Integrand size = 33, antiderivative size = 147 \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right ) \, dx=-\frac {593193 (1-4 x) \sqrt {3-x+2 x^2}}{1048576}-\frac {8597 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{65536}+\frac {1167 \left (3-x+2 x^2\right )^{5/2}}{14336}+\frac {125 x \left (3-x+2 x^2\right )^{5/2}}{3584}+\frac {23}{448} x^2 \left (3-x+2 x^2\right )^{5/2}+\frac {5}{16} x^3 \left (3-x+2 x^2\right )^{5/2}-\frac {13643439 \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )}{2097152 \sqrt {2}} \]
-8597/65536*(1-4*x)*(2*x^2-x+3)^(3/2)+1167/14336*(2*x^2-x+3)^(5/2)+125/358 4*x*(2*x^2-x+3)^(5/2)+23/448*x^2*(2*x^2-x+3)^(5/2)+5/16*x^3*(2*x^2-x+3)^(5 /2)-13643439/4194304*arcsinh(1/23*(1-4*x)*23^(1/2))*2^(1/2)-593193/1048576 *(1-4*x)*(2*x^2-x+3)^(1/2)
Time = 0.57 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.58 \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right ) \, dx=\frac {4 \sqrt {3-x+2 x^2} \left (-1663407+27845612 x+3845856 x^2+27023744 x^3-7497728 x^4+29335552 x^5-7667712 x^6+9175040 x^7\right )-95504073 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{29360128} \]
(4*Sqrt[3 - x + 2*x^2]*(-1663407 + 27845612*x + 3845856*x^2 + 27023744*x^3 - 7497728*x^4 + 29335552*x^5 - 7667712*x^6 + 9175040*x^7) - 95504073*Sqrt [2]*Log[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]])/29360128
Time = 0.40 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.17, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2192, 27, 2192, 27, 2192, 27, 1160, 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.
| \(\displaystyle \int \left (2 x^2-x+3\right )^{3/2} \left (5 x^4-x^3+3 x^2+x+2\right ) \, dx\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{16} \int \frac {1}{2} \left (2 x^2-x+3\right )^{3/2} \left (23 x^3+6 x^2+32 x+64\right )dx+\frac {5}{16} \left (2 x^2-x+3\right )^{5/2} x^3\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{32} \int \left (2 x^2-x+3\right )^{3/2} \left (23 x^3+6 x^2+32 x+64\right )dx+\frac {5}{16} \left (2 x^2-x+3\right )^{5/2} x^3\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{14} \int \frac {1}{2} \left (2 x^2-x+3\right )^{3/2} \left (375 x^2+620 x+1792\right )dx+\frac {23}{14} x^2 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {5}{16} \left (2 x^2-x+3\right )^{5/2} x^3\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{28} \int \left (2 x^2-x+3\right )^{3/2} \left (375 x^2+620 x+1792\right )dx+\frac {23}{14} x^2 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {5}{16} \left (2 x^2-x+3\right )^{5/2} x^3\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{28} \left (\frac {1}{12} \int \frac {3}{2} (5835 x+13586) \left (2 x^2-x+3\right )^{3/2}dx+\frac {125}{4} x \left (2 x^2-x+3\right )^{5/2}\right )+\frac {23}{14} x^2 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {5}{16} \left (2 x^2-x+3\right )^{5/2} x^3\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{28} \left (\frac {1}{8} \int (5835 x+13586) \left (2 x^2-x+3\right )^{3/2}dx+\frac {125}{4} x \left (2 x^2-x+3\right )^{5/2}\right )+\frac {23}{14} x^2 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {5}{16} \left (2 x^2-x+3\right )^{5/2} x^3\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{28} \left (\frac {1}{8} \left (\frac {60179}{4} \int \left (2 x^2-x+3\right )^{3/2}dx+\frac {1167}{2} \left (2 x^2-x+3\right )^{5/2}\right )+\frac {125}{4} x \left (2 x^2-x+3\right )^{5/2}\right )+\frac {23}{14} x^2 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {5}{16} \left (2 x^2-x+3\right )^{5/2} x^3\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{28} \left (\frac {1}{8} \left (\frac {60179}{4} \left (\frac {69}{32} \int \sqrt {2 