\(\int \frac {(1+x) \sqrt [4]{x^3+x^5}}{x (-1+x^3)} \, dx\) [2140]

Optimal result

Integrand size = 25, antiderivative size = 156 \[ \int \frac {(1+x) \sqrt [4]{x^3+x^5}}{x \left (-1+x^3\right )} \, dx=\frac {2}{3} \sqrt [4]{2} \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^5}}\right )-\frac {1}{3} \sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt [4]{x^3+x^5}}{-x^2+\sqrt {x^3+x^5}}\right )-\frac {2}{3} \sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^5}}\right )+\frac {1}{3} \sqrt {2} \text {arctanh}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {x^3+x^5}}{\sqrt {2}}}{x \sqrt [4]{x^3+x^5}}\right ) \]

[Out]

2/3*2^(1/4)*arctan(2^(1/4)*x/(x^5+x^3)^(1/4))-1/3*2^(1/2)*arctan(2^(1/2)*x*(x^5+x^3)^(1/4)/(-x^2+(x^5+x^3)^(1/
2)))-2/3*2^(1/4)*arctanh(2^(1/4)*x/(x^5+x^3)^(1/4))+1/3*2^(1/2)*arctanh((1/2*2^(1/2)*x^2+1/2*(x^5+x^3)^(1/2)*2
^(1/2))/x/(x^5+x^3)^(1/4))

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 0.50 (sec) , antiderivative size = 320, normalized size of antiderivative = 2.05, number of steps used = 18, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2081, 6857, 973, 477, 524} \[ \int \frac {(1+x) \sqrt [4]{x^3+x^5}}{x \left (-1+x^3\right )} \, dx=-\frac {4 \left (1-\sqrt [3]{-1}\right ) \sqrt [4]{x^5+x^3} \operatorname {AppellF1}\left (\frac {3}{8},-\frac {1}{4},1,\frac {11}{8},-x^2,-\sqrt [3]{-1} x^2\right )}{9 \sqrt [4]{x^2+1}}-\frac {4 \left (1+(-1)^{2/3}\right ) \sqrt [4]{x^5+x^3} \operatorname {AppellF1}\left (\frac {3}{8},-\frac {1}{4},1,\frac {11}{8},-x^2,(-1)^{2/3} x^2\right )}{9 \sqrt [4]{x^2+1}}-\frac {8 \sqrt [4]{x^5+x^3} \operatorname {AppellF1}\left (\frac {3}{8},1,-\frac {1}{4},\frac {11}{8},x^2,-x^2\right )}{9 \sqrt [4]{x^2+1}}-\frac {4 \left (1+(-1)^{2/3}\right ) x \sqrt [4]{x^5+x^3} \operatorname {AppellF1}\left (\frac {7}{8},-\frac {1}{4},1,\frac {15}{8},-x^2,-\sqrt [3]{-1} x^2\right )}{21 \sqrt [4]{x^2+1}}-\frac {4 \left (1-\sqrt [3]{-1}\right ) x \sqrt [4]{x^5+x^3} \operatorname {AppellF1}\left (\frac {7}{8},-\frac {1}{4},1,\frac {15}{8},-x^2,(-1)^{2/3} x^2\right )}{21 \sqrt [4]{x^2+1}}-\frac {8 x \sqrt [4]{x^5+x^3} \operatorname {AppellF1}\left (\frac {7}{8},1,-\frac {1}{4},\frac {15}{8},x^2,-x^2\right )}{21 \sqrt [4]{x^2+1}} \]

[In]

Int[((1 + x)*(x^3 + x^5)^(1/4))/(x*(-1 + x^3)),x]

[Out]

