\(\int \frac {b+a x^4}{x^4 (-2 b+a x^4) \sqrt [4]{b x^2+a x^4}} \, dx\) [1447]

Optimal result

Integrand size = 37, antiderivative size = 102 \[ \int \frac {b+a x^4}{x^4 \left (-2 b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\frac {\left (3 b-4 a x^2\right ) \left (b x^2+a x^4\right )^{3/4}}{21 b^2 x^5}+\frac {3 a \text {RootSum}\left [2 a^2-a b-4 a \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{16 b} \]

[Out]

Unintegrable

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(546\) vs. \(2(102)=204\).

Time = 1.02 (sec) , antiderivative size = 546, normalized size of antiderivative = 5.35, number of steps used = 19, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.297, Rules used = {2081, 6857, 277, 270, 1284, 1543, 1443, 385, 218, 212, 209} \[ \int \frac {b+a x^4}{x^4 \left (-2 b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=-\frac {3 a^{7/8} \sqrt {x} \sqrt [4]{a x^2+b} \arctan \left (\frac {\sqrt [8]{a} \sqrt {x} \sqrt [4]{\sqrt {2} \sqrt {a}-\sqrt {b}}}{\sqrt [8]{2} \sqrt [4]{a x^2+b}}\right )}{4\ 2^{7/8} b \sqrt [4]{\sqrt {2} \sqrt {a}-\sqrt {b}} \sqrt [4]{a x^4+b x^2}}-\frac {3 a^{7/8} \sqrt {x} \sqrt [4]{a x^2+b} \arctan \left (\frac {\sqrt [8]{a} \sqrt {x} \sqrt [4]{\sqrt {2} \sqrt {a}+\sqrt {b}}}{\sqrt [8]{2} \sqrt [4]{a x^2+b}}\right )}{4\ 2^{7/8} b \sqrt [4]{\sqrt {2} \sqrt {a}+\sqrt {b}} \sqrt [4]{a x^4+b x^2}}-\frac {3 a^{7/8} \sqrt {x} \sqrt [4]{a x^2+b} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt {x} \sqrt [4]{\sqrt {2} \sqrt {a}-\sqrt {b}}}{\sqrt [8]{2} \sqrt [4]{a x^2+b}}\right )}{4\ 2^{7/8} b \sqrt [4]{\sqrt {2} \sqrt {a}-\sqrt {b}} \sqrt [4]{a x^4+b x^2}}-\frac {3 a^{7/8} \sqrt {x} \sqrt [4]{a x^2+b} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt {x} \sqrt [4]{\sqrt {2} \sqrt {a}+\sqrt {b}}}{\sqrt [8]{2} \sqrt [4]{a x^2+b}}\right )}{4\ 2^{7/8} b \sqrt [4]{\sqrt {2} \sqrt {a}+\sqrt {b}} \sqrt [4]{a x^4+b x^2}}-\frac {4 a \left (a x^2+b\right )}{21 b^2 x \sqrt [4]{a x^4+b x^2}}+\frac {a x^2+b}{7 b x^3 \sqrt [4]{a x^4+b x^2}} \]

[In]

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

[Out]

(b + a*x^2)/(7*b*x^3*(b*x^2 + a*x^4)^(1/4)) - (4*a*(b + a*x^2))/(21*b^2*x*(b*x^2 + a*x^4)^(1/4)) - (3*a^(7/8)*
Sqrt[x]*(b + a*x^2)^(1/4)*ArcTan[(a^(1/8)*(Sqrt[2]*Sqrt[a] - Sqrt[b])^(1/4)*Sqrt[x])/(2^(1/8)*(b + a*x^2)^(1/4
))])/(4*2^(7/8)*(Sqrt[2]*Sqrt[a] - Sqrt[b])^(1/4)*b*(b*x^2 + a*x^4)^(1/4)) - (3*a^(7/8)*Sqrt[x]*(b + a*x^2)^(1
/4)*ArcTan[(a^(1/8)*(Sqrt[2]*Sqrt[a] + Sqrt[b])^(1/4)*Sqrt[x])/(2^(1/8)*(b + a*x^2)^(1/4))])/(4*2^(7/8)*(Sqrt[
2]*Sqrt[a] + Sqrt[b])^(1/4)*b*(b*x^2 + a*x^4)^(1/4)) - (3*a^(7/8)*Sqrt[x]*(b + a*x^2)^(1/4)*ArcTanh[(a^(1/8)*(
Sqrt[2]*Sqrt[a] - Sqrt[b])^(1/4)*Sqrt[x])/(2^(1/8)*(b + a*x^2)^(1/4))])/(4*2^(7/8)*(Sqrt[2]*Sqrt[a] - Sqrt[b])
^(1/4)*b*(b*x^2 + a*x^4)^(1/4)) - (3*a^(7/8)*Sqrt[x]*(b + a*x^2)^(1/4)*ArcTanh[(a^(1/8)*(Sqrt[2]*Sqrt[a] + Sqr
t[b])^(1/4)*Sqrt[x])/(2^(1/8)*(b + a*x^2)^(1/4))])/(4*2^(7/8)*(Sqrt[2]*Sqrt[a] + Sqrt[b])^(1/4)*b*(b*x^2 + a*x
^4)^(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 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[a/b, 0]

