3.22.20 \(\int \frac {\sqrt [4]{-x^3+x^4}}{x (1+x^3)} \, dx\)

Optimal. Leaf size=154 \[ -\frac {1}{3} \text {RootSum}\left [\text {$\#$1}^8-\text {$\#$1}^4+1\& ,\frac {-\text {$\#$1}^4 \log \left (\sqrt [4]{x^4-x^3}-\text {$\#$1} x\right )+\text {$\#$1}^4 \log (x)-\log \left (\sqrt [4]{x^4-x^3}-\text {$\#$1} x\right )+\log (x)}{2 \text {$\#$1}^7-\text {$\#$1}^3}\& \right ]+\frac {2}{3} \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4-x^3}}\right )-\frac {2}{3} \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4-x^3}}\right ) \]

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Rubi [B]  time = 0.65, antiderivative size = 663, normalized size of antiderivative = 4.31, number of steps used = 33, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2056, 6725, 105, 63, 240, 212, 206, 203, 93, 298} \begin {gather*} \frac {2 (-1)^{2/3} \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{3 \sqrt [4]{x-1} x^{3/4}}-\frac {2 \sqrt [3]{-1} \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{3 \sqrt [4]{x-1} x^{3/4}}+\frac {2 \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{3 \sqrt [4]{x-1} x^{3/4}}+\frac {2 \sqrt [4]{2} \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{3 \sqrt [4]{x-1} x^{3/4}}-\frac {2 \left (1-\sqrt [3]{-1}\right )^{5/4} \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{1-\sqrt [3]{-1}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{3 \sqrt [4]{x-1} x^{3/4}}-\frac {2 \sqrt [3]{-1} \sqrt [4]{1+(-1)^{2/3}} \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{1+(-1)^{2/3}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{3 \sqrt [4]{x-1} x^{3/4}}+\frac {2 (-1)^{2/3} \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{3 \sqrt [4]{x-1} x^{3/4}}-\frac {2 \sqrt [3]{-1} \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{3 \sqrt [4]{x-1} x^{3/4}}+\frac {2 \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{3 \sqrt [4]{x-1} x^{3/4}}-\frac {2 \sqrt [4]{2} \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{3 \sqrt [4]{x-1} x^{3/4}}+\frac {2 \left (1-\sqrt [3]{-1}\right )^{5/4} \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{1-\sqrt [3]{-1}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{3 \sqrt [4]{x-1} x^{3/4}}+\frac {2 \sqrt [3]{-1} \sqrt [4]{1+(-1)^{2/3}} \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{1+(-1)^{2/3}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{3 \sqrt [4]{x-1} x^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-x^3 + x^4)^(1/4)/(x*(1 + x^3)),x]

[Out]

