\(\int \frac {(2+x)^2 \sqrt [3]{-19+66 x-30 x^2+9 x^3}}{(-3+2 x)^2 (-5+6 x-6 x^2+x^3)} \, dx\) [2513]

Optimal result

Integrand size = 47, antiderivative size = 210 \[ \int \frac {(2+x)^2 \sqrt [3]{-19+66 x-30 x^2+9 x^3}}{(-3+2 x)^2 \left (-5+6 x-6 x^2+x^3\right )} \, dx=\frac {\sqrt [3]{-19+66 x-30 x^2+9 x^3}}{-3+2 x}+\frac {\sqrt [3]{2} \arctan \left (\frac {-3 \sqrt {3}+2 \sqrt {3} x}{-3+2 x+2^{2/3} \sqrt [3]{-19+66 x-30 x^2+9 x^3}}\right )}{\sqrt {3}}+\frac {1}{3} \sqrt [3]{2} \log \left (6-4 x+2^{2/3} \sqrt [3]{-19+66 x-30 x^2+9 x^3}\right )-\frac {\log \left (18-24 x+8 x^2+\left (-3 2^{2/3}+2\ 2^{2/3} x\right ) \sqrt [3]{-19+66 x-30 x^2+9 x^3}+\sqrt [3]{2} \left (-19+66 x-30 x^2+9 x^3\right )^{2/3}\right )}{3\ 2^{2/3}} \]

[Out]

(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)

Rubi [F]

\[ \int \frac {(2+x)^2 \sqrt [3]{-19+66 x-30 x^2+9 x^3}}{(-3+2 x)^2 \left (-5+6 x-6 x^2+x^3\right )} \, dx=\int \frac {(2+x)^2 \sqrt [3]{-19+66 x-30 x^2+9 x^3}}{(-3+2 x)^2 \left (-5+6 x-6 x^2+x^3\right )} \, dx \]

[In]

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]

[Out]

((-19 + 66*x - 30*x^2 + 9*x^3)^(1/3)*Defer[Subst][Defer[Int][((7 + 9*x)^(1/3)*(343 - 63*x + 81*x^2)^(1/3))/(-3
5/9 + x), x], x, -10/9 + x])/(63*3^(1/3)*(-1 + 3*x)^(1/3)*(19 - 9*x + 3*x^2)^(1/3)) - (2*(-19 + 66*x - 30*x^2
+ 9*x^3)^(1/3)*Defer[Subst][Defer[Int][((7 + 9*x)^(1/3)*(343 - 63*x + 81*x^2)^(1/3))/(-7/9 + 2*x)^2, x], x, -1
0/9 + x])/(3*3^(1/3)*(-1 + 3*x)^(1/3)*(19 - 9*x + 3*x^2)^(1/3)) + (2*(-19 + 66*x - 30*x^2 + 9*x^3)^(1/3)*Defer
[Subst][Defer[Int][((7 + 9*x)^(1/3)*(343 - 63*x + 81*x^2)^(1/3))/(-7/9 + 2*x), x], x, -10/9 + x])/(21*3^(1/3)*
(-1 + 3*x)^(1/3)*(19 - 9*x + 3*x^2)^(1/3)) - (2*(2 + I*Sqrt[3])*(-19 + 66*x - 30*x^2 + 9*x^3)^(1/3)*Defer[Subs
t][Defer[Int][((7 + 9*x)^(1/3)*(343 - 63*x + 81*x^2)^(1/3))/((60 + 27*(-1 - I*Sqrt[3]))/27 + 2*x), x], x, -10/
9 + x])/(63*3^(1/3)*(-1 + 3*x)^(1/3)*(19 - 9*x + 3*x^2)^(1/3)) - (2*(2 - I*Sqrt[3])*(-19 + 66*x - 30*x^2 + 9*x
^3)^(1/3)*Defer[Subst][Defer[Int][((7 + 9*x)^(1/3)*(343 - 63*x + 81*x^2)^(1/3))/((60 + 27*(-1 + I*Sqrt[3]))/27
+ 2*x), x], x, -10/9 + x])/(63*3^(1/3)*(-1 + 3*x)^(1/3)*(19 - 9*x + 3*x^2)^(1/3))