Rule 270

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

Rule 277

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

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 1284

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

Rule 1443

Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{r = Rt[(-a)*c, 2]}, Dist[-c/(2
*r), Int[(d + e*x^n)^q/(r - c*x^n), x], x] - Dist[c/(2*r), Int[(d + e*x^n)^q/(r + c*x^n), x], x]] /; FreeQ[{a,
c, d, e, n, q}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[q]

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 2081

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

Rule 6857

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

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \int \frac {b+a x^4}{x^{9/2} \sqrt [4]{b+a x^2} \left (-2 b+a x^4\right )} \, dx}{\sqrt [4]{b x^2+a x^4}} \\ & = \frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \int \left (\frac {1}{x^{9/2} \sqrt [4]{b+a x^2}}+\frac {3 b}{x^{9/2} \sqrt [4]{b+a x^2} \left (-2 b+a x^4\right )}\right ) \, dx}{\sqrt [4]{b x^2+a x^4}} \\ & = \frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \int \frac {1}{x^{9/2} \sqrt [4]{b+a x^2}} \, dx}{\sqrt [4]{b x^2+a x^4}}+\frac {\left (3 b \sqrt {x} \sqrt [4]{b+a x^2}\right ) \int \frac {1}{x^{9/2} \sqrt [4]{b+a x^2} \left (-2 b+a x^4\right )} \, dx}{\sqrt [4]{b x^2+a x^4}} \\ & = -\frac {2 \left (b+a x^2\right )}{7 b x^3 \sqrt [4]{b x^2+a x^4}}-\frac {\left (4 a \sqrt {x} \sqrt [4]{b+a x^2}\right ) \int \frac {1}{x^{5/2} \sqrt [4]{b+a x^2}} \, dx}{7 b \sqrt [4]{b x^2+a x^4}}+\frac {\left (6 b \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{x^8 \sqrt [4]{b+a x^4} \left (-2 b+a x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}} \\ & = -\frac {2 \left (b+a x^2\right )}{7 b x^3 \sqrt [4]{b x^2+a x^4}}+\frac {8 a \left (b+a x^2\right )}{21 b^2 x \sqrt [4]{b x^2+a x^4}}+\frac {\left (6 b \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \left (-\frac {1}{2 b x^8 \sqrt [4]{b+a x^4}}-\frac {a}{2 b \sqrt [4]{b+a x^4} \left (2 b-a x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}} \\ & = -\frac {2 \left (b+a x^2\right )}{7 b x^3 \sqrt [4]{b x^2+a x^4}}+\frac {8 a \left (b+a x^2\right )}{21 b^2 x \sqrt [4]{b x^2+a x^4}}-\frac {\left (3 \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{x^8 \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}-\frac {\left (3 a \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{b+a x^4} \left (2 b-a x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}} \\ & = \frac {b+a x^2}{7 b x^3 \sqrt [4]{b x^2+a x^4}}+\frac {8 a \left (b+a x^2\right )}{21 b^2 x \sqrt [4]{b x^2+a x^4}}+\frac {\left (12 a \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{x^4 \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{7 b \sqrt [4]{b x^2+a x^4}}-\frac {\left (3 a^{3/2} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {2} \sqrt {a} \sqrt {b}-a x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} \sqrt {b} \sqrt [4]{b x^2+a x^4}}-\frac {\left (3 a^{3/2} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {2} \sqrt {a} \sqrt {b}+a x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} \sqrt {b} \sqrt [4]{b x^2+a x^4}} \\ & = \frac {b+a x^2}{7 b x^3 \sqrt [4]{b x^2+a x^4}}-\frac {4 a \left (b+a x^2\right )}{21 b^2 x \sqrt [4]{b x^2+a x^4}}-\frac {\left (3 a^{3/2} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2} \sqrt {a} \sqrt {b}-\left (\sqrt {2} a^{3/2} \sqrt {b}-a b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt {2} \sqrt {b} \sqrt [4]{b x^2+a x^4}}-\frac {\left (3 a^{3/2} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2} \sqrt {a} \sqrt {b}-\left (\sqrt {2} a^{3/2} \sqrt {b}+a b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt {2} \sqrt {b} \sqrt [4]{b x^2+a x^4}} \\ & = \frac {b+a x^2}{7 b x^3 \sqrt [4]{b x^2+a x^4}}-\frac {4 a \left (b+a x^2\right )}{21 b^2 x \sqrt [4]{b x^2+a x^4}}-\frac {\left (3 a \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2}-\sqrt [4]{a} \sqrt {\sqrt {2} \sqrt {a}-\sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4\ 2^{3/4} b \sqrt [4]{b x^2+a x^4}}-\frac {\left (3 a \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2}+\sqrt [4]{a} \sqrt {\sqrt {2} \sqrt {a}-\sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4\ 2^{3/4} b \sqrt [4]{b x^2+a x^4}}-\frac {\left (3 a \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2}-\sqrt [4]{a} \sqrt {\sqrt {2} \sqrt {a}+\sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4\ 2^{3/4} b \sqrt [4]{b x^2+a x^4}}-\frac {\left (3 a \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2}+\sqrt [4]{a} \sqrt {\sqrt {2} \sqrt {a}+\sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4\ 2^{3/4} b \sqrt [4]{b x^2+a x^4}} \\ & = \frac {b+a x^2}{7 b x^3 \sqrt [4]{b x^2+a x^4}}-\frac {4 a \left (b+a x^2\right )}{21 b^2 x \sqrt [4]{b x^2+a x^4}}-\frac {3 a^{7/8} \sqrt {x} \sqrt [4]{b+a x^2} \arctan \left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {2} \sqrt {a}-\sqrt {b}} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{b+a x^2}}\right )}{4\ 2^{7/8} \sqrt [4]{\sqrt {2} \sqrt {a}-\sqrt {b}} b \sqrt [4]{b x^2+a x^4}}-\frac {3 a^{7/8} \sqrt {x} \sqrt [4]{b+a x^2} \arctan \left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {2} \sqrt {a}+\sqrt {b}} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{b+a x^2}}\right )}{4\ 2^{7/8} \sqrt [4]{\sqrt {2} \sqrt {a}+\sqrt {b}} b \sqrt [4]{b x^2+a x^4}}-\frac {3 a^{7/8} \sqrt {x} \sqrt [4]{b+a x^2} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {2} \sqrt {a}-\sqrt {b}} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{b+a x^2}}\right )}{4\ 2^{7/8} \sqrt [4]{\sqrt {2} \sqrt {a}-\sqrt {b}} b \sqrt [4]{b x^2+a x^4}}-\frac {3 a^{7/8} \sqrt {x} \sqrt [4]{b+a x^2} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {2} \sqrt {a}+\sqrt {b}} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{b+a x^2}}\right )}{4\ 2^{7/8} \sqrt [4]{\sqrt {2} \sqrt {a}+\sqrt {b}} b \sqrt [4]{b x^2+a x^4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.07 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.27 \[ \int \frac {b+a x^4}{x^4 \left (-2 b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\frac {-16 \left (-3 b^2+a b x^2+4 a^2 x^4\right )+63 a b x^{7/2} \sqrt [4]{b+a x^2} \text {RootSum}\left [2 a^2-a b-4 a \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right )+\log \left (\sqrt [4]{b+a x^2}-\sqrt {x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{336 b^2 x^3 \sqrt [4]{x^2 \left (b+a x^2\right )}} \]

