\(\int \frac {2+3 x}{\sqrt [3]{4+3 x^2} (-12+52 x+9 x^2)} \, dx\) [2432]
Optimal result
Integrand size = 29, antiderivative size = 197 \[
\int \frac {2+3 x}{\sqrt [3]{4+3 x^2} \left (-12+52 x+9 x^2\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\frac {10 \sqrt [3]{2}}{\sqrt {3} 7^{2/3}}-\frac {\sqrt [3]{2} \sqrt {3} x}{7^{2/3}}+\frac {\sqrt [3]{4+3 x^2}}{\sqrt {3}}}{\sqrt [3]{4+3 x^2}}\right )}{14 \sqrt [3]{14}}+\frac {\log \left (-10 \sqrt [3]{14}+3 \sqrt [3]{14} x+14 \sqrt [3]{4+3 x^2}\right )}{14 \sqrt [3]{14}}-\frac {\log \left (100\ 14^{2/3}-60\ 14^{2/3} x+9\ 14^{2/3} x^2+\left (140 \sqrt [3]{14}-42 \sqrt [3]{14} x\right ) \sqrt [3]{4+3 x^2}+196 \left (4+3 x^2\right )^{2/3}\right )}{28 \sqrt [3]{14}}
\]
[Out]
-1/196*3^(1/2)*arctan((10/21*2^(1/3)*3^(1/2)*7^(1/3)-1/7*2^(1/3)*3^(1/2)*x*7^(1/3)+1/3*(3*x^2+4)^(1/3)*3^(1/2)
)/(3*x^2+4)^(1/3))*14^(2/3)+1/196*ln(-10*14^(1/3)+3*14^(1/3)*x+14*(3*x^2+4)^(1/3))*14^(2/3)-1/392*ln(100*14^(2
/3)-60*14^(2/3)*x+9*14^(2/3)*x^2+(140*14^(1/3)-42*14^(1/3)*x)*(3*x^2+4)^(1/3)+196*(3*x^2+4)^(2/3))*14^(2/3)
Rubi [F]
\[
\int \frac {2+3 x}{\sqrt [3]{4+3 x^2} \left (-12+52 x+9 x^2\right )} \, dx=\int \frac {2+3 x}{\sqrt [3]{4+3 x^2} \left (-12+52 x+9 x^2\right )} \, dx
\]
[In]
Int[(2 + 3*x)/((4 + 3*x^2)^(1/3)*(-12 + 52*x + 9*x^2)),x]
[Out]
Defer[Int][(2 + 3*x)/((4 + 3*x^2)^(1/3)*(-12 + 52*x + 9*x^2)), x]
Rubi steps \begin{align*}
\text {integral}& = \int \frac {2+3 x}{\sqrt [3]{4+3 x^2} \left (-12+52 x+9 x^2\right )} \, dx \\
\end{align*}
Mathematica [A] (verified)
Time = 0.27 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.80
\[
\int \frac {2+3 x}{\sqrt [3]{4+3 x^2} \left (-12+52 x+9 x^2\right )} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {10 \sqrt [3]{14}-3 \sqrt [3]{14} x+7 \sqrt [3]{4+3 x^2}}{7 \sqrt {3} \sqrt [3]{4+3 x^2}}\right )-2 \log \left (-10 \sqrt [3]{14}+3 \sqrt [3]{14} x+14 \sqrt [3]{4+3 x^2}\right )+\log \left (100\ 14^{2/3}-60\ 14^{2/3} x+9\ 14^{2/3} x^2+196 \left (4+3 x^2\right )^{2/3}+14 (10-3 x) \sqrt [3]{56+42 x^2}\right )}{28 \sqrt [3]{14}}
\]
[In]
Integrate[(2 + 3*x)/((4 + 3*x^2)^(1/3)*(-12 + 52*x + 9*x^2)),x]
[Out]
-1/28*(2*Sqrt[3]*ArcTan[(10*14^(1/3) - 3*14^(1/3)*x + 7*(4 + 3*x^2)^(1/3))/(7*Sqrt[3]*(4 + 3*x^2)^(1/3))] - 2*
Log[-10*14^(1/3) + 3*14^(1/3)*x + 14*(4 + 3*x^2)^(1/3)] + Log[100*14^(2/3) - 60*14^(2/3)*x + 9*14^(2/3)*x^2 +
196*(4 + 3*x^2)^(2/3) + 14*(10 - 3*x)*(56 + 42*x^2)^(1/3)])/14^(1/3)
Maple [C] (warning: unable to verify)
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 13.46 (sec) , antiderivative size = 1411, normalized size of antiderivative =
7.16
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| method | result | size |
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\(\text {Expression too large to display}\) |
\(1411\) |
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[In]
int((2+3*x)/(3*x^2+4)^(1/3)/(9*x^2+52*x-12),x,method=_RETURNVERBOSE)
[Out]
1/2*RootOf(RootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3-196)+9604*_Z^2)*ln((68411000939778*RootOf(RootOf(_Z^3-196)^2+9
