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

Optimal. Leaf size=148 \[ \frac {1}{3} \text {RootSum}\left [\text {$\#$1}^8-\text {$\#$1}^4+1\& ,\frac {-\text {$\#$1}^4 \log \left (\sqrt [4]{x^4+x^3}-\text {$\#$1} x\right )+\text {$\#$1}^4 \log (x)+2 \log \left (\sqrt [4]{x^4+x^3}-\text {$\#$1} x\right )-2 \log (x)}{2 \text {$\#$1}^7-\text {$\#$1}^3}\& \right ]+\frac {4}{3} \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+x^3}}\right )-\frac {4}{3} \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+x^3}}\right ) \]

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Rubi [C]  time = 0.85, antiderivative size = 708, normalized size of antiderivative = 4.78, number of steps used = 54, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2056, 6725, 105, 50, 63, 331, 298, 203, 206, 93} \begin {gather*} \frac {1}{3} (-1)^{2/3} \sqrt [4]{x^4+x^3}-\frac {1}{3} \sqrt [3]{-1} \sqrt [4]{x^4+x^3}+\frac {1}{3} \sqrt [4]{x^4+x^3}-\frac {(-1)^{2/3} \sqrt [4]{x^4+x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{6 x^{3/4} \sqrt [4]{x+1}}+\frac {\sqrt [3]{-1} \sqrt [4]{x^4+x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{6 x^{3/4} \sqrt [4]{x+1}}-\frac {\sqrt [4]{x^4+x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{6 x^{3/4} \sqrt [4]{x+1}}+\frac {4 \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 x^{3/4} \sqrt [4]{x+1}}-\frac {2^{3/4} \sqrt [4]{1-i \sqrt {3}} \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 x^{3/4} \sqrt [4]{x+1}}+\frac {2 (-1)^{2/3} \left (1+(-1)^{2/3}\right )^{5/4} \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 x^{3/4} \sqrt [4]{x+1}}+\frac {(-1)^{2/3} \sqrt [4]{x^4+x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{6 x^{3/4} \sqrt [4]{x+1}}-\frac {\sqrt [3]{-1} \sqrt [4]{x^4+x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{6 x^{3/4} \sqrt [4]{x+1}}+\frac {\sqrt [4]{x^4+x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{6 x^{3/4} \sqrt [4]{x+1}}-\frac {4 \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 x^{3/4} \sqrt [4]{x+1}}+\frac {2^{3/4} \sqrt [4]{1-i \sqrt {3}} \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 x^{3/4} \sqrt [4]{x+1}}-\frac {2 (-1)^{2/3} \left (1+(-1)^{2/3}\right )^{5/4} \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 x^{3/4} \sqrt [4]{x+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^3 + x^4)^(1/4)/3 - ((-1)^(1/3)*(x^3 + x^4)^(1/4))/3 + ((-1)^(2/3)*(x^3 + x^4)^(1/4))/3 - ((x^3 + x^4)^(1/4)
*ArcTan[x^(1/4)/(1 + x)^(1/4)])/(6*x^(3/4)*(1 + x)^(1/4)) + ((-1)^(1/3)*(x^3 + x^4)^(1/4)*ArcTan[x^(1/4)/(1 +
x)^(1/4)])/(6*x^(3/4)*(1 + x)^(1/4)) - ((-1)^(2/3)*(x^3 + x^4)^(1/4)*ArcTan[x^(1/4)/(1 + x)^(1/4)])/(6*x^(3/4)
*(1 + x)^(1/4)) + (4*2^(1/4)*(x^3 + x^4)^(1/4)*ArcTan[(2^(1/4)*x^(1/4))/(1 + x)^(1/4)])/(3*x^(3/4)*(1 + x)^(1/
4)) - (2^(3/4)*(1 - I*Sqrt[3])^(1/4)*(x^3 + x^4)^(1/4)*ArcTan[((1 - (-1)^(1/3))^(1/4)*x^(1/4))/(1 + x)^(1/4)])
/(3*x^(3/4)*(1 + x)^(1/4)) + (2*(-1)^(2/3)*(1 + (-1)^(2/3))^(5/4)*(x^3 + x^4)^(1/4)*ArcTan[((1 + (-1)^(2/3))^(
1/4)*x^(1/4))/(1 + x)^(1/4)])/(3*x^(3/4)*(1 + x)^(1/4)) + ((x^3 + x^4)^(1/4)*ArcTanh[x^(1/4)/(1 + x)^(1/4)])/(
6*x^(3/4)*(1 + x)^(1/4)) - ((-1)^(1/3)*(x^3 + x^4)^(1/4)*ArcTanh[x^(1/4)/(1 + x)^(1/4)])/(6*x^(3/4)*(1 + x)^(1
/4)) + ((-1)^(2/3)*(x^3 + x^4)^(1/4)*ArcTanh[x^(1/4)/(1 + x)^(1/4)])/(6*x^(3/4)*(1 + x)^(1/4)) - (4*2^(1/4)*(x
^3 + x^4)^(1/4)*ArcTanh[(2^(1/4)*x^(1/4))/(1 + x)^(1/4)])/(3*x^(3/4)*(1 + x)^(1/4)) + (2^(3/4)*(1 - I*Sqrt[3])
^(1/4)*(x^3 + x^4)^(1/4)*ArcTanh[((1 - (-1)^(1/3))^(1/4)*x^(1/4))/(1 + x)^(1/4)])/(3*x^(3/4)*(1 + x)^(1/4)) -