x^2-x+3}dx-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )+\frac {1167}{2} \left (2 x^2-x+3\right )^{5/2}\right )+\frac {125}{4} x \left (2 x^2-x+3\right )^{5/2}\right )+\frac {23}{14} x^2 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {5}{16} \left (2 x^2-x+3\right )^{5/2} x^3\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{28} \left (\frac {1}{8} \left (\frac {60179}{4} \left (\frac {69}{32} \left (\frac {23}{16} \int \frac {1}{\sqrt {2 x^2-x+3}}dx-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )+\frac {1167}{2} \left (2 x^2-x+3\right )^{5/2}\right )+\frac {125}{4} x \left (2 x^2-x+3\right )^{5/2}\right )+\frac {23}{14} x^2 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {5}{16} \left (2 x^2-x+3\right )^{5/2} x^3\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{28} \left (\frac {1}{8} \left (\frac {60179}{4} \left (\frac {69}{32} \left (\frac {1}{16} \sqrt {\frac {23}{2}} \int \frac {1}{\sqrt {\frac {1}{23} (4 x-1)^2+1}}d(4 x-1)-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )+\frac {1167}{2} \left (2 x^2-x+3\right )^{5/2}\right )+\frac {125}{4} x \left (2 x^2-x+3\right )^{5/2}\right )+\frac {23}{14} x^2 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {5}{16} \left (2 x^2-x+3\right )^{5/2} x^3\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{28} \left (\frac {1}{8} \left (\frac {60179}{4} \left (\frac {69}{32} \left (\frac {23 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{16 \sqrt {2}}-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )+\frac {1167}{2} \left (2 x^2-x+3\right )^{5/2}\right )+\frac {125}{4} x \left (2 x^2-x+3\right )^{5/2}\right )+\frac {23}{14} x^2 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {5}{16} \left (2 x^2-x+3\right )^{5/2} x^3\) |
(5*x^3*(3 - x + 2*x^2)^(5/2))/16 + ((23*x^2*(3 - x + 2*x^2)^(5/2))/14 + (( 125*x*(3 - x + 2*x^2)^(5/2))/4 + ((1167*(3 - x + 2*x^2)^(5/2))/2 + (60179* (-1/16*((1 - 4*x)*(3 - x + 2*x^2)^(3/2)) + (69*(-1/8*((1 - 4*x)*Sqrt[3 - x + 2*x^2]) + (23*ArcSinh[(-1 + 4*x)/Sqrt[23]])/(16*Sqrt[2])))/32))/4)/8)/2 8)/32
3.4.35.3.1 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_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b *e)/(2*c) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(q + 2*p + 1))), x] + Simp[1/(c*(q + 2*p + 1)) Int[(a + b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b *e*(q + p)*x^(q - 1) - c*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c , p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && !LeQ[p, -1]
Time = 0.25 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.44
| method | result | size |
| risch | \(\frac {\left (9175040 x^{7}-7667712 x^{6}+29335552 x^{5}-7497728 x^{4}+27023744 x^{3}+3845856 x^{2}+27845612 x -1663407\right ) \sqrt {2 x^{2}-x +3}}{7340032}+\frac {13643439 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{4194304}\) | \(65\) |
| trager | \(\left (\frac {5}{4} x^{7}-\frac {117}{112} x^{6}+\frac {3581}{896} x^{5}-\frac {523}{512} x^{4}+\frac {211123}{57344} x^{3}+\frac {17169}{32768} x^{2}+\frac {6961403}{1835008} x -\frac {1663407}{7340032}\right ) \sqrt {2 x^{2}-x +3}-\frac {13643439 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )+4 \sqrt {2 x^{2}-x +3}\right )}{4194304}\) | \(89\) |
| default | \(\frac {1167 \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{14336}+\frac {8597 \left (4 x -1\right ) \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{65536}+\frac {593193 \sqrt {2 x^{2}-x +3}\, \left (4 x -1\right )}{1048576}+\frac {13643439 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{4194304}+\frac {5 x^{3} \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{16}+\frac {23 x^{2} \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{448}+\frac {125 x \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{3584}\) | \(117\) |
