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

Optimal. Leaf size=173 \[ -\frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^4+2\& ,\frac {-\text {$\#$1}^4 \log \left (\sqrt [4]{x^4+x^3}-\text {$\#$1} x\right )+\text {$\#$1}^4 \log (x)+2 \log \left (\sqrt [4]{x^4+x^3}-\text {$\#$1} x\right )-2 \log (x)}{\text {$\#$1}^7-\text {$\#$1}^3}\& \right ]-2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4+x^3}}\right )+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+x^3}}\right )}{2^{3/4}}+2 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4+x^3}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+x^3}}\right )}{2^{3/4}} \]

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Rubi [C]  time = 1.79, antiderivative size = 616, normalized size of antiderivative = 3.56, number of steps used = 67, number of rules used = 21, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.954, Rules used = {2056, 1586, 6733, 6742, 331, 298, 203, 206, 1240, 410, 237, 335, 275, 231, 407, 409, 1213, 537, 494, 6725, 1529} \begin {gather*} -\frac {2 \sqrt [4]{x^4+x^3} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2^{3/4} x^{3/4} \sqrt [4]{x+1}} \end {gather*}

Antiderivative was successfully verified.

[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 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 231

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

Rule 237

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 275

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 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 335

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 407

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 409

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 410

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 494

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 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), 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 1213

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 1240

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 1529

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 1586

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

Rule 2056

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 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]

Rule 6733

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 6742

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

Rubi steps

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

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Mathematica [C]  time = 0.08, size = 221, normalized size = 1.28 \begin {gather*} \frac {\sqrt [4]{x^3 (x+1)} \left (-7 \left (11 x^4 \, _2F_1\left (\frac {3}{4},\frac {15}{4};\frac {19}{4};-x\right )+11 x^3 \, _2F_1\left (\frac {3}{4},\frac {15}{4};\frac {19}{4};-x\right )-15 (x+1) x^2 \, _2F_1\left (-\frac {1}{4},\frac {11}{4};\frac {15}{4};-x\right )-165 (x+1) \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};-x\right )+(55-55 i) \sqrt [4]{x+1} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {(1-i) x}{x+1}\right )+(55+55 i) \sqrt [4]{x+1} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {(1+i) x}{x+1}\right )+110 \sqrt [4]{x+1} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {2 x}{x+1}\right )\right )+385 (x+1) \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};-x\right )-165 x (x+1) \, _2F_1\left (-\frac {1}{4},\frac {7}{4};\frac {11}{4};-x\right )\right )}{1155 (x+1)^{5/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((x^3*(1 + x))^(1/4)*(385*(1 + x)*Hypergeometric2F1[-1/4, 3/4, 7/4, -x] - 165*x*(1 + x)*Hypergeometric2F1[-1/4
, 7/4, 11/4, -x] - 7*(-15*x^2*(1 + x)*Hypergeometric2F1[-1/4, 11/4, 15/4, -x] - 165*(1 + x)*Hypergeometric2F1[
3/4, 3/4, 7/4, -x] + (55 - 55*I)*(1 + x)^(1/4)*Hypergeometric2F1[3/4, 1, 7/4, ((1 - I)*x)/(1 + x)] + (55 + 55*
I)*(1 + x)^(1/4)*Hypergeometric2F1[3/4, 1, 7/4, ((1 + I)*x)/(1 + x)] + 110*(1 + x)^(1/4)*Hypergeometric2F1[3/4
, 1, 7/4, (2*x)/(1 + x)] + 11*x^3*Hypergeometric2F1[3/4, 15/4, 19/4, -x] + 11*x^4*Hypergeometric2F1[3/4, 15/4,
19/4, -x])))/(1155*(1 + x)^(5/4))

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IntegrateAlgebraic [A]  time = 0.00, size = 173, normalized size = 1.00 \begin {gather*} -2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^4}}\right )}{2^{3/4}}+2 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )-\frac {\tanh ^{-1}\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 ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.84, size = 2015, normalized size = 11.65

