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

Optimal. Leaf size=113 \[ 2 \text {RootSum}\left [\text {$\#$1}^8-3 \text {$\#$1}^4+3\& ,\frac {\log \left (\sqrt [4]{x^4+x^3}-\text {$\#$1} x\right )-\log (x)}{2 \text {$\#$1}^7-3 \text {$\#$1}^3}\& \right ]+\frac {1}{8} \sqrt [4]{x^4+x^3} (4 x+1)+\frac {3}{16} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4+x^3}}\right )-\frac {3}{16} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4+x^3}}\right ) \]

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Rubi [C]  time = 1.02, antiderivative size = 530, normalized size of antiderivative = 4.69, number of steps used = 39, number of rules used = 14, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {2056, 1586, 6728, 50, 63, 331, 298, 203, 206, 911, 105, 93, 205, 208} \begin {gather*} \frac {1}{2} \sqrt [4]{x^4+x^3} x+\frac {1}{8} \sqrt [4]{x^4+x^3}+\frac {3 \sqrt [4]{x^4+x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{16 \sqrt [4]{x+1} x^{3/4}}+\frac {4 i \left (\frac {-\sqrt {3}+i}{-\sqrt {3}+3 i}\right )^{3/4} \sqrt [4]{x^4+x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{\frac {-\sqrt {3}+i}{-\sqrt {3}+3 i}} \sqrt [4]{x+1}}\right )}{\sqrt {3} \sqrt [4]{x+1} x^{3/4}}-\frac {2 \left (\sqrt {3}+i\right ) \sqrt [4]{-\frac {\sqrt {3}+i}{\sqrt {3}+3 i}} \sqrt [4]{x^4+x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{\frac {\sqrt {3}+i}{\sqrt {3}+3 i}} \sqrt [4]{x+1}}\right )}{3^{3/4} \sqrt [4]{x+1} x^{3/4}}-\frac {3 \sqrt [4]{x^4+x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{16 \sqrt [4]{x+1} x^{3/4}}-\frac {4 i \left (\frac {-\sqrt {3}+i}{-\sqrt {3}+3 i}\right )^{3/4} \sqrt [4]{x^4+x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{\frac {-\sqrt {3}+i}{-\sqrt {3}+3 i}} \sqrt [4]{x+1}}\right )}{\sqrt {3} \sqrt [4]{x+1} x^{3/4}}+\frac {2 \left (\sqrt {3}+i\right ) \sqrt [4]{-\frac {\sqrt {3}+i}{\sqrt {3}+3 i}} \sqrt [4]{x^4+x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{\frac {\sqrt {3}+i}{\sqrt {3}+3 i}} \sqrt [4]{x+1}}\right )}{3^{3/4} \sqrt [4]{x+1} x^{3/4}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(x^3 + x^4)^(1/4)/8 + (x*(x^3 + x^4)^(1/4))/2 + (3*(x^3 + x^4)^(1/4)*ArcTan[x^(1/4)/(1 + x)^(1/4)])/(16*x^(3/4
)*(1 + x)^(1/4)) + ((4*I)*((I - Sqrt[3])/(3*I - Sqrt[3]))^(3/4)*(x^3 + x^4)^(1/4)*ArcTan[x^(1/4)/(((I - Sqrt[3
])/(3*I - Sqrt[3]))^(1/4)*(1 + x)^(1/4))])/(Sqrt[3]*x^(3/4)*(1 + x)^(1/4)) - (2*(I + Sqrt[3])*(-((I + Sqrt[3])
/(3*I + Sqrt[3])))^(1/4)*(x^3 + x^4)^(1/4)*ArcTan[x^(1/4)/(((I + Sqrt[3])/(3*I + Sqrt[3]))^(1/4)*(1 + x)^(1/4)
)])/(3^(3/4)*x^(3/4)*(1 + x)^(1/4)) - (3*(x^3 + x^4)^(1/4)*ArcTanh[x^(1/4)/(1 + x)^(1/4)])/(16*x^(3/4)*(1 + x)
^(1/4)) - ((4*I)*((I - Sqrt[3])/(3*I - Sqrt[3]))^(3/4)*(x^3 + x^4)^(1/4)*ArcTanh[x^(1/4)/(((I - Sqrt[3])/(3*I
- Sqrt[3]))^(1/4)*(1 + x)^(1/4))])/(Sqrt[3]*x^(3/4)*(1 + x)^(1/4)) + (2*(I + Sqrt[3])*(-((I + Sqrt[3])/(3*I +
Sqrt[3])))^(1/4)*(x^3 + x^4)^(1/4)*ArcTanh[x^(1/4)/(((I + Sqrt[3])/(3*I + Sqrt[3]))^(1/4)*(1 + x)^(1/4))])/(3^
(3/4)*x^(3/4)*(1 + x)^(1/4))

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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 205

