Integrand size = 38, antiderivative size = 166 \[ \int (5+2 x) \left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right ) \, dx=-\frac {6398163 (1-4 x) \sqrt {3-x+2 x^2}}{2097152}-\frac {92727 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{131072}+\frac {69415 (5+2 x)^2 \left (3-x+2 x^2\right )^{5/2}}{32256}-\frac {1121 (5+2 x)^3 \left (3-x+2 x^2\right )^{5/2}}{2304}+\frac {5}{144} (5+2 x)^4 \left (3-x+2 x^2\right )^{5/2}-\frac {3 (661397+215900 x) \left (3-x+2 x^2\right )^{5/2}}{143360}-\frac {147157749 \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )}{4194304 \sqrt {2}} \]
-92727/131072*(1-4*x)*(2*x^2-x+3)^(3/2)+69415/32256*(5+2*x)^2*(2*x^2-x+3)^ (5/2)-1121/2304*(5+2*x)^3*(2*x^2-x+3)^(5/2)+5/144*(5+2*x)^4*(2*x^2-x+3)^(5 /2)-3/143360*(661397+215900*x)*(2*x^2-x+3)^(5/2)-147157749/8388608*arcsinh (1/23*(1-4*x)*23^(1/2))*2^(1/2)-6398163/2097152*(1-4*x)*(2*x^2-x+3)^(1/2)
Time = 0.68 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.54 \[ \int (5+2 x) \left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right ) \, dx=\frac {4 \sqrt {3-x+2 x^2} \left (1592737263+12357760788 x+4870637856 x^2+12669290112 x^3+379086848 x^4+12117893120 x^5+1033175040 x^6+2926837760 x^7+1468006400 x^8\right )-46354690935 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{2642411520} \]
(4*Sqrt[3 - x + 2*x^2]*(1592737263 + 12357760788*x + 4870637856*x^2 + 1266 9290112*x^3 + 379086848*x^4 + 12117893120*x^5 + 1033175040*x^6 + 292683776 0*x^7 + 1468006400*x^8) - 46354690935*Sqrt[2]*Log[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]])/2642411520
Time = 0.48 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.15, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.289, Rules used = {2184, 25, 2184, 27, 2184, 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.
| \(\displaystyle \int (2 x+5) \left (2 x^2-x+3\right )^{3/2} \left (5 x^4-x^3+3 x^2+x+2\right ) \, dx\) |
| \(\Big \downarrow \) 2184 |
| \(\displaystyle \frac {1}{288} \int -\left ((2 x+5) \left (2 x^2-x+3\right )^{3/2} \left (8968 x^3+15996 x^2+11262 x+2299\right )\right )dx+\frac {5}{144} \left (2 x^2-x+3\right )^{5/2} (2 x+5)^4\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5}{144} (2 x+5)^4 \left (2 x^2-x+3\right )^{5/2}-\frac {1}{288} \int (2 x+5) \left (2 x^2-x+3\right )^{3/2} \left (8968 x^3+15996 x^2+11262 x+2299\right )dx\) |
| \(\Big \downarrow \) 2184 |
| \(\displaystyle \frac {1}{288} \left (-\frac {1}{128} \int -8 (2 x+5) \left (2 x^2-x+3\right )^{3/2} \left (277660 x^2+281660 x+24871\right )dx-\frac {1121}{8} \left (2 x^2-x+3\right )^{5/2} (2 x+5)^3\right )+\frac {5}{144} \left (2 x^2-x+3\right )^{5/2} (2 x+5)^4\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{288} \left (\frac {1}{16} \int (2 x+5) \left (2 x^2-x+3\right )^{3/2} \left (277660 x^2+281660 x+24871\right )dx-\frac {1121}{8} (2 x+5)^3 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {5}{144} \left (2 x^2-x+3\right )^{5/2} (2 x+5)^4\) |
| \(\Big \downarrow \) 2184 |
| \(\displaystyle \frac {1}{288} \left (\frac {1}{16} \left (\frac {1}{56} \int 108 (15467-64770 x) (2 x+5) \left (2 x^2-x+3\right )^{3/2}dx+\frac {69415}{7} (2 x+5)^2 \left (2 x^2-x+3\right )^{5/2}\right )-\frac {1121}{8} (2 x+5)^3 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {5}{144} \left (2 x^2-x+3\right )^{5/2} (2 x+5)^4\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{288} \left (\frac {1}{16} \left (\frac {27}{14} \int (15467-64770 x) (2 x+5) \left (2 x^2-x+3\right )^{3/2}dx+\frac {69415}{7} (2 x+5)^2 \left (2 x^2-x+3\right )^{5/2}\right )-\frac {1121}{8} (2 x+5)^3 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {5}{144} \left (2 x^2-x+3\right )^{5/2} (2 x+5)^4\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle \frac {1}{288} \left (\frac {1}{16} \left (\frac {27}{14} \left (\frac {216363}{8} \int \left (2 x^2-x+3\right )^{3/2}dx-\frac {1}{20} (215900 x+661397) \left (2 x^2-x+3\right )^{5/2}\right )+\frac {69415}{7} (2 x+5)^2 \left (2 x^2-x+3\right )^{5/2}\right )-\frac {1121}{8} (2 x+5)^3 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {5}{144} \left (2 x^2-x+3\right )^{5/2} (2 