3.28.6 \(\int \frac {x^4 \sqrt [4]{b x^2+a x^4}}{b+a x^2+x^4} \, dx\) [2706]

Optimal. Leaf size=247 \[ \frac {1}{2} x \sqrt [4]{b x^2+a x^4}+\frac {\left (4 a^2-b\right ) \text {ArcTan}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right )}{4 a^{3/4}}+\frac {\left (-4 a^2+b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right )}{4 a^{3/4}}+\frac {1}{2} \text {RootSum}\left [b-a \text {$\#$1}^4+\text {$\#$1}^8\& ,\frac {a b \log (x)-a b \log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right )-a^2 \log (x) \text {$\#$1}^4+b \log (x) \text {$\#$1}^4+a^2 \log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4-b \log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}^3-2 \text {$\#$1}^7}\& \right ] \]

[Out]

Unintegrable

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(1403\) vs. \(2(247)=494\).
time = 3.52, antiderivative size = 1403, normalized size of antiderivative = 5.68, number of steps used = 30, number of rules used = 13, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {2081, 1283, 1530, 470, 338, 304, 209, 212, 6860, 1542, 508, 211, 214} \begin {gather*} \frac {1}{2} \sqrt [4]{a x^4+b x^2} x+\frac {\left (4 a^2-b\right ) \sqrt [4]{a x^4+b x^2} \text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{4 a^{3/4} \sqrt [4]{a x^2+b} \sqrt {x}}-\frac {2 \left (a^2-b\right ) b \sqrt [4]{a x^4+b x^2} \text {ArcTan}\left (\frac {\sqrt [4]{a^2-\sqrt {a^2-4 b} a-2 b} \sqrt {x}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{a x^2+b}}\right )}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt {a^2-4 b} \left (a^2-\sqrt {a^2-4 b} a-2 b\right )^{3/4} \sqrt [4]{a x^2+b} \sqrt {x}}+\frac {a \left (a-\sqrt {a^2-4 b}\right )^{3/4} \left (a^2-2 b\right ) \sqrt [4]{a x^4+b x^2} \text {ArcTan}\left (\frac {\sqrt [4]{a^2-\sqrt {a^2-4 b} a-2 b} \sqrt {x}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{a x^2+b}}\right )}{\sqrt {a^2-4 b} \left (a^2-\sqrt {a^2-4 b} a-2 b\right )^{3/4} \sqrt [4]{a x^2+b} \sqrt {x}}+\frac {2 \left (a^2-b\right ) b \sqrt [4]{a x^4+b x^2} \text {ArcTan}\left (\frac {\sqrt [4]{a^2+\sqrt {a^2-4 b} a-2 b} \sqrt {x}}{\sqrt [4]{a+\sqrt {a^2-4 b}} \sqrt [4]{a x^2+b}}\right )}{\sqrt [4]{a+\sqrt {a^2-4 b}} \sqrt {a^2-4 b} \left (a^2+\sqrt {a^2-4 b} a-2 b\right )^{3/4} \sqrt [4]{a x^2+b} \sqrt {x}}-\frac {a \left (a+\sqrt {a^2-4 b}\right )^{3/4} \left (a^2-2 b\right ) \sqrt [4]{a x^4+b x^2} \text {ArcTan}\left (\frac {\sqrt [4]{a^2+\sqrt {a^2-4 b} a-2 b} \sqrt {x}}{\sqrt [4]{a+\sqrt {a^2-4 b}} \sqrt [4]{a x^2+b}}\right )}{\sqrt {a^2-4 b} \left (a^2+\sqrt {a^2-4 b} a-2 b\right )^{3/4} \sqrt [4]{a x^2+b} \sqrt {x}}-\frac {\left (4 a^2-b\right ) \sqrt [4]{a x^4+b x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{4 a^{3/4} \sqrt [4]{a x^2+b} \sqrt {x}}+\frac {2 \left (a^2-b\right ) b \sqrt [4]{a x^4+b x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2-\sqrt {a^2-4 b} a-2 b} \sqrt {x}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{a x^2+b}}\right )}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt {a^2-4 b} \left (a^2-\sqrt {a^2-4 b} a-2 b\right )^{3/4} \sqrt [4]{a x^2+b} \sqrt {x}}-\frac {a \left (a-\sqrt {a^2-4 b}\right )^{3/4} \left (a^2-2 b\right ) \sqrt [4]{a x^4+b x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2-\sqrt {a^2-4 b} a-2 b} \sqrt {x}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{a x^2+b}}\right )}{\sqrt {a^2-4 b} \left (a^2-\sqrt {a^2-4 b} a-2 b\right )^{3/4} \sqrt [4]{a x^2+b} \sqrt {x}}-\frac {2 \left (a^2-b\right ) b \sqrt [4]{a x^4+b x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2+\sqrt {a^2-4 b} a-2 b} \sqrt {x}}{\sqrt [4]{a+\sqrt {a^2-4 b}} \sqrt [4]{a x^2+b}}\right )}{\sqrt [4]{a+\sqrt {a^2-4 b}} \sqrt {a^2-4 b} \left (a^2+\sqrt {a^2-4 b} a-2 b\right )^{3/4} \sqrt [4]{a x^2+b} \sqrt {x}}+\frac {a \left (a+\sqrt {a^2-4 b}\right )^{3/4} \left (a^2-2 b\right ) \sqrt [4]{a x^4+b x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2+\sqrt {a^2-4 b} a-2 b} \sqrt {x}}{\sqrt [4]{a+\sqrt {a^2-4 b}} \sqrt [4]{a x^2+b}}\right )}{\sqrt {a^2-4 b} \left (a^2+\sqrt {a^2-4 b} a-2 b\right )^{3/4} \sqrt [4]{a x^2+b} \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^4*(b*x^2 + a*x^4)^(1/4))/(b + a*x^2 + x^4),x]

