Integrand size = 47, antiderivative size = 210 \[ \int \frac {(2+x)^2 \sqrt [3]{-19+66 x-30 x^2+9 x^3}}{(-3+2 x)^2 \left (-5+6 x-6 x^2+x^3\right )} \, dx=\frac {\sqrt [3]{-19+66 x-30 x^2+9 x^3}}{-3+2 x}+\frac {\sqrt [3]{2} \arctan \left (\frac {-3 \sqrt {3}+2 \sqrt {3} x}{-3+2 x+2^{2/3} \sqrt [3]{-19+66 x-30 x^2+9 x^3}}\right )}{\sqrt {3}}+\frac {1}{3} \sqrt [3]{2} \log \left (6-4 x+2^{2/3} \sqrt [3]{-19+66 x-30 x^2+9 x^3}\right )-\frac {\log \left (18-24 x+8 x^2+\left (-3 2^{2/3}+2\ 2^{2/3} x\right ) \sqrt [3]{-19+66 x-30 x^2+9 x^3}+\sqrt [3]{2} \left (-19+66 x-30 x^2+9 x^3\right )^{2/3}\right )}{3\ 2^{2/3}} \]
(9*x^3-30*x^2+66*x-19)^(1/3)/(-3+2*x)+1/3*2^(1/3)*arctan((-3*3^(1/2)+2*x*3 ^(1/2))/(-3+2*x+2^(2/3)*(9*x^3-30*x^2+66*x-19)^(1/3)))*3^(1/2)+1/3*2^(1/3) *ln(6-4*x+2^(2/3)*(9*x^3-30*x^2+66*x-19)^(1/3))-1/6*ln(18-24*x+8*x^2+(-3*2 ^(2/3)+2*2^(2/3)*x)*(9*x^3-30*x^2+66*x-19)^(1/3)+2^(1/3)*(9*x^3-30*x^2+66* x-19)^(2/3))*2^(1/3)
Time = 0.88 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.94 \[ \int \frac {(2+x)^2 \sqrt [3]{-19+66 x-30 x^2+9 x^3}}{(-3+2 x)^2 \left (-5+6 x-6 x^2+x^3\right )} \, dx=\frac {\sqrt [3]{-19+66 x-30 x^2+9 x^3}}{-3+2 x}+\frac {\sqrt [3]{2} \arctan \left (\frac {\sqrt {3} (-3+2 x)}{-3+2 x+2^{2/3} \sqrt [3]{-19+66 x-30 x^2+9 x^3}}\right )}{\sqrt {3}}+\frac {1}{3} \sqrt [3]{2} \log \left (6-4 x+2^{2/3} \sqrt [3]{-19+66 x-30 x^2+9 x^3}\right )-\frac {\log \left (18-24 x+8 x^2+2^{2/3} (-3+2 x) \sqrt [3]{-19+66 x-30 x^2+9 x^3}+\sqrt [3]{2} \left (-19+66 x-30 x^2+9 x^3\right )^{2/3}\right )}{3\ 2^{2/3}} \]
Integrate[((2 + x)^2*(-19 + 66*x - 30*x^2 + 9*x^3)^(1/3))/((-3 + 2*x)^2*(- 5 + 6*x - 6*x^2 + x^3)),x]
(-19 + 66*x - 30*x^2 + 9*x^3)^(1/3)/(-3 + 2*x) + (2^(1/3)*ArcTan[(Sqrt[3]* (-3 + 2*x))/(-3 + 2*x + 2^(2/3)*(-19 + 66*x - 30*x^2 + 9*x^3)^(1/3))])/Sqr t[3] + (2^(1/3)*Log[6 - 4*x + 2^(2/3)*(-19 + 66*x - 30*x^2 + 9*x^3)^(1/3)] )/3 - Log[18 - 24*x + 8*x^2 + 2^(2/3)*(-3 + 2*x)*(-19 + 66*x - 30*x^2 + 9* x^3)^(1/3) + 2^(1/3)*(-19 + 66*x - 30*x^2 + 9*x^3)^(2/3)]/(3*2^(2/3))
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.
