3.12.49 \(\int \frac {1}{\sqrt [4]{b x^2+a x^4} (-2 b+a x^8)} \, dx\)

Optimal. Leaf size=85 \[ \frac {\text {RootSum}\left [2 \text {$\#$1}^{16}-8 \text {$\#$1}^{12} a+12 \text {$\#$1}^8 a^2-8 \text {$\#$1}^4 a^3+2 a^4-a b^3\& ,\frac {\log \left (\sqrt [4]{a x^4+b x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]}{16 b} \]

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Rubi [B]  time = 3.58, antiderivative size = 1065, normalized size of antiderivative = 12.53, number of steps used = 22, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2056, 6715, 6725, 1429, 377, 212, 206, 203} \begin {gather*} -\frac {\sqrt {x} \sqrt [4]{a x^2+b} \tan ^{-1}\left (\frac {\sqrt [16]{a} \sqrt [4]{\sqrt [4]{2} a^{3/4}-b^{3/4}} \sqrt {x}}{\sqrt [16]{2} \sqrt [4]{a x^2+b}}\right )}{4\ 2^{15/16} \sqrt [16]{a} \sqrt [4]{\sqrt [4]{2} a^{3/4}-b^{3/4}} b \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt {x} \sqrt [4]{a x^2+b} \tan ^{-1}\left (\frac {\sqrt [8]{-\sqrt {a}} \sqrt [4]{\frac {\sqrt [4]{2} a}{\sqrt {-\sqrt {a}}}-b^{3/4}} \sqrt {x}}{\sqrt [16]{2} \sqrt [4]{a x^2+b}}\right )}{4\ 2^{15/16} \sqrt [8]{-\sqrt {a}} \sqrt [4]{\frac {\sqrt [4]{2} a}{\sqrt {-\sqrt {a}}}-b^{3/4}} b \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt {x} \sqrt [4]{a x^2+b} \tan ^{-1}\left (\frac {\sqrt [16]{a} \sqrt [4]{\sqrt [4]{2} a^{3/4}+b^{3/4}} \sqrt {x}}{\sqrt [16]{2} \sqrt [4]{a x^2+b}}\right )}{4\ 2^{15/16} \sqrt [16]{a} \sqrt [4]{\sqrt [4]{2} a^{3/4}+b^{3/4}} b \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt {x} \sqrt [4]{a x^2+b} \tan ^{-1}\left (\frac {\sqrt [8]{-\sqrt {a}} \sqrt [4]{\frac {\sqrt [4]{2} a}{\sqrt {-\sqrt {a}}}+b^{3/4}} \sqrt {x}}{\sqrt [16]{2} \sqrt [4]{a x^2+b}}\right )}{4\ 2^{15/16} \sqrt [8]{-\sqrt {a}} \sqrt [4]{\frac {\sqrt [4]{2} a}{\sqrt {-\sqrt {a}}}+b^{3/4}} b \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt {x} \sqrt [4]{a x^2+b} \tanh ^{-1}\left (\frac {\sqrt [16]{a} \sqrt [4]{\sqrt [4]{2} a^{3/4}-b^{3/4}} \sqrt {x}}{\sqrt [16]{2} \sqrt [4]{a x^2+b}}\right )}{4\ 2^{15/16} \sqrt [16]{a} \sqrt [4]{\sqrt [4]{2} a^{3/4}-b^{3/4}} b \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt {x} \sqrt [4]{a x^2+b} \tanh ^{-1}\left (\frac {\sqrt [8]{-\sqrt {a}} \sqrt [4]{\frac {\sqrt [4]{2} a}{\sqrt {-\sqrt {a}}}-b^{3/4}} \sqrt {x}}{\sqrt [16]{2} \sqrt [4]{a x^2+b}}\right )}{4\ 2^{15/16} \sqrt [8]{-\sqrt {a}} \sqrt [4]{\frac {\sqrt [4]{2} a}{\sqrt {-\sqrt {a}}}-b^{3/4}} b \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt {x} \sqrt [4]{a x^2+b} \tanh ^{-1}\left (\frac {\sqrt [16]{a} \sqrt [4]{\sqrt [4]{2} a^{3/4}+b^{3/4}} \sqrt {x}}{\sqrt [16]{2} \sqrt [4]{a x^2+b}}\right )}{4\ 2^{15/16} \sqrt [16]{a} \sqrt [4]{\sqrt [4]{2} a^{3/4}+b^{3/4}} b \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt {x} \sqrt [4]{a x^2+b} \tanh ^{-1}\left (\frac {\sqrt [8]{-\sqrt {a}} \sqrt [4]{\frac {\sqrt [4]{2} a}{\sqrt {-\sqrt {a}}}+b^{3/4}} \sqrt {x}}{\sqrt [16]{2} \sqrt [4]{a x^2+b}}\right )}{4\ 2^{15/16} \sqrt [8]{-\sqrt {a}} \sqrt [4]{\frac {\sqrt [4]{2} a}{\sqrt {-\sqrt {a}}}+b^{3/4}} b \sqrt [4]{a x^4+b x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((b*x^2 + a*x^4)^(1/4)*(-2*b + a*x^8)),x]