8*_Z*RootOf(_Z^3-196)+9604*_Z^2)*RootOf(_Z^3-196)^3*x^3+4149815744330613*RootOf(RootOf(_Z^3-196)^2+98*_Z*RootO
f(_Z^3-196)+9604*_Z^2)^2*RootOf(_Z^3-196)^2*x^3+1216195572262720*RootOf(RootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3-1
96)+9604*_Z^2)*RootOf(_Z^3-196)^3*x^2+73774502121433120*RootOf(RootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3-196)+9604*
_Z^2)^2*RootOf(_Z^3-196)^2*x^2-1605300200763000*(3*x^2+4)^(2/3)*RootOf(RootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3-19
6)+9604*_Z^2)*RootOf(_Z^3-196)^2*x+2280366697992600*RootOf(RootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3-196)+9604*_Z^2
)*RootOf(_Z^3-196)^3*x+138327191477687100*RootOf(RootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3-196)+9604*_Z^2)^2*RootOf
(_Z^3-196)^2*x+5351000669210000*(3*x^2+4)^(2/3)*RootOf(RootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3-196)+9604*_Z^2)*Ro
otOf(_Z^3-196)^2-49141842880500*(3*x^2+4)^(1/3)*RootOf(_Z^3-196)^2*x^2-15109141162947363*(3*x^2+4)^(1/3)*RootO
f(_Z^3-196)*RootOf(RootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3-196)+9604*_Z^2)*x^2+327612285870000*(3*x^2+4)^(1/3)*Ro
otOf(_Z^3-196)^2*x+100727607752982420*(3*x^2+4)^(1/3)*RootOf(_Z^3-196)*RootOf(RootOf(_Z^3-196)^2+98*_Z*RootOf(
_Z^3-196)+9604*_Z^2)*x-174517859540250*RootOf(_Z^3-196)*x^3-10586264653904625*RootOf(RootOf(_Z^3-196)^2+98*_Z*
RootOf(_Z^3-196)+9604*_Z^2)*x^3-546020476450000*(3*x^2+4)^(1/3)*RootOf(_Z^3-196)^2-167879346254970700*(3*x^2+4
)^(1/3)*RootOf(_Z^3-196)*RootOf(RootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3-196)+9604*_Z^2)+8525468910767164*RootOf(_
Z^3-196)*x^2+517155495866035894*RootOf(RootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3-196)+9604*_Z^2)*x^2+68621603737722
42*(3*x^2+4)^(2/3)*x-5817261984675000*RootOf(_Z^3-196)*x-352875488463487500*RootOf(RootOf(_Z^3-196)^2+98*_Z*Ro
otOf(_Z^3-196)+9604*_Z^2)*x-22873867912574140*(3*x^2+4)^(2/3)+15504011514569552*RootOf(_Z^3-196)+9404743418398
61192*RootOf(RootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3-196)+9604*_Z^2))/(9*x-2)/(6+x)^2)+1/196*RootOf(_Z^3-196)*ln(
-(84690117231237*RootOf(RootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3-196)+9604*_Z^2)*RootOf(_Z^3-196)^3*x^3+1340855618
4196488*RootOf(RootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3-196)+9604*_Z^2)^2*RootOf(_Z^3-196)^2*x^3+1505602084110880*
RootOf(RootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3-196)+9604*_Z^2)*RootOf(_Z^3-196)^3*x^2+238374332163493120*RootOf(R
ootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3-196)+9604*_Z^2)^2*RootOf(_Z^3-196)^2*x^2-3210600401526000*(3*x^2+4)^(2/3)*
RootOf(RootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3-196)+9604*_Z^2)*RootOf(_Z^3-196)^2*x+2823003907707900*RootOf(RootO
f(_Z^3-196)^2+98*_Z*RootOf(_Z^3-196)+9604*_Z^2)*RootOf(_Z^3-196)^3*x+446951872806549600*RootOf(RootOf(_Z^3-196
)^2+98*_Z*RootOf(_Z^3-196)+9604*_Z^2)^2*RootOf(_Z^3-196)^2*x+10702001338420000*(3*x^2+4)^(2/3)*RootOf(RootOf(_