| \(\displaystyle \int \frac {(x+2)^2 \sqrt [3]{9 x^3-30 x^2+66 x-19}}{(2 x-3)^2 \left (x^3-6 x^2+6 x-5\right )} \, dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {(-x-4) \sqrt [3]{9 x^3-30 x^2+66 x-19} (x+2)^2}{21 (2 x-3)^2 \left (x^2-x+1\right )}+\frac {\sqrt [3]{9 x^3-30 x^2+66 x-19} (x+2)^2}{21 (x-5) (2 x-3)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt [3]{(9 x-10)^3-294 (10-9 x)+2401} \text {Subst}\left (\int \frac {\sqrt [3]{9 x+7} \sqrt [3]{81 x^2-63 x+343}}{x-\frac {35}{9}}dx,x,x-\frac {10}{9}\right )}{189\ 3^{2/3} \sqrt [3]{3 x-1} \sqrt [3]{3 x^2-9 x+19}}-\frac {2 \sqrt [3]{(9 x-10)^3-294 (10-9 x)+2401} \text {Subst}\left (\int \frac {\sqrt [3]{9 x+7} \sqrt [3]{81 x^2-63 x+343}}{\left (2 x-\frac {7}{9}\right )^2}dx,x,x-\frac {10}{9}\right )}{9\ 3^{2/3} \sqrt [3]{3 x-1} \sqrt [3]{3 x^2-9 x+19}}+\frac {2 \sqrt [3]{(9 x-10)^3-294 (10-9 x)+2401} \text {Subst}\left (\int \frac {\sqrt [3]{9 x+7} \sqrt [3]{81 x^2-63 x+343}}{2 x-\frac {7}{9}}dx,x,x-\frac {10}{9}\right )}{63\ 3^{2/3} \sqrt [3]{3 x-1} \sqrt [3]{3 x^2-9 x+19}}-\frac {2 \left (2+i \sqrt {3}\right ) \sqrt [3]{(9 x-10)^3-294 (10-9 x)+2401} \text {Subst}\left (\int \frac {\sqrt [3]{9 x+7} \sqrt [3]{81 x^2-63 x+343}}{2 x+\frac {1}{27} \left (60+27 \left (-1-i \sqrt {3}\right )\right )}dx,x,x-\frac {10}{9}\right )}{189\ 3^{2/3} \sqrt [3]{3 x-1} \sqrt [3]{3 x^2-9 x+19}}-\frac {2 \left (2-i \sqrt {3}\right ) \sqrt [3]{(9 x-10)^3-294 (10-9 x)+2401} \text {Subst}\left (\int \frac {\sqrt [3]{9 x+7} \sqrt [3]{81 x^2-63 x+343}}{2 x+\frac {1}{27} \left (60+27 \left (-1+i \sqrt {3}\right )\right )}dx,x,x-\frac {10}{9}\right )}{189\ 3^{2/3} \sqrt [3]{3 x-1} \sqrt [3]{3 x^2-9 x+19}}\) |
Int[((2 + x)^2*(-19 + 66*x - 30*x^2 + 9*x^3)^(1/3))/((-3 + 2*x)^2*(-5 + 6* x - 6*x^2 + x^3)),x]
3.26.13.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 17.49 (sec) , antiderivative size = 2422, normalized size of antiderivative = 11.53
| method | result | size |
| trager | \(\text {Expression too large to display}\) | \(2422\) |
| risch | \(\text {Expression too large to display}\) | \(5124\) |
int((2+x)^2*(9*x^3-30*x^2+66*x-19)^(1/3)/(-3+2*x)^2/(x^3-6*x^2+6*x-5),x,me thod=_RETURNVERBOSE)