(2*(-x^3 + x^4)^(1/4)*ArcTan[(-1 + x)^(1/4)/x^(1/4)])/(3*(-1 + x)^(1/4)*x^(3/4)) - (2*(-1)^(1/3)*(-x^3 + x^4)^
(1/4)*ArcTan[(-1 + x)^(1/4)/x^(1/4)])/(3*(-1 + x)^(1/4)*x^(3/4)) + (2*(-1)^(2/3)*(-x^3 + x^4)^(1/4)*ArcTan[(-1
+ x)^(1/4)/x^(1/4)])/(3*(-1 + x)^(1/4)*x^(3/4)) + (2*2^(1/4)*(-x^3 + x^4)^(1/4)*ArcTan[(2^(1/4)*x^(1/4))/(-1
+ x)^(1/4)])/(3*(-1 + x)^(1/4)*x^(3/4)) - (2*(1 - (-1)^(1/3))^(5/4)*(-x^3 + x^4)^(1/4)*ArcTan[((1 - (-1)^(1/3)
)^(1/4)*x^(1/4))/(-1 + x)^(1/4)])/(3*(-1 + x)^(1/4)*x^(3/4)) - (2*(-1)^(1/3)*(1 + (-1)^(2/3))^(1/4)*(-x^3 + x^
4)^(1/4)*ArcTan[((1 + (-1)^(2/3))^(1/4)*x^(1/4))/(-1 + x)^(1/4)])/(3*(-1 + x)^(1/4)*x^(3/4)) + (2*(-x^3 + x^4)
^(1/4)*ArcTanh[(-1 + x)^(1/4)/x^(1/4)])/(3*(-1 + x)^(1/4)*x^(3/4)) - (2*(-1)^(1/3)*(-x^3 + x^4)^(1/4)*ArcTanh[
(-1 + x)^(1/4)/x^(1/4)])/(3*(-1 + x)^(1/4)*x^(3/4)) + (2*(-1)^(2/3)*(-x^3 + x^4)^(1/4)*ArcTanh[(-1 + x)^(1/4)/
x^(1/4)])/(3*(-1 + x)^(1/4)*x^(3/4)) - (2*2^(1/4)*(-x^3 + x^4)^(1/4)*ArcTanh[(2^(1/4)*x^(1/4))/(-1 + x)^(1/4)]
)/(3*(-1 + x)^(1/4)*x^(3/4)) + (2*(1 - (-1)^(1/3))^(5/4)*(-x^3 + x^4)^(1/4)*ArcTanh[((1 - (-1)^(1/3))^(1/4)*x^
(1/4))/(-1 + x)^(1/4)])/(3*(-1 + x)^(1/4)*x^(3/4)) + (2*(-1)^(1/3)*(1 + (-1)^(2/3))^(1/4)*(-x^3 + x^4)^(1/4)*A
rcTanh[((1 + (-1)^(2/3))^(1/4)*x^(1/4))/(-1 + x)^(1/4)])/(3*(-1 + x)^(1/4)*x^(3/4))

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 105

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Dist[b/f, Int[(a
+ b*x)^(m - 1)*(c + d*x)^n, x], x] - Dist[(b*e - a*f)/f, Int[((a + b*x)^(m - 1)*(c + d*x)^n)/(e + f*x), x], x]
/; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[Simplify[m + n + 1], 0] && (GtQ[m, 0] || ( !RationalQ[m] && (Su
mSimplerQ[m, -1] ||  !SumSimplerQ[n, -1])))

Rule 203

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

Rule 206

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

Rule 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

Rule 240

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[1/(1 - b*x^n)^(p + 1/n + 1), x], x
, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p
+ 1/n]

Rule 298

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b),
2]]}, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &
&  !GtQ[a/b, 0]

Rule 2056

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

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

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Mathematica [A]  time = 0.15, size = 100, normalized size = 0.65 \begin {gather*} -\frac {4 \sqrt [4]{(x-1) x^3} \left ((-1)^{2/3} \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {1-x}{\left (-1+\sqrt [3]{-1}\right ) x}\right )+\, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {x-1}{2 x}\right )-\sqrt [3]{-1} \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {x-1}{\left (1+(-1)^{2/3}\right ) x}\right )\right )}{3 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-x^3 + x^4)^(1/4)/(x*(1 + x^3)),x]

[Out]

(-4*((-1 + x)*x^3)^(1/4)*((-1)^(2/3)*Hypergeometric2F1[1/4, 1, 5/4, (1 - x)/((-1 + (-1)^(1/3))*x)] + Hypergeom
etric2F1[1/4, 1, 5/4, (-1 + x)/(2*x)] - (-1)^(1/3)*Hypergeometric2F1[1/4, 1, 5/4, (-1 + x)/((1 + (-1)^(2/3))*x
)]))/(3*x)

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IntegrateAlgebraic [A]  time = 0.39, size = 154, normalized size = 1.00 \begin {gather*} \frac {2}{3} \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^3+x^4}}\right )-\frac {2}{3} \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^3+x^4}}\right )-\frac {1}{3} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x)-\log \left (\sqrt [4]{-x^3+x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-x^3 + x^4)^(1/4)/(x*(1 + x^3)),x]

[Out]

(2*2^(1/4)*ArcTan[(2^(1/4)*x)/(-x^3 + x^4)^(1/4)])/3 - (2*2^(1/4)*ArcTanh[(2^(1/4)*x)/(-x^3 + x^4)^(1/4)])/3 -
RootSum[1 - #1^4 + #1^8 & , (Log[x] - Log[(-x^3 + x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-x^3 + x^4)^(1/4) -
x*#1]*#1^4)/(-#1^3 + 2*#1^7) & ]/3