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\sqrt [3]{-19+66 x-30 x^2+9 x^3}}{21 (-5+x)}-\frac {2 \sqrt [3]{-19+66 x-30 x^2+9 x^3}}{(-3+2 x)^2}+\frac {2 \sqrt [3]{-19+66 x-30 x^2+9 x^3}}{7 (-3+2 x)}+\frac {(5-4 x) \sqrt [3]{-19+66 x-30 x^2+9 x^3}}{21 \left (1-x+x^2\right )}\right ) \, dx \\ & = \frac {1}{21} \int \frac {\sqrt [3]{-19+66 x-30 x^2+9 x^3}}{-5+x} \, dx+\frac {1}{21} \int \frac {(5-4 x) \sqrt [3]{-19+66 x-30 x^2+9 x^3}}{1-x+x^2} \, dx+\frac {2}{7} \int \frac {\sqrt [3]{-19+66 x-30 x^2+9 x^3}}{-3+2 x} \, dx-2 \int \frac {\sqrt [3]{-19+66 x-30 x^2+9 x^3}}{(-3+2 x)^2} \, dx \\ & = \frac {1}{21} \int \left (\frac {\left (-4-2 i \sqrt {3}\right ) \sqrt [3]{-19+66 x-30 x^2+9 x^3}}{-1-i \sqrt {3}+2 x}+\frac {\left (-4+2 i \sqrt {3}\right ) \sqrt [3]{-19+66 x-30 x^2+9 x^3}}{-1+i \sqrt {3}+2 x}\right ) \, dx+\frac {1}{21} \text {Subst}\left (\int \frac {\sqrt [3]{\frac {2401}{81}+\frac {98 x}{3}+9 x^3}}{-\frac {35}{9}+x} \, dx,x,-\frac {10}{9}+x\right )+\frac {2}{7} \text {Subst}\left (\int \frac {\sqrt [3]{\frac {2401}{81}+\frac {98 x}{3}+9 x^3}}{-\frac {7}{9}+2 x} \, dx,x,-\frac {10}{9}+x\right )-2 \text {Subst}\left (\int \frac {\sqrt [3]{\frac {2401}{81}+\frac {98 x}{3}+9 x^3}}{\left (-\frac {7}{9}+2 x\right )^2} \, dx,x,-\frac {10}{9}+x\right ) \\ & = -\left (\frac {1}{21} \left (2 \left (2-i \sqrt {3}\right )\right ) \int \frac {\sqrt [3]{-19+66 x-30 x^2+9 x^3}}{-1+i \sqrt {3}+2 x} \, dx\right )-\frac {1}{21} \left (2 \left (2+i \sqrt {3}\right )\right ) \int \frac {\sqrt [3]{-19+66 x-30 x^2+9 x^3}}{-1-i \sqrt {3}+2 x} \, dx+\frac {\sqrt [3]{-19+66 x-30 x^2+9 x^3} \text {Subst}\left (\int \frac {\sqrt [3]{7+9 x} \sqrt [3]{343-63 x+81 x^2}}{-\frac {35}{9}+x} \, dx,x,-\frac {10}{9}+x\right )}{63 \sqrt [3]{-3+9 x} \sqrt [3]{19-9 x+3 x^2}}+\frac {\left (2 \sqrt [3]{-19+66 x-30 x^2+9 x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{7+9 x} \sqrt [3]{343-63 x+81 x^2}}{-\frac {7}{9}+2 x} \, dx,x,-\frac {10}{9}+x\right )}{21 \sqrt [3]{-3+9 x} \sqrt [3]{19-9 x+3 x^2}}-\frac {\left (2 \sqrt [3]{-19+66 x-30 x^2+9 x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{7+9 x} \sqrt [3]{343-63 x+81 x^2}}{\left (-\frac {7}{9}+2 x\right )^2} \, dx,x,-\frac {10}{9}+x\right )}{3 \sqrt [3]{-3+9 x} \sqrt [3]{19-9 x+3 x^2}} \\ & = -\left (\frac {1}{21} \left (2 \left (2-i \sqrt {3}\right )\right ) \text {Subst}\left (\int \frac {\sqrt [3]{\frac {2401}{81}+\frac {98 x}{3}+9 x^3}}{\frac {1}{27} \left (60+27 \left (-1+i \sqrt {3}\right )\right )+2 x} \, dx,x,-\frac {10}{9}+x\right )\right )-\frac {1}{21} \left (2 \left (2+i \sqrt {3}\right )\right ) \text {Subst}\left (\int \frac {\sqrt [3]{\frac {2401}{81}+\frac {98 x}{3}+9 x^3}}{\frac {1}{27} \left (60+27 \left (-1-i \sqrt {3}\right )\right )+2 x} \, dx,x,-\frac {10}{9}+x\right )+\frac {\sqrt [3]{-19+66 x-30 x^2+9 x^3} \text {Subst}\left (\int \frac {\sqrt [3]{7+9 x} \sqrt [3]{343-63 x+81 x^2}}{-\frac {35}{9}+x} \, dx,x,-\frac {10}{9}+x\right )}{63 \sqrt [3]{-3+9 x} \sqrt [3]{19-9 x+3 x^2}}+\frac {\left (2 \sqrt [3]{-19+66 x-30 x^2+9 x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{7+9 x} \sqrt [3]{343-63 x+81 x^2}}{-\frac {7}{9}+2 x} \, dx,x,-\frac {10}{9}+x\right )}{21 \sqrt [3]{-3+9 x} \sqrt [3]{19-9 x+3 x^2}}-\frac {\left (2 \sqrt [3]{-19+66 x-30 x^2+9 x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{7+9 x} \sqrt [3]{343-63 x+81 x^2}}{\left (-\frac {7}{9}+2 x\right )^2} \, dx,x,-\frac {10}{9}+x\right )}{3 \sqrt [3]{-3+9 x} \sqrt [3]{19-9 x+3 x^2}} \\ & = \frac {\sqrt [3]{-19+66 x-30 x^2+9 x^3} \text {Subst}\left (\int \frac {\sqrt [3]{7+9 x} \sqrt [3]{343-63 x+81 x^2}}{-\frac {35}{9}+x} \, dx,x,-\frac {10}{9}+x\right )}{63 \sqrt [3]{-3+9 x} \sqrt [3]{19-9 x+3 x^2}}+\frac {\left (2 \sqrt [3]{-19+66 x-30 x^2+9 x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{7+9 x} \sqrt [3]{343-63 x+81 x^2}}{-\frac {7}{9}+2 x} \, dx,x,-\frac {10}{9}+x\right )}{21 \sqrt [3]{-3+9 x} \sqrt [3]{19-9 x+3 x^2}}-\frac {\left (2 \sqrt [3]{-19+66 x-30 x^2+9 x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{7+9 x} \sqrt [3]{343-63 x+81 x^2}}{\left (-\frac {7}{9}+2 x\right )^2} \, dx,x,-\frac {10}{9}+x\right )}{3 \sqrt [3]{-3+9 x} \sqrt [3]{19-9 x+3 x^2}}-\frac {\left (2 \left (2-i \sqrt {3}\right ) \sqrt [3]{-19+66 x-30 x^2+9 x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{7+9 x} \sqrt [3]{343-63 x+81 x^2}}{\frac {1}{27} \left (60+27 \left (-1+i \sqrt {3}\right )\right )+2 x} \, dx,x,-\frac {10}{9}+x\right )}{63 \sqrt [3]{-3+9 x} \sqrt [3]{19-9 x+3 x^2}}-\frac {\left (2 \left (2+i \sqrt {3}\right ) \sqrt [3]{-19+66 x-30 x^2+9 x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{7+9 x} \sqrt [3]{343-63 x+81 x^2}}{\frac {1}{27} \left (60+27 \left (-1-i \sqrt {3}\right )\right )+2 x} \, dx,x,-\frac {10}{9}+x\right )}{63 \sqrt [3]{-3+9 x} \sqrt [3]{19-9 x+3 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.88 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.94 \[ \int \frac {(2+x)^2 \sqrt [3]{-19+66 x-30 x^2+9 x^3}}{(-3+2 x)^2 \left (-5+6 x-6 x^2+x^3\right )} \, dx=\frac {\sqrt [3]{-19+66 x-30 x^2+9 x^3}}{-3+2 x}+\frac {\sqrt [3]{2} \arctan \left (\frac {\sqrt {3} (-3+2 x)}{-3+2 x+2^{2/3} \sqrt [3]{-19+66 x-30 x^2+9 x^3}}\right )}{\sqrt {3}}+\frac {1}{3} \sqrt [3]{2} \log \left (6-4 x+2^{2/3} \sqrt [3]{-19+66 x-30 x^2+9 x^3}\right )-\frac {\log \left (18-24 x+8 x^2+2^{2/3} (-3+2 x) \sqrt [3]{-19+66 x-30 x^2+9 x^3}+\sqrt [3]{2} \left (-19+66 x-30 x^2+9 x^3\right )^{2/3}\right )}{3\ 2^{2/3}} \]

