\(\int \frac {x^2 \sqrt [4]{x^3+x^4}}{-1+x^4} \, dx\) [2275]

Optimal result

Integrand size = 22, antiderivative size = 173 \[ \int \frac {x^2 \sqrt [4]{x^3+x^4}}{-1+x^4} \, dx=-2 \arctan \left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )+\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^4}}\right )}{2^{3/4}}+2 \text {arctanh}\left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^4}}\right )}{2^{3/4}}-\frac {1}{4} \text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 \log (x)+2 \log \left (\sqrt [4]{x^3+x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+\text {$\#$1}^7}\&\right ] \]

[Out]

Unintegrable

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.06 (sec) , antiderivative size = 616, normalized size of antiderivative = 3.56, number of steps used = 67, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.955, Rules used = {2081, 1600, 6865, 6874, 338, 304, 209, 212, 1254, 419, 243, 342, 281, 237, 416, 418, 1227, 551, 508, 6857, 1543} \[ \int \frac {x^2 \sqrt [4]{x^3+x^4}}{-1+x^4} \, dx=-\frac {2 \sqrt [4]{x^4+x^3} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{x^{3/4} \sqrt [4]{x+1}}+\frac {(1-i)^{5/4} \sqrt [4]{x^4+x^3} \arctan \left (\frac {\sqrt [4]{1-i} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{4 x^{3/4} \sqrt [4]{x+1}}+\frac {\sqrt [4]{x^4+x^3} \arctan \left (\frac {\sqrt [4]{1-i} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2 (1-i)^{3/4} x^{3/4} \sqrt [4]{x+1}}+\frac {(1+i)^{5/4} \sqrt [4]{x^4+x^3} \arctan \left (\frac {\sqrt [4]{1+i} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{4 x^{3/4} \sqrt [4]{x+1}}+\frac {\sqrt [4]{x^4+x^3} \arctan \left (\frac {\sqrt [4]{1+i} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2 (1+i)^{3/4} x^{3/4} \sqrt [4]{x+1}}+\frac {\sqrt [4]{x^4+x^3} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2^{3/4} x^{3/4} \sqrt [4]{x+1}}+\frac {2 \sqrt [4]{x^4+x^3} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{x^{3/4} \sqrt [4]{x+1}}-\frac {(1-i)^{5/4} \sqrt [4]{x^4+x^3} \text {arctanh}\left (\frac {\sqrt [4]{1-i} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{4 x^{3/4} \sqrt [4]{x+1}}-\frac {\sqrt [4]{x^4+x^3} \text {arctanh}\left (\frac {\sqrt [4]{1-i} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2 (1-i)^{3/4} x^{3/4} \sqrt [4]{x+1}}-\frac {(1+i)^{5/4} \sqrt [4]{x^4+x^3} \text {arctanh}\left (\frac {\sqrt [4]{1+i} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{4 x^{3/4} \sqrt [4]{x+1}}-\frac {\sqrt [4]{x^4+x^3} \text {arctanh}\left (\frac {\sqrt [4]{1+i} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2 (1+i)^{3/4} x^{3/4} \sqrt [4]{x+1}}-\frac {\sqrt [4]{x^4+x^3} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2^{3/4} x^{3/4} \sqrt [4]{x+1}} \]

[In]

Int[(x^2*(x^3 + x^4)^(1/4))/(-1 + x^4),x]

[Out]

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

Rule 209

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

Rule 212

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

Rule 237

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2/(a^(3/4)*Rt[b/a, 2]))*EllipticF[(1/2)*ArcTan[Rt[b/a, 2]
*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b/a]

Rule 243

Int[((a_) + (b_.)*(x_)^4)^(-3/4), x_Symbol] :> Dist[x^3*((1 + a/(b*x^4))^(3/4)/(a + b*x^4)^(3/4)), Int[1/(x^3*
(1 + a/(b*x^4))^(3/4)), x], x] /; FreeQ[{a, b}, x]

