3.21.60 \(\int \frac {(1+x) \sqrt [4]{x^3+x^4}}{x (-1+x^3)} \, dx\) [2060]

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

[Out]

Unintegrable

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Rubi [A]
time = 0.33, antiderivative size = 125, normalized size of antiderivative = 0.84, number of steps used = 9, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2081, 6857, 129, 524} \begin {gather*} -\frac {4 \sqrt [4]{x^4+x^3} F_1\left (\frac {3}{4};-\frac {5}{4},1;\frac {7}{4};-x,-\sqrt [3]{-1} x\right )}{9 \sqrt [4]{x+1}}-\frac {4 \sqrt [4]{x^4+x^3} F_1\left (\frac {3}{4};-\frac {5}{4},1;\frac {7}{4};-x,(-1)^{2/3} x\right )}{9 \sqrt [4]{x+1}}-\frac {4 \sqrt [4]{x^4+x^3} F_1\left (\frac {3}{4};1,-\frac {5}{4};\frac {7}{4};x,-x\right )}{9 \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]

(-4*(x^3 + x^4)^(1/4)*AppellF1[3/4, -5/4, 1, 7/4, -x, -((-1)^(1/3)*x)])/(9*(1 + x)^(1/4)) - (4*(x^3 + x^4)^(1/
4)*AppellF1[3/4, -5/4, 1, 7/4, -x, (-1)^(2/3)*x])/(9*(1 + x)^(1/4)) - (4*(x^3 + x^4)^(1/4)*AppellF1[3/4, 1, -5
/4, 7/4, x, -x])/(9*(1 + x)^(1/4))

Rule 129

Int[((e_.)*(x_))^(p_)*((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> With[{k = Denominator[p]
}, Dist[k/e, Subst[Int[x^(k*(p + 1) - 1)*(a + b*(x^k/e))^m*(c + d*(x^k/e))^n, x], x, (e*x)^(1/k)], x]] /; Free
Q[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && FractionQ[p] && IntegerQ[m]

Rule 524

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*
((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 2081

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 6857

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} \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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} \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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} \text {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} \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 [A]
time = 0.28, size = 161, normalized size = 1.09 \begin {gather*} \frac {x^{9/4} (1+x)^{3/4} \left (16 \sqrt [4]{2} \left (\text {ArcTan}\left (\sqrt [4]{2} \sqrt [4]{\frac {x}{1+x}}\right )-\tanh ^{-1}\left (\sqrt [4]{2} \sqrt [4]{\frac {x}{1+x}}\right )\right )+\text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 \log (x)+8 \log \left (\sqrt [4]{1+x}-\sqrt [4]{x} \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-4 \log \left (\sqrt [4]{1+x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ]\right )}{12 \left (x^3 (1+x)\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]

(x^(9/4)*(1 + x)^(3/4)*(16*2^(1/4)*(ArcTan[2^(1/4)*(x/(1 + x))^(1/4)] - ArcTanh[2^(1/4)*(x/(1 + x))^(1/4)]) +
RootSum[1 - #1^4 + #1^8 & , (-2*Log[x] + 8*Log[(1 + x)^(1/4) - x^(1/4)*#1] + Log[x]*#1^4 - 4*Log[(1 + x)^(1/4)
- x^(1/4)*#1]*#1^4)/(-#1^3 + 2*#1^7) & ]))/(12*(x^3*(1 + x))^(3/4))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 1.
time = 59.41, size = 16619, normalized size = 112.29

method	result	size
trager	\(\text {Expression too large to display}\)	\(16619\)

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*} \text {Failed to integrate} \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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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 0.39, size = 833, normalized size = 5.63 \begin {gather*} \frac {1}{12} \, {\left (\sqrt {3} + 2\right )} \sqrt {-4 \, \sqrt {3} + 8} \log \left (\frac {2 \, {\left (2 \, x^{2} + {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} {\left (\sqrt {3} x + 2 \, x\right )} \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 {2 \, {\left (2 \, x^{2} - {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} {\left (\sqrt {3} x + 2 \, x\right )} \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 {4 \, {\left (x^{2} + {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} {\left (\sqrt {3} x - 2 \, x\right )} \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 {4 \, {\left (x^{2} - {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} {\left (\sqrt {3} x - 2 \, x\right )} \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} {\left (\sqrt {3} x + 2 \, x\right )} \sqrt {-4 \, \sqrt {3} + 8} \sqrt {\frac {2 \, x^{2} + {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} {\left (\sqrt {3} x + 2 \, x\right )} \sqrt {-4 \, \sqrt {3} + 8} + 2 \, \sqrt {x^{4} + x^{3}}}{x^{2}}} - 2 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} {\left (\sqrt {3} + 2\right )} \sqrt {-4 \, \sqrt {3} + 8} - 2 \, \sqrt {3} x - 4 \, x}{2 \, x}\right ) - \frac {1}{3} \, \sqrt {-4 \, \sqrt {3} + 8} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {3} x + 2 \, x\right )} \sqrt {-4 \, \sqrt {3} + 8} \sqrt {\frac {2 \, x^{2} - {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} {\left (\sqrt {3} x + 2 \, x\right )} \sqrt {-4 \, \sqrt {3} + 8} + 2 \, \sqrt {x^{4} + x^{3}}}{x^{2}}} - 2 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} {\left (\sqrt {3} + 2\right )} \sqrt {-4 \, \sqrt {3} + 8} + 2 \, \sqrt {3} x + 4 \, x}{2 \, x}\right ) + \frac {2}{3} \, \sqrt {\sqrt {3} + 2} \arctan \left (\frac {2 \, {\left (\sqrt {3} x - 2 \, x\right )} \sqrt {\sqrt {3} + 2} \sqrt {\frac {x^{2} + {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} {\left (\sqrt {3} x - 2 \, x\right )} \sqrt {\sqrt {3} + 2} + \sqrt {x^{4} + x^{3}}}{x^{2}}} - 2 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} \sqrt {\sqrt {3} + 2} {\left (\sqrt {3} - 2\right )} + \sqrt {3} x - 2 \, x}{x}\right ) + \frac {2}{3} \, \sqrt {\sqrt {3} + 2} \arctan \left (\frac {2 \, {\left (\sqrt {3} x - 2 \, x\right )} \sqrt {\sqrt {3} + 2} \sqrt {\frac {x^{2} - {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} {\left (\sqrt {3} x - 2 \, x\right )} \sqrt {\sqrt {3} + 2} + \sqrt {x^{4} + x^{3}}}{x^{2}}} - 2 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} \sqrt {\sqrt {3} + 2} {\left (\sqrt {3} - 2\right )} - \sqrt {3} x + 2 \, x}{x}\right ) + \frac {8}{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 {2}{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 {2}{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((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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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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Giac [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 0.45, 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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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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