(2*(-1)^(2/3)*(1 + (-1)^(2/3))^(5/4)*(x^3 + x^4)^(1/4)*ArcTanh[((1 + (-1)^(2/3))^(1/4)*x^(1/4))/(1 + x)^(1/4)]
)/(3*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 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 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 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 {(1+x) \sqrt [4]{x^3+x^4}}{x \left (-1+x^3\right )} \, dx &=\frac {\sqrt [4]{x^3+x^4} \int \frac {(1+x)^{5/4}}{\sqrt [4]{x} \left (-1+x^3\right )} \, dx}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {\sqrt [4]{x^3+x^4} \int \left (-\frac {(1+x)^{5/4}}{3 (1-x) \sqrt [4]{x}}-\frac {(1+x)^{5/4}}{3 \sqrt [4]{x} \left (1+\sqrt [3]{-1} x\right )}-\frac {(1+x)^{5/4}}{3 \sqrt [4]{x} \left (1-(-1)^{2/3} x\right )}\right ) \, dx}{x^{3/4} \sqrt [4]{1+x}}\\ &=-\frac {\sqrt [4]{x^3+x^4} \int \frac {(1+x)^{5/4}}{(1-x) \sqrt [4]{x}} \, dx}{3 x^{3/4} \sqrt [4]{1+x}}-\frac {\sqrt [4]{x^3+x^4} \int \frac {(1+x)^{5/4}}{\sqrt [4]{x} \left (1+\sqrt [3]{-1} x\right )} \, dx}{3 x^{3/4} \sqrt [4]{1+x}}-\frac {\sqrt [4]{x^3+x^4} \int \frac {(1+x)^{5/4}}{\sqrt [4]{x} \left (1-(-1)^{2/3} x\right )} \, dx}{3 x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {\sqrt [4]{x^3+x^4} \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{x}} \, dx}{3 x^{3/4} \sqrt [4]{1+x}}-\frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \int \frac {\sqrt [4]{1+x}}{(1-x) \sqrt [4]{x}} \, dx}{3 x^{3/4} \sqrt [4]{1+x}}-\frac {\left (\sqrt [3]{-1} \sqrt [4]{x^3+x^4}\right ) \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{x}} \, dx}{3 x^{3/4} \sqrt [4]{1+x}}+\frac {\left ((-1)^{2/3} \sqrt [4]{x^3+x^4}\right ) \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{x}} \, dx}{3 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (-1+\sqrt [3]{-1}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{x} \left (1-(-1)^{2/3} x\right )} \, dx}{3 x^{3/4} \sqrt [4]{1+x}}-\frac {\left (\left (1+(-1)^{2/3}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{x} \left (1+\sqrt [3]{-1} x\right )} \, dx}{3 x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {1}{3} \sqrt [4]{x^3+x^4}-\frac {1}{3} \sqrt [3]{-1} \sqrt [4]{x^3+x^4}+\frac {1}{3} (-1)^{2/3} \sqrt [4]{x^3+x^4}+\frac {\sqrt [4]{x^3+x^4} \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4}} \, dx}{12 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4}} \, dx}{3 x^{3/4} \sqrt [4]{1+x}}-\frac {\left (4 \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{(1-x) \sqrt [4]{x} (1+x)^{3/4}} \, dx}{3 x^{3/4} \sqrt [4]{1+x}}-\frac {\left (\sqrt [3]{-1} \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4}} \, dx}{12 x^{3/4} \sqrt [4]{1+x}}+\frac {\left ((-1)^{2/3} \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4}} \, dx}{12 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\sqrt [3]{-1} \left (-1+\sqrt [3]{-1}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4}} \, dx}{3 x^{3/4} \sqrt [4]{1+x}}-\frac {\left (\left (-1+\sqrt [3]{-1}\right )^2 \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4} \left (1-(-1)^{2/3} x\right )} \, dx}{3 x^{3/4} \sqrt [4]{1+x}}+\frac {\left ((-1)^{2/3} \left (1+(-1)^{2/3}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4}} \, dx}{3 x^{3/4} \sqrt [4]{1+x}}-\frac {\left (\left (1+(-1)^{2/3}\right )^2 \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4} \left (1+\sqrt [3]{-1} x\right )} \, dx}{3 x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {1}{3} \sqrt [4]{x^3+x^4}-\frac {1}{3} \sqrt [3]{-1} \sqrt [4]{x^3+x^4}+\frac {1}{3} (-1)^{2/3} \sqrt [4]{x^3+x^4}+\frac {\sqrt [4]{x^3+x^4} \operatorname {Subst}\left (\int \frac {x^2}{\left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{3 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (8 \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 )}{3 x^{3/4} \sqrt [4]{1+x}}-\frac {\left (16 \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 x^{3/4} \sqrt [4]{1+x}}-\frac {\left (\sqrt [3]{-1} \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 )}{3 x^{3/4} \sqrt [4]{1+x}}+\frac {\left ((-1)^{2/3} \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 )}{3 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (4 \sqrt [3]{-1} \left (-1+\sqrt [3]{-1}\right ) \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 )}{3 x^{3/4} \sqrt [4]{1+x}}-\frac {\left (4 \left (-1+\sqrt [3]{-1}\right )^2 \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 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (4 (-1)^{2/3} \left (1+(-1)^{2/3}\right ) \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 )}{3 x^{3/4} \sqrt [4]{1+x}}-\frac {\left (4 \left (1+(-1)^{2/3}\right )^2 \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 x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {1}{3} \sqrt [4]{x^3+x^4}-\frac {1}{3} \sqrt [3]{-1} \sqrt [4]{x^3+x^4}+\frac {1}{3} (-1)^{2/3} \sqrt [4]{x^3+x^4}+\frac {\sqrt [4]{x^3+x^4} \operatorname {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{3 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (8 \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 )}{3 x^{3/4} \sqrt [4]{1+x}}-\frac {\left (\sqrt [3]{-1} \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 )}{3 x^{3/4} \sqrt [4]{1+x}}+\frac {\left ((-1)^{2/3} \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 )}{3 x^{3/4} \sqrt [4]{1+x}}-\frac {\left (4 \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 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (4 \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 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (4 \sqrt [3]{-1} \left (-1+\sqrt [3]{-1}\right ) \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 )}{3 x^{3/4} \sqrt [4]{1+x}}-\frac {\left (2 \left (-1+\sqrt [3]{-1}\right )^2 \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}} x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \left (-1+\sqrt [3]{-1}\right )^2 \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}} x^{3/4} \sqrt [4]{1+x}}+\frac {\left (4 (-1)^{2/3} \left (1+(-1)^{2/3}\right ) \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 )}{3 x^{3/4} \sqrt [4]{1+x}}-\frac {\left (2 \left (1+(-1)^{2/3}\right )^2 \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}} x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \left (1+(-1)^{2/3}\right )^2 \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}} x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {1}{3} \sqrt [4]{x^3+x^4}-\frac {1}{3} \sqrt [3]{-1} \sqrt [4]{x^3+x^4}+\frac {1}{3} (-1)^{2/3} \sqrt [4]{x^3+x^4}+\frac {4 \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 x^{3/4} \sqrt [4]{1+x}}-\frac {2 \sqrt [3]{-1} \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 x^{3/4} \sqrt [4]{1+x}}+\frac {2 (-1)^{2/3} \left (1+(-1)^{2/3}\right )^{5/4} \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 x^{3/4} \sqrt [4]{1+x}}-\frac {4 \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 x^{3/4} \sqrt [4]{1+x}}+\frac {2 \sqrt [3]{-1} \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 x^{3/4} \sqrt [4]{1+x}}-\frac {2 (-1)^{2/3} \left (1+(-1)^{2/3}\right )^{5/4} \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 