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

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 208

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

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 331

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(
p + (m + 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)] && IntegersQ[m, p + (m + 1)/n]

Rule 911

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n, 1/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x
] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[m] &&  !IntegerQ[n]

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 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 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\left (-1+x^3\right ) \sqrt [4]{x^3+x^4}}{1+x^3} \, dx &=\frac {\sqrt [4]{x^3+x^4} \int \frac {x^{3/4} \sqrt [4]{1+x} \left (-1+x^3\right )}{1+x^3} \, dx}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {\sqrt [4]{x^3+x^4} \int \frac {x^{3/4} \left (-1+x^3\right )}{(1+x)^{3/4} \left (1-x+x^2\right )} \, dx}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {\sqrt [4]{x^3+x^4} \int \left (\frac {x^{3/4}}{(1+x)^{3/4}}+\frac {x^{7/4}}{(1+x)^{3/4}}-\frac {2 x^{3/4}}{(1+x)^{3/4} \left (1-x+x^2\right )}\right ) \, dx}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {\sqrt [4]{x^3+x^4} \int \frac {x^{3/4}}{(1+x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \int \frac {x^{7/4}}{(1+x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{1+x}}-\frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \int \frac {x^{3/4}}{(1+x)^{3/4} \left (1-x+x^2\right )} \, dx}{x^{3/4} \sqrt [4]{1+x}}\\ &=\sqrt [4]{x^3+x^4}+\frac {1}{2} x \sqrt [4]{x^3+x^4}-\frac {\left (3 \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4}} \, dx}{4 x^{3/4} \sqrt [4]{1+x}}-\frac {\left (7 \sqrt [4]{x^3+x^4}\right ) \int \frac {x^{3/4}}{(1+x)^{3/4}} \, dx}{8 x^{3/4} \sqrt [4]{1+x}}-\frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \int \left (\frac {2 i x^{3/4}}{\sqrt {3} \left (1+i \sqrt {3}-2 x\right ) (1+x)^{3/4}}+\frac {2 i x^{3/4}}{\sqrt {3} (1+x)^{3/4} \left (-1+i \sqrt {3}+2 x\right )}\right ) \, dx}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {1}{8} \sqrt [4]{x^3+x^4}+\frac {1}{2} x \sqrt [4]{x^3+x^4}+\frac {\left (21 \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4}} \, dx}{32 x^{3/4} \sqrt [4]{1+x}}-\frac {\left (3 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}-\frac {\left (4 i \sqrt [4]{x^3+x^4}\right ) \int \frac {x^{3/4}}{\left (1+i \sqrt {3}-2 x\right ) (1+x)^{3/4}} \, dx}{\sqrt {3} x^{3/4} \sqrt [4]{1+x}}-\frac {\left (4 i \sqrt [4]{x^3+x^4}\right ) \int \frac {x^{3/4}}{(1+x)^{3/4} \left (-1+i \sqrt {3}+2 x\right )} \, dx}{\sqrt {3} x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {1}{8} \sqrt [4]{x^3+x^4}+\frac {1}{2} x \sqrt [4]{x^3+x^4}+\frac {\left (21 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{8 x^{3/4} \sqrt [4]{1+x}}-\frac {\left (3 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 i \left (-1-i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\left (1+i \sqrt {3}-2 x\right ) \sqrt [4]{x} (1+x)^{3/4}} \, dx}{\sqrt {3} x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 i \left (-1+i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4} \left (-1+i \sqrt {3}+2 x\right )} \, dx}{\sqrt {3} x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {1}{8} \sqrt [4]{x^3+x^4}+\frac {1}{2} x \sqrt [4]{x^3+x^4}-\frac {\left (3 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (3 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (21 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{8 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (8 i \left (-1-i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1+i \sqrt {3}-\left (3+i \sqrt {3}\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{\sqrt {3} x^{3/4} \sqrt [4]{1+x}}+\frac {\left (8 i \left (-1+i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{-1+i \sqrt {3}-\left (-3+i \sqrt {3}\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{\sqrt {3} x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {1}{8} \sqrt [4]{x^3+x^4}+\frac {1}{2} x \sqrt [4]{x^3+x^4}+\frac {3 \sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 x^{3/4} \sqrt [4]{1+x}}-\frac {3 \sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (21 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{16 x^{3/4} \sqrt [4]{1+x}}-\frac {\left (21 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{16 x^{3/4} \sqrt [4]{1+x}}-\frac {\left (4 \left (-1-i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {i-\sqrt {3}}-\sqrt {3 i-\sqrt {3}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{\sqrt {3 \left (3 i-\sqrt {3}\right )} x^{3/4} \sqrt [4]{1+x}}+\frac {\left (4 \left (-1-i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {i-\sqrt {3}}+\sqrt {3 i-\sqrt {3}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{\sqrt {3 \left (3 i-\sqrt {3}\right )} x^{3/4} \sqrt [4]{1+x}}+\frac {\left (4 \left (-1+i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {i+\sqrt {3}}-\sqrt {3 i+\sqrt {3}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{\sqrt {3 \left (3 i+\sqrt {3}\right )} x^{3/4} \sqrt [4]{1+x}}-\frac {\left (4 \left (-1+i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {i+\sqrt {3}}+\sqrt {3 i+\sqrt {3}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{\sqrt {3 \left (3 i+\sqrt {3}\right )} x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {1}{8} \sqrt [4]{x^3+x^4}+\frac {1}{2} x \sqrt [4]{x^3+x^4}+\frac {3 \sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{16 x^{3/4} \sqrt [4]{1+x}}-\frac {4 \sqrt [4]{\frac {i-\sqrt {3}}{3 i-\sqrt {3}}} \left (1+i \sqrt {3}\right ) \sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{\frac {i-\sqrt {3}}{3 i-\sqrt {3}}} \sqrt [4]{1+x}}\right )}{\sqrt {3 \left (i-\sqrt {3}\right )} \sqrt {3 i-\sqrt {3}} x^{3/4} \sqrt [4]{1+x}}-\frac {4 i \left (\frac {i+\sqrt {3}}{3 i+\sqrt {3}}\right )^{3/4} \sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{\frac {i+\sqrt {3}}{3 i+\sqrt {3}}} \sqrt [4]{1+x}}\right )}{\sqrt {3} x^{3/4} \sqrt [4]{1+x}}-\frac {3 \sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{16 x^{3/4} \sqrt [4]{1+x}}+\frac {4 \sqrt [4]{\frac {i-\sqrt {3}}{3 i-\sqrt {3}}} \left (1+i \sqrt {3}\right ) \sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{\frac {i-\sqrt {3}}{3 i-\sqrt {3}}} \sqrt [4]{1+x}}\right )}{\sqrt {3 \left (i-\sqrt {3}\right )} \sqrt {3 i-\sqrt {3}} x^{3/4} \sqrt [4]{1+x}}+\frac {4 i \left (\frac {i+\sqrt {3}}{3 i+\sqrt {3}}\right )^{3/4} \sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{\frac {i+\sqrt {3}}{3 i+\sqrt {3}}} \sqrt [4]{1+x}}\right )}{\sqrt {3} x^{3/4} \sqrt [4]{1+x}}\\ \end {align*}

