3.20.79 \(\int \frac {2+3 x}{\sqrt [3]{4+3 x^2} (-12+52 x+9 x^2)} \, dx\)
Optimal. Leaf size=197 \[ \frac {\log \left (14 \sqrt [3]{3 x^2+4}+3 \sqrt [3]{14} x-10 \sqrt [3]{14}\right )}{14 \sqrt [3]{14}}-\frac {\log \left (9\ 14^{2/3} x^2+196 \left (3 x^2+4\right )^{2/3}+\left (140 \sqrt [3]{14}-42 \sqrt [3]{14} x\right ) \sqrt [3]{3 x^2+4}-60\ 14^{2/3} x+100\ 14^{2/3}\right )}{28 \sqrt [3]{14}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {\sqrt [3]{3 x^2+4}}{\sqrt {3}}-\frac {\sqrt [3]{2} \sqrt {3} x}{7^{2/3}}+\frac {10 \sqrt [3]{2}}{\sqrt {3} 7^{2/3}}}{\sqrt [3]{3 x^2+4}}\right )}{14 \sqrt [3]{14}} \]
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Rubi [F] time = 0.01, antiderivative size = 0, normalized size of antiderivative = 0.00,
number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used =
{} \begin {gather*} \int \frac {2+3 x}{\sqrt [3]{4+3 x^2} \left (-12+52 x+9 x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[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*} \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\\ \end {align*}
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Mathematica [F] time = 1.22, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2+3 x}{\sqrt [3]{4+3 x^2} \left (-12+52 x+9 x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Integrate[(2 + 3*x)/((4 + 3*x^2)^(1/3)*(-12 + 52*x + 9*x^2)),x]
[Out]
Integrate[(2 + 3*x)/((4 + 3*x^2)^(1/3)*(-12 + 52*x + 9*x^2)), x]
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IntegrateAlgebraic [A] time = 0.30, size = 197, normalized size = 1.00 \begin {gather*} \frac {\log \left (14 \sqrt [3]{3 x^2+4}+3 \sqrt [3]{14} x-10 \sqrt [3]{14}\right )}{14 \sqrt [3]{14}}-\frac {\log \left (9\ 14^{2/3} x^2+196 \left (3 x^2+4\right )^{2/3}+\left (140 \sqrt [3]{14}-42 \sqrt [3]{14} x\right ) \sqrt [3]{3 x^2+4}-60\ 14^{2/3} x+100\ 14^{2/3}\right )}{28 \sqrt [3]{14}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {\sqrt [3]{3 x^2+4}}{\sqrt {3}}-\frac {\sqrt [3]{2} \sqrt {3} x}{7^{2/3}}+\frac {10 \sqrt [3]{2}}{\sqrt {3} 7^{2/3}}}{\sqrt [3]{3 x^2+4}}\right )}{14 \sqrt [3]{14}} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(2 + 3*x)/((4 + 3*x^2)^(1/3)*(-12 + 52*x + 9*x^2)),x]
[Out]
-1/14*(Sqrt[3]*ArcTan[((10*2^(1/3))/(Sqrt[3]*7^(2/3)) - (2^(1/3)*Sqrt[3]*x)/7^(2/3) + (4 + 3*x^2)^(1/3)/Sqrt[3
])/(4 + 3*x^2)^(1/3)])/14^(1/3) + Log[-10*14^(1/3) + 3*14^(1/3)*x + 14*(4 + 3*x^2)^(1/3)]/(14*14^(1/3)) - Log[
100*14^(2/3) - 60*14^(2/3)*x + 9*14^(2/3)*x^2 + (140*14^(1/3) - 42*14^(1/3)*x)*(4 + 3*x^2)^(1/3) + 196*(4 + 3*
x^2)^(2/3)]/(28*14^(1/3))
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 \, x + 2}{{\left (9 \, x^{2} + 52 \, x - 12\right )} {\left (3 \, x^{2} + 4\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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maple [C] time = 16.64, size = 1411, normalized size = 7.16 \begin {gather*} \text {Expression too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((2+3*x)/(3*x^2+4)^(1/3)/(9*x^2+52*x-12),x)
[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
+98*_Z*RootOf(_Z^3-196)+9604*_Z^2)*RootOf(_Z^3-196)^3*x^3-4149815744330613*RootOf(RootOf(_Z^3-196)^2+98*_Z*Roo
tOf(_Z^3-196)+9604*_Z^2)^2*RootOf(_Z^3-196)^2*x^3+1605300200763000*(3*x^2+4)^(2/3)*RootOf(RootOf(_Z^3-196)^2+9
8*_Z*RootOf(_Z^3-196)+9604*_Z^2)*RootOf(_Z^3-196)^2*x-1216195572262720*RootOf(RootOf(_Z^3-196)^2+98*_Z*RootOf(
_Z^3-196)+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^2-5351000669210000*(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+49141842880500*(3*x^2+4)^(1/3)*RootOf(_Z^3-196)^2*x^2+15109141162947363
*(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^2-228036669799
2600*RootOf(RootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3-196)+9604*_Z^2)*RootOf(_Z^3-196)^3*x-138327191477687100*RootO
f(RootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3-196)+9604*_Z^2)^2*RootOf(_Z^3-196)^2*x-327612285870000*(3*x^2+4)^(1/3)*
RootOf(_Z^3-196)^2*x-100727607752982420*(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)*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^3-6862160373772242*(3*x^2+4)^(2/3)*x+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+22873867912574140*(3*x^2+4)^(2/3)+5817261984675000*RootOf(_Z^3-196)*x+35287548846348750
0*RootOf(RootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3-196)+9604*_Z^2)*x-15504011514569552*RootOf(_Z^3-196)-94047434183
9861192*RootOf(RootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3-196)+9604*_Z^2))/(6+x)^2/(9*x-2))+1/196*RootOf(_Z^3-196)*l
n((-84690117231237*RootOf(RootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3-196)+9604*_Z^2)*RootOf(_Z^3-196)^3*x^3-13408556
184196488*RootOf(RootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3-196)+9604*_Z^2)^2*RootOf(_Z^3-196)^2*x^3+321060040152600
0*(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-15056020841
10880*RootOf(RootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3-196)+9604*_Z^2)*RootOf(_Z^3-196)^3*x^2-238374332163493120*Ro
otOf(RootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3-196)+9604*_Z^2)^2*RootOf(_Z^3-196)^2*x^2-10702001338420000*(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)+9604*_Z^2)*x^2-2823003907707900*RootOf(RootOf(_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-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^3-6
1022612838281976*RootOf(RootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3-196)+9604*_Z^2)*x^3+20145521550596484*(3*x^2+4)^(
2/3)*x+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*RootOf(_Z^3-196)+9604*_Z^2)+7542989624962646*RootOf(_Z^3-196)*x^2+11942432421
83377904*RootOf(RootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3-196)+9604*_Z^2)*x^2-67151738501988280*(3*x^2+4)^(2/3)-128
47548396303300*RootOf(_Z^3-196)*x-2034087094609399200*RootOf(RootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3-196)+9604*_Z
^2)*x+19193353915099208*RootOf(_Z^3-196)+3038786256855632192*RootOf(RootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3-196)+
9604*_Z^2))/(6+x)^2/(9*x-2))
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 \, x + 2}{{\left (9 \, x^{2} + 52 \, x - 12\right )} {\left (3 \, x^{2} + 4\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {3\,x+2}{{\left (3\,x^2+4\right )}^{1/3}\,\left (9\,x^2+52\,x-12\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 x + 2}{\left (x + 6\right ) \left (9 x - 2\right ) \sqrt [3]{3 x^{2} + 4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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