x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{6 x^{3/4} \sqrt [4]{1+x}}-\frac {\sqrt [4]{x^3+x^4} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{6 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (4 \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 )}{3 x^{3/4} \sqrt [4]{1+x}}-\frac {\left (4 \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 )}{3 x^{3/4} \sqrt [4]{1+x}}-\frac {\left (\sqrt [3]{-1} \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 )}{6 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\sqrt [3]{-1} \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 )}{6 x^{3/4} \sqrt [4]{1+x}}+\frac {\left ((-1)^{2/3} \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 )}{6 x^{3/4} \sqrt [4]{1+x}}-\frac {\left ((-1)^{2/3} \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 )}{6 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \sqrt [3]{-1} \left (-1+\sqrt [3]{-1}\right ) \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 )}{3 x^{3/4} \sqrt [4]{1+x}}-\frac {\left (2 \sqrt [3]{-1} \left (-1+\sqrt [3]{-1}\right ) \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 )}{3 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 (-1)^{2/3} \left (1+(-1)^{2/3}\right ) \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 )}{3 x^{3/4} \sqrt [4]{1+x}}-\frac {\left (2 (-1)^{2/3} \left (1+(-1)^{2/3}\right ) \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 )}{3 x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {1}{3} \sqrt [4]{x^3+x^4}-\frac {1}{3} \sqrt [3]{-1} \sqrt [4]{x^3+x^4}+\frac {1}{3} (-1)^{2/3} \sqrt [4]{x^3+x^4}-\frac {\sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{6 x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [3]{-1} \sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{6 x^{3/4} \sqrt [4]{1+x}}-\frac {(-1)^{2/3} \sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{6 x^{3/4} \sqrt [4]{1+x}}+\frac {4 \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 x^{3/4} \sqrt [4]{1+x}}-\frac {2 \sqrt [3]{-1} \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 x^{3/4} \sqrt [4]{1+x}}+\frac {2 (-1)^{2/3} \left (1+(-1)^{2/3}\right )^{5/4} \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 x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{6 x^{3/4} \sqrt [4]{1+x}}-\frac {\sqrt [3]{-1} \sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{6 x^{3/4} \sqrt [4]{1+x}}+\frac {(-1)^{2/3} \sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{6 x^{3/4} \sqrt [4]{1+x}}-\frac {4 \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 x^{3/4} \sqrt [4]{1+x}}+\frac {2 \sqrt [3]{-1} \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 x^{3/4} \sqrt [4]{1+x}}-\frac {2 (-1)^{2/3} \left (1+(-1)^{2/3}\right )^{5/4} \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 x^{3/4} \sqrt [4]{1+x}}\\ \end {align*}

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Mathematica [C]  time = 0.19, size = 117, normalized size = 0.79 \begin {gather*} \frac {2 x^3 \left (-8 \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {2 x}{x+1}\right )+\left (1-i \sqrt {3}\right ) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {x-i \sqrt {3} x}{2 x+2}\right )+\left (1+i \sqrt {3}\right ) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {i \sqrt {3} x+x}{2 x+2}\right )\right )}{9 \left (x^3 (x+1)\right )^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^3*(-8*Hypergeometric2F1[3/4, 1, 7/4, (2*x)/(1 + x)] + (1 - I*Sqrt[3])*Hypergeometric2F1[3/4, 1, 7/4, (x -
I*Sqrt[3]*x)/(2 + 2*x)] + (1 + I*Sqrt[3])*Hypergeometric2F1[3/4, 1, 7/4, (x + I*Sqrt[3]*x)/(2 + 2*x)]))/(9*(x