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Mathematica [C]  time = 0.22, size = 119, normalized size = 1.05 \begin {gather*} \frac {4 \sqrt [4]{x^3 (x+1)} \left (9 x (x+1) \, _2F_1\left (-\frac {1}{4},\frac {7}{4};\frac {11}{4};-x\right )-14 i \sqrt {3} \sqrt [4]{x+1} \left (\, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {\left (3-i \sqrt {3}\right ) x}{2 (x+1)}\right )-\, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {\left (3+i \sqrt {3}\right ) x}{2 (x+1)}\right )\right )\right )}{63 (x+1)^{5/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(x^3*(1 + x))^(1/4)*(9*x*(1 + x)*Hypergeometric2F1[-1/4, 7/4, 11/4, -x] - (14*I)*Sqrt[3]*(1 + x)^(1/4)*(Hyp
ergeometric2F1[3/4, 1, 7/4, ((3 - I*Sqrt[3])*x)/(2*(1 + x))] - Hypergeometric2F1[3/4, 1, 7/4, ((3 + I*Sqrt[3])
*x)/(2*(1 + x))])))/(63*(1 + x)^(5/4))

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IntegrateAlgebraic [A]  time = 0.41, size = 113, normalized size = 1.00 \begin {gather*} \frac {1}{8} (1+4 x) \sqrt [4]{x^3+x^4}+\frac {3}{16} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )-\frac {3}{16} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )+2 \text {RootSum}\left [3-3 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{x^3+x^4}-x \text {$\#$1}\right )}{-3 \text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((1 + 4*x)*(x^3 + x^4)^(1/4))/8 + (3*ArcTan[x/(x^3 + x^4)^(1/4)])/16 - (3*ArcTanh[x/(x^3 + x^4)^(1/4)])/16 + 2
*RootSum[3 - 3*#1^4 + #1^8 & , (-Log[x] + Log[(x^3 + x^4)^(1/4) - x*#1])/(-3*#1^3 + 2*#1^7) & ]

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fricas [B]  time = 0.75, size = 2796, normalized size = 24.74

