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

Optimal. Leaf size=140 \[ \frac {1}{4} \text {RootSum}\left [\text {$\#$1}^{16}-4 \text {$\#$1}^{12} a+6 \text {$\#$1}^8 a^2-4 \text {$\#$1}^4 a^3+a^4-a b^3\& ,\frac {\log \left (\sqrt [4]{a x^4-b x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b x^2}}\right )}{\sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b x^2}}\right )}{\sqrt [4]{a}} \]

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Rubi [B]  time = 2.61, antiderivative size = 1043, normalized size of antiderivative = 7.45, number of steps used = 28, number of rules used = 9, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {2056, 6715, 6725, 240, 212, 206, 203, 1429, 377} \begin {gather*} \frac {\sqrt {x} \sqrt [4]{a x^2-b} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{\sqrt [4]{a} \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]{a^{3/4}-b^{3/4}} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{2 \sqrt [16]{a} \sqrt [4]{a^{3/4}-b^{3/4}} \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 {a}{\sqrt {-\sqrt {a}}}-b^{3/4}} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{2 \sqrt [8]{-\sqrt {a}} \sqrt [4]{\frac {a}{\sqrt {-\sqrt {a}}}-b^{3/4}} \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]{a^{3/4}+b^{3/4}} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{2 \sqrt [16]{a} \sqrt [4]{a^{3/4}+b^{3/4}} \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 {a}{\sqrt {-\sqrt {a}}}+b^{3/4}} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{2 \sqrt [8]{-\sqrt {a}} \sqrt [4]{\frac {a}{\sqrt {-\sqrt {a}}}+b^{3/4}} \sqrt [4]{a x^4-b x^2}}+\frac {\sqrt {x} \sqrt [4]{a x^2-b} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{\sqrt [4]{a} \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]{a^{3/4}-b^{3/4}} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{2 \sqrt [16]{a} \sqrt [4]{a^{3/4}-b^{3/4}} \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 {a}{\sqrt {-\sqrt {a}}}-b^{3/4}} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{2 \sqrt [8]{-\sqrt {a}} \sqrt [4]{\frac {a}{\sqrt {-\sqrt {a}}}-b^{3/4}} \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]{a^{3/4}+b^{3/4}} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{2 \sqrt [16]{a} \sqrt [4]{a^{3/4}+b^{3/4}} \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 {a}{\sqrt {-\sqrt {a}}}+b^{3/4}} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{2 \sqrt [8]{-\sqrt {a}} \sqrt [4]{\frac {a}{\sqrt {-\sqrt {a}}}+b^{3/4}} \sqrt [4]{a x^4-b x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x]*(-b + a*x^2)^(1/4)*ArcTan[(a^(1/4)*Sqrt[x])/(-b + a*x^2)^(1/4)])/(a^(1/4)*(-(b*x^2) + a*x^4)^(1/4)) -
(Sqrt[x]*(-b + a*x^2)^(1/4)*ArcTan[(a^(1/16)*(a^(3/4) - b^(3/4))^(1/4)*Sqrt[x])/(-b + a*x^2)^(1/4)])/(2*a^(1/
16)*(a^(3/4) - b^(3/4))^(1/4)*(-(b*x^2) + a*x^4)^(1/4)) - (Sqrt[x]*(-b + a*x^2)^(1/4)*ArcTan[((-Sqrt[a])^(1/8)
*(a/Sqrt[-Sqrt[a]] - b^(3/4))^(1/4)*Sqrt[x])/(-b + a*x^2)^(1/4)])/(2*(-Sqrt[a])^(1/8)*(a/Sqrt[-Sqrt[a]] - b^(3
/4))^(1/4)*(-(b*x^2) + a*x^4)^(1/4)) - (Sqrt[x]*(-b + a*x^2)^(1/4)*ArcTan[(a^(1/16)*(a^(3/4) + b^(3/4))^(1/4)*
Sqrt[x])/(-b + a*x^2)^(1/4)])/(2*a^(1/16)*(a^(3/4) + b^(3/4))^(1/4)*(-(b*x^2) + a*x^4)^(1/4)) - (Sqrt[x]*(-b +
a*x^2)^(1/4)*ArcTan[((-Sqrt[a])^(1/8)*(a/Sqrt[-Sqrt[a]] + b^(3/4))^(1/4)*Sqrt[x])/(-b + a*x^2)^(1/4)])/(2*(-S
qrt[a])^(1/8)*(a/Sqrt[-Sqrt[a]] + b^(3/4))^(1/4)*(-(b*x^2) + a*x^4)^(1/4)) + (Sqrt[x]*(-b + a*x^2)^(1/4)*ArcTa
nh[(a^(1/4)*Sqrt[x])/(-b + a*x^2)^(1/4)])/(a^(1/4)*(-(b*x^2) + a*x^4)^(1/4)) - (Sqrt[x]*(-b + a*x^2)^(1/4)*Arc
Tanh[(a^(1/16)*(a^(3/4) - b^(3/4))^(1/4)*Sqrt[x])/(-b + a*x^2)^(1/4)])/(2*a^(1/16)*(a^(3/4) - b^(3/4))^(1/4)*(
-(b*x^2) + a*x^4)^(1/4)) - (Sqrt[x]*(-b + a*x^2)^(1/4)*ArcTanh[((-Sqrt[a])^(1/8)*(a/Sqrt[-Sqrt[a]] - b^(3/4))^
(1/4)*Sqrt[x])/(-b + a*x^2)^(1/4)])/(2*(-Sqrt[a])^(1/8)*(a/Sqrt[-Sqrt[a]] - b^(3/4))^(1/4)*(-(b*x^2) + a*x^4)^
(1/4)) - (Sqrt[x]*(-b + a*x^2)^(1/4)*ArcTanh[(a^(1/16)*(a^(3/4) + b^(3/4))^(1/4)*Sqrt[x])/(-b + a*x^2)^(1/4)])
/(2*a^(1/16)*(a^(3/4) + b^(3/4))^(1/4)*(-(b*x^2) + a*x^4)^(1/4)) - (Sqrt[x]*(-b + a*x^2)^(1/4)*ArcTanh[((-Sqrt
[a])^(1/8)*(a/Sqrt[-Sqrt[a]] + b^(3/4))^(1/4)*Sqrt[x])/(-b + a*x^2)^(1/4)])/(2*(-Sqrt[a])^(1/8)*(a/Sqrt[-Sqrt[
a]] + b^(3/4))^(1/4)*(-(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 240

