Integrand size = 32, antiderivative size = 124 \[ \int \frac {\left (2-x+3 x^2\right )^{3/2} \left (1+3 x+4 x^2\right )}{1+2 x} \, dx=\frac {(869+402 x) \sqrt {2-x+3 x^2}}{1152}+\frac {1}{144} (7+30 x) \left (2-x+3 x^2\right )^{3/2}+\frac {2}{15} \left (2-x+3 x^2\right )^{5/2}+\frac {2203 \text {arcsinh}\left (\frac {1-6 x}{\sqrt {23}}\right )}{2304 \sqrt {3}}-\frac {13}{32} \sqrt {13} \text {arctanh}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {2-x+3 x^2}}\right ) \]
1/144*(7+30*x)*(3*x^2-x+2)^(3/2)+2/15*(3*x^2-x+2)^(5/2)+2203/6912*arcsinh( 1/23*(1-6*x)*23^(1/2))*3^(1/2)-13/32*arctanh(1/26*(9-8*x)*13^(1/2)/(3*x^2- x+2)^(1/2))*13^(1/2)+1/1152*(869+402*x)*(3*x^2-x+2)^(1/2)
Time = 0.38 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.92 \[ \int \frac {\left (2-x+3 x^2\right )^{3/2} \left (1+3 x+4 x^2\right )}{1+2 x} \, dx=\frac {6 \sqrt {2-x+3 x^2} \left (7977+1058 x+9624 x^2-1008 x^3+6912 x^4\right )+28080 \sqrt {13} \text {arctanh}\left (\frac {\sqrt {3}+2 \sqrt {3} x-2 \sqrt {2-x+3 x^2}}{\sqrt {13}}\right )+11015 \sqrt {3} \log \left (1-6 x+2 \sqrt {6-3 x+9 x^2}\right )}{34560} \]
(6*Sqrt[2 - x + 3*x^2]*(7977 + 1058*x + 9624*x^2 - 1008*x^3 + 6912*x^4) + 28080*Sqrt[13]*ArcTanh[(Sqrt[3] + 2*Sqrt[3]*x - 2*Sqrt[2 - x + 3*x^2])/Sqr t[13]] + 11015*Sqrt[3]*Log[1 - 6*x + 2*Sqrt[6 - 3*x + 9*x^2]])/34560
Time = 0.39 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.09, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.344, Rules used = {2184, 27, 1231, 27, 1231, 27, 1269, 1090, 222, 1154, 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.
| \(\displaystyle \int \frac {\left (3 x^2-x+2\right )^{3/2} \left (4 x^2+3 x+1\right )}{2 x+1} \, dx\) |
| \(\Big \downarrow \) 2184 |
| \(\displaystyle \frac {1}{60} \int \frac {20 (5 x+4) \left (3 x^2-x+2\right )^{3/2}}{2 x+1}dx+\frac {2}{15} \left (3 x^2-x+2\right )^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \int \frac {(5 x+4) \left (3 x^2-x+2\right )^{3/2}}{2 x+1}dx+\frac {2}{15} \left (3 x^2-x+2\right )^{5/2}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{48} (30 x+7) \left (3 x^2-x+2\right )^{3/2}-\frac {1}{96} \int -\frac {3 (134 x+223) \sqrt {3 x^2-x+2}}{2 x+1}dx\right )+\frac {2}{15} \left (3 x^2-x+2\right )^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{32} \int \frac {(134 x+223) \sqrt {3 x^2-x+2}}{2 x+1}dx+\frac {1}{48} (30 x+7) \left (3 x^2-x+2\right )^{3/2}\right )+\frac {2}{15} \left (3 x^2-x+2\right )^{5/2}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{32} \left (\frac {1}{12} (402 x+869) \sqrt {3 x^2-x+2}-\frac {1}{48} \int -\frac {2 (9965-4406 x)}{(2 x+1) \sqrt {3 x^2-x+2}}dx\right )+\frac {1}{48} (30 x+7) \left (3 x^2-x+2\right )^{3/2}\right )+\frac {2}{15} \left (3 x^2-x+2\right )^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{32} \left (\frac {1}{24} \int \frac {9965-4406 x}{(2 