(9*x^3-30*x^2+66*x-19)^(1/3)/(-3+2*x)+6*RootOf(RootOf(_Z^3-2)^2+18*_Z*Root Of(_Z^3-2)+324*_Z^2)*ln(-(2992717796974542*RootOf(RootOf(_Z^3-2)^2+18*_Z*R ootOf(_Z^3-2)+324*_Z^2)^2*RootOf(_Z^3-2)^3*x^3+112287225960*RootOf(RootOf( _Z^3-2)^2+18*_Z*RootOf(_Z^3-2)+324*_Z^2)*RootOf(_Z^3-2)^4*x^3-414376310350 3212*RootOf(RootOf(_Z^3-2)^2+18*_Z*RootOf(_Z^3-2)+324*_Z^2)^2*RootOf(_Z^3- 2)^3*x^2-155474620560*RootOf(RootOf(_Z^3-2)^2+18*_Z*RootOf(_Z^3-2)+324*_Z^ 2)*RootOf(_Z^3-2)^4*x^2+336264686537256*RootOf(_Z^3-2)^2*RootOf(RootOf(_Z^ 3-2)^2+18*_Z*RootOf(_Z^3-2)+324*_Z^2)*(9*x^3-30*x^2+66*x-19)^(2/3)*x+24862 578621019272*RootOf(RootOf(_Z^3-2)^2+18*_Z*RootOf(_Z^3-2)+324*_Z^2)^2*Root Of(_Z^3-2)^3*x+932847723360*RootOf(RootOf(_Z^3-2)^2+18*_Z*RootOf(_Z^3-2)+3 24*_Z^2)*RootOf(_Z^3-2)^4*x-504397029805884*RootOf(_Z^3-2)^2*RootOf(RootOf (_Z^3-2)^2+18*_Z*RootOf(_Z^3-2)+324*_Z^2)*(9*x^3-30*x^2+66*x-19)^(2/3)+277 8952240047789*RootOf(RootOf(_Z^3-2)^2+18*_Z*RootOf(_Z^3-2)+324*_Z^2)*RootO f(_Z^3-2)*x^3+104266709820*RootOf(_Z^3-2)^2*x^3+1345058746149024*RootOf(Ro otOf(_Z^3-2)^2+18*_Z*RootOf(_Z^3-2)+324*_Z^2)*(9*x^3-30*x^2+66*x-19)^(1/3) *x^2-28351223523420*RootOf(_Z^3-2)*(9*x^3-30*x^2+66*x-19)^(1/3)*x^2-111048 46623938502*RootOf(RootOf(_Z^3-2)^2+18*_Z*RootOf(_Z^3-2)+324*_Z^2)*RootOf( _Z^3-2)*x^2-416655530760*RootOf(_Z^3-2)^2*x^2-14175611761710*(9*x^3-30*x^2 +66*x-19)^(2/3)*x-4035176238447072*RootOf(RootOf(_Z^3-2)^2+18*_Z*RootOf(_Z ^3-2)+324*_Z^2)*(9*x^3-30*x^2+66*x-19)^(1/3)*x+85053670570260*RootOf(_Z...
Leaf count of result is larger than twice the leaf count of optimal. 532 vs. \(2 (175) = 350\).
Time = 14.16 (sec) , antiderivative size = 532, normalized size of antiderivative = 2.53 \[ \int \frac {(2+x)^2 \sqrt [3]{-19+66 x-30 x^2+9 x^3}}{(-3+2 x)^2 \left (-5+6 x-6 x^2+x^3\right )} \, dx=\frac {2 \, \sqrt {3} 2^{\frac {1}{3}} {\left (2 \, x - 3\right )} \arctan \left (-\frac {6 \, \sqrt {3} 2^{\frac {2}{3}} {\left (5380 \, x^{8} - 59100 \, x^{7} + 301161 \, x^{6} - 909412 \, x^{5} + 1740060 \, x^{4} - 2110416 \, x^{3} + 1545376 \, x^{2} - 606864 \, x + 94131\right )} {\left (9 \, x^{3} - 30 \, x^{2} + 66 \, x - 19\right )}^{\frac {1}{3}} - 42 \, \sqrt {3} 2^{\frac {1}{3}} {\left (82 \, x^{7} - 963 \, x^{6} + 4404 \, x^{5} - 10852 \, x^{4} + 15852 \, x^{3} - 14316 \, x^{2} + 7786 \, x - 1905\right )} {\left (9 \, x^{3} - 30 \, x^{2} + 66 \, x - 19\right )}^{\frac {2}{3}} + \sqrt {3} {\left (43721 \, x^{9} - 510066 \, x^{8} + 2889414 \, x^{7} - 10065027 \, x^{6} + 23187528 \, x^{5} - 35703864 \, x^{4} + 35637567 \, x^{3} - 21385926 \, x^{2} + 6711858 \, x - 806653\right )}}{3 \, {\left (62551 \, x^{9} - 773406 \, x^{8} + 4465170 \, x^{7} - 15587817 \, x^{6} + 35620200 \, x^{5} - 54275256 \, x^{4} + 54133401 \, x^{3} - 33459498 \, x^{2} + 11334294 \, x - 1538783\right )}}\right ) - 2^{\frac {1}{3}} {\left (2 \, x - 3\right )} \log \left (\frac {3 \cdot 2^{\frac {2}{3}} {\left (82 \, x^{4} - 471 \, x^{3} + 1086 \, x^{2} - 1100 \, x + 381\right )} {\left (9 \, x^{3} - 30 \, x^{2} + 66 \, x - 19\right )}^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (1345 \, x^{6} - 10740 \, x^{5} + 40044 \, x^{4} - 83056 \, x^{3} + 95748 \, x^{2} - 53484 \, x + 10459\right )} + 12 \, {\left (68 \, x^{5} - 468 \, x^{4} + 1425 \, x^{3} - 2218 \, x^{2} + 1632 \, x - 414\right )} {\left (9 \, x^{3} - 30 \, x^{2} + 66 \, x - 19\right )}^{\frac {1}{3}}}{x^{6} - 12 \, x^{5} + 48 \, x^{4} - 82 \, x^{3} + 96 \, x^{2} - 60 \, x + 25}\right ) + 2 \cdot 2^{\frac {1}{3}} {\left (2 \, x - 3\right )} \log \left (\frac {7 \cdot 2^{\frac {2}{3}} {\left (x^{3} - 6 \, x^{2} + 6 \, x - 5\right )} - 6 \cdot 2^{\frac {1}{3}} {\left (9 \, x^{3} - 30 \, x^{2} + 66 \, x - 19\right )}^{\frac {1}{3}} {\left (4 \, x^{2} - 12 \, x + 9\right )} + 6 \, {\left (9 \, x^{3} - 30 \, x^{2} + 66 \, x - 19\right )}^{\frac {2}{3}} {\left (2 \, x - 3\right )}}{x^{3} - 6 \, x^{2} + 6 \, x - 5}\right ) + 18 \, {\left (9 \, x^{3} - 30 \, x^{2} + 66 \, x - 19\right )}^{\frac {1}{3}}}{18 \, {\left (2 \, x - 3\right )}} \]
integrate((2+x)^2*(9*x^3-30*x^2+66*x-19)^(1/3)/(-3+2*x)^2/(x^3-6*x^2+6*x-5 ),x, algorithm="fricas")
1/18*(2*sqrt(3)*2^(1/3)*(2*x - 3)*arctan(-1/3*(6*sqrt(3)*2^(2/3)*(5380*x^8 - 59100*x^7 + 301161*x^6 - 909412*x^5 + 1740060*x^4 - 2110416*x^3 + 15453 76*x^2 - 606864*x + 94131)*(9*x^3 - 30*x^2 + 66*x - 19)^(1/3) - 42*sqrt(3) *2^(1/3)*(82*x^7 - 963*x^6 + 4404*x^5 - 10852*x^4 + 15852*x^3 - 14316*x^2 + 7786*x - 1905)*(9*x^3 - 30*x^2 + 66*x - 19)^(2/3) + sqrt(3)*(43721*x^9 - 510066*x^8 + 2889414*x^7 - 10065027*x^6 + 23187528*x^5 - 35703864*x^4 + 3 5637567*x^3 - 21385926*x^2 + 6711858*x - 806653))/(62551*x^9 - 773406*x^8 + 4465170*x^7 - 15587817*x^6 + 35620200*x^5 - 54275256*x^4 + 54133401*x^3 - 33459498*x^2 + 11334294*x - 