Rule 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 304

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 338

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 342

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

Rule 416

Int[((a_) + (b_.)*(x_)^4)^(1/4)/((c_) + (d_.)*(x_)^4), x_Symbol] :> Dist[Sqrt[a + b*x^4]*Sqrt[a/(a + b*x^4)],
Subst[Int[1/(Sqrt[1 - b*x^4]*(c - (b*c - a*d)*x^4)), x], x, x/(a + b*x^4)^(1/4)], x] /; FreeQ[{a, b, c, d}, x]
&& NeQ[b*c - a*d, 0]

Rule 418

Int[1/(Sqrt[(a_) + (b_.)*(x_)^4]*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Dist[1/(2*c), Int[1/(Sqrt[a + b*x^4]*(1
- Rt[-d/c, 2]*x^2)), x], x] + Dist[1/(2*c), Int[1/(Sqrt[a + b*x^4]*(1 + Rt[-d/c, 2]*x^2)), x], x] /; FreeQ[{a,
b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 419

Int[1/(((a_) + (b_.)*(x_)^4)^(3/4)*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x^4)^
(3/4), x], x] - Dist[d/(b*c - a*d), Int[(a + b*x^4)^(1/4)/(c + d*x^4), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ
[b*c - a*d, 0]

Rule 508

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{k = Denominato
r[p]}, Dist[k*(a^(p + (m + 1)/n)/n), Subst[Int[x^(k*((m + 1)/n) - 1)*((c - (b*c - a*d)*x^k)^q/(1 - b*x^k)^(p +
q + (m + 1)/n + 1)), x], x, x^(n/k)/(a + b*x^n)^(1/k)], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && Ration
alQ[m, p] && IntegersQ[p + (m + 1)/n, q] && LtQ[-1, p, 0]

Rule 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b*(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-f/e, -d/c])

Rule 1227

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[(-a)*c, 2]}, Dist[Sqrt[-c],
Int[1/((d + e*x^2)*Sqrt[q + c*x^2]*Sqrt[q - c*x^2]), x], x]] /; FreeQ[{a, c, d, e}, x] && GtQ[a, 0] && LtQ[c,
0]

Rule 1254

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + c*x^4)^p, (d/
(d^2 - e^2*x^4) - e*(x^2/(d^2 - e^2*x^4)))^(-q), x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[c*d^2 + a*e^2, 0]
&&  !IntegerQ[p] && ILtQ[q, 0]

Rule 1543

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))^(q_))/((a_) + (c_.)*(x_)^(n2_.)), x_Symbol] :> Int[ExpandInte
grand[(d + e*x^n)^q, (f*x)^m/(a + c*x^(2*n)), x], x] /; FreeQ[{a, c, d, e, f, q, n}, x] && EqQ[n2, 2*n] && IGt
Q[n, 0] &&  !IntegerQ[q] && IntegerQ[m]

Rule 1600

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 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]