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/2187*2187^(7/8)*sqrt(-sqrt(2) + 2)*arctan(-1/4374*(4374*x*sqrt(sqrt(2) + 2)*sqrt(-sqrt(2) + 2) + sqrt(2)*(21
87^(5/8)*x*sqrt(sqrt(2) + 2) - 27*2187^(1/8)*x*sqrt(-sqrt(2) + 2))*sqrt((162*27^(3/4)*x^2 + (x^4 + x^3)^(1/4)*
(2187^(7/8)*x*sqrt(sqrt(2) + 2) - 81*2187^(3/8)*x*sqrt(-sqrt(2) + 2)) + 1458*sqrt(x^4 + x^3))/x^2) - 4374*sqrt
(3)*x - 54*(x^4 + x^3)^(1/4)*(2187^(5/8)*sqrt(sqrt(2) + 2) - 27*2187^(1/8)*sqrt(-sqrt(2) + 2)))/(x*(sqrt(2) +
2) - x)) + 2/2187*2187^(7/8)*sqrt(-sqrt(2) + 2)*arctan(1/4374*(4374*x*sqrt(sqrt(2) + 2)*sqrt(-sqrt(2) + 2) - s
qrt(2)*(2187^(5/8)*x*sqrt(sqrt(2) + 2) - 27*2187^(1/8)*x*sqrt(-sqrt(2) + 2))*sqrt((162*27^(3/4)*x^2 - (x^4 + x
^3)^(1/4)*(2187^(7/8)*x*sqrt(sqrt(2) + 2) - 81*2187^(3/8)*x*sqrt(-sqrt(2) + 2)) + 1458*sqrt(x^4 + x^3))/x^2) -
4374*sqrt(3)*x + 54*(x^4 + x^3)^(1/4)*(2187^(5/8)*sqrt(sqrt(2) + 2) - 27*2187^(1/8)*sqrt(-sqrt(2) + 2)))/(x*(
sqrt(2) + 2) - x)) - 2/2187*2187^(7/8)*sqrt(sqrt(2) + 2)*arctan(-1/4374*(4374*x*sqrt(sqrt(2) + 2)*sqrt(-sqrt(2
) + 2) - sqrt(2)*(2187^(5/8)*x*sqrt(-sqrt(2) + 2) + 27*2187^(1/8)*x*sqrt(sqrt(2) + 2))*sqrt((162*27^(3/4)*x^2
+ (x^4 + x^3)^(1/4)*(2187^(7/8)*x*sqrt(-sqrt(2) + 2) + 81*2187^(3/8)*x*sqrt(sqrt(2) + 2)) + 1458*sqrt(x^4 + x^
3))/x^2) + 4374*sqrt(3)*x + 54*(x^4 + x^3)^(1/4)*(2187^(5/8)*sqrt(-sqrt(2) + 2) + 27*2187^(1/8)*sqrt(sqrt(2) +
2)))/(x*(sqrt(2) + 2) - 3*x)) - 2/2187*2187^(7/8)*sqrt(sqrt(2) + 2)*arctan(1/4374*(4374*x*sqrt(sqrt(2) + 2)*s
qrt(-sqrt(2) + 2) + sqrt(2)*(2187^(5/8)*x*sqrt(-sqrt(2) + 2) + 27*2187^(1/8)*x*sqrt(sqrt(2) + 2))*sqrt((162*27
^(3/4)*x^2 - (x^4 + x^3)^(1/4)*(2187^(7/8)*x*sqrt(-sqrt(2) + 2) + 81*2187^(3/8)*x*sqrt(sqrt(2) + 2)) + 1458*sq
rt(x^4 + x^3))/x^2) + 4374*sqrt(3)*x - 54*(x^4 + x^3)^(1/4)*(2187^(5/8)*sqrt(-sqrt(2) + 2) + 27*2187^(1/8)*sqr
t(sqrt(2) + 2)))/(x*(sqrt(2) + 2) - 3*x)) + 1/4374*2187^(7/8)*sqrt(sqrt(2) + 2)*log(1458*(162*27^(3/4)*x^2 + (
x^4 + x^3)^(1/4)*(2187^(7/8)*x*sqrt(sqrt(2) + 2) - 81*2187^(3/8)*x*sqrt(-sqrt(2) + 2)) + 1458*sqrt(x^4 + x^3))
/x^2) - 1/4374*2187^(7/8)*sqrt(sqrt(2) + 2)*log(1458*(162*27^(3/4)*x^2 - (x^4 + x^3)^(1/4)*(2187^(7/8)*x*sqrt(
sqrt(2) + 2) - 81*2187^(3/8)*x*sqrt(-sqrt(2) + 2)) + 1458*sqrt(x^4 + x^3))/x^2) + 1/4374*2187^(7/8)*sqrt(-sqrt