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*2^(5/8)*sqrt(2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8)*sqrt(-2*sqrt(2) + 4)*arctan(1/4*(2^(3
/8)*(sqrt(2)*(sqrt(2)*x + x)*sqrt(-2*sqrt(2) + 4) - sqrt(2)*(sqrt(2)*x + x))*sqrt(2*(2*sqrt(2) + 3)*sqrt(-2*sq
rt(2) + 4) + 4*sqrt(2) + 8)*sqrt((4*2^(1/4)*x^2 + 2^(1/8)*((x^4 + x^3)^(1/4)*(sqrt(2)*x + 2*x)*sqrt(-2*sqrt(2)
+ 4) - 4*(x^4 + x^3)^(1/4)*x)*sqrt(2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8) + 4*sqrt(x^4 + x^3
))/x^2) - 2*2^(3/8)*(sqrt(2)*(x^4 + x^3)^(1/4)*(sqrt(2) + 1)*sqrt(-2*sqrt(2) + 4) - sqrt(2)*(x^4 + x^3)^(1/4)*
(sqrt(2) + 1))*sqrt(2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8) - 4*sqrt(2)*x + 4*(sqrt(2)*x + x)*
sqrt(-2*sqrt(2) + 4) - 4*x)/x) + 1/8*2^(5/8)*sqrt(2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8)*sqrt
(-2*sqrt(2) + 4)*arctan(1/4*(2^(3/8)*(sqrt(2)*(sqrt(2)*x + x)*sqrt(-2*sqrt(2) + 4) - sqrt(2)*(sqrt(2)*x + x))*
sqrt(2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8)*sqrt((4*2^(1/4)*x^2 - 2^(1/8)*((x^4 + x^3)^(1/4)*
(sqrt(2)*x + 2*x)*sqrt(-2*sqrt(2) + 4) - 4*(x^4 + x^3)^(1/4)*x)*sqrt(2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) +
4*sqrt(2) + 8) + 4*sqrt(x^4 + x^3))/x^2) - 2*2^(3/8)*(sqrt(2)*(x^4 + x^3)^(1/4)*(sqrt(2) + 1)*sqrt(-2*sqrt(2)
+ 4) - sqrt(2)*(x^4 + x^3)^(1/4)*(sqrt(2) + 1))*sqrt(2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8) +
4*sqrt(2)*x - 4*(sqrt(2)*x + x)*sqrt(-2*sqrt(2) + 4) + 4*x)/x) + 1/8*2^(5/8)*sqrt(-2*(2*sqrt(2) + 3)*sqrt(-2*
sqrt(2) + 4) + 4*sqrt(2) + 8)*sqrt(-2*sqrt(2) + 4)*arctan(1/4*(2^(3/8)*(sqrt(2)*(sqrt(2)*x + x)*sqrt(-2*sqrt(2
) + 4) + sqrt(2)*(sqrt(2)*x + x))*sqrt(-2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8)*sqrt((4*2^(1/4
)*x^2 + 2^(1/8)*((x^4 + x^3)^(1/4)*(sqrt(2)*x + 2*x)*sqrt(-2*sqrt(2) + 4) + 4*(x^4 + x^3)^(1/4)*x)*sqrt(-2*(2*
sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8) + 4*sqrt(x^4 + x^3))/x^2) - 2*2^(3/8)*(sqrt(2)*(x^4 + x^3)^
(1/4)*(sqrt(2) + 1)*sqrt(-2*sqrt(2) + 4) + sqrt(2)*(x^4 + x^3)^(1/4)*(sqrt(2) + 1))*sqrt(-2*(2*sqrt(2) + 3)*sq
rt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8) - 4*sqrt(2)*x - 4*(sqrt(2)*x + x)*sqrt(-2*sqrt(2) + 4) - 4*x)/x) + 1/8*2^(