[In]

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]

[Out]

(-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))])/Sqrt[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))

Maple [C] (warning: unable to verify)

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	\(\text {Expression too large to display}\)	\(5124\)

[In]

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,method=_RETURNVERBOSE)

[Out]

(9*x^3-30*x^2+66*x-19)^(1/3)/(-3+2*x)+6*RootOf(RootOf(_Z^3-2)^2+18*_Z*RootOf(_Z^3-2)+324*_Z^2)*ln(-(2992717796
974542*RootOf(RootOf(_Z^3-2)^2+18*_Z*RootOf(_Z^3-2)+324*_Z^2)^2*RootOf(_Z^3-2)^3*x^3+112287225960*RootOf(RootO
f(_Z^3-2)^2+18*_Z*RootOf(_Z^3-2)+324*_Z^2)*RootOf(_Z^3-2)^4*x^3-4143763103503212*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+24862578621019272*RootOf(RootOf(_Z^3-2)^2+18*_Z*RootOf(_Z^3-2)+324*_Z^2)^
2*RootOf(_Z^3-2)^3*x+932847723360*RootOf(RootOf(_Z^3-2)^2+18*_Z*RootOf(_Z^3-2)+324*_Z^2)*RootOf(_Z^3-2)^4*x-50
4397029805884*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)+2778952240047789*RootOf(RootOf(_Z^3-2)^2+18*_Z*RootOf(_Z^3-2)+324*_Z^2)*RootOf(_Z^3-2)*x^3+104266709820*R
ootOf(_Z^3-2)^2*x^3+1345058746149024*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^2-28351223523420*RootOf(_Z^3-2)*(9*x^3-30*x^2+66*x-19)^(1/3)*x^2-11104846623938502*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^3-3
0*x^2+66*x-19)^(1/3)*x+85053670570260*RootOf(_Z^3-2)*(9*x^3-30*x^2+66*x-19)^(1/3)*x+19458146848460850*RootOf(R
ootOf(_Z^3-2)^2+18*_Z*RootOf(_Z^3-2)+324*_Z^2)*RootOf(_Z^3-2)*x+730072623000*RootOf(_Z^3-2)^2*x+21263417642565
*(9*x^3-30*x^2+66*x-19)^(2/3)+3026382178835304*RootOf(RootOf(_Z^3-2)^2+18*_Z*RootOf(_Z^3-2)+324*_Z^2)*(9*x^3-3
0*x^2+66*x-19)^(1/3)-63790252927695*RootOf(_Z^3-2)*(9*x^3-30*x^2+66*x-19)^(1/3)-7861822149195027*RootOf(RootOf
(_Z^3-2)^2+18*_Z*RootOf(_Z^3-2)+324*_Z^2)*RootOf(_Z^3-2)-294976760260*RootOf(_Z^3-2)^2)/(-5+x)/(x^2-x+1))-1/3*
ln((-5985435593949084*RootOf(RootOf(_Z^3-2)^2+18*_Z*RootOf(_Z^3-2)+324*_Z^2)^2*RootOf(_Z^3-2)^3*x^3-3322996252
11918*RootOf(RootOf(_Z^3-2)^2+18*_Z*RootOf(_Z^3-2)+324*_Z^2)*RootOf(_Z^3-2)^4*x^3+8287526207006424*RootOf(Root
Of(_Z^3-2)^2+18*_Z*RootOf(_Z^3-2)+324*_Z^2)^2*RootOf(_Z^3-2)^3*x^2+460107173370348*RootOf(RootOf(_Z^3-2)^2+18*
_Z*RootOf(_Z^3-2)+324*_Z^2)*RootOf(_Z^3-2)^4*x^2+672529373074512*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-49725157242038544*RootOf(RootOf(_Z^3-2)^2+18*_Z*Root
Of(_Z^3-2)+324*_Z^2)^2*RootOf(_Z^3-2)^3*x-2760643040222088*RootOf(RootOf(_Z^3-2)^2+18*_Z*RootOf(_Z^3-2)+324*_Z
^2)*RootOf(_Z^3-2)^4*x-1008794059611768*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)+4892856080767902*RootOf(RootOf(_Z^3-2)^2+18*_Z*RootOf(_Z^3-2)+324*_Z^2)*RootOf(