(2) + 2)*log(1458*(162*27^(3/4)*x^2 + (x^4 + x^3)^(1/4)*(2187^(7/8)*x*sqrt(-sqrt(2) + 2) + 81*2187^(3/8)*x*sqr
t(sqrt(2) + 2)) + 1458*sqrt(x^4 + x^3))/x^2) - 1/4374*2187^(7/8)*sqrt(-sqrt(2) + 2)*log(1458*(162*27^(3/4)*x^2
- (x^4 + x^3)^(1/4)*(2187^(7/8)*x*sqrt(-sqrt(2) + 2) + 81*2187^(3/8)*x*sqrt(sqrt(2) + 2)) + 1458*sqrt(x^4 + x
^3))/x^2) - 1/2187*(2187^(7/8)*sqrt(2)*sqrt(sqrt(2) + 2) - 2187^(7/8)*sqrt(2)*sqrt(-sqrt(2) + 2))*arctan(1/874
8*(8748*x*(sqrt(2) + 2) - 8748*sqrt(3)*x + ((2187^(5/8)*sqrt(2)*x - 27*2187^(1/8)*sqrt(2)*x)*sqrt(sqrt(2) + 2)
+ (2187^(5/8)*sqrt(2)*x + 27*2187^(1/8)*sqrt(2)*x)*sqrt(-sqrt(2) + 2))*sqrt((324*27^(3/4)*x^2 + (x^4 + x^3)^(
1/4)*((2187^(7/8)*sqrt(2)*x - 81*2187^(3/8)*sqrt(2)*x)*sqrt(sqrt(2) + 2) + (2187^(7/8)*sqrt(2)*x + 81*2187^(3/
8)*sqrt(2)*x)*sqrt(-sqrt(2) + 2)) + 2916*sqrt(x^4 + x^3))/x^2) - 54*(x^4 + x^3)^(1/4)*((2187^(5/8)*sqrt(2) - 2
7*2187^(1/8)*sqrt(2))*sqrt(sqrt(2) + 2) + (2187^(5/8)*sqrt(2) + 27*2187^(1/8)*sqrt(2))*sqrt(-sqrt(2) + 2)) - 1
7496*x)/(x*sqrt(sqrt(2) + 2)*sqrt(-sqrt(2) + 2) + x)) - 1/2187*(2187^(7/8)*sqrt(2)*sqrt(sqrt(2) + 2) - 2187^(7
/8)*sqrt(2)*sqrt(-sqrt(2) + 2))*arctan(-1/8748*(8748*x*(sqrt(2) + 2) - 8748*sqrt(3)*x - ((2187^(5/8)*sqrt(2)*x
- 27*2187^(1/8)*sqrt(2)*x)*sqrt(sqrt(2) + 2) + (2187^(5/8)*sqrt(2)*x + 27*2187^(1/8)*sqrt(2)*x)*sqrt(-sqrt(2)
+ 2))*sqrt((324*27^(3/4)*x^2 - (x^4 + x^3)^(1/4)*((2187^(7/8)*sqrt(2)*x - 81*2187^(3/8)*sqrt(2)*x)*sqrt(sqrt(
2) + 2) + (2187^(7/8)*sqrt(2)*x + 81*2187^(3/8)*sqrt(2)*x)*sqrt(-sqrt(2) + 2)) + 2916*sqrt(x^4 + x^3))/x^2) +
54*(x^4 + x^3)^(1/4)*((2187^(5/8)*sqrt(2) - 27*2187^(1/8)*sqrt(2))*sqrt(sqrt(2) + 2) + (2187^(5/8)*sqrt(2) + 2
7*2187^(1/8)*sqrt(2))*sqrt(-sqrt(2) + 2)) - 17496*x)/(x*sqrt(sqrt(2) + 2)*sqrt(-sqrt(2) + 2) + x)) + 1/2187*(2
187^(7/8)*sqrt(2)*sqrt(sqrt(2) + 2) + 2187^(7/8)*sqrt(2)*sqrt(-sqrt(2) + 2))*arctan(1/8748*(8748*x*(sqrt(2) +
2) + 8748*sqrt(3)*x - ((2187^(5/8)*sqrt(2)*x + 27*2187^(1/8)*sqrt(2)*x)*sqrt(sqrt(2) + 2) - (2187^(5/8)*sqrt(2
)*x - 27*2187^(1/8)*sqrt(2)*x)*sqrt(-sqrt(2) + 2))*sqrt((324*27^(3/4)*x^2 + (x^4 + x^3)^(1/4)*((2187^(7/8)*sqr
t(2)*x + 81*2187^(3/8)*sqrt(2)*x)*sqrt(sqrt(2) + 2) - (2187^(7/8)*sqrt(2)*x - 81*2187^(3/8)*sqrt(2)*x)*sqrt(-s
qrt(2) + 2)) + 2916*sqrt(x^4 + x^3))/x^2) + 54*(x^4 + x^3)^(1/4)*((2187^(5/8)*sqrt(2) + 27*2187^(1/8)*sqrt(2))