^3*(1 + x))^(3/4))

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IntegrateAlgebraic [A]  time = 0.41, size = 148, normalized size = 1.00 \begin {gather*} \frac {4}{3} \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^4}}\right )-\frac {4}{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 {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 \log (x)+2 \log \left (\sqrt [4]{x^3+x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*2^(1/4)*ArcTan[(2^(1/4)*x)/(x^3 + x^4)^(1/4)])/3 - (4*2^(1/4)*ArcTanh[(2^(1/4)*x)/(x^3 + x^4)^(1/4)])/3 + R
ootSum[1 - #1^4 + #1^8 & , (-2*Log[x] + 2*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 = 0.54, size = 833, normalized size = 5.63

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(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(2*(2*x^2 + (x^4 + x^3)^(1/4)*(sqrt(3)*x + 2*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(2*(2*x^2 - (x^4 + x^3)^(1/4)*(sqrt(3
)*x + 2*x)*sqrt(-4*sqrt(3) + 8) + 2*sqrt(x^4 + x^3))/x^2) + 1/6*sqrt(sqrt(3) + 2)*(sqrt(3) - 2)*log(4*(x^2 + (
x^4 + x^3)^(1/4)*(sqrt(3)*x - 2*x)*sqrt(sqrt(3) + 2) + sqrt(x^4 + x^3))/x^2) - 1/6*sqrt(sqrt(3) + 2)*(sqrt(3)
- 2)*log(4*(x^2 - (x^4 + x^3)^(1/4)*(sqrt(3)*x - 2*x)*sqrt(sqrt(3) + 2) + sqrt(x^4 + x^3))/x^2) - 1/3*sqrt(-4*
sqrt(3) + 8)*arctan(1/2*(sqrt(2)*(sqrt(3)*x + 2*x)*sqrt(-4*sqrt(3) + 8)*sqrt((2*x^2 + (x^4 + x^3)^(1/4)*(sqrt(
3)*x + 2*x)*sqrt(-4*sqrt(3) + 8) + 2*sqrt(x^4 + x^3))/x^2) - 2*(x^4 + x^3)^(1/4)*(sqrt(3) + 2)*sqrt(-4*sqrt(3)
+ 8) - 2*sqrt(3)*x - 4*x)/x) - 1/3*sqrt(-4*sqrt(3) + 8)*arctan(1/2*(sqrt(2)*(sqrt(3)*x + 2*x)*sqrt(-4*sqrt(3)
+ 8)*sqrt((2*x^2 - (x^4 + x^3)^(1/4)*(sqrt(3)*x + 2*x)*sqrt(-4*sqrt(3) + 8) + 2*sqrt(x^4 + x^3))/x^2) - 2*(x^
4 + x^3)^(1/4)*(sqrt(3) + 2)*sqrt(-4*sqrt(3) + 8) + 2*sqrt(3)*x + 4*x)/x) + 2/3*sqrt(sqrt(3) + 2)*arctan((2*(s
qrt(3)*x - 2*x)*sqrt(sqrt(3) + 2)*sqrt((x^2 + (x^4 + x^3)^(1/4)*(sqrt(3)*x - 2*x)*sqrt(sqrt(3) + 2) + sqrt(x^4
+ x^3))/x^2) - 2*(x^4 + x^3)^(1/4)*sqrt(sqrt(3) + 2)*(sqrt(3) - 2) + sqrt(3)*x - 2*x)/x) + 2/3*sqrt(sqrt(3) +
2)*arctan((2*(sqrt(3)*x - 2*x)*sqrt(sqrt(3) + 2)*sqrt((x^2 - (x^4 + x^3)^(1/4)*(sqrt(3)*x - 2*x)*sqrt(sqrt(3)
+ 2) + sqrt(x^4 + x^3))/x^2) - 2*(x^4 + x^3)^(1/4)*sqrt(sqrt(3) + 2)*(sqrt(3) - 2) - sqrt(3)*x + 2*x)/x) + 8/
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) - 2/
3*2^(1/4)*log((2^(1/4)*x + (x^4 + x^3)^(1/4))/x) + 2/3*2^(1/4)*log(-(2^(1/4)*x - (x^4 + x^3)^(1/4))/x)

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giac [B]  time = 0.69, size = 362, normalized size = 2.45 \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 {1}{3} \cdot 8^{\frac {3}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \frac {2}{3} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} + {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \frac {2}{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((1+x)*(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) + s
qrt(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))*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) - 1/3*8^(3/4)*arctan(1/2
*2^(3/4)*(1/x + 1)^(1/4)) - 2/3*2^(1/4)*log(2^(1/4) + (1/x + 1)^(1/4)) + 2/3*2^(1/4)*log(abs(-2^(1/4) + (1/x +
1)^(1/4)))

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maple [B]  time = 101.64, size = 16793, normalized size = 113.47

method	result	size
trager	\(\text {Expression too large to display}\)	\(16793\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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