_Z^3-2)*x^3+271641757117679*RootOf(_Z^3-2)^2*x^3+2690117492298048*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^2+206153418841176*RootOf(_Z^3-2)*(9*x^3-30*x^2+66*x-19)^(1/3)*x^2-21
288857002654068*RootOf(RootOf(_Z^3-2)^2+18*_Z*RootOf(_Z^3-2)+324*_Z^2)*RootOf(_Z^3-2)*x^2-1181915516779386*Roo
tOf(_Z^3-2)^2*x^2+103076709420588*(9*x^3-30*x^2+66*x-19)^(2/3)*x-8070352476894144*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-618460256523528*RootOf(_Z^3-2)*(9*x^3-30*x^2+66*x-19
)^(1/3)*x+33391276225584084*RootOf(RootOf(_Z^3-2)^2+18*_Z*RootOf(_Z^3-2)+324*_Z^2)*RootOf(_Z^3-2)*x+1853818055
669418*RootOf(_Z^3-2)^2*x-154615064130882*(9*x^3-30*x^2+66*x-19)^(2/3)+6052764357670608*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)[phone]*RootOf(_Z^3-2)*(9*x^3-30*x^2+66*
x-19)^(1/3)-15723644298390054*RootOf(RootOf(_Z^3-2)^2+18*_Z*RootOf(_Z^3-2)+324*_Z^2)*RootOf(_Z^3-2)-8729458408
34483*RootOf(_Z^3-2)^2)/(-5+x)/(x^2-x+1))*RootOf(_Z^3-2)-6*ln((-5985435593949084*RootOf(RootOf(_Z^3-2)^2+18*_Z
*RootOf(_Z^3-2)+324*_Z^2)^2*RootOf(_Z^3-2)^3*x^3-332299625211918*RootOf(RootOf(_Z^3-2)^2+18*_Z*RootOf(_Z^3-2)+
324*_Z^2)*RootOf(_Z^3-2)^4*x^3+8287526207006424*RootOf(RootOf(_Z^3-2)^2+18*_Z*RootOf(_Z^3-2)+324*_Z^2)^2*RootO
f(_Z^3-2)^3*x^2+460107173370348*RootOf(RootOf(_Z^3-2)^2+18*_Z*RootOf(_Z^3-2)+324*_Z^2)*RootOf(_Z^3-2)^4*x^2+67
2529373074512*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-49725157242038544*RootOf(RootOf(_Z^3-2)^2+18*_Z*RootOf(_Z^3-2)+324*_Z^2)^2*RootOf(_Z^3-2)^3*x-276064304
0222088*RootOf(RootOf(_Z^3-2)^2+18*_Z*RootOf(_Z^3-2)+324*_Z^2)*RootOf(_Z^3-2)^4*x-1008794059611768*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)+4892856080767902*Roo
tOf(RootOf(_Z^3-2)^2+18*_Z*RootOf(_Z^3-2)+324*_Z^2)*RootOf(_Z^3-2)*x^3+271641757117679*RootOf(_Z^3-2)^2*x^3+26
90117492298048*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^2+2061534
18841176*RootOf(_Z^3-2)*(9*x^3-30*x^2+66*x-19)^(1/3)*x^2-21288857002654068*RootOf(RootOf(_Z^3-2)^2+18*_Z*RootO
f(_Z^3-2)+324*_Z^2)*RootOf(_Z^3-2)*x^2-1181915516779386*RootOf(_Z^3-2)^2*x^2+103076709420588*(9*x^3-30*x^2+66*
x-19)^(2/3)*x-8070352476894144*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-618460256523528*RootOf(_Z^3-2)*(9*x^3-30*x^2+66*x-19)^(1/3)*x+33391276225584084*RootOf(RootOf(_Z^3-2)^2
+18*_Z*RootOf(_Z^3-2)+324*_Z^2)*RootOf(_Z^3-2)*x+1853818055669418*RootOf(_Z^3-2)^2*x-154615064130882*(9*x^3-30
*x^2+66*x-19)^(2/3)+6052764357670608*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)[phone]*RootOf(_Z^3-2)*(9*x^3-30*x^2+66*x-19)^(1/3)-15723644298390054*RootOf(RootOf(_Z^3-2)
^2+18*_Z*RootOf(_Z^3-2)+324*_Z^2)*RootOf(_Z^3-2)-872945840834483*RootOf(_Z^3-2)^2)/(-5+x)/(x^2-x+1))*RootOf(Ro
otOf(_Z^3-2)^2+18*_Z*RootOf(_Z^3-2)+324*_Z^2)