(-4*(1 - (-1)^(1/3))*(x^3 + x^5)^(1/4)*AppellF1[3/8, -1/4, 1, 11/8, -x^2, -((-1)^(1/3)*x^2)])/(9*(1 + x^2)^(1/
4)) - (4*(1 + (-1)^(2/3))*(x^3 + x^5)^(1/4)*AppellF1[3/8, -1/4, 1, 11/8, -x^2, (-1)^(2/3)*x^2])/(9*(1 + x^2)^(
1/4)) - (8*(x^3 + x^5)^(1/4)*AppellF1[3/8, 1, -1/4, 11/8, x^2, -x^2])/(9*(1 + x^2)^(1/4)) - (4*(1 + (-1)^(2/3)
)*x*(x^3 + x^5)^(1/4)*AppellF1[7/8, -1/4, 1, 15/8, -x^2, -((-1)^(1/3)*x^2)])/(21*(1 + x^2)^(1/4)) - (4*(1 - (-
1)^(1/3))*x*(x^3 + x^5)^(1/4)*AppellF1[7/8, -1/4, 1, 15/8, -x^2, (-1)^(2/3)*x^2])/(21*(1 + x^2)^(1/4)) - (8*x*
(x^3 + x^5)^(1/4)*AppellF1[7/8, 1, -1/4, 15/8, x^2, -x^2])/(21*(1 + x^2)^(1/4))

Rule 477

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/e^n))^p*(c + d*(x^(k*n)/e^n))^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

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 973

Int[(((g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_)^2)^(p_))/((d_) + (e_.)*(x_)), x_Symbol] :> Dist[d*((g*x)^n/x^n), In
t[(x^n*(a + c*x^2)^p)/(d^2 - e^2*x^2), x], x] - Dist[e*((g*x)^n/x^n), Int[(x^(n + 1)*(a + c*x^2)^p)/(d^2 - e^2
*x^2), x], x] /; FreeQ[{a, c, d, e, g, n, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] &&  !IntegersQ[n, 2
*p]