x+1) \sqrt {3 x^2-x+2}}dx+\frac {1}{12} \sqrt {3 x^2-x+2} (402 x+869)\right )+\frac {1}{48} (30 x+7) \left (3 x^2-x+2\right )^{3/2}\right )+\frac {2}{15} \left (3 x^2-x+2\right )^{5/2}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{32} \left (\frac {1}{24} \left (12168 \int \frac {1}{(2 x+1) \sqrt {3 x^2-x+2}}dx-2203 \int \frac {1}{\sqrt {3 x^2-x+2}}dx\right )+\frac {1}{12} \sqrt {3 x^2-x+2} (402 x+869)\right )+\frac {1}{48} (30 x+7) \left (3 x^2-x+2\right )^{3/2}\right )+\frac {2}{15} \left (3 x^2-x+2\right )^{5/2}\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{32} \left (\frac {1}{24} \left (12168 \int \frac {1}{(2 x+1) \sqrt {3 x^2-x+2}}dx-\frac {2203 \int \frac {1}{\sqrt {\frac {1}{23} (6 x-1)^2+1}}d(6 x-1)}{\sqrt {69}}\right )+\frac {1}{12} \sqrt {3 x^2-x+2} (402 x+869)\right )+\frac {1}{48} (30 x+7) \left (3 x^2-x+2\right )^{3/2}\right )+\frac {2}{15} \left (3 x^2-x+2\right )^{5/2}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{32} \left (\frac {1}{24} \left (12168 \int \frac {1}{(2 x+1) \sqrt {3 x^2-x+2}}dx-\frac {2203 \text {arcsinh}\left (\frac {6 x-1}{\sqrt {23}}\right )}{\sqrt {3}}\right )+\frac {1}{12} \sqrt {3 x^2-x+2} (402 x+869)\right )+\frac {1}{48} (30 x+7) \left (3 x^2-x+2\right )^{3/2}\right )+\frac {2}{15} \left (3 x^2-x+2\right )^{5/2}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{32} \left (\frac {1}{24} \left (-24336 \int \frac {1}{52-\frac {(9-8 x)^2}{3 x^2-x+2}}d\frac {9-8 x}{\sqrt {3 x^2-x+2}}-\frac {2203 \text {arcsinh}\left (\frac {6 x-1}{\sqrt {23}}\right )}{\sqrt {3}}\right )+\frac {1}{12} \sqrt {3 x^2-x+2} (402 x+869)\right )+\frac {1}{48} (30 x+7) \left (3 x^2-x+2\right )^{3/2}\right )+\frac {2}{15} \left (3 x^2-x+2\right )^{5/2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{32} \left (\frac {1}{24} \left (-\frac {2203 \text {arcsinh}\left (\frac {6 x-1}{\sqrt {23}}\right )}{\sqrt {3}}-936 \sqrt {13} \text {arctanh}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {3 x^2-x+2}}\right )\right )+\frac {1}{12} \sqrt {3 x^2-x+2} (402 x+869)\right )+\frac {1}{48} (30 x+7) \left (3 x^2-x+2\right )^{3/2}\right )+\frac {2}{15} \left (3 x^2-x+2\right )^{5/2}\) |
(2*(2 - x + 3*x^2)^(5/2))/15 + (((7 + 30*x)*(2 - x + 3*x^2)^(3/2))/48 + (( (869 + 402*x)*Sqrt[2 - x + 3*x^2])/12 + ((-2203*ArcSinh[(-1 + 6*x)/Sqrt[23 ]])/Sqrt[3] - 936*Sqrt[13]*ArcTanh[(9 - 8*x)/(2*Sqrt[13]*Sqrt[2 - x + 3*x^ 2])])/24)/32)/3
3.3.17.3.1 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[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[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[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ (c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)) Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c ^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x ] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || !R ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Integer Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 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 = 0.66 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.65
