3.25.91 \(\int \frac {1}{\sqrt [3]{-8+12 x+54 x^2-135 x^3+81 x^4}} \, dx\)

Optimal. Leaf size=206 \[ \frac {\log \left (3^{2/3} \sqrt [3]{81 x^4-135 x^3+54 x^2+12 x-8}-9 x+6\right )}{3 \sqrt [3]{3}}-\frac {\log \left (27 x^2+\sqrt [3]{3} \left (81 x^4-135 x^3+54 x^2+12 x-8\right )^{2/3}+\left (3\ 3^{2/3} x-2\ 3^{2/3}\right ) \sqrt [3]{81 x^4-135 x^3+54 x^2+12 x-8}-36 x+12\right )}{6 \sqrt [3]{3}}-\frac {\tan ^{-1}\left (\frac {3\ 3^{5/6} x-2\ 3^{5/6}}{2 \sqrt [3]{81 x^4-135 x^3+54 x^2+12 x-8}+3 \sqrt [3]{3} x-2 \sqrt [3]{3}}\right )}{3^{5/6}} \]

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Rubi [A]  time = 0.07, antiderivative size = 170, normalized size of antiderivative = 0.83, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6688, 6719, 55, 617, 204, 31} \begin {gather*} \frac {(2-3 x) \sqrt [3]{3 x+1} \log (2-3 x)}{6 \sqrt [3]{3} \sqrt [3]{-(2-3 x)^3 (3 x+1)}}-\frac {(2-3 x) \sqrt [3]{3 x+1} \log \left (\sqrt [3]{3}-\sqrt [3]{3 x+1}\right )}{2 \sqrt [3]{3} \sqrt [3]{-(2-3 x)^3 (3 x+1)}}-\frac {(2-3 x) \sqrt [3]{3 x+1} \tan ^{-1}\left (\frac {2 \sqrt [3]{3 x+1}+\sqrt [3]{3}}{3^{5/6}}\right )}{3^{5/6} \sqrt [3]{-(2-3 x)^3 (3 x+1)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-8 + 12*x + 54*x^2 - 135*x^3 + 81*x^4)^(-1/3),x]

[Out]

-(((2 - 3*x)*(1 + 3*x)^(1/3)*ArcTan[(3^(1/3) + 2*(1 + 3*x)^(1/3))/3^(5/6)])/(3^(5/6)*(-((2 - 3*x)^3*(1 + 3*x))
)^(1/3))) + ((2 - 3*x)*(1 + 3*x)^(1/3)*Log[2 - 3*x])/(6*3^(1/3)*(-((2 - 3*x)^3*(1 + 3*x)))^(1/3)) - ((2 - 3*x)
*(1 + 3*x)^(1/3)*Log[3^(1/3) - (1 + 3*x)^(1/3)])/(2*3^(1/3)*(-((2 - 3*x)^3*(1 + 3*x)))^(1/3))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 55

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, -Simp[L
og[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/
3)], x] - Dist[3/(2*b*q), Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && PosQ
[(b*c - a*d)/b]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6719