Rule 6865

Int[(u_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k, Subst[Int[x^(k*(m + 1) - 1)*(u /. x -> x^k
), x], x, x^(1/k)], x]] /; FractionQ[m]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [4]{x^3+x^4} \int \frac {x^{11/4} \sqrt [4]{1+x}}{-1+x^4} \, dx}{x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {\sqrt [4]{x^3+x^4} \int \frac {x^{11/4}}{(1+x)^{3/4} \left (-1+x-x^2+x^3\right )} \, dx}{x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {\left (4 \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^{14}}{\left (1+x^4\right )^{3/4} \left (-1+x^4-x^8+x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {\left (4 \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \left (\frac {x^2}{\left (1+x^4\right )^{3/4}}+\frac {x^2 \left (1-x^4+x^8\right )}{\left (1+x^4\right )^{3/4} \left (-1+x^4-x^8+x^{12}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {\left (4 \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 )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (4 \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2 \left (1-x^4+x^8\right )}{\left (1+x^4\right )^{3/4} \left (-1+x^4-x^8+x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {\left (4 \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 )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (4 \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \left (\frac {1}{4 \left (-1+x^2\right ) \left (1+x^4\right )^{3/4}}+\frac {1}{4 \left (1+x^2\right ) \left (1+x^4\right )^{3/4}}+\frac {x^2 \left (-1+x^4\right )}{2 \left (1+x^4\right )^{3/4} \left (1+x^8\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {\sqrt [4]{x^3+x^4} \text {Subst}\left (\int \frac {1}{\left (-1+x^2\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \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 )}{x^{3/4} \sqrt [4]{1+x}}-\frac {\left (2 \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 )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2 \left (-1+x^4\right )}{\left (1+x^4\right )^{3/4} \left (1+x^8\right )} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}} \\ & = -\frac {2 \sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {2 \sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \text {Subst}\left (\int \left (\frac {1}{\left (1-x^4\right ) \left (1+x^4\right )^{3/4}}+\frac {x^2}{\left (-1+x^4\right ) \left (1+x^4\right )^{3/4}}\right ) \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \text {Subst}\left (\int \left (\frac {1}{\left (-1+x^4\right ) \left (1+x^4\right )^{3/4}}+\frac {x^2}{\left (-1+x^4\right ) \left (1+x^4\right )^{3/4}}\right ) \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \left (-\frac {x^2}{\left (1+x^4\right )^{3/4} \left (1+x^8\right )}+\frac {x^6}{\left (1+x^4\right )^{3/4} \left (1+x^8\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}} \\ & = -\frac {2 \sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {2 \sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \text {Subst}\left (\int \frac {1}{\left (1-x^4\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \text {Subst}\left (\int \frac {1}{\left (-1+x^4\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+2 \frac {\sqrt [4]{x^3+x^4} \text {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}-\frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (1+x^4\right )^{3/4} \left (1+x^8\right )} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^6}{\left (1+x^4\right )^{3/4} \left (1+x^8\right )} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}} \\ & = -\frac {2 \sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {2 \sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \text {Subst}\left (\int \frac {\sqrt [4]{1+x^4}}{1-x^4} \, dx,x,\sqrt [4]{x}\right )}{2 x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \text {Subst}\left (\int \frac {\sqrt [4]{1+x^4}}{-1+x^4} \, dx,x,\sqrt [4]{x}\right )}{2 x^{3/4} \sqrt [4]{1+x}}+2 \frac {\sqrt [4]{x^3+x^4} \text {Subst}\left (\int \frac {x^2}{-1+2 x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}-\frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \left (-\frac {i x^2}{2 \left (-i+x^4\right ) \left (1+x^4\right )^{3/4}}+\frac {i x^2}{2 \left (i+x^4\right ) \left (1+x^4\right )^{3/4}}\right ) \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \left (\frac {x^2}{2 \left (-i+x^4\right ) \left (1+x^4\right )^{3/4}}+\frac {x^2}{2 \left (i+x^4\right ) \left (1+x^4\right )^{3/4}}\right ) \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}} \\ & = -\frac {2 \sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {2 \sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (i \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-i+x^4\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}-\frac {\left (i \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (i+x^4\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \text {Subst}\left (\int \frac {x^2}{\left (-i+x^4\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \text {Subst}\left (\int \frac {x^2}{\left (i+x^4\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+2 \left (-\frac {\sqrt [4]{x^3+x^4} \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 \sqrt {2} x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 \sqrt {2} x^{3/4} \sqrt [4]{1+x}}\right )+\frac {\left (\sqrt {\frac {1}{1+x}} \sqrt [4]{1+x} \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{\left (1-2 x^4\right ) \sqrt {1-x^4}} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 x^{3/4}}+\frac {\left (\sqrt {\frac {1}{1+x}} \sqrt [4]{1+x} \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^4} \left (-1+2 x^4\right )} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 x^{3/4}} \\ & = -\frac {2 \sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {2 \sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+2 \left (\frac {\sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2\ 2^{3/4} x^{3/4} \sqrt [4]{1+x}}-\frac {\sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2\ 2^{3/4} x^{3/4} \sqrt [4]{1+x}}\right )-\frac {\left (i \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{i+(1-i) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (i \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{-i+(1+i) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \text {Subst}\left (\int \frac {x^2}{i+(1-i) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \text {Subst}\left (\int \frac {x^2}{-i+(1+i) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}} \\ & = -\frac {2 \sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {2 \sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+2 \left (\frac {\sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2\ 2^{3/4} x^{3/4} \sqrt [4]{1+x}}-\frac {\sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2\ 2^{3/4} x^{3/4} \sqrt [4]{1+x}}\right )-\frac {\left (i (1-i)^{3/2} \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {1-i} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (i (1-i)^{3/2} \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {1-i} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4 x^{3/4} \sqrt [4]{1+x}}-\frac {\left ((1-i)^{3/2} \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {1-i} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4 x^{3/4} \sqrt [4]{1+x}}+\frac {\left ((1-i)^{3/2} \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {1-i} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (i (1+i)^{3/2} \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {1+i} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4 x^{3/4} \sqrt [4]{1+x}}-\frac {\left (i (1+i)^{3/2} \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {1+i} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4 x^{3/4} \sqrt [4]{1+x}}-\frac {\left ((1+i)^{3/2} \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {1+i} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4 x^{3/4} \sqrt [4]{1+x}}+\frac {\left ((1+i)^{3/2} \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {1+i} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4 x^{3/4} \sqrt [4]{1+x}} \\ & = -\frac {2 \sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{1-i} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 (1-i)^{3/4} x^{3/4} \sqrt [4]{1+x}}+\frac {(1-i)^{5/4} \sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{1-i} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4 x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{1+i} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 (1+i)^{3/4} x^{3/4} \sqrt [4]{1+x}}+\frac {(1+i)^{5/4} \sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{1+i} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4 x^{3/4} \sqrt [4]{1+x}}+\frac {2 \sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}-\frac {\sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{1-i} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 (1-i)^{3/4} x^{3/4} \sqrt [4]{1+x}}-\frac {(1-i)^{5/4} \sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{1-i} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4 x^{3/4} \sqrt [4]{1+x}}-\frac {\sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{1+i} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 (1+i)^{3/4} x^{3/4} \sqrt [4]{1+x}}-\frac {(1+i)^{5/4} \sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{1+i} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4 x^{3/4} \sqrt [4]{1+x}}+2 \left (\frac {\sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2\ 2^{3/4} x^{3/4} \sqrt [4]{1+x}}-\frac {\sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2\ 2^{3/4} x^{3/4} \sqrt [4]{1+x}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.12 \[ \int \frac {x^2 \sqrt [4]{x^3+x^4}}{-1+x^4} \, dx=-\frac {x^{9/4} (1+x)^{3/4} \left (8 \left (4 \arctan \left (\sqrt [4]{\frac {x}{1+x}}\right )-\sqrt [4]{2} \arctan \left (\sqrt [4]{2} \sqrt [4]{\frac {x}{1+x}}\right )-4 \text {arctanh}\left (\sqrt [4]{\frac {x}{1+x}}\right )+\sqrt [4]{2} \text {arctanh}\left (\sqrt [4]{2} \sqrt [4]{\frac {x}{1+x}}\right )\right )+\text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 \log (x)+8 \log \left (\sqrt [4]{1+x}-\sqrt [4]{x} \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-4 \log \left (\sqrt [4]{1+x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+\text {$\#$1}^7}\&\right ]\right )}{16 \left (x^3 (1+x)\right )^{3/4}} \]