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*} \text {integral}& = \frac {\sqrt [4]{x^3+x^5} \int \frac {(1+x) \sqrt [4]{1+x^2}}{\sqrt [4]{x} \left (-1+x^3\right )} \, dx}{x^{3/4} \sqrt [4]{1+x^2}} \\ & = \frac {\sqrt [4]{x^3+x^5} \int \left (-\frac {2 \sqrt [4]{1+x^2}}{3 (1-x) \sqrt [4]{x}}-\frac {\left (1+(-1)^{2/3}\right ) \sqrt [4]{1+x^2}}{3 \sqrt [4]{x} \left (1+\sqrt [3]{-1} x\right )}-\frac {\left (1-\sqrt [3]{-1}\right ) \sqrt [4]{1+x^2}}{3 \sqrt [4]{x} \left (1-(-1)^{2/3} x\right )}\right ) \, dx}{x^{3/4} \sqrt [4]{1+x^2}} \\ & = -\frac {\left (2 \sqrt [4]{x^3+x^5}\right ) \int \frac {\sqrt [4]{1+x^2}}{(1-x) \sqrt [4]{x}} \, dx}{3 x^{3/4} \sqrt [4]{1+x^2}}+\frac {\left (\left (-1+\sqrt [3]{-1}\right ) \sqrt [4]{x^3+x^5}\right ) \int \frac {\sqrt [4]{1+x^2}}{\sqrt [4]{x} \left (1-(-1)^{2/3} x\right )} \, dx}{3 x^{3/4} \sqrt [4]{1+x^2}}+\frac {\left (\left (-1-(-1)^{2/3}\right ) \sqrt [4]{x^3+x^5}\right ) \int \frac {\sqrt [4]{1+x^2}}{\sqrt [4]{x} \left (1+\sqrt [3]{-1} x\right )} \, dx}{3 x^{3/4} \sqrt [4]{1+x^2}} \\ & = -\frac {\left (2 \sqrt [4]{x^3+x^5}\right ) \int \frac {\sqrt [4]{1+x^2}}{\sqrt [4]{x} \left (1-x^2\right )} \, dx}{3 x^{3/4} \sqrt [4]{1+x^2}}-\frac {\left (2 \sqrt [4]{x^3+x^5}\right ) \int \frac {x^{3/4} \sqrt [4]{1+x^2}}{1-x^2} \, dx}{3 x^{3/4} \sqrt [4]{1+x^2}}+\frac {\left (\left (-1+\sqrt [3]{-1}\right ) \sqrt [4]{x^3+x^5}\right ) \int \frac {\sqrt [4]{1+x^2}}{\sqrt [4]{x} \left (1+\sqrt [3]{-1} x^2\right )} \, dx}{3 x^{3/4} \sqrt [4]{1+x^2}}+\frac {\left ((-1)^{2/3} \left (-1+\sqrt [3]{-1}\right ) \sqrt [4]{x^3+x^5}\right ) \int \frac {x^{3/4} \sqrt [4]{1+x^2}}{1+\sqrt [3]{-1} x^2} \, dx}{3 x^{3/4} \sqrt [4]{1+x^2}}+\frac {\left (\left (-1-(-1)^{2/3}\right ) \sqrt [4]{x^3+x^5}\right ) \int \frac {\sqrt [4]{1+x^2}}{\sqrt [4]{x} \left (1-(-1)^{2/3} x^2\right )} \, dx}{3 x^{3/4} \sqrt [4]{1+x^2}}-\frac {\left (\sqrt [3]{-1} \left (-1-(-1)^{2/3}\right ) \sqrt [4]{x^3+x^5}\right ) \int \frac {x^{3/4} \sqrt [4]{1+x^2}}{1-(-1)^{2/3} x^2} \, dx}{3 x^{3/4} \sqrt [4]{1+x^2}} \\ & = -\frac {\left (8 \sqrt [4]{x^3+x^5}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{1+x^8}}{1-x^8} \, dx,x,\sqrt [4]{x}\right )}{3 x^{3/4} \sqrt [4]{1+x^2}}-\frac {\left (8 \sqrt [4]{x^3+x^5}\right ) \text {Subst}\left (\int \frac {x^6 \sqrt [4]{1+x^8}}{1-x^8} \, dx,x,\sqrt [4]{x}\right )}{3 x^{3/4} \sqrt [4]{1+x^2}}+\frac {\left (4 \left (-1+\sqrt [3]{-1}\right ) \sqrt [4]{x^3+x^5}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{1+x^8}}{1+\sqrt [3]{-1} x^8} \, dx,x,\sqrt [4]{x}\right )}{3 x^{3/4} \sqrt [4]{1+x^2}}+\frac {\left (4 (-1)^{2/3} \left (-1+\sqrt [3]{-1}\right ) \sqrt [4]{x^3+x^5}\right ) \text {Subst}\left (\int \frac {x^6 \sqrt [4]{1+x^8}}{1+\sqrt [3]{-1} x^8} \, dx,x,\sqrt [4]{x}\right )}{3 x^{3/4} \sqrt [4]{1+x^2}}+\frac {\left (4 \left (-1-(-1)^{2/3}\right ) \sqrt [4]{x^3+x^5}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{1+x^8}}{1-(-1)^{2/3} x^8} \, dx,x,\sqrt [4]{x}\right )}{3 x^{3/4} \sqrt [4]{1+x^2}}-\frac {\left (4 \sqrt [3]{-1} \left (-1-(-1)^{2/3}\right ) \sqrt [4]{x^3+x^5}\right ) \text {Subst}\left (\int \frac {x^6 \sqrt [4]{1+x^8}}{1-(-1)^{2/3} x^8} \, dx,x,\sqrt [4]{x}\right )}{3 x^{3/4} \sqrt [4]{1+x^2}} \\ & = -\frac {4 \left (1-\sqrt [3]{-1}\right ) \sqrt [4]{x^3+x^5} \operatorname {AppellF1}\left (\frac {3}{8},-\frac {1}{4},1,\frac {11}{8},-x^2,-\sqrt [3]{-1} x^2\right )}{9 \sqrt [4]{1+x^2}}-\frac {4 \left (1+(-1)^{2/3}\right ) \sqrt [4]{x^3+x^5} \operatorname {AppellF1}\left (\frac {3}{8},-\frac {1}{4},1,\frac {11}{8},-x^2,(-1)^{2/3} x^2\right )}{9 \sqrt [4]{1+x^2}}-\frac {8 \sqrt [4]{x^3+x^5} \operatorname {AppellF1}\left (\frac {3}{8},1,-\frac {1}{4},\frac {11}{8},x^2,-x^2\right )}{9 \sqrt [4]{1+x^2}}-\frac {4 (-1)^{2/3} \left (1-\sqrt [3]{-1}\right ) x \sqrt [4]{x^3+x^5} \operatorname {AppellF1}\left (\frac {7}{8},-\frac {1}{4},1,\frac {15}{8},-x^2,-\sqrt [3]{-1} x^2\right )}{21 \sqrt [4]{1+x^2}}-\frac {4 \left (1-\sqrt [3]{-1}\right ) x \sqrt [4]{x^3+x^5} \operatorname {AppellF1}\left (\frac {7}{8},-\frac {1}{4},1,\frac {15}{8},-x^2,(-1)^{2/3} x^2\right )}{21 \sqrt [4]{1+x^2}}-\frac {8 x \sqrt [4]{x^3+x^5} \operatorname {AppellF1}\left (\frac {7}{8},1,-\frac {1}{4},\frac {15}{8},x^2,-x^2\right )}{21 \sqrt [4]{1+x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.84 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.10 \[ \int \frac {(1+x) \sqrt [4]{x^3+x^5}}{x \left (-1+x^3\right )} \, dx=\frac {\sqrt [4]{2} \sqrt [4]{x^3+x^5} \left (2 \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x^2}}\right )+\sqrt [4]{2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{x} \sqrt [4]{1+x^2}}{\sqrt {x}-\sqrt {1+x^2}}\right )-2 \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x^2}}\right )+\sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{x} \sqrt [4]{1+x^2}}{\sqrt {x}+\sqrt {1+x^2}}\right )\right )}{3 x^{3/4} \sqrt [4]{1+x^2}} \]