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fricas [B]  time = 0.50, size = 765, normalized size = 4.97 \begin {gather*} \frac {1}{12} \, {\left (\sqrt {3} + 2\right )} \sqrt {-4 \, \sqrt {3} + 8} \log \left (\frac {8 \, {\left (2 \, x^{2} + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} x \sqrt {-4 \, \sqrt {3} + 8} + 2 \, \sqrt {x^{4} - x^{3}}\right )}}{x^{2}}\right ) - \frac {1}{12} \, {\left (\sqrt {3} + 2\right )} \sqrt {-4 \, \sqrt {3} + 8} \log \left (\frac {8 \, {\left (2 \, x^{2} - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} x \sqrt {-4 \, \sqrt {3} + 8} + 2 \, \sqrt {x^{4} - x^{3}}\right )}}{x^{2}}\right ) - \frac {1}{6} \, \sqrt {\sqrt {3} + 2} {\left (\sqrt {3} - 2\right )} \log \left (\frac {16 \, {\left (x^{2} + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} x \sqrt {\sqrt {3} + 2} + \sqrt {x^{4} - x^{3}}\right )}}{x^{2}}\right ) + \frac {1}{6} \, \sqrt {\sqrt {3} + 2} {\left (\sqrt {3} - 2\right )} \log \left (\frac {16 \, {\left (x^{2} - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} x \sqrt {\sqrt {3} + 2} + \sqrt {x^{4} - x^{3}}\right )}}{x^{2}}\right ) - \frac {1}{3} \, \sqrt {-4 \, \sqrt {3} + 8} \arctan \left (\frac {\sqrt {2} x \sqrt {-4 \, \sqrt {3} + 8} \sqrt {\frac {2 \, x^{2} + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} x \sqrt {-4 \, \sqrt {3} + 8} + 2 \, \sqrt {x^{4} - x^{3}}}{x^{2}}} + 2 \, \sqrt {3} x - 4 \, x - 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} \sqrt {-4 \, \sqrt {3} + 8}}{2 \, x}\right ) - \frac {1}{3} \, \sqrt {-4 \, \sqrt {3} + 8} \arctan \left (\frac {\sqrt {2} x \sqrt {-4 \, \sqrt {3} + 8} \sqrt {\frac {2 \, x^{2} - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} x \sqrt {-4 \, \sqrt {3} + 8} + 2 \, \sqrt {x^{4} - x^{3}}}{x^{2}}} - 2 \, \sqrt {3} x + 4 \, x - 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} \sqrt {-4 \, \sqrt {3} + 8}}{2 \, x}\right ) - \frac {2}{3} \, \sqrt {\sqrt {3} + 2} \arctan \left (\frac {2 \, x \sqrt {\sqrt {3} + 2} \sqrt {\frac {x^{2} + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} x \sqrt {\sqrt {3} + 2} + \sqrt {x^{4} - x^{3}}}{x^{2}}} - \sqrt {3} x - 2 \, x - 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} \sqrt {\sqrt {3} + 2}}{x}\right ) - \frac {2}{3} \, \sqrt {\sqrt {3} + 2} \arctan \left (\frac {2 \, x \sqrt {\sqrt {3} + 2} \sqrt {\frac {x^{2} - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} x \sqrt {\sqrt {3} + 2} + \sqrt {x^{4} - x^{3}}}{x^{2}}} + \sqrt {3} x + 2 \, x - 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} \sqrt {\sqrt {3} + 2}}{x}\right ) + \frac {4}{3} \cdot 2^{\frac {1}{4}} \arctan \left (\frac {2^{\frac {3}{4}} x \sqrt {\frac {\sqrt {2} x^{2} + \sqrt {x^{4} - x^{3}}}{x^{2}}} - 2^{\frac {3}{4}} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{2 \, x}\right ) - \frac {1}{3} \cdot 2^{\frac {1}{4}} \log \left (\frac {2^{\frac {1}{4}} x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{3} \cdot 2^{\frac {1}{4}} \log \left (-\frac {2^{\frac {1}{4}} x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-x^3)^(1/4)/x/(x^3+1),x, algorithm="fricas")

[Out]