[In]

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

[Out]

(-16*(-3*b^2 + a*b*x^2 + 4*a^2*x^4) + 63*a*b*x^(7/2)*(b + a*x^2)^(1/4)*RootSum[2*a^2 - a*b - 4*a*#1^4 + 2*#1^8
& , (-Log[Sqrt[x]] + Log[(b + a*x^2)^(1/4) - Sqrt[x]*#1])/#1 & ])/(336*b^2*x^3*(x^2*(b + a*x^2))^(1/4))

Maple [N/A]

Time = 0.45 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.02

method	result	size
pseudoelliptic	\(\frac {63 a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{8}-4 a \,\textit {\_Z}^{4}+2 a^{2}-a b \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}}\right ) b \,x^{5}-64 \left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {3}{4}} a \,x^{2}+48 b \left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {3}{4}}}{336 x^{5} b^{2}}\)	\(104\)

[In]

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

[Out]

1/336/x^5*(63*a*sum(ln((-_R*x+(x^2*(a*x^2+b))^(1/4))/x)/_R,_R=RootOf(2*_Z^8-4*_Z^4*a+2*a^2-a*b))*b*x^5-64*(x^2
*(a*x^2+b))^(3/4)*a*x^2+48*b*(x^2*(a*x^2+b))^(3/4))/b^2

Fricas [F(-1)]

Timed out. \[ \int \frac {b+a x^4}{x^4 \left (-2 b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [N/A]

Not integrable

Time = 24.56 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.31 \[ \int \frac {b+a x^4}{x^4 \left (-2 b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int \frac {a x^{4} + b}{x^{4} \sqrt [4]{x^{2} \left (a x^{2} + b\right )} \left (a x^{4} - 2 b\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.36 \[ \int \frac {b+a x^4}{x^4 \left (-2 b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int { \frac {a x^{4} + b}{{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} - 2 \, b\right )} x^{4}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 2.90 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.36 \[ \int \frac {b+a x^4}{x^4 \left (-2 b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int { \frac {a x^{4} + b}{{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} - 2 \, b\right )} x^{4}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 7.40 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.39 \[ \int \frac {b+a x^4}{x^4 \left (-2 b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=-\int \frac {a\,x^4+b}{x^4\,{\left (a\,x^4+b\,x^2\right )}^{1/4}\,\left (2\,b-a\,x^4\right )} \,d x \]

[In]

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

[Out]

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