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n)^FracPart[p])/(v^(m*F
racPart[p])*w^(n*FracPart[p])), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !
FreeQ[v, x] &&  !FreeQ[w, x]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt [3]{-8+12 x+54 x^2-135 x^3+81 x^4}} \, dx &=\int \frac {1}{\sqrt [3]{(-2+3 x)^3 (1+3 x)}} \, dx\\ &=\frac {\left ((-2+3 x) \sqrt [3]{1+3 x}\right ) \int \frac {1}{(-2+3 x) \sqrt [3]{1+3 x}} \, dx}{\sqrt [3]{(-2+3 x)^3 (1+3 x)}}\\ &=\frac {(2-3 x) \sqrt [3]{1+3 x} \log (2-3 x)}{6 \sqrt [3]{3} \sqrt [3]{-(2-3 x)^3 (1+3 x)}}+\frac {\left ((-2+3 x) \sqrt [3]{1+3 x}\right ) \operatorname {Subst}\left (\int \frac {1}{3^{2/3}+\sqrt [3]{3} x+x^2} \, dx,x,\sqrt [3]{1+3 x}\right )}{2 \sqrt [3]{(-2+3 x)^3 (1+3 x)}}-\frac {\left ((-2+3 x) \sqrt [3]{1+3 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{3}-x} \, dx,x,\sqrt [3]{1+3 x}\right )}{2 \sqrt [3]{3} \sqrt [3]{(-2+3 x)^3 (1+3 x)}}\\ &=\frac {(2-3 x) \sqrt [3]{1+3 x} \log (2-3 x)}{6 \sqrt [3]{3} \sqrt [3]{-(2-3 x)^3 (1+3 x)}}-\frac {(2-3 x) \sqrt [3]{1+3 x} \log \left (\sqrt [3]{3}-\sqrt [3]{1+3 x}\right )}{2 \sqrt [3]{3} \sqrt [3]{-(2-3 x)^3 (1+3 x)}}-\frac {\left ((-2+3 x) \sqrt [3]{1+3 x}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{\frac {1}{3}+x}\right )}{\sqrt [3]{3} \sqrt [3]{(-2+3 x)^3 (1+3 x)}}\\ &=-\frac {(2-3 x) \sqrt [3]{1+3 x} \tan ^{-1}\left (\frac {1}{3} \left (\sqrt {3}+2 \sqrt [6]{3} \sqrt [3]{1+3 x}\right )\right )}{3^{5/6} \sqrt [3]{-(2-3 x)^3 (1+3 x)}}+\frac {(2-3 x) \sqrt [3]{1+3 x} \log (2-3 x)}{6 \sqrt [3]{3} \sqrt [3]{-(2-3 x)^3 (1+3 x)}}-\frac {(2-3 x) \sqrt [3]{1+3 x} \log \left (\sqrt [3]{3}-\sqrt [3]{1+3 x}\right )}{2 \sqrt [3]{3} \sqrt [3]{-(2-3 x)^3 (1+3 x)}}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 100, normalized size = 0.49 \begin {gather*} -\frac {(3 x-2) \sqrt [3]{3 x+1} \left (\sqrt {3} \left (\log (2-3 x)-3 \log \left (\sqrt [3]{3}-\sqrt [3]{3 x+1}\right )\right )-6 \tan ^{-1}\left (\frac {2 \sqrt [3]{3 x+1}+\sqrt [3]{3}}{3^{5/6}}\right )\right )}{6\ 3^{5/6} \sqrt [3]{(3 x-2)^3 (3 x+1)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-8 + 12*x + 54*x^2 - 135*x^3 + 81*x^4)^(-1/3),x]

[Out]

-1/6*((-2 + 3*x)*(1 + 3*x)^(1/3)*(-6*ArcTan[(3^(1/3) + 2*(1 + 3*x)^(1/3))/3^(5/6)] + Sqrt[3]*(Log[2 - 3*x] - 3
*Log[3^(1/3) - (1 + 3*x)^(1/3)])))/(3^(5/6)*((-2 + 3*x)^3*(1 + 3*x))^(1/3))

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IntegrateAlgebraic [A]  time = 0.27, size = 206, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {-2 3^{5/6}+3\ 3^{5/6} x}{-2 \sqrt [3]{3}+3 \sqrt [3]{3} x+2 \sqrt [3]{-8+12 x+54 x^2-135 x^3+81 x^4}}\right )}{3^{5/6}}+\frac {\log \left (6-9 x+3^{2/3} \sqrt [3]{-8+12 x+54 x^2-135 x^3+81 x^4}\right )}{3 \sqrt [3]{3}}-\frac {\log \left (12-36 x+27 x^2+\left (-2 3^{2/3}+3\ 3^{2/3} x\right ) \sqrt [3]{-8+12 x+54 x^2-135 x^3+81 x^4}+\sqrt [3]{3} \left (-8+12 x+54 x^2-135 x^3+81 x^4\right )^{2/3}\right )}{6 \sqrt [3]{3}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-8 + 12*x + 54*x^2 - 135*x^3 + 81*x^4)^(-1/3),x]

[Out]

-(ArcTan[(-2*3^(5/6) + 3*3^(5/6)*x)/(-2*3^(1/3) + 3*3^(1/3)*x + 2*(-8 + 12*x + 54*x^2 - 135*x^3 + 81*x^4)^(1/3
))]/3^(5/6)) + Log[6 - 9*x + 3^(2/3)*(-8 + 12*x + 54*x^2 - 135*x^3 + 81*x^4)^(1/3)]/(3*3^(1/3)) - Log[12 - 36*
x + 27*x^2 + (-2*3^(2/3) + 3*3^(2/3)*x)*(-8 + 12*x + 54*x^2 - 135*x^3 + 81*x^4)^(1/3) + 3^(1/3)*(-8 + 12*x + 5
4*x^2 - 135*x^3 + 81*x^4)^(2/3)]/(6*3^(1/3))