1/12*(sqrt(3) + 2)*sqrt(-4*sqrt(3) + 8)*log(8*(2*x^2 + (x^4 - x^3)^(1/4)*x*sqrt(-4*sqrt(3) + 8) + 2*sqrt(x^4 -
x^3))/x^2) - 1/12*(sqrt(3) + 2)*sqrt(-4*sqrt(3) + 8)*log(8*(2*x^2 - (x^4 - x^3)^(1/4)*x*sqrt(-4*sqrt(3) + 8)
+ 2*sqrt(x^4 - x^3))/x^2) - 1/6*sqrt(sqrt(3) + 2)*(sqrt(3) - 2)*log(16*(x^2 + (x^4 - x^3)^(1/4)*x*sqrt(sqrt(3)
+ 2) + sqrt(x^4 - x^3))/x^2) + 1/6*sqrt(sqrt(3) + 2)*(sqrt(3) - 2)*log(16*(x^2 - (x^4 - x^3)^(1/4)*x*sqrt(sqr
t(3) + 2) + sqrt(x^4 - x^3))/x^2) - 1/3*sqrt(-4*sqrt(3) + 8)*arctan(1/2*(sqrt(2)*x*sqrt(-4*sqrt(3) + 8)*sqrt((
2*x^2 + (x^4 - x^3)^(1/4)*x*sqrt(-4*sqrt(3) + 8) + 2*sqrt(x^4 - x^3))/x^2) + 2*sqrt(3)*x - 4*x - 2*(x^4 - x^3)
^(1/4)*sqrt(-4*sqrt(3) + 8))/x) - 1/3*sqrt(-4*sqrt(3) + 8)*arctan(1/2*(sqrt(2)*x*sqrt(-4*sqrt(3) + 8)*sqrt((2*
x^2 - (x^4 - x^3)^(1/4)*x*sqrt(-4*sqrt(3) + 8) + 2*sqrt(x^4 - x^3))/x^2) - 2*sqrt(3)*x + 4*x - 2*(x^4 - x^3)^(
1/4)*sqrt(-4*sqrt(3) + 8))/x) - 2/3*sqrt(sqrt(3) + 2)*arctan((2*x*sqrt(sqrt(3) + 2)*sqrt((x^2 + (x^4 - x^3)^(1
/4)*x*sqrt(sqrt(3) + 2) + sqrt(x^4 - x^3))/x^2) - sqrt(3)*x - 2*x - 2*(x^4 - x^3)^(1/4)*sqrt(sqrt(3) + 2))/x)
- 2/3*sqrt(sqrt(3) + 2)*arctan((2*x*sqrt(sqrt(3) + 2)*sqrt((x^2 - (x^4 - x^3)^(1/4)*x*sqrt(sqrt(3) + 2) + sqrt
(x^4 - x^3))/x^2) + sqrt(3)*x + 2*x - 2*(x^4 - x^3)^(1/4)*sqrt(sqrt(3) + 2))/x) + 4/3*2^(1/4)*arctan(1/2*(2^(3
/4)*x*sqrt((sqrt(2)*x^2 + sqrt(x^4 - x^3))/x^2) - 2^(3/4)*(x^4 - x^3)^(1/4))/x) - 1/3*2^(1/4)*log((2^(1/4)*x +
(x^4 - x^3)^(1/4))/x) + 1/3*2^(1/4)*log(-(2^(1/4)*x - (x^4 - x^3)^(1/4))/x)