[Out]

-1/4*(Sqrt[x]*(b + a*x^2)^(1/4)*ArcTan[(a^(1/16)*(2^(1/4)*a^(3/4) - b^(3/4))^(1/4)*Sqrt[x])/(2^(1/16)*(b + a*x
^2)^(1/4))])/(2^(15/16)*a^(1/16)*(2^(1/4)*a^(3/4) - b^(3/4))^(1/4)*b*(b*x^2 + a*x^4)^(1/4)) - (Sqrt[x]*(b + a*
x^2)^(1/4)*ArcTan[((-Sqrt[a])^(1/8)*((2^(1/4)*a)/Sqrt[-Sqrt[a]] - b^(3/4))^(1/4)*Sqrt[x])/(2^(1/16)*(b + a*x^2
)^(1/4))])/(4*2^(15/16)*(-Sqrt[a])^(1/8)*((2^(1/4)*a)/Sqrt[-Sqrt[a]] - b^(3/4))^(1/4)*b*(b*x^2 + a*x^4)^(1/4))
- (Sqrt[x]*(b + a*x^2)^(1/4)*ArcTan[(a^(1/16)*(2^(1/4)*a^(3/4) + b^(3/4))^(1/4)*Sqrt[x])/(2^(1/16)*(b + a*x^2
)^(1/4))])/(4*2^(15/16)*a^(1/16)*(2^(1/4)*a^(3/4) + b^(3/4))^(1/4)*b*(b*x^2 + a*x^4)^(1/4)) - (Sqrt[x]*(b + a*
x^2)^(1/4)*ArcTan[((-Sqrt[a])^(1/8)*((2^(1/4)*a)/Sqrt[-Sqrt[a]] + b^(3/4))^(1/4)*Sqrt[x])/(2^(1/16)*(b + a*x^2
)^(1/4))])/(4*2^(15/16)*(-Sqrt[a])^(1/8)*((2^(1/4)*a)/Sqrt[-Sqrt[a]] + b^(3/4))^(1/4)*b*(b*x^2 + a*x^4)^(1/4))
- (Sqrt[x]*(b + a*x^2)^(1/4)*ArcTanh[(a^(1/16)*(2^(1/4)*a^(3/4) - b^(3/4))^(1/4)*Sqrt[x])/(2^(1/16)*(b + a*x^
2)^(1/4))])/(4*2^(15/16)*a^(1/16)*(2^(1/4)*a^(3/4) - b^(3/4))^(1/4)*b*(b*x^2 + a*x^4)^(1/4)) - (Sqrt[x]*(b + a
*x^2)^(1/4)*ArcTanh[((-Sqrt[a])^(1/8)*((2^(1/4)*a)/Sqrt[-Sqrt[a]] - b^(3/4))^(1/4)*Sqrt[x])/(2^(1/16)*(b + a*x
^2)^(1/4))])/(4*2^(15/16)*(-Sqrt[a])^(1/8)*((2^(1/4)*a)/Sqrt[-Sqrt[a]] - b^(3/4))^(1/4)*b*(b*x^2 + a*x^4)^(1/4
)) - (Sqrt[x]*(b + a*x^2)^(1/4)*ArcTanh[(a^(1/16)*(2^(1/4)*a^(3/4) + b^(3/4))^(1/4)*Sqrt[x])/(2^(1/16)*(b + a*
x^2)^(1/4))])/(4*2^(15/16)*a^(1/16)*(2^(1/4)*a^(3/4) + b^(3/4))^(1/4)*b*(b*x^2 + a*x^4)^(1/4)) - (Sqrt[x]*(b +
a*x^2)^(1/4)*ArcTanh[((-Sqrt[a])^(1/8)*((2^(1/4)*a)/Sqrt[-Sqrt[a]] + b^(3/4))^(1/4)*Sqrt[x])/(2^(1/16)*(b + a
*x^2)^(1/4))])/(4*2^(15/16)*(-Sqrt[a])^(1/8)*((2^(1/4)*a)/Sqrt[-Sqrt[a]] + b^(3/4))^(1/4)*b*(b*x^2 + a*x^4)^(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 212