[In]

Integrate[(x^2*(x^3 + x^4)^(1/4))/(-1 + x^4),x]

[Out]

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

Maple [N/A] (verified)

Time = 15.86 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.99

method	result	size
pseudoelliptic	\(-\frac {\ln \left (\frac {-2^{\frac {1}{4}} x -\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}\right ) 2^{\frac {1}{4}}}{4}-\frac {\arctan \left (\frac {2^{\frac {3}{4}} \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{2 x}\right ) 2^{\frac {1}{4}}}{2}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2 \textit {\_Z}^{4}+2\right )}{\sum }\frac {\left (\textit {\_R}^{4}-2\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (\textit {\_R}^{4}-1\right )}\right )}{4}+\ln \left (\frac {x +\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x}\right )-\ln \left (\frac {\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}-x}{x}\right )+2 \arctan \left (\frac {\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x}\right )\)	\(172\)
trager	\(\text {Expression too large to display}\)	\(3931\)

[In]

int(x^2*(x^4+x^3)^(1/4)/(x^4-1),x,method=_RETURNVERBOSE)

[Out]

-1/4*ln((-2^(1/4)*x-(x^3*(1+x))^(1/4))/(2^(1/4)*x-(x^3*(1+x))^(1/4)))*2^(1/4)-1/2*arctan(1/2*2^(3/4)/x*(x^3*(1
+x))^(1/4))*2^(1/4)+1/4*sum((_R^4-2)*ln((-_R*x+(x^3*(1+x))^(1/4))/x)/_R^3/(_R^4-1),_R=RootOf(_Z^8-2*_Z^4+2))+l
n((x+(x^3*(1+x))^(1/4))/x)-ln(((x^3*(1+x))^(1/4)-x)/x)+2*arctan((x^3*(1+x))^(1/4)/x)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.31 (sec) , antiderivative size = 409, normalized size of antiderivative = 2.36 \[ \int \frac {x^2 \sqrt [4]{x^3+x^4}}{-1+x^4} \, dx=-\frac {1}{16} \cdot 8^{\frac {3}{4}} \log \left (\frac {8^{\frac {3}{4}} x + 4 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{16} \cdot 8^{\frac {3}{4}} \log \left (-\frac {8^{\frac {3}{4}} x - 4 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{16} i \cdot 8^{\frac {3}{4}} \log \left (\frac {i \cdot 8^{\frac {3}{4}} x + 4 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{16} i \cdot 8^{\frac {3}{4}} \log \left (\frac {-i \cdot 8^{\frac {3}{4}} x + 4 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{4} \, \sqrt {-\sqrt {i + 1}} \log \left (\frac {x \sqrt {-\sqrt {i + 1}} + {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{4} \, \sqrt {-\sqrt {i + 1}} \log \left (-\frac {x \sqrt {-\sqrt {i + 1}} - {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{4} \, \sqrt {-\sqrt {-i + 1}} \log \left (\frac {x \sqrt {-\sqrt {-i + 1}} + {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{4} \, \sqrt {-\sqrt {-i + 1}} \log \left (-\frac {x \sqrt {-\sqrt {-i + 1}} - {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{4} \, \left (i + 1\right )^{\frac {1}{4}} \log \left (\frac {\left (i + 1\right )^{\frac {1}{4}} x + {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{4} \, \left (i + 1\right )^{\frac {1}{4}} \log \left (-\frac {\left (i + 1\right )^{\frac {1}{4}} x - {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{4} \, \left (-i + 1\right )^{\frac {1}{4}} \log \left (\frac {\left (-i + 1\right )^{\frac {1}{4}} x + {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{4} \, \left (-i + 1\right )^{\frac {1}{4}} \log \left (-\frac {\left (-i + 1\right )^{\frac {1}{4}} x - {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + 2 \, \arctan \left (\frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \log \left (\frac {x + {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \log \left (-\frac {x - {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \]

[In]

integrate(x^2*(x^4+x^3)^(1/4)/(x^4-1),x, algorithm="fricas")

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.54 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.15 \[ \int \frac {x^2 \sqrt [4]{x^3+x^4}}{-1+x^4} \, dx=\int \frac {x^{2} \sqrt [4]{x^{3} \left (x + 1\right )}}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.13 \[ \int \frac {x^2 \sqrt [4]{x^3+x^4}}{-1+x^4} \, dx=\int { \frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} x^{2}}{x^{4} - 1} \,d x } \]