| method | result | size |
| risch | \(\frac {\left (6912 x^{4}-1008 x^{3}+9624 x^{2}+1058 x +7977\right ) \sqrt {3 x^{2}-x +2}}{5760}-\frac {2203 \sqrt {3}\, \operatorname {arcsinh}\left (\frac {6 \sqrt {23}\, \left (x -\frac {1}{6}\right )}{23}\right )}{6912}-\frac {13 \sqrt {13}\, \operatorname {arctanh}\left (\frac {2 \left (\frac {9}{2}-4 x \right ) \sqrt {13}}{13 \sqrt {12 \left (x +\frac {1}{2}\right )^{2}-16 x +5}}\right )}{32}\) | \(80\) |
| trager | \(\left (\frac {6}{5} x^{4}-\frac {7}{40} x^{3}+\frac {401}{240} x^{2}+\frac {529}{2880} x +\frac {2659}{1920}\right ) \sqrt {3 x^{2}-x +2}+\frac {13 \operatorname {RootOf}\left (\textit {\_Z}^{2}-13\right ) \ln \left (\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-13\right ) x +26 \sqrt {3 x^{2}-x +2}-9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-13\right )}{1+2 x}\right )}{32}+\frac {2203 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+6 \sqrt {3 x^{2}-x +2}\right )}{6912}\) | \(123\) |
| default | \(\frac {5 \left (-1+6 x \right ) \left (3 x^{2}-x +2\right )^{\frac {3}{2}}}{144}+\frac {115 \left (-1+6 x \right ) \sqrt {3 x^{2}-x +2}}{1152}-\frac {2203 \sqrt {3}\, \operatorname {arcsinh}\left (\frac {6 \sqrt {23}\, \left (x -\frac {1}{6}\right )}{23}\right )}{6912}+\frac {2 \left (3 x^{2}-x +2\right )^{\frac {5}{2}}}{15}+\frac {\left (3 \left (x +\frac {1}{2}\right )^{2}-4 x +\frac {5}{4}\right )^{\frac {3}{2}}}{12}-\frac {\left (-1+6 x \right ) \sqrt {3 \left (x +\frac {1}{2}\right )^{2}-4 x +\frac {5}{4}}}{24}+\frac {13 \sqrt {12 \left (x +\frac {1}{2}\right )^{2}-16 x +5}}{32}-\frac {13 \sqrt {13}\, \operatorname {arctanh}\left (\frac {2 \left (\frac {9}{2}-4 x \right ) \sqrt {13}}{13 \sqrt {12 \left (x +\frac {1}{2}\right )^{2}-16 x +5}}\right )}{32}\) | \(151\) |
1/5760*(6912*x^4-1008*x^3+9624*x^2+1058*x+7977)*(3*x^2-x+2)^(1/2)-2203/691 2*3^(1/2)*arcsinh(6/23*23^(1/2)*(x-1/6))-13/32*13^(1/2)*arctanh(2/13*(9/2- 4*x)*13^(1/2)/(12*(x+1/2)^2-16*x+5)^(1/2))
Time = 0.27 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.01 \[ \int \frac {\left (2-x+3 x^2\right )^{3/2} \left (1+3 x+4 x^2\right )}{1+2 x} \, dx=\frac {1}{5760} \, {\left (6912 \, x^{4} - 1008 \, x^{3} + 9624 \, x^{2} + 1058 \, x + 7977\right )} \sqrt {3 \, x^{2} - x + 2} + \frac {2203}{13824} \, \sqrt {3} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} - x + 2} {\left (6 \, x - 1\right )} - 72 \, x^{2} + 24 \, x - 25\right ) + \frac {13}{64} \, \sqrt {13} \log \left (-\frac {4 \, \sqrt {13} \sqrt {3 \, x^{2} - x + 2} {\left (8 \, x - 9\right )} + 220 \, x^{2} - 196 \, x + 185}{4 \, x^{2} + 4 \, x + 1}\right ) \]