x+5)^4\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{288} \left (\frac {1}{16} \left (\frac {27}{14} \left (\frac {216363}{8} \left (\frac {69}{32} \int \sqrt {2 x^2-x+3}dx-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )-\frac {1}{20} (215900 x+661397) \left (2 x^2-x+3\right )^{5/2}\right )+\frac {69415}{7} (2 x+5)^2 \left (2 x^2-x+3\right )^{5/2}\right )-\frac {1121}{8} (2 x+5)^3 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {5}{144} \left (2 x^2-x+3\right )^{5/2} (2 x+5)^4\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{288} \left (\frac {1}{16} \left (\frac {27}{14} \left (\frac {216363}{8} \left (\frac {69}{32} \left (\frac {23}{16} \int \frac {1}{\sqrt {2 x^2-x+3}}dx-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )-\frac {1}{20} (215900 x+661397) \left (2 x^2-x+3\right )^{5/2}\right )+\frac {69415}{7} (2 x+5)^2 \left (2 x^2-x+3\right )^{5/2}\right )-\frac {1121}{8} (2 x+5)^3 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {5}{144} \left (2 x^2-x+3\right )^{5/2} (2 x+5)^4\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle \frac {1}{288} \left (\frac {1}{16} \left (\frac {27}{14} \left (\frac {216363}{8} \left (\frac {69}{32} \left (\frac {1}{16} \sqrt {\frac {23}{2}} \int \frac {1}{\sqrt {\frac {1}{23} (4 x-1)^2+1}}d(4 x-1)-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )-\frac {1}{20} (215900 x+661397) \left (2 x^2-x+3\right )^{5/2}\right )+\frac {69415}{7} (2 x+5)^2 \left (2 x^2-x+3\right )^{5/2}\right )-\frac {1121}{8} (2 x+5)^3 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {5}{144} \left (2 x^2-x+3\right )^{5/2} (2 x+5)^4\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {1}{288} \left (\frac {1}{16} \left (\frac {27}{14} \left (\frac {216363}{8} \left (\frac {69}{32} \left (\frac {23 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{16 \sqrt {2}}-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )-\frac {1}{20} (215900 x+661397) \left (2 x^2-x+3\right )^{5/2}\right )+\frac {69415}{7} (2 x+5)^2 \left (2 x^2-x+3\right )^{5/2}\right )-\frac {1121}{8} (2 x+5)^3 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {5}{144} \left (2 x^2-x+3\right )^{5/2} (2 x+5)^4\) |
(5*(5 + 2*x)^4*(3 - x + 2*x^2)^(5/2))/144 + ((-1121*(5 + 2*x)^3*(3 - x + 2 *x^2)^(5/2))/8 + ((69415*(5 + 2*x)^2*(3 - x + 2*x^2)^(5/2))/7 + (27*(-1/20 *((661397 + 215900*x)*(3 - x + 2*x^2)^(5/2)) + (216363*(-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*Ar cSinh[(-1 + 4*x)/Sqrt[23]])/(16*Sqrt[2])))/32))/8))/14)/16)/288
3.4.34.3.1 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[(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 = 0.44 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.42
| method | result | size |
| risch | \(\frac {\left (1468006400 x^{8}+2926837760 x^{7}+1033175040 x^{6}+12117893120 x^{5}+379086848 x^{4}+12669290112 x^{3}+4870637856 x^{2}+12357760788 x +1592737263\right ) \sqrt {2 x^{2}-x +3}}{660602880}+\frac {147157749 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{8388608}\) | \(70\) |
| trager | \(\left (\frac {20}{9} x^{8}+\frac {319}{72} x^{7}+\frac {1051}{672} x^{6}+\frac {295847}{16128} x^{5}+\frac {26443}{46080} x^{4}+\frac {32992943}{1720320} x^{3}+\frac {2415991}{327680} x^{2}+\frac {343271133}{18350080} x +\frac {176970807}{73400320}\right ) \sqrt {2 x^{2}-x +3}-\frac {147157749 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )+4 \sqrt {2 x^{2}-x +3}\right )}{8388608}\) | \(94\) |
| default | \(\frac {92727 \left (4 x -1\right ) \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{131072}+\frac {6398163 \sqrt {2 x^{2}-x +3}\, \left (4 x -1\right )}{2097152}+\frac {147157749 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{8388608}+\frac {2005 x^{2} \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{8064}+\frac {5645 x \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{21504}+\frac {120809 \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{143360}+\frac {5 x^{4} \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{9}+\frac {479 x^{3} \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{288}\) | \(134\) |