1538783)) - 2^(1/3)*(2*x - 3)*log((3*2^(2/3) *(82*x^4 - 471*x^3 + 1086*x^2 - 1100*x + 381)*(9*x^3 - 30*x^2 + 66*x - 19) ^(2/3) + 2^(1/3)*(1345*x^6 - 10740*x^5 + 40044*x^4 - 83056*x^3 + 95748*x^2 - 53484*x + 10459) + 12*(68*x^5 - 468*x^4 + 1425*x^3 - 2218*x^2 + 1632*x - 414)*(9*x^3 - 30*x^2 + 66*x - 19)^(1/3))/(x^6 - 12*x^5 + 48*x^4 - 82*x^3 + 96*x^2 - 60*x + 25)) + 2*2^(1/3)*(2*x - 3)*log((7*2^(2/3)*(x^3 - 6*x^2 + 6*x - 5) - 6*2^(1/3)*(9*x^3 - 30*x^2 + 66*x - 19)^(1/3)*(4*x^2 - 12*x + 9) + 6*(9*x^3 - 30*x^2 + 66*x - 19)^(2/3)*(2*x - 3))/(x^3 - 6*x^2 + 6*x - 5)) + 18*(9*x^3 - 30*x^2 + 66*x - 19)^(1/3))/(2*x - 3)
\[ \int \frac {(2+x)^2 \sqrt [3]{-19+66 x-30 x^2+9 x^3}}{(-3+2 x)^2 \left (-5+6 x-6 x^2+x^3\right )} \, dx=\int \frac {\sqrt [3]{\left (3 x - 1\right ) \left (3 x^{2} - 9 x + 19\right )} \left (x + 2\right )^{2}}{\left (x - 5\right ) \left (2 x - 3\right )^{2} \left (x^{2} - x + 1\right )}\, dx \]
Integral(((3*x - 1)*(3*x**2 - 9*x + 19))**(1/3)*(x + 2)**2/((x - 5)*(2*x - 3)**2*(x**2 - x + 1)), x)
\[ \int \frac {(2+x)^2 \sqrt [3]{-19+66 x-30 x^2+9 x^3}}{(-3+2 x)^2 \left (-5+6 x-6 x^2+x^3\right )} \, dx=\int { \frac {{\left (9 \, x^{3} - 30 \, x^{2} + 66 \, x - 19\right )}^{\frac {1}{3}} {\left (x + 2\right )}^{2}}{{\left (x^{3} - 6 \, x^{2} + 6 \, x - 5\right )} {\left (2 \, x - 3\right )}^{2}} \,d x } \]
integrate((2+x)^2*(9*x^3-30*x^2+66*x-19)^(1/3)/(-3+2*x)^2/(x^3-6*x^2+6*x-5 ),x, algorithm="maxima")
integrate((9*x^3 - 30*x^2 + 66*x - 19)^(1/3)*(x + 2)^2/((x^3 - 6*x^2 + 6*x - 5)*(2*x - 3)^2), x)
\[ \int \frac {(2+x)^2 \sqrt [3]{-19+66 x-30 x^2+9 x^3}}{(-3+2 x)^2 \left (-5+6 x-6 x^2+x^3\right )} \, dx=\int { \frac {{\left (9 \, x^{3} - 30 \, x^{2} + 66 \, x - 19\right )}^{\frac {1}{3}} {\left (x + 2\right )}^{2}}{{\left (x^{3} - 6 \, x^{2} + 6 \, x - 5\right )} {\left (2 \, x - 3\right )}^{2}} \,d x } \]
integrate((2+x)^2*(9*x^3-30*x^2+66*x-19)^(1/3)/(-3+2*x)^2/(x^3-6*x^2+6*x-5 ),x, algorithm="giac")
integrate((9*x^3 - 30*x^2 + 66*x - 19)^(1/3)*(x + 2)^2/((x^3 - 6*x^2 + 6*x - 5)*(2*x - 3)^2), x)
Timed out. \[ \int \frac {(2+x)^2 \sqrt [3]{-19+66 x-30 x^2+9 x^3}}{(-3+2 x)^2 \left (-5+6 x-6 x^2+x^3\right )} \, dx=\int \frac {{\left (x+2\right )}^2\,{\left (9\,x^3-30\,x^2+66\,x-19\right )}^{1/3}}{{\left (2\,x-3\right )}^2\,\left (x^3-6\,x^2+6\,x-5\right )} \,d x \]