5/8)*sqrt(-2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8)*sqrt(-2*sqrt(2) + 4)*arctan(1/4*(2^(3/8)*(s
qrt(2)*(sqrt(2)*x + x)*sqrt(-2*sqrt(2) + 4) + sqrt(2)*(sqrt(2)*x + x))*sqrt(-2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2)
+ 4) + 4*sqrt(2) + 8)*sqrt((4*2^(1/4)*x^2 - 2^(1/8)*((x^4 + x^3)^(1/4)*(sqrt(2)*x + 2*x)*sqrt(-2*sqrt(2) + 4)
+ 4*(x^4 + x^3)^(1/4)*x)*sqrt(-2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8) + 4*sqrt(x^4 + x^3))/x
^2) - 2*2^(3/8)*(sqrt(2)*(x^4 + x^3)^(1/4)*(sqrt(2) + 1)*sqrt(-2*sqrt(2) + 4) + sqrt(2)*(x^4 + x^3)^(1/4)*(sqr
t(2) + 1))*sqrt(-2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8) + 4*sqrt(2)*x + 4*(sqrt(2)*x + x)*sqr
t(-2*sqrt(2) + 4) + 4*x)/x) - 1/32*2^(1/8)*sqrt(2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8)*((sqrt
(2) + 2)*sqrt(-2*sqrt(2) + 4) - 4)*log(1/4*(4*2^(1/4)*x^2 + 2^(1/8)*((x^4 + x^3)^(1/4)*(sqrt(2)*x + 2*x)*sqrt(
-2*sqrt(2) + 4) - 4*(x^4 + x^3)^(1/4)*x)*sqrt(2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8) + 4*sqrt
(x^4 + x^3))/x^2) + 1/32*2^(1/8)*sqrt(2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8)*((sqrt(2) + 2)*s
qrt(-2*sqrt(2) + 4) - 4)*log(1/4*(4*2^(1/4)*x^2 - 2^(1/8)*((x^4 + x^3)^(1/4)*(sqrt(2)*x + 2*x)*sqrt(-2*sqrt(2)
+ 4) - 4*(x^4 + x^3)^(1/4)*x)*sqrt(2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8) + 4*sqrt(x^4 + x^3
))/x^2) - 1/32*2^(1/8)*sqrt(-2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8)*((sqrt(2) + 2)*sqrt(-2*sq
rt(2) + 4) + 4)*log(1/4*(4*2^(1/4)*x^2 + 2^(1/8)*((x^4 + x^3)^(1/4)*(sqrt(2)*x + 2*x)*sqrt(-2*sqrt(2) + 4) + 4
*(x^4 + x^3)^(1/4)*x)*sqrt(-2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8) + 4*sqrt(x^4 + x^3))/x^2)
+ 1/32*2^(1/8)*sqrt(-2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8)*((sqrt(2) + 2)*sqrt(-2*sqrt(2) +
4) + 4)*log(1/4*(4*2^(1/4)*x^2 - 2^(1/8)*((x^4 + x^3)^(1/4)*(sqrt(2)*x + 2*x)*sqrt(-2*sqrt(2) + 4) + 4*(x^4 +
x^3)^(1/4)*x)*sqrt(-2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8) + 4*sqrt(x^4 + x^3))/x^2) + 1/4*8^
(3/4)*arctan(1/2*(8^(1/4)*x*sqrt((sqrt(2)*x^2 + sqrt(x^4 + x^3))/x^2) - 8^(1/4)*(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*8^(3/4)*log(-(8^(3/4)*x - 4*(x^4 + x^3)^(1/4))/x) + 2*a
rctan((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)