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[1/(1 - b*x^n)^(p + 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)] && IntegerQ[p
+ 1/n]

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

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Mathematica [F]  time = 1.99, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {b+a x^8}{\sqrt [4]{-b x^2+a x^4} \left (-b+a x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 2.65, size = 140, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )}{\sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )}{\sqrt [4]{a}}+\frac {1}{4} \text {RootSum}\left [a^4-a b^3-4 a^3 \text {$\#$1}^4+6 a^2 \text {$\#$1}^8-4 a \text {$\#$1}^{12}+\text {$\#$1}^{16}\&,\frac {-\log (x)+\log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)]/a^(1/4) + ArcTanh[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)]/a^(1/4) +
RootSum[a^4 - a*b^3 - 4*a^3*#1^4 + 6*a^2*#1^8 - 4*a*#1^12 + #1^16 & , (-Log[x] + Log[(-(b*x^2) + a*x^4)^(1/4)
- x*#1])/#1 & ]/4

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

[Out]

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

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maple [F]  time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{8}+b}{\left (a \,x^{4}-b \,x^{2}\right )^{\frac {1}{4}} \left (a \,x^{8}-b \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

integrate((a*x^8 + b)/((a*x^8 - 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 {a\,x^8+b}{\left (b-a\,x^8\right )\,{\left (a\,x^4-b\,x^2\right )}^{1/4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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