1/5760*(6912*x^4 - 1008*x^3 + 9624*x^2 + 1058*x + 7977)*sqrt(3*x^2 - x + 2 ) + 2203/13824*sqrt(3)*log(4*sqrt(3)*sqrt(3*x^2 - x + 2)*(6*x - 1) - 72*x^ 2 + 24*x - 25) + 13/64*sqrt(13)*log(-(4*sqrt(13)*sqrt(3*x^2 - x + 2)*(8*x - 9) + 220*x^2 - 196*x + 185)/(4*x^2 + 4*x + 1))
\[ \int \frac {\left (2-x+3 x^2\right )^{3/2} \left (1+3 x+4 x^2\right )}{1+2 x} \, dx=\int \frac {\left (3 x^{2} - x + 2\right )^{\frac {3}{2}} \cdot \left (4 x^{2} + 3 x + 1\right )}{2 x + 1}\, dx \]
Time = 0.29 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.01 \[ \int \frac {\left (2-x+3 x^2\right )^{3/2} \left (1+3 x+4 x^2\right )}{1+2 x} \, dx=\frac {2}{15} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {5}{2}} + \frac {5}{24} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}} x + \frac {7}{144} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}} + \frac {67}{192} \, \sqrt {3 \, x^{2} - x + 2} x - \frac {2203}{6912} \, \sqrt {3} \operatorname {arsinh}\left (\frac {6}{23} \, \sqrt {23} x - \frac {1}{23} \, \sqrt {23}\right ) + \frac {13}{32} \, \sqrt {13} \operatorname {arsinh}\left (\frac {8 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 1 \right |}} - \frac {9 \, \sqrt {23}}{23 \, {\left | 2 \, x + 1 \right |}}\right ) + \frac {869}{1152} \, \sqrt {3 \, x^{2} - x + 2} \]
2/15*(3*x^2 - x + 2)^(5/2) + 5/24*(3*x^2 - x + 2)^(3/2)*x + 7/144*(3*x^2 - x + 2)^(3/2) + 67/192*sqrt(3*x^2 - x + 2)*x - 2203/6912*sqrt(3)*arcsinh(6 /23*sqrt(23)*x - 1/23*sqrt(23)) + 13/32*sqrt(13)*arcsinh(8/23*sqrt(23)*x/a bs(2*x + 1) - 9/23*sqrt(23)/abs(2*x + 1)) + 869/1152*sqrt(3*x^2 - x + 2)
Time = 0.39 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.10 \[ \int \frac {\left (2-x+3 x^2\right )^{3/2} \left (1+3 x+4 x^2\right )}{1+2 x} \, dx=\frac {1}{5760} \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (48 \, x - 7\right )} x + 401\right )} x + 529\right )} x + 7977\right )} \sqrt {3 \, x^{2} - x + 2} + \frac {2203}{6912} \, \sqrt {3} \log \left (-6 \, \sqrt {3} x + \sqrt {3} + 6 \, \sqrt {3 \, x^{2} - x + 2}\right ) + \frac {13}{32} \, \sqrt {13} \log \left (-\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {13} - 2 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} - x + 2} \right |}}{2 \, {\left (2 \, \sqrt {3} x - \sqrt {13} + \sqrt {3} - 2 \, \sqrt {3 \, x^{2} - x + 2}\right )}}\right ) \]
1/5760*(2*(12*(6*(48*x - 7)*x + 401)*x + 529)*x + 7977)*sqrt(3*x^2 - x + 2 ) + 2203/6912*sqrt(3)*log(-6*sqrt(3)*x + sqrt(3) + 6*sqrt(3*x^2 - x + 2)) + 13/32*sqrt(13)*log(-1/2*abs(-4*sqrt(3)*x - 2*sqrt(13) - 2*sqrt(3) + 4*sq rt(3*x^2 - x + 2))/(2*sqrt(3)*x - sqrt(13) + sqrt(3) - 2*sqrt(3*x^2 - x + 2)))
Timed out. \[ \int \frac {\left (2-x+3 x^2\right )^{3/2} \left (1+3 x+4 x^2\right )}{1+2 x} \, dx=\int \frac {{\left (3\,x^2-x+2\right )}^{3/2}\,\left (4\,x^2+3\,x+1\right )}{2\,x+1} \,d x \]