*sqrt(sqrt(2) + 2) - (2187^(5/8)*sqrt(2) - 27*2187^(1/8)*sqrt(2))*sqrt(-sqrt(2) + 2)) - 17496*x)/(x*sqrt(sqrt(
2) + 2)*sqrt(-sqrt(2) + 2) - x)) + 1/2187*(2187^(7/8)*sqrt(2)*sqrt(sqrt(2) + 2) + 2187^(7/8)*sqrt(2)*sqrt(-sqr
t(2) + 2))*arctan(-1/8748*(8748*x*(sqrt(2) + 2) + 8748*sqrt(3)*x + ((2187^(5/8)*sqrt(2)*x + 27*2187^(1/8)*sqrt
(2)*x)*sqrt(sqrt(2) + 2) - (2187^(5/8)*sqrt(2)*x - 27*2187^(1/8)*sqrt(2)*x)*sqrt(-sqrt(2) + 2))*sqrt((324*27^(
3/4)*x^2 - (x^4 + x^3)^(1/4)*((2187^(7/8)*sqrt(2)*x + 81*2187^(3/8)*sqrt(2)*x)*sqrt(sqrt(2) + 2) - (2187^(7/8)
*sqrt(2)*x - 81*2187^(3/8)*sqrt(2)*x)*sqrt(-sqrt(2) + 2)) + 2916*sqrt(x^4 + x^3))/x^2) - 54*(x^4 + x^3)^(1/4)*
((2187^(5/8)*sqrt(2) + 27*2187^(1/8)*sqrt(2))*sqrt(sqrt(2) + 2) - (2187^(5/8)*sqrt(2) - 27*2187^(1/8)*sqrt(2))
*sqrt(-sqrt(2) + 2)) - 17496*x)/(x*sqrt(sqrt(2) + 2)*sqrt(-sqrt(2) + 2) - x)) + 1/8748*(2187^(7/8)*sqrt(2)*sqr
t(sqrt(2) + 2) - 2187^(7/8)*sqrt(2)*sqrt(-sqrt(2) + 2))*log(2916*(324*27^(3/4)*x^2 + (x^4 + x^3)^(1/4)*((2187^
(7/8)*sqrt(2)*x + 81*2187^(3/8)*sqrt(2)*x)*sqrt(sqrt(2) + 2) - (2187^(7/8)*sqrt(2)*x - 81*2187^(3/8)*sqrt(2)*x
)*sqrt(-sqrt(2) + 2)) + 2916*sqrt(x^4 + x^3))/x^2) - 1/8748*(2187^(7/8)*sqrt(2)*sqrt(sqrt(2) + 2) - 2187^(7/8)
*sqrt(2)*sqrt(-sqrt(2) + 2))*log(2916*(324*27^(3/4)*x^2 - (x^4 + x^3)^(1/4)*((2187^(7/8)*sqrt(2)*x + 81*2187^(
3/8)*sqrt(2)*x)*sqrt(sqrt(2) + 2) - (2187^(7/8)*sqrt(2)*x - 81*2187^(3/8)*sqrt(2)*x)*sqrt(-sqrt(2) + 2)) + 291
6*sqrt(x^4 + x^3))/x^2) + 1/8748*(2187^(7/8)*sqrt(2)*sqrt(sqrt(2) + 2) + 2187^(7/8)*sqrt(2)*sqrt(-sqrt(2) + 2)
)*log(2916*(324*27^(3/4)*x^2 + (x^4 + x^3)^(1/4)*((2187^(7/8)*sqrt(2)*x - 81*2187^(3/8)*sqrt(2)*x)*sqrt(sqrt(2
) + 2) + (2187^(7/8)*sqrt(2)*x + 81*2187^(3/8)*sqrt(2)*x)*sqrt(-sqrt(2) + 2)) + 2916*sqrt(x^4 + x^3))/x^2) - 1
/8748*(2187^(7/8)*sqrt(2)*sqrt(sqrt(2) + 2) + 2187^(7/8)*sqrt(2)*sqrt(-sqrt(2) + 2))*log(2916*(324*27^(3/4)*x^
2 - (x^4 + x^3)^(1/4)*((2187^(7/8)*sqrt(2)*x - 81*2187^(3/8)*sqrt(2)*x)*sqrt(sqrt(2) + 2) + (2187^(7/8)*sqrt(2
)*x + 81*2187^(3/8)*sqrt(2)*x)*sqrt(-sqrt(2) + 2)) + 2916*sqrt(x^4 + x^3))/x^2) + 1/8*(x^4 + x^3)^(1/4)*(4*x +
1) - 3/16*arctan((x^4 + x^3)^(1/4)/x) - 3/32*log((x + (x^4 + x^3)^(1/4))/x) + 3/32*log(-(x - (x^4 + x^3)^(1/4
))/x)