Z^3-196)^2+98*_Z*RootOf(_Z^3-196)+9604*_Z^2)*RootOf(_Z^3-196)^2-98283685761000*(3*x^2+4)^(1/3)*RootOf(_Z^3-196
)^2*x^2+20586481121316726*(3*x^2+4)^(1/3)*RootOf(_Z^3-196)*RootOf(RootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3-196)+96
04*_Z^2)*x^2+655224571740000*(3*x^2+4)^(1/3)*RootOf(_Z^3-196)^2*x-137243207475444840*(3*x^2+4)^(1/3)*RootOf(_Z
^3-196)*RootOf(RootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3-196)+9604*_Z^2)*x+385426451889099*RootOf(_Z^3-196)*x^3+610
22612838281976*RootOf(RootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3-196)+9604*_Z^2)*x^3-1092040952900000*(3*x^2+4)^(1/3
)*RootOf(_Z^3-196)^2+228738679125741400*(3*x^2+4)^(1/3)*RootOf(_Z^3-196)*RootOf(RootOf(_Z^3-196)^2+98*_Z*RootO
f(_Z^3-196)+9604*_Z^2)-7542989624962646*RootOf(_Z^3-196)*x^2-1194243242183377904*RootOf(RootOf(_Z^3-196)^2+98*
_Z*RootOf(_Z^3-196)+9604*_Z^2)*x^2-20145521550596484*(3*x^2+4)^(2/3)*x+12847548396303300*RootOf(_Z^3-196)*x+20
34087094609399200*RootOf(RootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3-196)+9604*_Z^2)*x+67151738501988280*(3*x^2+4)^(2
/3)-19193353915099208*RootOf(_Z^3-196)-3038786256855632192*RootOf(RootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3-196)+96
04*_Z^2))/(9*x-2)/(6+x)^2)
Fricas [F(-2)]
Exception generated. \[
\int \frac {2+3 x}{\sqrt [3]{4+3 x^2} \left (-12+52 x+9 x^2\right )} \, dx=\text {Exception raised: TypeError}
\]
[In]
integrate((2+3*x)/(3*x^2+4)^(1/3)/(9*x^2+52*x-12),x, algorithm="fricas")
[Out]
Exception raised: TypeError >> Error detected within library code: integrate: implementation incomplete (re
sidue poly has multiple non-linear factors)
Sympy [F]
\[
\int \frac {2+3 x}{\sqrt [3]{4+3 x^2} \left (-12+52 x+9 x^2\right )} \, dx=\int \frac {3 x + 2}{\left (x + 6\right ) \left (9 x - 2\right ) \sqrt [3]{3 x^{2} + 4}}\, dx
\]
[In]
integrate((2+3*x)/(3*x**2+4)**(1/3)/(9*x**2+52*x-12),x)
[Out]
Integral((3*x + 2)/((x + 6)*(9*x - 2)*(3*x**2 + 4)**(1/3)), x)
Maxima [F]
\[
\int \frac {2+3 x}{\sqrt [3]{4+3 x^2} \left (-12+52 x+9 x^2\right )} \, dx=\int { \frac {3 \, x + 2}{{\left (9 \, x^{2} + 52 \, x - 12\right )} {\left (3 \, x^{2} + 4\right )}^{\frac {1}{3}}} \,d x }
\]
[In]
integrate((2+3*x)/(3*x^2+4)^(1/3)/(9*x^2+52*x-12),x, algorithm="maxima")
[Out]
integrate((3*x + 2)/((9*x^2 + 52*x - 12)*(3*x^2 + 4)^(1/3)), x)
Giac [F]
\[
\int \frac {2+3 x}{\sqrt [3]{4+3 x^2} \left (-12+52 x+9 x^2\right )} \, dx=\int { \frac {3 \, x + 2}{{\left (9 \, x^{2} + 52 \, x - 12\right )} {\left (3 \, x^{2} + 4\right )}^{\frac {1}{3}}} \,d x }
\]
[In]
integrate((2+3*x)/(3*x^2+4)^(1/3)/(9*x^2+52*x-12),x, algorithm="giac")
[Out]
integrate((3*x + 2)/((9*x^2 + 52*x - 12)*(3*x^2 + 4)^(1/3)), x)
Mupad [F(-1)]
Timed out. \[
\int \frac {2+3 x}{\sqrt [3]{4+3 x^2} \left (-12+52 x+9 x^2\right )} \, dx=\int \frac {3\,x+2}{{\left (3\,x^2+4\right )}^{1/3}\,\left (9\,x^2+52\,x-12\right )} \,d x
\]
[In]
int((3*x + 2)/((3*x^2 + 4)^(1/3)*(52*x + 9*x^2 - 12)),x)
[Out]
int((3*x + 2)/((3*x^2 + 4)^(1/3)*(52*x + 9*x^2 - 12)), x)