x^{2} - x + 2} \]
32/21*(3*x^2 - x + 2)^(3/2)*x^4 + 844/189*(3*x^2 - x + 2)^(3/2)*x^3 + 1594 /315*(3*x^2 - x + 2)^(3/2)*x^2 + 7849/3780*(3*x^2 - x + 2)^(3/2)*x - 45739 /68040*(3*x^2 - x + 2)^(3/2) - 5393/2592*sqrt(3*x^2 - x + 2)*x - 124039/93 312*sqrt(3)*arcsinh(1/23*sqrt(23)*(6*x - 1)) + 5393/15552*sqrt(3*x^2 - x + 2)
Time = 0.28 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.55 \[ \int (1+2 x)^3 \sqrt {2-x+3 x^2} \left (1+3 x+4 x^2\right ) \, dx=\frac {1}{544320} \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (8 \, {\left (30 \, {\left (72 \, x + 187\right )} x + 6503\right )} x + 38179\right )} x + 136703\right )} x + 746947\right )} x - 543069\right )} \sqrt {3 \, x^{2} - x + 2} + \frac {124039}{93312} \, \sqrt {3} \log \left (-2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - x + 2}\right )} + 1\right ) \]
1/544320*(2*(12*(6*(8*(30*(72*x + 187)*x + 6503)*x + 38179)*x + 136703)*x + 746947)*x - 543069)*sqrt(3*x^2 - x + 2) + 124039/93312*sqrt(3)*log(-2*sq rt(3)*(sqrt(3)*x - sqrt(3*x^2 - x + 2)) + 1)
Time = 14.57 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.19 \[ \int (1+2 x)^3 \sqrt {2-x+3 x^2} \left (1+3 x+4 x^2\right ) \, dx=\frac {1594\,x^2\,{\left (3\,x^2-x+2\right )}^{3/2}}{315}+\frac {844\,x^3\,{\left (3\,x^2-x+2\right )}^{3/2}}{189}+\frac {32\,x^4\,{\left (3\,x^2-x+2\right )}^{3/2}}{21}-\frac {137057\,\sqrt {3}\,\ln \left (\sqrt {3\,x^2-x+2}+\frac {\sqrt {3}\,\left (3\,x-\frac {1}{2}\right )}{3}\right )}{136080}-\frac {5959\,\left (\frac {x}{2}-\frac {1}{12}\right )\,\sqrt {3\,x^2-x+2}}{1890}-\frac {45739\,\sqrt {3\,x^2-x+2}\,\left (72\,x^2-6\,x+45\right )}{1632960}+\frac {7849\,x\,{\left (3\,x^2-x+2\right )}^{3/2}}{3780}-\frac {1051997\,\sqrt {3}\,\ln \left (2\,\sqrt {3\,x^2-x+2}+\frac {\sqrt {3}\,\left (6\,x-1\right )}{3}\right )}{3265920} \]
(1594*x^2*(3*x^2 - x + 2)^(3/2))/315 + (844*x^3*(3*x^2 - x + 2)^(3/2))/189 + (32*x^4*(3*x^2 - x + 2)^(3/2))/21 - (137057*3^(1/2)*log((3*x^2 - x + 2) ^(1/2) + (3^(1/2)*(3*x - 1/2))/3))/136080 - (5959*(x/2 - 1/12)*(3*x^2 - x + 2)^(1/2))/1890 - (45739*(3*x^2 - x + 2)^(1/2)*(72*x^2 - 6*x + 45))/16329 60 + (7849*x*(3*x^2 - x + 2)^(3/2))/3780 - (1051997*3^(1/2)*log(2*(3*x^2 - x + 2)^(1/2) + (3^(1/2)*(6*x - 1))/3))/3265920