1/7340032*(9175040*x^7-7667712*x^6+29335552*x^5-7497728*x^4+27023744*x^3+3 845856*x^2+27845612*x-1663407)*(2*x^2-x+3)^(1/2)+13643439/4194304*2^(1/2)* arcsinh(4/23*23^(1/2)*(x-1/4))
Time = 0.25 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.60 \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right ) \, dx=\frac {1}{7340032} \, {\left (9175040 \, x^{7} - 7667712 \, x^{6} + 29335552 \, x^{5} - 7497728 \, x^{4} + 27023744 \, x^{3} + 3845856 \, x^{2} + 27845612 \, x - 1663407\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {13643439}{8388608} \, \sqrt {2} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) \]
1/7340032*(9175040*x^7 - 7667712*x^6 + 29335552*x^5 - 7497728*x^4 + 270237 44*x^3 + 3845856*x^2 + 27845612*x - 1663407)*sqrt(2*x^2 - x + 3) + 1364343 9/8388608*sqrt(2)*log(-4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25)
Time = 0.50 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.56 \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right ) \, dx=\sqrt {2 x^{2} - x + 3} \cdot \left (\frac {5 x^{7}}{4} - \frac {117 x^{6}}{112} + \frac {3581 x^{5}}{896} - \frac {523 x^{4}}{512} + \frac {211123 x^{3}}{57344} + \frac {17169 x^{2}}{32768} + \frac {6961403 x}{1835008} - \frac {1663407}{7340032}\right ) + \frac {13643439 \sqrt {2} \operatorname {asinh}{\left (\frac {4 \sqrt {23} \left (x - \frac {1}{4}\right )}{23} \right )}}{4194304} \]
sqrt(2*x**2 - x + 3)*(5*x**7/4 - 117*x**6/112 + 3581*x**5/896 - 523*x**4/5 12 + 211123*x**3/57344 + 17169*x**2/32768 + 6961403*x/1835008 - 1663407/73 40032) + 13643439*sqrt(2)*asinh(4*sqrt(23)*(x - 1/4)/23)/4194304
Time = 0.29 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.94 \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right ) \, dx=\frac {5}{16} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x^{3} + \frac {23}{448} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x^{2} + \frac {125}{3584} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x + \frac {1167}{14336} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {8597}{16384} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {8597}{65536} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {593193}{262144} \, \sqrt {2 \, x^{2} - x + 3} x + \frac {13643439}{4194304} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {593193}{1048576} \, \sqrt {2 \, x^{2} - x + 3} \]
5/16*(2*x^2 - x + 3)^(5/2)*x^3 + 23/448*(2*x^2 - x + 3)^(5/2)*x^2 + 125/35 84*(2*x^2 - x + 3)^(5/2)*x + 1167/14336*(2*x^2 - x + 3)^(5/2) + 8597/16384 *(2*x^2 - x + 3)^(3/2)*x - 8597/65536*(2*x^2 - x + 3)^(3/2) + 593193/26214 4*sqrt(2*x^2 - x + 3)*x + 13643439/4194304*sqrt(2)*arcsinh(1/23*sqrt(23)*( 4*x - 1)) - 593193/1048576*sqrt(2*x^2 - x + 3)
Time = 0.29 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.56 \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right ) \, dx=\frac {1}{7340032} \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (4 \, {\left (8 \, {\left (140 \, x - 117\right )} x + 3581\right )} x - 3661\right )} x + 211123\right )} x + 120183\right )} x + 6961403\right )} x - 1663407\right )} \sqrt {2 \, x^{2} - x + 3} - \frac {13643439}{4194304} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) \]
1/7340032*(4*(8*(4*(16*(4*(8*(140*x - 117)*x + 3581)*x - 3661)*x + 211123) *x + 120183)*x + 6961403)*x - 1663407)*sqrt(2*x^2 - x + 3) - 13643439/4194 304*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1)
Timed out. \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right ) \, dx=\int {\left (2\,x^2-x+3\right )}^{3/2}\,\left (5\,x^4-x^3+3\,x^2+x+2\right ) \,d x \]