[Out]

(x*(b*x^2 + a*x^4)^(1/4))/2 + ((4*a^2 - b)*(b*x^2 + a*x^4)^(1/4)*ArcTan[(a^(1/4)*Sqrt[x])/(b + a*x^2)^(1/4)])/
(4*a^(3/4)*Sqrt[x]*(b + a*x^2)^(1/4)) + (a*(a - Sqrt[a^2 - 4*b])^(3/4)*(a^2 - 2*b)*(b*x^2 + a*x^4)^(1/4)*ArcTa
n[((a^2 - a*Sqrt[a^2 - 4*b] - 2*b)^(1/4)*Sqrt[x])/((a - Sqrt[a^2 - 4*b])^(1/4)*(b + a*x^2)^(1/4))])/(Sqrt[a^2
- 4*b]*(a^2 - a*Sqrt[a^2 - 4*b] - 2*b)^(3/4)*Sqrt[x]*(b + a*x^2)^(1/4)) - (2*(a^2 - b)*b*(b*x^2 + a*x^4)^(1/4)
*ArcTan[((a^2 - a*Sqrt[a^2 - 4*b] - 2*b)^(1/4)*Sqrt[x])/((a - Sqrt[a^2 - 4*b])^(1/4)*(b + a*x^2)^(1/4))])/((a
- Sqrt[a^2 - 4*b])^(1/4)*Sqrt[a^2 - 4*b]*(a^2 - a*Sqrt[a^2 - 4*b] - 2*b)^(3/4)*Sqrt[x]*(b + a*x^2)^(1/4)) - (a
*(a + Sqrt[a^2 - 4*b])^(3/4)*(a^2 - 2*b)*(b*x^2 + a*x^4)^(1/4)*ArcTan[((a^2 + a*Sqrt[a^2 - 4*b] - 2*b)^(1/4)*S
qrt[x])/((a + Sqrt[a^2 - 4*b])^(1/4)*(b + a*x^2)^(1/4))])/(Sqrt[a^2 - 4*b]*(a^2 + a*Sqrt[a^2 - 4*b] - 2*b)^(3/
4)*Sqrt[x]*(b + a*x^2)^(1/4)) + (2*(a^2 - b)*b*(b*x^2 + a*x^4)^(1/4)*ArcTan[((a^2 + a*Sqrt[a^2 - 4*b] - 2*b)^(
1/4)*Sqrt[x])/((a + Sqrt[a^2 - 4*b])^(1/4)*(b + a*x^2)^(1/4))])/((a + Sqrt[a^2 - 4*b])^(1/4)*Sqrt[a^2 - 4*b]*(
a^2 + a*Sqrt[a^2 - 4*b] - 2*b)^(3/4)*Sqrt[x]*(b + a*x^2)^(1/4)) - ((4*a^2 - b)*(b*x^2 + a*x^4)^(1/4)*ArcTanh[(
a^(1/4)*Sqrt[x])/(b + a*x^2)^(1/4)])/(4*a^(3/4)*Sqrt[x]*(b + a*x^2)^(1/4)) - (a*(a - Sqrt[a^2 - 4*b])^(3/4)*(a
^2 - 2*b)*(b*x^2 + a*x^4)^(1/4)*ArcTanh[((a^2 - a*Sqrt[a^2 - 4*b] - 2*b)^(1/4)*Sqrt[x])/((a - Sqrt[a^2 - 4*b])
^(1/4)*(b + a*x^2)^(1/4))])/(Sqrt[a^2 - 4*b]*(a^2 - a*Sqrt[a^2 - 4*b] - 2*b)^(3/4)*Sqrt[x]*(b + a*x^2)^(1/4))
+ (2*(a^2 - b)*b*(b*x^2 + a*x^4)^(1/4)*ArcTanh[((a^2 - a*Sqrt[a^2 - 4*b] - 2*b)^(1/4)*Sqrt[x])/((a - Sqrt[a^2