[In]

Integrate[((1 + x)*(x^3 + x^5)^(1/4))/(x*(-1 + x^3)),x]

[Out]

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

Maple [A] (verified)

Time = 12.35 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.30

method	result	size
pseudoelliptic	\(-\frac {\ln \left (\frac {-2^{\frac {1}{4}} x -\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}\right ) 2^{\frac {1}{4}}}{3}-\frac {2 \arctan \left (\frac {2^{\frac {3}{4}} \left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{2 x}\right ) 2^{\frac {1}{4}}}{3}+\frac {\ln \left (\frac {\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{3} \left (x^{2}+1\right )}}{-\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{3} \left (x^{2}+1\right )}}\right ) \sqrt {2}}{6}+\frac {\arctan \left (\frac {\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}} \sqrt {2}+x}{x}\right ) \sqrt {2}}{3}+\frac {\arctan \left (\frac {\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}} \sqrt {2}-x}{x}\right ) \sqrt {2}}{3}\)	\(203\)
trager	\(\text {Expression too large to display}\)	\(737\)

[In]

int((1+x)*(x^5+x^3)^(1/4)/x/(x^3-1),x,method=_RETURNVERBOSE)

[Out]

-1/3*ln((-2^(1/4)*x-(x^3*(x^2+1))^(1/4))/(2^(1/4)*x-(x^3*(x^2+1))^(1/4)))*2^(1/4)-2/3*arctan(1/2*2^(3/4)/x*(x^
3*(x^2+1))^(1/4))*2^(1/4)+1/6*ln(((x^3*(x^2+1))^(1/4)*2^(1/2)*x+x^2+(x^3*(x^2+1))^(1/2))/(-(x^3*(x^2+1))^(1/4)
*2^(1/2)*x+x^2+(x^3*(x^2+1))^(1/2)))*2^(1/2)+1/3*arctan(((x^3*(x^2+1))^(1/4)*2^(1/2)+x)/x)*2^(1/2)+1/3*arctan(
((x^3*(x^2+1))^(1/4)*2^(1/2)-x)/x)*2^(1/2)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 3.00 (sec) , antiderivative size = 656, normalized size of antiderivative = 4.21 \[ \int \frac {(1+x) \sqrt [4]{x^3+x^5}}{x \left (-1+x^3\right )} \, dx=\left (\frac {1}{12} i - \frac {1}{12}\right ) \, \sqrt {2} \log \left (\frac {4 i \, {\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} x^{2} - \left (2 i - 2\right ) \, \sqrt {2} \sqrt {x^{5} + x^{3}} x + \sqrt {2} {\left (\left (i + 1\right ) \, x^{4} - \left (i + 1\right ) \, x^{3} + \left (i + 1\right ) \, x^{2}\right )} - 4 \, {\left (x^{5} + x^{3}\right )}^{\frac {3}{4}}}{x^{4} + x^{3} + x^{2}}\right ) - \left (\frac {1}{12} i - \frac {1}{12}\right ) \, \sqrt {2} \log \left (\frac {4 i \, {\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} x^{2} + \left (2 i - 2\right ) \, \sqrt {2} \sqrt {x^{5} + x^{3}} x + \sqrt {2} {\left (-\left (i + 1\right ) \, x^{4} + \left (i + 1\right ) \, x^{3} - \left (i + 1\right ) \, x^{2}\right )} - 4 \, {\left (x^{5} + x^{3}\right )}^{\frac {3}{4}}}{x^{4} + x^{3} + x^{2}}\right ) - \left (\frac {1}{12} i + \frac {1}{12}\right ) \, \sqrt {2} \log \left (\frac {-4 i \, {\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} x^{2} + \left (2 i + 2\right ) \, \sqrt {2} \sqrt {x^{5} + x^{3}} x + \sqrt {2} {\left (-\left (i - 1\right ) \, x^{4} + \left (i - 1\right ) \, x^{3} - \left (i - 1\right ) \, x^{2}\right )} - 4 \, {\left (x^{5} + x^{3}\right )}^{\frac {3}{4}}}{x^{4} + x^{3} + x^{2}}\right ) + \left (\frac {1}{12} i + \frac {1}{12}\right ) \, \sqrt {2} \log \left (\frac {-4 i \, {\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} x^{2} - \left (2 i + 2\right ) \, \sqrt {2} \sqrt {x^{5} + x^{3}} x + \sqrt {2} {\left (\left (i - 1\right ) \, x^{4} - \left (i - 1\right ) \, x^{3} + \left (i - 1\right ) \, x^{2}\right )} - 4 \, {\left (x^{5} + x^{3}\right )}^{\frac {3}{4}}}{x^{4} + x^{3} + x^{2}}\right ) - \frac {1}{6} \cdot 2^{\frac {1}{4}} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} x^{2} + 2^{\frac {3}{4}} {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} + 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{5} + x^{3}} x + 4 \, {\left (x^{5} + x^{3}\right )}^{\frac {3}{4}}}{x^{4} - 2 \, x^{3} + x^{2}}\right ) + \frac {1}{6} \cdot 2^{\frac {1}{4}} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} x^{2} - 2^{\frac {3}{4}} {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} - 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{5} + x^{3}} x + 4 \, {\left (x^{5} + x^{3}\right )}^{\frac {3}{4}}}{x^{4} - 2 \, x^{3} + x^{2}}\right ) - \frac {1}{6} i \cdot 2^{\frac {1}{4}} \log \left (\frac {4 \, \sqrt {2} {\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} x^{2} + 2^{\frac {3}{4}} {\left (i \, x^{4} + 2 i \, x^{3} + i \, x^{2}\right )} - 4 i \cdot 2^{\frac {1}{4}} \sqrt {x^{5} + x^{3}} x - 4 \, {\left (x^{5} + x^{3}\right )}^{\frac {3}{4}}}{x^{4} - 2 \, x^{3} + x^{2}}\right ) + \frac {1}{6} i \cdot 2^{\frac {1}{4}} \log \left (\frac {4 \, \sqrt {2} {\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} x^{2} + 2^{\frac {3}{4}} {\left (-i \, x^{4} - 2 i \, x^{3} - i \, x^{2}\right )} + 4 i \cdot 2^{\frac {1}{4}} \sqrt {x^{5} + x^{3}} x - 4 \, {\left (x^{5} + x^{3}\right )}^{\frac {3}{4}}}{x^{4} - 2 \, x^{3} + x^{2}}\right ) \]

[In]

integrate((1+x)*(x^5+x^3)^(1/4)/x/(x^3-1),x, algorithm="fricas")

[Out]