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fricas [A]  time = 0.58, size = 189, normalized size = 0.92 \begin {gather*} -\frac {1}{18} \cdot 3^{\frac {2}{3}} \log \left (\frac {3^{\frac {2}{3}} {\left (9 \, x^{2} - 12 \, x + 4\right )} + 3^{\frac {1}{3}} {\left (81 \, x^{4} - 135 \, x^{3} + 54 \, x^{2} + 12 \, x - 8\right )}^{\frac {1}{3}} {\left (3 \, x - 2\right )} + {\left (81 \, x^{4} - 135 \, x^{3} + 54 \, x^{2} + 12 \, x - 8\right )}^{\frac {2}{3}}}{9 \, x^{2} - 12 \, x + 4}\right ) + \frac {1}{9} \cdot 3^{\frac {2}{3}} \log \left (-\frac {3^{\frac {1}{3}} {\left (3 \, x - 2\right )} - {\left (81 \, x^{4} - 135 \, x^{3} + 54 \, x^{2} + 12 \, x - 8\right )}^{\frac {1}{3}}}{3 \, x - 2}\right ) + \frac {1}{3} \cdot 3^{\frac {1}{6}} \arctan \left (\frac {3^{\frac {1}{6}} {\left (3^{\frac {1}{3}} {\left (3 \, x - 2\right )} + 2 \, {\left (81 \, x^{4} - 135 \, x^{3} + 54 \, x^{2} + 12 \, x - 8\right )}^{\frac {1}{3}}\right )}}{3 \, {\left (3 \, x - 2\right )}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(81*x^4-135*x^3+54*x^2+12*x-8)^(1/3),x, algorithm="fricas")

[Out]

-1/18*3^(2/3)*log((3^(2/3)*(9*x^2 - 12*x + 4) + 3^(1/3)*(81*x^4 - 135*x^3 + 54*x^2 + 12*x - 8)^(1/3)*(3*x - 2)
+ (81*x^4 - 135*x^3 + 54*x^2 + 12*x - 8)^(2/3))/(9*x^2 - 12*x + 4)) + 1/9*3^(2/3)*log(-(3^(1/3)*(3*x - 2) - (
81*x^4 - 135*x^3 + 54*x^2 + 12*x - 8)^(1/3))/(3*x - 2)) + 1/3*3^(1/6)*arctan(1/3*3^(1/6)*(3^(1/3)*(3*x - 2) +
2*(81*x^4 - 135*x^3 + 54*x^2 + 12*x - 8)^(1/3))/(3*x - 2))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (81 \, x^{4} - 135 \, x^{3} + 54 \, x^{2} + 12 \, x - 8\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(81*x^4-135*x^3+54*x^2+12*x-8)^(1/3),x, algorithm="giac")

[Out]

integrate((81*x^4 - 135*x^3 + 54*x^2 + 12*x - 8)^(-1/3), x)

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maple [C]  time = 2.85, size = 1368, normalized size = 6.64

method	result	size
trager	\(\text {Expression too large to display}\)	\(1368\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(81*x^4-135*x^3+54*x^2+12*x-8)^(1/3),x,method=_RETURNVERBOSE)

[Out]