- 4*b])^(1/4)*(b + a*x^2)^(1/4))])/((a - Sqrt[a^2 - 4*b])^(1/4)*Sqrt[a^2 - 4*b]*(a^2 - a*Sqrt[a^2 - 4*b] - 2*b
)^(3/4)*Sqrt[x]*(b + a*x^2)^(1/4)) + (a*(a + Sqrt[a^2 - 4*b])^(3/4)*(a^2 - 2*b)*(b*x^2 + a*x^4)^(1/4)*ArcTanh[
((a^2 + a*Sqrt[a^2 - 4*b] - 2*b)^(1/4)*Sqrt[x])/((a + Sqrt[a^2 - 4*b])^(1/4)*(b + a*x^2)^(1/4))])/(Sqrt[a^2 -
4*b]*(a^2 + a*Sqrt[a^2 - 4*b] - 2*b)^(3/4)*Sqrt[x]*(b + a*x^2)^(1/4)) - (2*(a^2 - b)*b*(b*x^2 + a*x^4)^(1/4)*A
rcTanh[((a^2 + a*Sqrt[a^2 - 4*b] - 2*b)^(1/4)*Sqrt[x])/((a + Sqrt[a^2 - 4*b])^(1/4)*(b + a*x^2)^(1/4))])/((a +
Sqrt[a^2 - 4*b])^(1/4)*Sqrt[a^2 - 4*b]*(a^2 + a*Sqrt[a^2 - 4*b] - 2*b)^(3/4)*Sqrt[x]*(b + a*x^2)^(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 211

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

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 214

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

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 470

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*(e*x)^(m +
1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p + 1) + 1))), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m +
n*(p + 1) + 1)), Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0]
&& NeQ[m + n*(p + 1) + 1, 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 1283

Int[((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With
[{k = Denominator[m]}, Dist[k/f, Subst[Int[x^(k*(m + 1) - 1)*(d + e*(x^(2*k)/f^2))^q*(a + b*(x^(2*k)/f^k) + c*
(x^(4*k)/f^4))^p, x], x, (f*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, f, p, q}, x] && NeQ[b^2 - 4*a*c, 0] && Fra
ctionQ[m] && IntegerQ[p]