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] &&
!GtQ[a/b, 0]

Rule 377

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 1429

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 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 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

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]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt [4]{b x^2+a x^4} \left (-2 b+a x^8\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt [4]{b+a x^2} \left (-2 b+a x^8\right )} \, dx}{\sqrt [4]{b x^2+a x^4}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{b+a x^4} \left (-2 b+a x^{16}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{2 \sqrt {2} \sqrt {b} \sqrt [4]{b+a x^4} \left (\sqrt {2} \sqrt {b}-\sqrt {a} x^8\right )}-\frac {1}{2 \sqrt {2} \sqrt {b} \sqrt [4]{b+a x^4} \left (\sqrt {2} \sqrt {b}+\sqrt {a} x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}\\ &=-\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{b+a x^4} \left (\sqrt {2} \sqrt {b}-\sqrt {a} x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {2} \sqrt {b} \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{b+a x^4} \left (\sqrt {2} \sqrt {b}+\sqrt {a} x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {2} \sqrt {b} \sqrt [4]{b x^2+a x^4}}\\ &=-\frac {\left (\sqrt {-\sqrt {a}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt [4]{2} \sqrt {-\sqrt {a}} \sqrt [4]{b}-\sqrt {a} x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{2\ 2^{3/4} b^{3/4} \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt {-\sqrt {a}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt [4]{2} \sqrt {-\sqrt {a}} \sqrt [4]{b}+\sqrt {a} x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{2\ 2^{3/4} b^{3/4} \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt [4]{a} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{b}-\sqrt {a} x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{2\ 2^{3/4} b^{3/4} \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt [4]{a} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{b}+\sqrt {a} x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{2\ 2^{3/4} b^{3/4} \sqrt [4]{b x^2+a x^4}}\\ &=-\frac {\left (\sqrt {-\sqrt {a}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2} \sqrt {-\sqrt {a}} \sqrt [4]{b}-\left (\sqrt [4]{2} \sqrt {-\sqrt {a}} a \sqrt [4]{b}-\sqrt {a} b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2\ 2^{3/4} b^{3/4} \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt {-\sqrt {a}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2} \sqrt {-\sqrt {a}} \sqrt [4]{b}-\left (\sqrt [4]{2} \sqrt {-\sqrt {a}} a \sqrt [4]{b}+\sqrt {a} b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2\ 2^{3/4} b^{3/4} \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt [4]{a} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{b}-\left (\sqrt [4]{2} a^{5/4} \sqrt [4]{b}-\sqrt {a} b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2\ 2^{3/4} b^{3/4} \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt [4]{a} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{b}-\left (\sqrt [4]{2} a^{5/4} \sqrt [4]{b}+\sqrt {a} b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2\ 2^{3/4} b^{3/4} \sqrt [4]{b x^2+a x^4}}\\ &=-\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [8]{2}-\sqrt [4]{-\sqrt {a}} \sqrt {-\sqrt [4]{2} \sqrt {-\sqrt {a}} \sqrt {a}-b^{3/4}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4\ 2^{7/8} b \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [8]{2}+\sqrt [4]{-\sqrt {a}} \sqrt {-\sqrt [4]{2} \sqrt {-\sqrt {a}} \sqrt {a}-b^{3/4}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4\ 2^{7/8} b \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [8]{2}-\sqrt [8]{a} \sqrt {\sqrt [4]{2} a^{3/4}-b^{3/4}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4\ 2^{7/8} b \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [8]{2}+\sqrt [8]{a} \sqrt {\sqrt [4]{2} a^{3/4}-b^{3/4}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4\ 2^{7/8} b \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [8]{2}-\sqrt [4]{-\sqrt {a}} \sqrt {-\sqrt [4]{2} \sqrt {-\sqrt {a}} \sqrt {a}+b^{3/4}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4\ 2^{7/8} b \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [8]{2}+\sqrt [4]{-\sqrt {a}} \sqrt {-\sqrt [4]{2} \sqrt {-\sqrt {a}} \sqrt {a}+b^{3/4}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4\ 2^{7/8} b \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [8]{2}-\sqrt [8]{a} \sqrt {\sqrt [4]{2} a^{3/4}+b^{3/4}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4\ 2^{7/8} b \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [8]{2}+\sqrt [8]{a} \sqrt {\sqrt [4]{2} a^{3/4}+b^{3/4}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4\ 2^{7/8} b \sqrt [4]{b x^2+a x^4}}\\ &=-\frac {\sqrt {x} \sqrt [4]{b+a x^2} \tan ^{-1}\left (\frac {\sqrt [16]{a} \sqrt [4]{\sqrt [4]{2} a^{3/4}-b^{3/4}} \sqrt {x}}{\sqrt [16]{2} \sqrt [4]{b+a x^2}}\right )}{4\ 2^{15/16} \sqrt [16]{a} \sqrt [4]{\sqrt [4]{2} a^{3/4}-b^{3/4}} b \sqrt [4]{b x^2+a x^4}}-\frac {\sqrt {x} \sqrt [4]{b+a x^2} \tan ^{-1}\left (\frac {\sqrt [8]{-\sqrt {a}} \sqrt [4]{\frac {\sqrt [4]{2} a}{\sqrt {-\sqrt {a}}}-b^{3/4}} \sqrt {x}}{\sqrt [16]{2} \sqrt [4]{b+a x^2}}\right )}{4\ 2^{15/16} \sqrt [8]{-\sqrt {a}} \sqrt [4]{\frac {\sqrt [4]{2} a}{\sqrt {-\sqrt {a}}}-b^{3/4}} b \sqrt [4]{b x^2+a x^4}}-\frac {\sqrt {x} \sqrt [4]{b+a x^2} \tan ^{-1}\left (\frac {\sqrt [16]{a} \sqrt [4]{\sqrt [4]{2} a^{3/4}+b^{3/4}} \sqrt {x}}{\sqrt [16]{2} \sqrt [4]{b+a x^2}}\right )}{4\ 2^{15/16} \sqrt [16]{a} \sqrt [4]{\sqrt [4]{2} a^{3/4}+b^{3/4}} b \sqrt [4]{b x^2+a x^4}}-\frac {\sqrt {x} \sqrt [4]{b+a x^2} \tan ^{-1}\left (\frac {\sqrt [8]{-\sqrt {a}} \sqrt [4]{\frac {\sqrt [4]{2} a}{\sqrt {-\sqrt {a}}}+b^{3/4}} \sqrt {x}}{\sqrt [16]{2} \sqrt [4]{b+a x^2}}\right )}{4\ 2^{15/16} \sqrt [8]{-\sqrt {a}} \sqrt [4]{\frac {\sqrt [4]{2} a}{\sqrt {-\sqrt {a}}}+b^{3/4}} b \sqrt [4]{b x^2+a x^4}}-\frac {\sqrt {x} \sqrt [4]{b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [16]{a} \sqrt [4]{\sqrt [4]{2} a^{3/4}-b^{3/4}} \sqrt {x}}{\sqrt [16]{2} \sqrt [4]{b+a x^2}}\right )}{4\ 2^{15/16} \sqrt [16]{a} \sqrt [4]{\sqrt [4]{2} a^{3/4}-b^{3/4}} b \sqrt [4]{b x^2+a x^4}}-\frac {\sqrt {x} \sqrt [4]{b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [8]{-\sqrt {a}} \sqrt [4]{\frac {\sqrt [4]{2} a}{\sqrt {-\sqrt {a}}}-b^{3/4}} \sqrt {x}}{\sqrt [16]{2} \sqrt [4]{b+a x^2}}\right )}{4\ 2^{15/16} \sqrt [8]{-\sqrt {a}} \sqrt [4]{\frac {\sqrt [4]{2} a}{\sqrt {-\sqrt {a}}}-b^{3/4}} b \sqrt [4]{b x^2+a x^4}}-\frac {\sqrt {x} \sqrt [4]{b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [16]{a} \sqrt [4]{\sqrt [4]{2} a^{3/4}+b^{3/4}} \sqrt {x}}{\sqrt [16]{2} \sqrt [4]{b+a x^2}}\right )}{4\ 2^{15/16} \sqrt [16]{a} \sqrt [4]{\sqrt [4]{2} a^{3/4}+b^{3/4}} b \sqrt [4]{b x^2+a x^4}}-\frac {\sqrt {x} \sqrt [4]{b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [8]{-\sqrt {a}} \sqrt [4]{\frac {\sqrt [4]{2} a}{\sqrt {-\sqrt {a}}}+b^{3/4}} \sqrt {x}}{\sqrt [16]{2} \sqrt [4]{b+a x^2}}\right )}{4\ 2^{15/16} \sqrt [8]{-\sqrt {a}} \sqrt [4]{\frac {\sqrt [4]{2} a}{\sqrt {-\sqrt {a}}}+b^{3/4}} b \sqrt [4]{b x^2+a x^4}}\\ \end {align*}