(1/12*I - 1/12)*sqrt(2)*log((4*I*(x^5 + x^3)^(1/4)*x^2 - (2*I - 2)*sqrt(2)*sqrt(x^5 + x^3)*x + sqrt(2)*((I + 1
)*x^4 - (I + 1)*x^3 + (I + 1)*x^2) - 4*(x^5 + x^3)^(3/4))/(x^4 + x^3 + x^2)) - (1/12*I - 1/12)*sqrt(2)*log((4*
I*(x^5 + x^3)^(1/4)*x^2 + (2*I - 2)*sqrt(2)*sqrt(x^5 + x^3)*x + sqrt(2)*(-(I + 1)*x^4 + (I + 1)*x^3 - (I + 1)*
x^2) - 4*(x^5 + x^3)^(3/4))/(x^4 + x^3 + x^2)) - (1/12*I + 1/12)*sqrt(2)*log((-4*I*(x^5 + x^3)^(1/4)*x^2 + (2*
I + 2)*sqrt(2)*sqrt(x^5 + x^3)*x + sqrt(2)*(-(I - 1)*x^4 + (I - 1)*x^3 - (I - 1)*x^2) - 4*(x^5 + x^3)^(3/4))/(
x^4 + x^3 + x^2)) + (1/12*I + 1/12)*sqrt(2)*log((-4*I*(x^5 + x^3)^(1/4)*x^2 - (2*I + 2)*sqrt(2)*sqrt(x^5 + x^3
)*x + sqrt(2)*((I - 1)*x^4 - (I - 1)*x^3 + (I - 1)*x^2) - 4*(x^5 + x^3)^(3/4))/(x^4 + x^3 + x^2)) - 1/6*2^(1/4
)*log(-(4*sqrt(2)*(x^5 + x^3)^(1/4)*x^2 + 2^(3/4)*(x^4 + 2*x^3 + x^2) + 4*2^(1/4)*sqrt(x^5 + x^3)*x + 4*(x^5 +
x^3)^(3/4))/(x^4 - 2*x^3 + x^2)) + 1/6*2^(1/4)*log(-(4*sqrt(2)*(x^5 + x^3)^(1/4)*x^2 - 2^(3/4)*(x^4 + 2*x^3 +
x^2) - 4*2^(1/4)*sqrt(x^5 + x^3)*x + 4*(x^5 + x^3)^(3/4))/(x^4 - 2*x^3 + x^2)) - 1/6*I*2^(1/4)*log((4*sqrt(2)
*(x^5 + x^3)^(1/4)*x^2 + 2^(3/4)*(I*x^4 + 2*I*x^3 + I*x^2) - 4*I*2^(1/4)*sqrt(x^5 + x^3)*x - 4*(x^5 + x^3)^(3/
4))/(x^4 - 2*x^3 + x^2)) + 1/6*I*2^(1/4)*log((4*sqrt(2)*(x^5 + x^3)^(1/4)*x^2 + 2^(3/4)*(-I*x^4 - 2*I*x^3 - I*
x^2) + 4*I*2^(1/4)*sqrt(x^5 + x^3)*x - 4*(x^5 + x^3)^(3/4))/(x^4 - 2*x^3 + x^2))

Sympy [F]

\[ \int \frac {(1+x) \sqrt [4]{x^3+x^5}}{x \left (-1+x^3\right )} \, dx=\int \frac {\sqrt [4]{x^{3} \left (x^{2} + 1\right )} \left (x + 1\right )}{x \left (x - 1\right ) \left (x^{2} + x + 1\right )}\, dx \]

[In]

integrate((1+x)*(x**5+x**3)**(1/4)/x/(x**3-1),x)

[Out]

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

Maxima [F]

\[ \int \frac {(1+x) \sqrt [4]{x^3+x^5}}{x \left (-1+x^3\right )} \, dx=\int { \frac {{\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} {\left (x + 1\right )}}{{\left (x^{3} - 1\right )} x} \,d x } \]

[In]

integrate((1+x)*(x^5+x^3)^(1/4)/x/(x^3-1),x, algorithm="maxima")

[Out]

integrate((x^5 + x^3)^(1/4)*(x + 1)/((x^3 - 1)*x), x)

Giac [F]

\[ \int \frac {(1+x) \sqrt [4]{x^3+x^5}}{x \left (-1+x^3\right )} \, dx=\int { \frac {{\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} {\left (x + 1\right )}}{{\left (x^{3} - 1\right )} x} \,d x } \]

[In]

integrate((1+x)*(x^5+x^3)^(1/4)/x/(x^3-1),x, algorithm="giac")

[Out]

integrate((x^5 + x^3)^(1/4)*(x + 1)/((x^3 - 1)*x), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(1+x) \sqrt [4]{x^3+x^5}}{x \left (-1+x^3\right )} \, dx=\int \frac {{\left (x^5+x^3\right )}^{1/4}\,\left (x+1\right )}{x\,\left (x^3-1\right )} \,d x \]

[In]

int(((x^3 + x^5)^(1/4)*(x + 1))/(x*(x^3 - 1)),x)

[Out]

int(((x^3 + x^5)^(1/4)*(x + 1))/(x*(x^3 - 1)), x)