Rule 1530

Int[(((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^(n_))^(q_))/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symbol
] :> Dist[f^(2*n)/c^2, Int[(f*x)^(m - 2*n)*(c*d - b*e + c*e*x^n)*(d + e*x^n)^(q - 1), x], x] - Dist[f^(2*n)/c^
2, Int[(f*x)^(m - 2*n)*(d + e*x^n)^(q - 1)*(Simp[a*(c*d - b*e) + (b*c*d - b^2*e + a*c*e)*x^n, x]/(a + b*x^n +
c*x^(2*n))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] &&  !
IntegerQ[q] && GtQ[q, 0] && GtQ[m, 2*n - 1]

Rule 1542

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))^(q_))/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symbol]
:> Int[ExpandIntegrand[(d + e*x^n)^q, (f*x)^m/(a + b*x^n + c*x^(2*n)), x], x] /; FreeQ[{a, b, c, d, e, f, q,
n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] &&  !IntegerQ[q] && IntegerQ[m]

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 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {x^4 \sqrt [4]{b x^2+a x^4}}{b+a x^2+x^4} \, dx &=\frac {\sqrt [4]{b x^2+a x^4} \int \frac {x^{9/2} \sqrt [4]{b+a x^2}}{b+a x^2+x^4} \, dx}{\sqrt {x} \sqrt [4]{b+a x^2}}\\ &=\frac {\left (2 \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^{10} \sqrt [4]{b+a x^4}}{b+a x^4+x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{b+a x^2}}\\ &=\frac {\left (2 \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2 \left (-a^2+b+a x^4\right )}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (2 \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2 \left (-\left (\left (a^2-b\right ) b\right )-a \left (a^2-2 b\right ) x^4\right )}{\left (b+a x^4\right )^{3/4} \left (b+a x^4+x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{b+a x^2}}\\ &=\frac {1}{2} x \sqrt [4]{b x^2+a x^4}-\frac {\left (2 \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \left (-\frac {\left (a^2-b\right ) b x^2}{\left (b+a x^4\right )^{3/4} \left (b+a x^4+x^8\right )}-\frac {a \left (a^2-2 b\right ) x^6}{\left (b+a x^4\right )^{3/4} \left (b+a x^4+x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (\left (4 a^2-b\right ) \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt [4]{b+a x^2}}\\ &=\frac {1}{2} x \sqrt [4]{b x^2+a x^4}+\frac {\left (2 a \left (a^2-2 b\right ) \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^6}{\left (b+a x^4\right )^{3/4} \left (b+a x^4+x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (\left (4 a^2-b\right ) \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (2 \left (a^2-b\right ) b \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4} \left (b+a x^4+x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{b+a x^2}}\\ &=\frac {1}{2} x \sqrt [4]{b x^2+a x^4}+\frac {\left (2 a \left (a^2-2 b\right ) \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \left (-\frac {\left (-a+\sqrt {a^2-4 b}\right ) x^2}{\sqrt {a^2-4 b} \left (-a+\sqrt {a^2-4 b}-2 x^4\right ) \left (b+a x^4\right )^{3/4}}+\frac {\left (a+\sqrt {a^2-4 b}\right ) x^2}{\sqrt {a^2-4 b} \left (a+\sqrt {a^2-4 b}+2 x^4\right ) \left (b+a x^4\right )^{3/4}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (\left (4 a^2-b\right ) \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4 \sqrt {a} \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (\left (4 a^2-b\right ) \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4 \sqrt {a} \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (2 \left (a^2-b\right ) b \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \left (\frac {2 x^2}{\sqrt {a^2-4 b} \left (a-\sqrt {a^2-4 b}+2 x^4\right ) \left (b+a x^4\right )^{3/4}}-\frac {2 x^2}{\sqrt {a^2-4 b} \left (a+\sqrt {a^2-4 b}+2 x^4\right ) \left (b+a x^4\right )^{3/4}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{b+a x^2}}\\ &=\frac {1}{2} x \sqrt [4]{b x^2+a x^4}+\frac {\left (4 a^2-b\right ) \sqrt [4]{b x^2+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4 a^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (4 a^2-b\right ) \sqrt [4]{b x^2+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4 a^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (2 a \left (1-\frac {a}{\sqrt {a^2-4 b}}\right ) \left (a^2-2 b\right ) \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-a+\sqrt {a^2-4 b}-2 x^4\right ) \left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (2 a \left (1+\frac {a}{\sqrt {a^2-4 b}}\right ) \left (a^2-2 b\right ) \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (a+\sqrt {a^2-4 b}+2 x^4\right ) \left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (4 \left (a^2-b\right ) b \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (a-\sqrt {a^2-4 b}+2 x^4\right ) \left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{\sqrt {a^2-4 b} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (4 \left (a^2-b\right ) b \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (a+\sqrt {a^2-4 b}+2 x^4\right ) \left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{\sqrt {a^2-4 b} \sqrt {x} \sqrt [4]{b+a x^2}}\\ &=\frac {1}{2} x \sqrt [4]{b x^2+a x^4}+\frac {\left (4 a^2-b\right ) \sqrt [4]{b x^2+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4 a^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (4 a^2-b\right ) \sqrt [4]{b x^2+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4 a^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (2 a \left (1-\frac {a}{\sqrt {a^2-4 b}}\right ) \left (a^2-2 b\right ) \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{-a+\sqrt {a^2-4 b}-\left (a \left (-a+\sqrt {a^2-4 b}\right )+2 b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (2 a \left (1+\frac {a}{\sqrt {a^2-4 b}}\right ) \left (a^2-2 b\right ) \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{a+\sqrt {a^2-4 b}-\left (a \left (a+\sqrt {a^2-4 b}\right )-2 b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (4 \left (a^2-b\right ) b \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{a-\sqrt {a^2-4 b}-\left (a \left (a-\sqrt {a^2-4 b}\right )-2 b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {a^2-4 b} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (4 \left (a^2-b\right ) b \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{a+\sqrt {a^2-4 b}-\left (a \left (a+\sqrt {a^2-4 b}\right )-2 b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {a^2-4 b} \sqrt {x} \sqrt [4]{b+a x^2}}\\ &=\frac {1}{2} x \sqrt [4]{b x^2+a x^4}+\frac {\left (4 a^2-b\right ) \sqrt [4]{b x^2+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4 a^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (4 a^2-b\right ) \sqrt [4]{b x^2+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4 a^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (a \left (1-\frac {a}{\sqrt {a^2-4 b}}\right ) \left (a^2-2 b\right ) \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2-4 b}}-\sqrt {a^2-a \sqrt {a^2-4 b}-2 b} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {a^2-a \sqrt {a^2-4 b}-2 b} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (a \left (1-\frac {a}{\sqrt {a^2-4 b}}\right ) \left (a^2-2 b\right ) \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2-4 b}}+\sqrt {a^2-a \sqrt {a^2-4 b}-2 b} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {a^2-a \sqrt {a^2-4 b}-2 b} \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (a \left (1+\frac {a}{\sqrt {a^2-4 b}}\right ) \left (a^2-2 b\right ) \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2-4 b}}-\sqrt {a^2+a \sqrt {a^2-4 b}-2 b} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {a^2+a \sqrt {a^2-4 b}-2 b} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (a \left (1+\frac {a}{\sqrt {a^2-4 b}}\right ) \left (a^2-2 b\right ) \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2-4 b}}+\sqrt {a^2+a \sqrt {a^2-4 b}-2 b} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {a^2+a \sqrt {a^2-4 b}-2 b} \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (2 \left (a^2-b\right ) b \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2-4 b}}-\sqrt {a^2-a \sqrt {a^2-4 b}-2 b} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {a^2-4 b} \sqrt {a^2-a \sqrt {a^2-4 b}-2 b} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (2 \left (a^2-b\right ) b \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2-4 b}}+\sqrt {a^2-a \sqrt {a^2-4 b}-2 b} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {a^2-4 b} \sqrt {a^2-a \sqrt {a^2-4 b}-2 b} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (2 \left (a^2-b\right ) b \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2-4 b}}-\sqrt {a^2+a \sqrt {a^2-4 b}-2 b} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {a^2-4 