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giac [B]  time = 0.29, size = 392, normalized size = 2.55 \begin {gather*} -\frac {1}{6} \, {\left (\sqrt {6} - \sqrt {2}\right )} \arctan \left (\frac {\sqrt {6} - \sqrt {2} + 4 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {6} + \sqrt {2}}\right ) - \frac {1}{6} \, {\left (\sqrt {6} - \sqrt {2}\right )} \arctan \left (-\frac {\sqrt {6} - \sqrt {2} - 4 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {6} + \sqrt {2}}\right ) - \frac {1}{6} \, {\left (\sqrt {6} + \sqrt {2}\right )} \arctan \left (\frac {\sqrt {6} + \sqrt {2} + 4 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {6} - \sqrt {2}}\right ) - \frac {1}{6} \, {\left (\sqrt {6} + \sqrt {2}\right )} \arctan \left (-\frac {\sqrt {6} + \sqrt {2} - 4 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {6} - \sqrt {2}}\right ) - \frac {1}{12} \, {\left (\sqrt {6} - \sqrt {2}\right )} \log \left (\frac {1}{2} \, {\left (\sqrt {6} + \sqrt {2}\right )} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {-\frac {1}{x} + 1} + 1\right ) + \frac {1}{12} \, {\left (\sqrt {6} - \sqrt {2}\right )} \log \left (-\frac {1}{2} \, {\left (\sqrt {6} + \sqrt {2}\right )} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {-\frac {1}{x} + 1} + 1\right ) - \frac {1}{12} \, {\left (\sqrt {6} + \sqrt {2}\right )} \log \left (\frac {1}{2} \, {\left (\sqrt {6} - \sqrt {2}\right )} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {-\frac {1}{x} + 1} + 1\right ) + \frac {1}{12} \, {\left (\sqrt {6} + \sqrt {2}\right )} \log \left (-\frac {1}{2} \, {\left (\sqrt {6} - \sqrt {2}\right )} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {-\frac {1}{x} + 1} + 1\right ) + \frac {2}{3} \cdot 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{3} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{3} \cdot 2^{\frac {1}{4}} \log \left ({\left | -2^{\frac {1}{4}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-x^3)^(1/4)/x/(x^3+1),x, algorithm="giac")

[Out]

-1/6*(sqrt(6) - sqrt(2))*arctan((sqrt(6) - sqrt(2) + 4*(-1/x + 1)^(1/4))/(sqrt(6) + sqrt(2))) - 1/6*(sqrt(6) -
sqrt(2))*arctan(-(sqrt(6) - sqrt(2) - 4*(-1/x + 1)^(1/4))/(sqrt(6) + sqrt(2))) - 1/6*(sqrt(6) + sqrt(2))*arct
an((sqrt(6) + sqrt(2) + 4*(-1/x + 1)^(1/4))/(sqrt(6) - sqrt(2))) - 1/6*(sqrt(6) + sqrt(2))*arctan(-(sqrt(6) +
sqrt(2) - 4*(-1/x + 1)^(1/4))/(sqrt(6) - sqrt(2))) - 1/12*(sqrt(6) - sqrt(2))*log(1/2*(sqrt(6) + sqrt(2))*(-1/
x + 1)^(1/4) + sqrt(-1/x + 1) + 1) + 1/12*(sqrt(6) - sqrt(2))*log(-1/2*(sqrt(6) + sqrt(2))*(-1/x + 1)^(1/4) +
sqrt(-1/x + 1) + 1) - 1/12*(sqrt(6) + sqrt(2))*log(1/2*(sqrt(6) - sqrt(2))*(-1/x + 1)^(1/4) + sqrt(-1/x + 1) +
1) + 1/12*(sqrt(6) + sqrt(2))*log(-1/2*(sqrt(6) - sqrt(2))*(-1/x + 1)^(1/4) + sqrt(-1/x + 1) + 1) + 2/3*2^(1/
4)*arctan(1/2*2^(3/4)*(-1/x + 1)^(1/4)) + 1/3*2^(1/4)*log(2^(1/4) + (-1/x + 1)^(1/4)) - 1/3*2^(1/4)*log(abs(-2
^(1/4) + (-1/x + 1)^(1/4)))

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maple [B]  time = 93.54, size = 16978, normalized size = 110.25

method	result	size
trager	\(\text {Expression too large to display}\)	\(16978\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4-x^3)^(1/4)/x/(x^3+1),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-x^3)^(1/4)/x/(x^3+1),x, algorithm="maxima")

[Out]

integrate((x^4 - x^3)^(1/4)/((x^3 + 1)*x), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x^4-x^3\right )}^{1/4}}{x\,\left (x^3+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4 - x^3)^(1/4)/(x*(x^3 + 1)),x)

[Out]

int((x^4 - x^3)^(1/4)/(x*(x^3 + 1)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{x^{3} \left (x - 1\right )}}{x \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4-x**3)**(1/4)/x/(x**3+1),x)

[Out]

Integral((x**3*(x - 1))**(1/4)/(x*(x + 1)*(x**2 - x + 1)), x)

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