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giac [B]  time = 0.29, size = 257, normalized size = 1.49

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [B]  time = 12.33, size = 1934, normalized size = 11.18

method	result	size
trager	\(\text {Expression too large to display}\)	\(1934\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootOf(_Z^2+1)*ln((2*RootOf(_Z^2+1)*(x^4+x^3)^(1/2)*x-2*RootOf(_Z^2+1)*x^3+2*(x^4+x^3)^(3/4)-2*x^2*(x^4+x^3)^(
1/4)-RootOf(_Z^2+1)*x^2)/x^2)-ln((2*(x^4+x^3)^(3/4)-2*(x^4+x^3)^(1/2)*x+2*x^2*(x^4+x^3)^(1/4)-2*x^3-x^2)/x^2)-
1/4*RootOf(_Z^2+1)*RootOf(_Z^4+RootOf(_Z^2+1)-1)*ln(-(33*RootOf(_Z^4+RootOf(_Z^2+1)-1)^3*RootOf(_Z^2+1)^2*x^3-
66*RootOf(_Z^4+RootOf(_Z^2+1)-1)^3*RootOf(_Z^2+1)^2*x^2-415*RootOf(_Z^4+RootOf(_Z^2+1)-1)^3*RootOf(_Z^2+1)*x^3
-127*RootOf(_Z^4+RootOf(_Z^2+1)-1)^3*RootOf(_Z^2+1)*x^2-68*(x^4+x^3)^(1/4)*RootOf(_Z^2+1)*RootOf(_Z^4+RootOf(_
Z^2+1)-1)^2*x^2+228*RootOf(_Z^4+RootOf(_Z^2+1)-1)^3*x^3+576*RootOf(_Z^4+RootOf(_Z^2+1)-1)*RootOf(_Z^2+1)*(x^4+
x^3)^(1/2)*x+95*RootOf(_Z^4+RootOf(_Z^2+1)-1)^3*x^2-576*(x^4+x^3)^(1/4)*RootOf(_Z^4+RootOf(_Z^2+1)-1)^2*x^2+68
*RootOf(_Z^2+1)*(x^4+x^3)^(3/4)-68*RootOf(_Z^4+RootOf(_Z^2+1)-1)*(x^4+x^3)^(1/2)*x+576*(x^4+x^3)^(3/4))/(RootO
f(_Z^2+1)*x-2*RootOf(_Z^2+1)+2*x+1)/x^2)-1/4*RootOf(_Z^4+RootOf(_Z^2+1)-1)*ln((19*RootOf(_Z^4+RootOf(_Z^2+1)-1
)^3*RootOf(_Z^2+1)^2*x^3-38*RootOf(_Z^4+RootOf(_Z^2+1)-1)^3*RootOf(_Z^2+1)^2*x^2-195*RootOf(_Z^4+RootOf(_Z^2+1
)-1)^3*RootOf(_Z^2+1)*x^3-161*RootOf(_Z^4+RootOf(_Z^2+1)-1)^3*RootOf(_Z^2+1)*x^2-68*(x^4+x^3)^(1/4)*RootOf(_Z^
2+1)*RootOf(_Z^4+RootOf(_Z^2+1)-1)^2*x^2-396*RootOf(_Z^4+RootOf(_Z^2+1)-1)^3*x^3-68*RootOf(_Z^4+RootOf(_Z^2+1)
-1)*RootOf(_Z^2+1)*(x^4+x^3)^(1/2)*x-165*RootOf(_Z^4+RootOf(_Z^2+1)-1)^3*x^2-576*(x^4+x^3)^(1/4)*RootOf(_Z^4+R
ootOf(_Z^2+1)-1)^2*x^2-68*RootOf(_Z^2+1)*(x^4+x^3)^(3/4)-576*RootOf(_Z^4+RootOf(_Z^2+1)-1)*(x^4+x^3)^(1/2)*x-5
76*(x^4+x^3)^(3/4))/(RootOf(_Z^2+1)*x-2*RootOf(_Z^2+1)+2*x+1)/x^2)-1/4*RootOf(_Z^4-RootOf(_Z^2+1)-1)*ln((19*Ro
otOf(_Z^4-RootOf(_Z^2+1)-1)^3*RootOf(_Z^2+1)^2*x^3-38*RootOf(_Z^4-RootOf(_Z^2+1)-1)^3*RootOf(_Z^2+1)^2*x^2+195
*RootOf(_Z^4-RootOf(_Z^2+1)-1)^3*RootOf(_Z^2+1)*x^3+68*(x^4+x^3)^(1/4)*RootOf(_Z^2+1)*RootOf(_Z^4-RootOf(_Z^2+