b} \sqrt {a^2+a \sqrt {a^2-4 b}-2 b} \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (2 \left (a^2-b\right ) b \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2-4 b}}+\sqrt {a^2+a \sqrt {a^2-4 b}-2 b} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {a^2-4 b} \sqrt {a^2+a \sqrt {a^2-4 b}-2 b} \sqrt {x} \sqrt [4]{b+a x^2}}\\ &=\frac {1}{2} x \sqrt [4]{b x^2+a x^4}+\frac {\left (4 a^2-b\right ) \sqrt [4]{b x^2+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4 a^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {a \left (1-\frac {a}{\sqrt {a^2-4 b}}\right ) \left (a^2-2 b\right ) \sqrt [4]{b x^2+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a^2-a \sqrt {a^2-4 b}-2 b} \sqrt {x}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{a-\sqrt {a^2-4 b}} \left (a^2-a \sqrt {a^2-4 b}-2 b\right )^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {2 \left (a^2-b\right ) b \sqrt [4]{b x^2+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a^2-a \sqrt {a^2-4 b}-2 b} \sqrt {x}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt {a^2-4 b} \left (a^2-a \sqrt {a^2-4 b}-2 b\right )^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {a \left (a+\sqrt {a^2-4 b}\right )^{3/4} \left (a^2-2 b\right ) \sqrt [4]{b x^2+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a^2+a \sqrt {a^2-4 b}-2 b} \sqrt {x}}{\sqrt [4]{a+\sqrt {a^2-4 b}} \sqrt [4]{b+a x^2}}\right )}{\sqrt {a^2-4 b} \left (a^2+a \sqrt {a^2-4 b}-2 b\right )^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {2 \left (a^2-b\right ) b \sqrt [4]{b x^2+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a^2+a \sqrt {a^2-4 b}-2 b} \sqrt {x}}{\sqrt [4]{a+\sqrt {a^2-4 b}} \sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{a+\sqrt {a^2-4 b}} \sqrt {a^2-4 b} \left (a^2+a \sqrt {a^2-4 b}-2 b\right )^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (4 a^2-b\right ) \sqrt [4]{b x^2+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4 a^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {a \left (1-\frac {a}{\sqrt {a^2-4 b}}\right ) \left (a^2-2 b\right ) \sqrt [4]{b x^2+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2-a \sqrt {a^2-4 b}-2 b} \sqrt {x}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{a-\sqrt {a^2-4 b}} \left (a^2-a \sqrt {a^2-4 b}-2 b\right )^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {2 \left (a^2-b\right ) b \sqrt [4]{b x^2+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2-a \sqrt {a^2-4 b}-2 b} \sqrt {x}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt {a^2-4 b} \left (a^2-a \sqrt {a^2-4 b}-2 b\right )^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {a \left (a+\sqrt {a^2-4 b}\right )^{3/4} \left (a^2-2 b\right ) \sqrt [4]{b x^2+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2+a \sqrt {a^2-4 b}-2 b} \sqrt {x}}{\sqrt [4]{a+\sqrt {a^2-4 b}} \sqrt [4]{b+a x^2}}\right )}{\sqrt {a^2-4 b} \left (a^2+a \sqrt {a^2-4 b}-2 b\right )^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {2 \left (a^2-b\right ) b \sqrt [4]{b x^2+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2+a \sqrt {a^2-4 b}-2 b} \sqrt {x}}{\sqrt [4]{a+\sqrt {a^2-4 b}} \sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{a+\sqrt {a^2-4 b}} \sqrt {a^2-4 b} \left (a^2+a \sqrt {a^2-4 b}-2 b\right )^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.55, size = 300, normalized size = 1.21 \begin {gather*} \frac {x^{3/2} \left (b+a x^2\right )^{3/4} \left (2 a^{3/4} x^{3/2} \sqrt [4]{b+a x^2}+4 a^2 \text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )-b \text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )+\left (-4 a^2+b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )-a^{3/4} \text {RootSum}\left [b-a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-a b \log (x)+2 a b \log \left (\sqrt [4]{b+a x^2}-\sqrt {x} \text {$\#$1}\right )+a^2 \log (x) \text {$\#$1}^4-b \log (x) \text {$\#$1}^4-2 a^2 \log \left (\sqrt [4]{b+a x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4+2 b \log \left (\sqrt [4]{b+a x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}^3-2 \text {$\#$1}^7}\&\right ]\right )}{4 a^{3/4} \left (x^2 \left (b+a x^2\right )\right )^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^4*(b*x^2 + a*x^4)^(1/4))/(b + a*x^2 + x^4),x]