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giac [B]  time = 0.20, size = 57, normalized size = 0.50 \begin {gather*} \frac {1}{8} \, {\left ({\left (\frac {1}{x} + 1\right )}^{\frac {5}{4}} + 3 \, {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )} x^{2} - \frac {3}{16} \, \arctan \left ({\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \frac {3}{32} \, \log \left ({\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {3}{32} \, \log \left ({\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*((1/x + 1)^(5/4) + 3*(1/x + 1)^(1/4))*x^2 - 3/16*arctan((1/x + 1)^(1/4)) - 3/32*log((1/x + 1)^(1/4) + 1) +
3/32*log((1/x + 1)^(1/4) - 1)

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maple [B]  time = 23.85, size = 1783, normalized size = 15.78

method	result	size
trager	\(\text {Expression too large to display}\)	\(1783\)
risch	\(\text {Expression too large to display}\)	\(3761\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1/8+1/2*x)*(x^4+x^3)^(1/4)+3/32*ln((2*(x^4+x^3)^(3/4)-2*(x^4+x^3)^(1/2)*x+2*x^2*(x^4+x^3)^(1/4)-2*x^3-x^2)/x^
2)-3/32*RootOf(_Z^2+1)*ln((2*(x^4+x^3)^(1/2)*RootOf(_Z^2+1)*x-2*RootOf(_Z^2+1)*x^3+2*(x^4+x^3)^(3/4)-2*x^2*(x^
4+x^3)^(1/4)-RootOf(_Z^2+1)*x^2)/x^2)-2/3*RootOf(_Z^8+3)*ln(-(-5*x^3*RootOf(_Z^8+3)^11+5*RootOf(_Z^8+3)^11*x^2
+86*RootOf(_Z^8+3)^7*x^3+96*(x^4+x^3)^(1/4)*RootOf(_Z^8+3)^6*x^2+44*RootOf(_Z^8+3)^7*x^2-180*(x^4+x^3)^(1/2)*R
ootOf(_Z^8+3)^5*x+84*(x^4+x^3)^(3/4)*RootOf(_Z^8+3)^4+171*RootOf(_Z^8+3)^3*x^3-264*(x^4+x^3)^(1/4)*RootOf(_Z^8
+3)^2*x^2+63*RootOf(_Z^8+3)^3*x^2-12*(x^4+x^3)^(1/2)*RootOf(_Z^8+3)*x+276*(x^4+x^3)^(3/4))/(x*RootOf(_Z^8+3)^4
-RootOf(_Z^8+3)^4+x+1)/x^2)+2/3*RootOf(_Z^2+RootOf(_Z^8+3)^2)*ln(-(-5*RootOf(_Z^2+RootOf(_Z^8+3)^2)*RootOf(_Z^
8+3)^10*x^3+5*RootOf(_Z^2+RootOf(_Z^8+3)^2)*RootOf(_Z^8+3)^10*x^2+86*RootOf(_Z^8+3)^6*RootOf(_Z^2+RootOf(_Z^8+
3)^2)*x^3+44*RootOf(_Z^8+3)^6*RootOf(_Z^2+RootOf(_Z^8+3)^2)*x^2-96*(x^4+x^3)^(1/4)*RootOf(_Z^8+3)^6*x^2+180*(x
^4+x^3)^(1/2)*RootOf(_Z^8+3)^4*RootOf(_Z^2+RootOf(_Z^8+3)^2)*x+84*(x^4+x^3)^(3/4)*RootOf(_Z^8+3)^4+171*RootOf(
_Z^2+RootOf(_Z^8+3)^2)*RootOf(_Z^8+3)^2*x^3+63*RootOf(_Z^2+RootOf(_Z^8+3)^2)*RootOf(_Z^8+3)^2*x^2+264*(x^4+x^3
)^(1/4)*RootOf(_Z^8+3)^2*x^2+12*RootOf(_Z^2+RootOf(_Z^8+3)^2)*(x^4+x^3)^(1/2)*x+276*(x^4+x^3)^(3/4))/(x*RootOf
(_Z^8+3)^4-RootOf(_Z^8+3)^4+x+1)/x^2)-2/3*RootOf(_Z^2+RootOf(_Z^8+3)*RootOf(_Z^2+RootOf(_Z^8+3)^2))*ln(-(5*Roo
tOf(_Z^2+RootOf(_Z^8+3)^2)*RootOf(_Z^2+RootOf(_Z^8+3)*RootOf(_Z^2+RootOf(_Z^8+3)^2))*RootOf(_Z^8+3)^9*x^3-5*Ro
otOf(_Z^2+RootOf(_Z^8+3)^2)*RootOf(_Z^2+RootOf(_Z^8+3)*RootOf(_Z^2+RootOf(_Z^8+3)^2))*RootOf(_Z^8+3)^9*x^2+86*