Fricas [B] (verification not implemented)

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 )}} \]

[In]

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")

[Out]

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 + 1545376*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
+ 35637567*x^3 - 21385926*x^2 + 6711858*x - 806653))/(62551*x^9 - 773406*x^8 + 4465170*x^7 - 15587817*x^6 + 3
5620200*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 - 1
0740*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)

Sympy [F]

\[ \int \frac {(2+x)^2 \sqrt [3]{-19+66 x-30 x^2+9 x^3}}{(-3+2 x)^2 \left (-5+6 x-6 x^2+x^3\right )} \, dx=\int \frac {\sqrt [3]{\left (3 x - 1\right ) \left (3 x^{2} - 9 x + 19\right )} \left (x + 2\right )^{2}}{\left (x - 5\right ) \left (2 x - 3\right )^{2} \left (x^{2} - x + 1\right )}\, dx \]

[In]

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)

[Out]

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)

Maxima [F]

\[ \int \frac {(2+x)^2 \sqrt [3]{-19+66 x-30 x^2+9 x^3}}{(-3+2 x)^2 \left (-5+6 x-6 x^2+x^3\right )} \, dx=\int { \frac {{\left (9 \, x^{3} - 30 \, x^{2} + 66 \, x - 19\right )}^{\frac {1}{3}} {\left (x + 2\right )}^{2}}{{\left (x^{3} - 6 \, x^{2} + 6 \, x - 5\right )} {\left (2 \, x - 3\right )}^{2}} \,d x } \]

[In]

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")

[Out]

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)

Giac [F]

\[ \int \frac {(2+x)^2 \sqrt [3]{-19+66 x-30 x^2+9 x^3}}{(-3+2 x)^2 \left (-5+6 x-6 x^2+x^3\right )} \, dx=\int { \frac {{\left (9 \, x^{3} - 30 \, x^{2} + 66 \, x - 19\right )}^{\frac {1}{3}} {\left (x + 2\right )}^{2}}{{\left (x^{3} - 6 \, x^{2} + 6 \, x - 5\right )} {\left (2 \, x - 3\right )}^{2}} \,d x } \]

[In]

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")

[Out]

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)

Mupad [F(-1)]

Timed out. \[ \int \frac {(2+x)^2 \sqrt [3]{-19+66 x-30 x^2+9 x^3}}{(-3+2 x)^2 \left (-5+6 x-6 x^2+x^3\right )} \, dx=\int \frac {{\left (x+2\right )}^2\,{\left (9\,x^3-30\,x^2+66\,x-19\right )}^{1/3}}{{\left (2\,x-3\right )}^2\,\left (x^3-6\,x^2+6\,x-5\right )} \,d x \]

[In]

int(((x + 2)^2*(66*x - 30*x^2 + 9*x^3 - 19)^(1/3))/((2*x - 3)^2*(6*x - 6*x^2 + x^3 - 5)),x)

[Out]

int(((x + 2)^2*(66*x - 30*x^2 + 9*x^3 - 19)^(1/3))/((2*x - 3)^2*(6*x - 6*x^2 + x^3 - 5)), x)