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Mathematica [B]  time = 3.23, size = 665, normalized size = 7.82 \begin {gather*} -\frac {x \sqrt [4]{a+\frac {b}{x^2}} \left (a^3 \text {RootSum}\left [2 \text {$\#$1}^4-8 \text {$\#$1}^3 a+12 \text {$\#$1}^2 a^2-8 \text {$\#$1} a^3+2 a^4-a b^3\&,\frac {\frac {\log \left (\sqrt [4]{\text {$\#$1}}-\sqrt [4]{a+\frac {b}{x^2}}\right )}{\sqrt [4]{\text {$\#$1}}}-\frac {\log \left (\sqrt [4]{\text {$\#$1}}+\sqrt [4]{a+\frac {b}{x^2}}\right )}{\sqrt [4]{\text {$\#$1}}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{\text {$\#$1}}}\right )}{\sqrt [4]{\text {$\#$1}}}}{\text {$\#$1}^3-3 \text {$\#$1}^2 a+3 \text {$\#$1} a^2-a^3}\&\right ]-3 a^2 \text {RootSum}\left [2 \text {$\#$1}^4-8 \text {$\#$1}^3 a+12 \text {$\#$1}^2 a^2-8 \text {$\#$1} a^3+2 a^4-a b^3\&,\frac {\text {$\#$1}^{3/4} \log \left (\sqrt [4]{\text {$\#$1}}-\sqrt [4]{a+\frac {b}{x^2}}\right )-\text {$\#$1}^{3/4} \log \left (\sqrt [4]{\text {$\#$1}}+\sqrt [4]{a+\frac {b}{x^2}}\right )+2 \text {$\#$1}^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{\text {$\#$1}}}\right )}{\text {$\#$1}^3-3 \text {$\#$1}^2 a+3 \text {$\#$1} a^2-a^3}\&\right ]+3 a \text {RootSum}\left [2 \text {$\#$1}^4-8 \text {$\#$1}^3 a+12 \text {$\#$1}^2 a^2-8 \text {$\#$1} a^3+2 a^4-a b^3\&,\frac {\text {$\#$1}^{7/4} \log \left (\sqrt [4]{\text {$\#$1}}-\sqrt [4]{a+\frac {b}{x^2}}\right )-\text {$\#$1}^{7/4} \log \left (\sqrt [4]{\text {$\#$1}}+\sqrt [4]{a+\frac {b}{x^2}}\right )+2 \text {$\#$1}^{7/4} \tan ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{\text {$\#$1}}}\right )}{\text {$\#$1}^3-3 \text {$\#$1}^2 a+3 \text {$\#$1} a^2-a^3}\&\right ]-\text {RootSum}\left [2 \text {$\#$1}^4-8 \text {$\#$1}^3 a+12 \text {$\#$1}^2 a^2-8 \text {$\#$1} a^3+2 a^4-a b^3\&,\frac {\text {$\#$1}^{11/4} \log \left (\sqrt [4]{\text {$\#$1}}-\sqrt [4]{a+\frac {b}{x^2}}\right )-\text {$\#$1}^{11/4} \log \left (\sqrt [4]{\text {$\#$1}}+\sqrt [4]{a+\frac {b}{x^2}}\right )+2 \text {$\#$1}^{11/4} \tan ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{\text {$\#$1}}}\right )}{\text {$\#$1}^3-3 \text {$\#$1}^2 a+3 \text {$\#$1} a^2-a^3}\&\right ]\right )}{16 b \sqrt [4]{x^2 \left (a x^2+b\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((b*x^2 + a*x^4)^(1/4)*(-2*b + a*x^8)),x]

[Out]