[Out]

(x^(3/2)*(b + a*x^2)^(3/4)*(2*a^(3/4)*x^(3/2)*(b + a*x^2)^(1/4) + 4*a^2*ArcTan[(a^(1/4)*Sqrt[x])/(b + a*x^2)^(
1/4)] - b*ArcTan[(a^(1/4)*Sqrt[x])/(b + a*x^2)^(1/4)] + (-4*a^2 + b)*ArcTanh[(a^(1/4)*Sqrt[x])/(b + a*x^2)^(1/
4)] - a^(3/4)*RootSum[b - a*#1^4 + #1^8 & , (-(a*b*Log[x]) + 2*a*b*Log[(b + a*x^2)^(1/4) - Sqrt[x]*#1] + a^2*L
og[x]*#1^4 - b*Log[x]*#1^4 - 2*a^2*Log[(b + a*x^2)^(1/4) - Sqrt[x]*#1]*#1^4 + 2*b*Log[(b + a*x^2)^(1/4) - Sqrt
[x]*#1]*#1^4)/(a*#1^3 - 2*#1^7) & ]))/(4*a^(3/4)*(x^2*(b + a*x^2))^(3/4))

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {x^{4} \left (a \,x^{4}+b \,x^{2}\right )^{\frac {1}{4}}}{x^{4}+a \,x^{2}+b}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a*x^4+b*x^2)^(1/4)/(x^4+a*x^2+b),x)

[Out]

int(x^4*(a*x^4+b*x^2)^(1/4)/(x^4+a*x^2+b),x)

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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(x^4*(a*x^4+b*x^2)^(1/4)/(x^4+a*x^2+b),x, algorithm="maxima")

[Out]

integrate((a*x^4 + b*x^2)^(1/4)*x^4/(x^4 + a*x^2 + b), x)

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} \sqrt [4]{x^{2} \left (a x^{2} + b\right )}}{a x^{2} + b + x^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(a*x**4+b*x**2)**(1/4)/(x**4+a*x**2+b),x)

[Out]

Integral(x**4*(x**2*(a*x**2 + b))**(1/4)/(a*x**2 + b + x**4), x)

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Giac [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 7.14, size = 224, normalized size = 0.91 \begin {gather*} \frac {1}{2} \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} x^{2} + \frac {\sqrt {2} {\left (4 \, a^{2} - b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{8 \, \left (-a\right )^{\frac {3}{4}}} + \frac {\sqrt {2} {\left (4 \, a^{2} - b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{8 \, \left (-a\right )^{\frac {3}{4}}} + \frac {\sqrt {2} {\left (4 \, a^{2} - b\right )} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x^{2}}}\right )}{16 \, \left (-a\right )^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (4 \, a^{2} - b\right )} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x^{2}}}\right )}{16 \, \left (-a\right )^{\frac {3}{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*(a*x^4 + b*x^2)^(1/4))/(b + a*x^2 + x^4),x)

[Out]

int((x^4*(a*x^4 + b*x^2)^(1/4))/(b + a*x^2 + x^4), x)

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