RootOf(_Z^2+RootOf(_Z^8+3)^2)*RootOf(_Z^8+3)^5*RootOf(_Z^2+RootOf(_Z^8+3)*RootOf(_Z^2+RootOf(_Z^8+3)^2))*x^3+4
4*RootOf(_Z^2+RootOf(_Z^8+3)^2)*RootOf(_Z^8+3)^5*RootOf(_Z^2+RootOf(_Z^8+3)*RootOf(_Z^2+RootOf(_Z^8+3)^2))*x^2
+96*RootOf(_Z^2+RootOf(_Z^8+3)^2)*(x^4+x^3)^(1/4)*RootOf(_Z^8+3)^5*x^2+180*(x^4+x^3)^(1/2)*RootOf(_Z^8+3)^4*Ro
otOf(_Z^2+RootOf(_Z^8+3)*RootOf(_Z^2+RootOf(_Z^8+3)^2))*x-84*(x^4+x^3)^(3/4)*RootOf(_Z^8+3)^4-171*RootOf(_Z^2+
RootOf(_Z^8+3)^2)*RootOf(_Z^2+RootOf(_Z^8+3)*RootOf(_Z^2+RootOf(_Z^8+3)^2))*RootOf(_Z^8+3)*x^3-63*RootOf(_Z^2+
RootOf(_Z^8+3)^2)*RootOf(_Z^2+RootOf(_Z^8+3)*RootOf(_Z^2+RootOf(_Z^8+3)^2))*RootOf(_Z^8+3)*x^2+264*RootOf(_Z^2
+RootOf(_Z^8+3)^2)*(x^4+x^3)^(1/4)*RootOf(_Z^8+3)*x^2-12*RootOf(_Z^2+RootOf(_Z^8+3)*RootOf(_Z^2+RootOf(_Z^8+3)
^2))*(x^4+x^3)^(1/2)*x+276*(x^4+x^3)^(3/4))/(x*RootOf(_Z^8+3)^4-RootOf(_Z^8+3)^4-x-1)/x^2)+2/9*RootOf(_Z^8+3)^
7*RootOf(_Z^2+RootOf(_Z^8+3)^2)*RootOf(_Z^2+RootOf(_Z^8+3)*RootOf(_Z^2+RootOf(_Z^8+3)^2))*ln(-(5*RootOf(_Z^2+R
ootOf(_Z^8+3)*RootOf(_Z^2+RootOf(_Z^8+3)^2))*RootOf(_Z^8+3)^10*x^3-5*RootOf(_Z^2+RootOf(_Z^8+3)*RootOf(_Z^2+Ro
otOf(_Z^8+3)^2))*RootOf(_Z^8+3)^10*x^2+4*RootOf(_Z^2+RootOf(_Z^8+3)*RootOf(_Z^2+RootOf(_Z^8+3)^2))*RootOf(_Z^2
+RootOf(_Z^8+3)^2)*(x^4+x^3)^(1/2)*RootOf(_Z^8+3)^7*x+86*RootOf(_Z^8+3)^6*RootOf(_Z^2+RootOf(_Z^8+3)*RootOf(_Z
^2+RootOf(_Z^8+3)^2))*x^3+44*RootOf(_Z^8+3)^6*RootOf(_Z^2+RootOf(_Z^8+3)*RootOf(_Z^2+RootOf(_Z^8+3)^2))*x^2-96
*RootOf(_Z^2+RootOf(_Z^8+3)^2)*(x^4+x^3)^(1/4)*RootOf(_Z^8+3)^5*x^2+180*(x^4+x^3)^(1/2)*RootOf(_Z^2+RootOf(_Z^
8+3)*RootOf(_Z^2+RootOf(_Z^8+3)^2))*RootOf(_Z^2+RootOf(_Z^8+3)^2)*RootOf(_Z^8+3)^3*x-84*(x^4+x^3)^(3/4)*RootOf
(_Z^8+3)^4-171*RootOf(_Z^2+RootOf(_Z^8+3)*RootOf(_Z^2+RootOf(_Z^8+3)^2))*RootOf(_Z^8+3)^2*x^3-63*RootOf(_Z^2+R
ootOf(_Z^8+3)*RootOf(_Z^2+RootOf(_Z^8+3)^2))*RootOf(_Z^8+3)^2*x^2-264*RootOf(_Z^2+RootOf(_Z^8+3)^2)*(x^4+x^3)^
(1/4)*RootOf(_Z^8+3)*x^2+276*(x^4+x^3)^(3/4))/(x*RootOf(_Z^8+3)^4-RootOf(_Z^8+3)^4-x-1)/x^2)

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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^{3} + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x^4 + x^3)^(1/4)*(x^3 - 1)/(x^3 + 1), 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}\,\left (x^3-1\right )}{x^3+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(((x^3 + x^4)^(1/4)*(x^3 - 1))/(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 )} \left (x - 1\right ) \left (x^{2} + x + 1\right )}{\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**3-1)*(x**4+x**3)**(1/4)/(x**3+1),x)

[Out]

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

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