1/660602880*(1468006400*x^8+2926837760*x^7+1033175040*x^6+12117893120*x^5+ 379086848*x^4+12669290112*x^3+4870637856*x^2+12357760788*x+1592737263)*(2* x^2-x+3)^(1/2)+147157749/8388608*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))
Time = 0.27 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.56 \[ \int (5+2 x) \left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right ) \, dx=\frac {1}{660602880} \, {\left (1468006400 \, x^{8} + 2926837760 \, x^{7} + 1033175040 \, x^{6} + 12117893120 \, x^{5} + 379086848 \, x^{4} + 12669290112 \, x^{3} + 4870637856 \, x^{2} + 12357760788 \, x + 1592737263\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {147157749}{16777216} \, \sqrt {2} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) \]
1/660602880*(1468006400*x^8 + 2926837760*x^7 + 1033175040*x^6 + 1211789312 0*x^5 + 379086848*x^4 + 12669290112*x^3 + 4870637856*x^2 + 12357760788*x + 1592737263)*sqrt(2*x^2 - x + 3) + 147157749/16777216*sqrt(2)*log(-4*sqrt( 2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25)
Time = 0.71 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.54 \[ \int (5+2 x) \left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right ) \, dx=\sqrt {2 x^{2} - x + 3} \cdot \left (\frac {20 x^{8}}{9} + \frac {319 x^{7}}{72} + \frac {1051 x^{6}}{672} + \frac {295847 x^{5}}{16128} + \frac {26443 x^{4}}{46080} + \frac {32992943 x^{3}}{1720320} + \frac {2415991 x^{2}}{327680} + \frac {343271133 x}{18350080} + \frac {176970807}{73400320}\right ) + \frac {147157749 \sqrt {2} \operatorname {asinh}{\left (\frac {4 \sqrt {23} \left (x - \frac {1}{4}\right )}{23} \right )}}{8388608} \]
sqrt(2*x**2 - x + 3)*(20*x**8/9 + 319*x**7/72 + 1051*x**6/672 + 295847*x** 5/16128 + 26443*x**4/46080 + 32992943*x**3/1720320 + 2415991*x**2/327680 + 343271133*x/18350080 + 176970807/73400320) + 147157749*sqrt(2)*asinh(4*sq rt(23)*(x - 1/4)/23)/8388608
Time = 0.27 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.93 \[ \int (5+2 x) \left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right ) \, dx=\frac {5}{9} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x^{4} + \frac {479}{288} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x^{3} + \frac {2005}{8064} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x^{2} + \frac {5645}{21504} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x + \frac {120809}{143360} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {92727}{32768} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {92727}{131072} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {6398163}{524288} \, \sqrt {2 \, x^{2} - x + 3} x + \frac {147157749}{8388608} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {6398163}{2097152} \, \sqrt {2 \, x^{2} - x + 3} \]
5/9*(2*x^2 - x + 3)^(5/2)*x^4 + 479/288*(2*x^2 - x + 3)^(5/2)*x^3 + 2005/8 064*(2*x^2 - x + 3)^(5/2)*x^2 + 5645/21504*(2*x^2 - x + 3)^(5/2)*x + 12080 9/143360*(2*x^2 - x + 3)^(5/2) + 92727/32768*(2*x^2 - x + 3)^(3/2)*x - 927 27/131072*(2*x^2 - x + 3)^(3/2) + 6398163/524288*sqrt(2*x^2 - x + 3)*x + 1 47157749/8388608*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) - 6398163/209715 2*sqrt(2*x^2 - x + 3)
Time = 0.29 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.53 \[ \int (5+2 x) \left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right ) \, dx=\frac {1}{660602880} \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (20 \, {\left (8 \, {\left (28 \, {\left (160 \, x + 319\right )} x + 3153\right )} x + 295847\right )} x + 185101\right )} x + 98978829\right )} x + 152207433\right )} x + 3089440197\right )} x + 1592737263\right )} \sqrt {2 \, x^{2} - x + 3} - \frac {147157749}{8388608} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) \]
1/660602880*(4*(8*(4*(16*(20*(8*(28*(160*x + 319)*x + 3153)*x + 295847)*x + 185101)*x + 98978829)*x + 152207433)*x + 3089440197)*x + 1592737263)*sqr t(2*x^2 - x + 3) - 147157749/8388608*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - s qrt(2*x^2 - x + 3)) + 1)
Timed out. \[ \int (5+2 x) \left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right ) \, dx=\int \left (2\,x+5\right )\,{\left (2\,x^2-x+3\right )}^{3/2}\,\left (5\,x^4-x^3+3\,x^2+x+2\right ) \,d x \]