[In]

integrate(x^2*(x^4+x^3)^(1/4)/(x^4-1),x, algorithm="maxima")

[Out]

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

Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.32 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.49 \[ \int \frac {x^2 \sqrt [4]{x^3+x^4}}{-1+x^4} \, dx=\text {Too large to display} \]

[In]

integrate(x^2*(x^4+x^3)^(1/4)/(x^4-1),x, algorithm="giac")

[Out]

-1/2*2^(1/4)*arctan(1/2*2^(3/4)*(1/x + 1)^(1/4)) - 1/4*2^(1/4)*log(2^(1/4) + (1/x + 1)^(1/4)) + 2*(-1/4096*I +
1/4096)^(1/4)*log(I*(98079714615416886934934209737619787751599303819750539264*I - 980797146154168869349342097
37619787751599303819750539264)^(1/4) - (70368744177664*I - 70368744177664)*(1/x + 1)^(1/4)) - 2*(-1/4096*I + 1
/4096)^(1/4)*log(I*(98079714615416886934934209737619787751599303819750539264*I - 98079714615416886934934209737
619787751599303819750539264)^(1/4) + (70368744177664*I - 70368744177664)*(1/x + 1)^(1/4)) - 2*I*(1/4096*I + 1/
4096)^(1/4)*log(I*(-98079714615416886934934209737619787751599303819750539264*I - 98079714615416886934934209737
619787751599303819750539264)^(1/4) - (70368744177664*I - 70368744177664)*(1/x + 1)^(1/4)) + 2*I*(1/4096*I + 1/
4096)^(1/4)*log(I*(-98079714615416886934934209737619787751599303819750539264*I - 98079714615416886934934209737
619787751599303819750539264)^(1/4) + (70368744177664*I - 70368744177664)*(1/x + 1)^(1/4)) + 2*(1/4096*I + 1/40
96)^(1/4)*log(-I*(-98079714615416886934934209737619787751599303819750539264*I - 980797146154168869349342097376
19787751599303819750539264)^(1/4) + (70368744177664*I + 70368744177664)*(1/x + 1)^(1/4)) - 2*(1/4096*I + 1/409
6)^(1/4)*log(-I*(-98079714615416886934934209737619787751599303819750539264*I - 9807971461541688693493420973761
9787751599303819750539264)^(1/4) - (70368744177664*I + 70368744177664)*(1/x + 1)^(1/4)) + 16*I*(-1/16777216*I
+ 1/16777216)^(1/4)*log(-I*(1503067252975253265849267581945175697520436831301324717252666221780613776073349403
81676735896625196994043838464*I - 1503067252975253265849267581945175697520436831301324717252666221780613776073
34940381676735896625196994043838464)^(1/4) + (2475880078570760549798248448*I + 2475880078570760549798248448)*(
1/x + 1)^(1/4)) - 16*I*(-1/16777216*I + 1/16777216)^(1/4)*log(-I*(15030672529752532658492675819451756975204368
3130132471725266622178061377607334940381676735896625196994043838464*I - 15030672529752532658492675819451756975
2043683130132471725266622178061377607334940381676735896625196994043838464)^(1/4) - (24758800785707605497982484
48*I + 2475880078570760549798248448)*(1/x + 1)^(1/4)) + 1/4*2^(1/4)*log(abs(-2^(1/4) + (1/x + 1)^(1/4))) + 2*a
rctan((1/x + 1)^(1/4)) + log((1/x + 1)^(1/4) + 1) - log(abs((1/x + 1)^(1/4) - 1))

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.13 \[ \int \frac {x^2 \sqrt [4]{x^3+x^4}}{-1+x^4} \, dx=\int \frac {x^2\,{\left (x^4+x^3\right )}^{1/4}}{x^4-1} \,d x \]

[In]

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

[Out]

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