-1/16*((a + b/x^2)^(1/4)*x*(a^3*RootSum[2*a^4 - a*b^3 - 8*a^3*#1 + 12*a^2*#1^2 - 8*a*#1^3 + 2*#1^4 & , ((2*Arc
Tan[(a + b/x^2)^(1/4)/#1^(1/4)])/#1^(1/4) + Log[-(a + b/x^2)^(1/4) + #1^(1/4)]/#1^(1/4) - Log[(a + b/x^2)^(1/4
) + #1^(1/4)]/#1^(1/4))/(-a^3 + 3*a^2*#1 - 3*a*#1^2 + #1^3) & ] - 3*a^2*RootSum[2*a^4 - a*b^3 - 8*a^3*#1 + 12*
a^2*#1^2 - 8*a*#1^3 + 2*#1^4 & , (2*ArcTan[(a + b/x^2)^(1/4)/#1^(1/4)]*#1^(3/4) + Log[-(a + b/x^2)^(1/4) + #1^
(1/4)]*#1^(3/4) - Log[(a + b/x^2)^(1/4) + #1^(1/4)]*#1^(3/4))/(-a^3 + 3*a^2*#1 - 3*a*#1^2 + #1^3) & ] + 3*a*Ro
otSum[2*a^4 - a*b^3 - 8*a^3*#1 + 12*a^2*#1^2 - 8*a*#1^3 + 2*#1^4 & , (2*ArcTan[(a + b/x^2)^(1/4)/#1^(1/4)]*#1^
(7/4) + Log[-(a + b/x^2)^(1/4) + #1^(1/4)]*#1^(7/4) - Log[(a + b/x^2)^(1/4) + #1^(1/4)]*#1^(7/4))/(-a^3 + 3*a^
2*#1 - 3*a*#1^2 + #1^3) & ] - RootSum[2*a^4 - a*b^3 - 8*a^3*#1 + 12*a^2*#1^2 - 8*a*#1^3 + 2*#1^4 & , (2*ArcTan
[(a + b/x^2)^(1/4)/#1^(1/4)]*#1^(11/4) + Log[-(a + b/x^2)^(1/4) + #1^(1/4)]*#1^(11/4) - Log[(a + b/x^2)^(1/4)
+ #1^(1/4)]*#1^(11/4))/(-a^3 + 3*a^2*#1 - 3*a*#1^2 + #1^3) & ]))/(b*(x^2*(b + a*x^2))^(1/4))

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IntegrateAlgebraic [A]  time = 2.57, size = 85, normalized size = 1.00 \begin {gather*} \frac {\text {RootSum}\left [2 a^4-a b^3-8 a^3 \text {$\#$1}^4+12 a^2 \text {$\#$1}^8-8 a \text {$\#$1}^{12}+2 \text {$\#$1}^{16}\&,\frac {-\log (x)+\log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{16 b} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((b*x^2 + a*x^4)^(1/4)*(-2*b + a*x^8)),x]

[Out]

RootSum[2*a^4 - a*b^3 - 8*a^3*#1^4 + 12*a^2*#1^8 - 8*a*#1^12 + 2*#1^16 & , (-Log[x] + Log[(b*x^2 + a*x^4)^(1/4
) - x*#1])/#1 & ]/(16*b)

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fricas [F(-1)]  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(1/(a*x^4+b*x^2)^(1/4)/(a*x^8-2*b),x, algorithm="fricas")

[Out]

Timed out

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x^4+b*x^2)^(1/4)/(a*x^8-2*b),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Eval
uation time: 82.77Not invertible Error: Bad Argument Value

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a \,x^{4}+b \,x^{2}\right )^{\frac {1}{4}} \left (a \,x^{8}-2 b \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x^4+b*x^2)^(1/4)/(a*x^8-2*b),x)

[Out]

int(1/(a*x^4+b*x^2)^(1/4)/(a*x^8-2*b),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (a x^{8} - 2 \, b\right )} {\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x^4+b*x^2)^(1/4)/(a*x^8-2*b),x, algorithm="maxima")

[Out]

integrate(1/((a*x^8 - 2*b)*(a*x^4 + b*x^2)^(1/4)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {1}{{\left (a\,x^4+b\,x^2\right )}^{1/4}\,\left (2\,b-a\,x^8\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/((a*x^4 + b*x^2)^(1/4)*(2*b - a*x^8)),x)

[Out]

-int(1/((a*x^4 + b*x^2)^(1/4)*(2*b - a*x^8)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [4]{x^{2} \left (a x^{2} + b\right )} \left (a x^{8} - 2 b\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x**4+b*x**2)**(1/4)/(a*x**8-2*b),x)

[Out]

Integral(1/((x**2*(a*x**2 + b))**(1/4)*(a*x**8 - 2*b)), x)

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