1)-1)^2*x^2+161*RootOf(_Z^4-RootOf(_Z^2+1)-1)^3*RootOf(_Z^2+1)*x^2-396*RootOf(_Z^4-RootOf(_Z^2+1)-1)^3*x^3+68*
RootOf(_Z^4-RootOf(_Z^2+1)-1)*RootOf(_Z^2+1)*(x^4+x^3)^(1/2)*x-576*(x^4+x^3)^(1/4)*RootOf(_Z^4-RootOf(_Z^2+1)-
1)^2*x^2-165*RootOf(_Z^4-RootOf(_Z^2+1)-1)^3*x^2+68*RootOf(_Z^2+1)*(x^4+x^3)^(3/4)-576*RootOf(_Z^4-RootOf(_Z^2
+1)-1)*(x^4+x^3)^(1/2)*x-576*(x^4+x^3)^(3/4))/(RootOf(_Z^2+1)*x-2*RootOf(_Z^2+1)-2*x-1)/x^2)-1/4*RootOf(_Z^2+1
)*RootOf(_Z^4-RootOf(_Z^2+1)-1)*ln((33*RootOf(_Z^4-RootOf(_Z^2+1)-1)^3*RootOf(_Z^2+1)^2*x^3-66*RootOf(_Z^4-Roo
tOf(_Z^2+1)-1)^3*RootOf(_Z^2+1)^2*x^2+415*RootOf(_Z^4-RootOf(_Z^2+1)-1)^3*RootOf(_Z^2+1)*x^3+127*RootOf(_Z^4-R
ootOf(_Z^2+1)-1)^3*RootOf(_Z^2+1)*x^2-68*(x^4+x^3)^(1/4)*RootOf(_Z^2+1)*RootOf(_Z^4-RootOf(_Z^2+1)-1)^2*x^2+22
8*RootOf(_Z^4-RootOf(_Z^2+1)-1)^3*x^3-576*RootOf(_Z^4-RootOf(_Z^2+1)-1)*RootOf(_Z^2+1)*(x^4+x^3)^(1/2)*x+95*Ro
otOf(_Z^4-RootOf(_Z^2+1)-1)^3*x^2+576*(x^4+x^3)^(1/4)*RootOf(_Z^4-RootOf(_Z^2+1)-1)^2*x^2+68*RootOf(_Z^2+1)*(x
^4+x^3)^(3/4)-68*RootOf(_Z^4-RootOf(_Z^2+1)-1)*(x^4+x^3)^(1/2)*x-576*(x^4+x^3)^(3/4))/(RootOf(_Z^2+1)*x-2*Root
Of(_Z^2+1)-2*x-1)/x^2)-1/4*RootOf(_Z^4+RootOf(_Z^2+1)-1)*RootOf(_Z^4-RootOf(_Z^2+1)-1)*ln((3*RootOf(_Z^4+RootO
f(_Z^2+1)-1)^3*RootOf(_Z^4-RootOf(_Z^2+1)-1)^3*x^3+RootOf(_Z^4+RootOf(_Z^2+1)-1)^3*RootOf(_Z^4-RootOf(_Z^2+1)-
1)^3*x^2+4*(x^4+x^3)^(1/4)*RootOf(_Z^4+RootOf(_Z^2+1)-1)^2*RootOf(_Z^4-RootOf(_Z^2+1)-1)^2*x^2+4*(x^4+x^3)^(1/
2)*RootOf(_Z^4-RootOf(_Z^2+1)-1)*RootOf(_Z^4+RootOf(_Z^2+1)-1)*x+4*(x^4+x^3)^(3/4))/(-1+x)/x^2)+1/4*RootOf(_Z^
2+1)*RootOf(_Z^4-RootOf(_Z^2+1)-1)*RootOf(_Z^4+RootOf(_Z^2+1)-1)*ln((3*RootOf(_Z^2+1)*RootOf(_Z^4+RootOf(_Z^2+
1)-1)^3*RootOf(_Z^4-RootOf(_Z^2+1)-1)^3*x^3+RootOf(_Z^2+1)*RootOf(_Z^4+RootOf(_Z^2+1)-1)^3*RootOf(_Z^4-RootOf(
_Z^2+1)-1)^3*x^2-4*(x^4+x^3)^(1/4)*RootOf(_Z^4+RootOf(_Z^2+1)-1)^2*RootOf(_Z^4-RootOf(_Z^2+1)-1)^2*x^2-4*(x^4+
x^3)^(1/2)*RootOf(_Z^4-RootOf(_Z^2+1)-1)*RootOf(_Z^4+RootOf(_Z^2+1)-1)*RootOf(_Z^2+1)*x+4*(x^4+x^3)^(3/4))/(-1
+x)/x^2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} x^{2}}{x^{4} - 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2\,{\left (x^4+x^3\right )}^{1/4}}{x^4-1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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