1/4*RootOf(16*RootOf(_Z^3-9)^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2)*ln(-(648*RootOf(16*RootOf(_Z^3-9)^2+36*_Z*RootOf(
_Z^3-9)+81*_Z^2)*RootOf(_Z^3-9)^3*x^3-3645*RootOf(16*RootOf(_Z^3-9)^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2)^2*RootOf(_
Z^3-9)^2*x^3-864*RootOf(16*RootOf(_Z^3-9)^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2)*RootOf(_Z^3-9)^3*x^2+4860*RootOf(16*
RootOf(_Z^3-9)^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2)^2*RootOf(_Z^3-9)^2*x^2+288*RootOf(16*RootOf(_Z^3-9)^2+36*_Z*Roo
tOf(_Z^3-9)+81*_Z^2)*RootOf(_Z^3-9)^3*x+540*(81*x^4-135*x^3+54*x^2+12*x-8)^(2/3)*RootOf(_Z^3-9)^2*RootOf(16*Ro
otOf(_Z^3-9)^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2)-1620*RootOf(16*RootOf(_Z^3-9)^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2)^2*R
ootOf(_Z^3-9)^2*x-2160*(81*x^4-135*x^3+54*x^2+12*x-8)^(1/3)*RootOf(_Z^3-9)^2*x-2592*(81*x^4-135*x^3+54*x^2+12*
x-8)^(1/3)*RootOf(16*RootOf(_Z^3-9)^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2)*RootOf(_Z^3-9)*x+3456*RootOf(_Z^3-9)*x^3-1
9440*RootOf(16*RootOf(_Z^3-9)^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2)*x^3+1440*(81*x^4-135*x^3+54*x^2+12*x-8)^(1/3)*Ro
otOf(_Z^3-9)^2+1728*(81*x^4-135*x^3+54*x^2+12*x-8)^(1/3)*RootOf(16*RootOf(_Z^3-9)^2+36*_Z*RootOf(_Z^3-9)+81*_Z
^2)*RootOf(_Z^3-9)-864*RootOf(_Z^3-9)*x^2+4860*RootOf(16*RootOf(_Z^3-9)^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2)*x^2-34
56*RootOf(_Z^3-9)*x+1008*(81*x^4-135*x^3+54*x^2+12*x-8)^(2/3)+19440*RootOf(16*RootOf(_Z^3-9)^2+36*_Z*RootOf(_Z
^3-9)+81*_Z^2)*x+1664*RootOf(_Z^3-9)-9360*RootOf(16*RootOf(_Z^3-9)^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2))/(-2+3*x)^3
)+1/9*RootOf(_Z^3-9)*ln(-(810*RootOf(16*RootOf(_Z^3-9)^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2)*RootOf(_Z^3-9)^3*x^3-72
9*RootOf(16*RootOf(_Z^3-9)^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2)^2*RootOf(_Z^3-9)^2*x^3-1080*RootOf(16*RootOf(_Z^3-9
)^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2)*RootOf(_Z^3-9)^3*x^2+972*RootOf(16*RootOf(_Z^3-9)^2+36*_Z*RootOf(_Z^3-9)+81*
_Z^2)^2*RootOf(_Z^3-9)^2*x^2+360*RootOf(16*RootOf(_Z^3-9)^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2)*RootOf(_Z^3-9)^3*x-2
70*(81*x^4-135*x^3+54*x^2+12*x-8)^(2/3)*RootOf(_Z^3-9)^2*RootOf(16*RootOf(_Z^3-9)^2+36*_Z*RootOf(_Z^3-9)+81*_Z
^2)-324*RootOf(16*RootOf(_Z^3-9)^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2)^2*RootOf(_Z^3-9)^2*x+1080*(81*x^4-135*x^3+54*
x^2+12*x-8)^(1/3)*RootOf(_Z^3-9)^2*x+1134*(81*x^4-135*x^3+54*x^2+12*x-8)^(1/3)*RootOf(16*RootOf(_Z^3-9)^2+36*_
Z*RootOf(_Z^3-9)+81*_Z^2)*RootOf(_Z^3-9)*x-1080*RootOf(_Z^3-9)*x^3+972*RootOf(16*RootOf(_Z^3-9)^2+36*_Z*RootOf
(_Z^3-9)+81*_Z^2)*x^3-720*(81*x^4-135*x^3+54*x^2+12*x-8)^(1/3)*RootOf(_Z^3-9)^2-756*(81*x^4-135*x^3+54*x^2+12*
x-8)^(1/3)*RootOf(16*RootOf(_Z^3-9)^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2)*RootOf(_Z^3-9)-3240*RootOf(_Z^3-9)*x^2+291
6*RootOf(16*RootOf(_Z^3-9)^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2)*x^2+5760*RootOf(_Z^3-9)*x-576*(81*x^4-135*x^3+54*x^
2+12*x-8)^(2/3)-5184*RootOf(16*RootOf(_Z^3-9)^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2)*x-2080*RootOf(_Z^3-9)+1872*RootO
f(16*RootOf(_Z^3-9)^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2))/(-2+3*x)^3)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (81 \, x^{4} - 135 \, x^{3} + 54 \, x^{2} + 12 \, x - 8\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(81*x^4-135*x^3+54*x^2+12*x-8)^(1/3),x, algorithm="maxima")

[Out]

integrate((81*x^4 - 135*x^3 + 54*x^2 + 12*x - 8)^(-1/3), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (81\,x^4-135\,x^3+54\,x^2+12\,x-8\right )}^{1/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(12*x + 54*x^2 - 135*x^3 + 81*x^4 - 8)^(1/3),x)

[Out]

int(1/(12*x + 54*x^2 - 135*x^3 + 81*x^4 - 8)^(1/3), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{81 x^{4} - 135 x^{3} + 54 x^{2} + 12 x - 8}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(81*x**4-135*x**3+54*x**2+12*x-8)**(1/3),x)

[Out]

Integral((81*x**4 - 135*x**3 + 54*x**2 + 12*x - 8)**(-1/3), x)

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