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

Optimal. Leaf size=243 \[ \frac {1}{2} \text {RootSum}\left [\text {$\#$1}^8-3 \text {$\#$1}^4 a+2 a^2-b\& ,\frac {-\text {$\#$1}^4 a \log \left (\sqrt [4]{a x^4-b x^2}-\text {$\#$1} x\right )-\text {$\#$1}^4 \log \left (\sqrt [4]{a x^4-b x^2}-\text {$\#$1} x\right )+\text {$\#$1}^4 a \log (x)+\text {$\#$1}^4 \log (x)+2 a^2 \log \left (\sqrt [4]{a x^4-b x^2}-\text {$\#$1} x\right )+a \log \left (\sqrt [4]{a x^4-b x^2}-\text {$\#$1} x\right )-2 a^2 \log (x)-a \log (x)}{3 \text {$\#$1} a-2 \text {$\#$1}^5}\& \right ]+a^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b x^2}}\right )+a^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b x^2}}\right ) \]

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Rubi [B]  time = 2.77, antiderivative size = 805, normalized size of antiderivative = 3.31, number of steps used = 18, number of rules used = 10, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.256, Rules used = {2056, 6715, 6728, 240, 212, 206, 203, 377, 208, 205} \begin {gather*} \frac {a^{3/4} \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 x^4-b x^2}}-\frac {\left (a^2-\frac {a^3+2 b a+2 b}{\sqrt {a^2+4 b}}\right ) \sqrt {x} \sqrt [4]{a x^2-b} \tan ^{-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 )}{\left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-\sqrt {a^2+4 b} a-2 b} \sqrt [4]{a x^4-b x^2}}-\frac {\left (a^2+\frac {a^3+2 b a+2 b}{\sqrt {a^2+4 b}}\right ) \sqrt {x} \sqrt [4]{a x^2-b} \tan ^{-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 )}{\left (a+\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2+\sqrt {a^2+4 b} a-2 b} \sqrt [4]{a x^4-b x^2}}+\frac {a^{3/4} \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 x^4-b x^2}}-\frac {\left (a^2-\frac {a^3+2 b a+2 b}{\sqrt {a^2+4 b}}\right ) \sqrt {x} \sqrt [4]{a x^2-b} \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 )}{\left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-\sqrt {a^2+4 b} a-2 b} \sqrt [4]{a x^4-b x^2}}-\frac {\left (a^2+\frac {a^3+2 b a+2 b}{\sqrt {a^2+4 b}}\right ) \sqrt {x} \sqrt [4]{a x^2-b} \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 )}{\left (a+\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2+\sqrt {a^2+4 b} a-2 b} \sqrt [4]{a x^4-b x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^(3/4)*Sqrt[x]*(-b + a*x^2)^(1/4)*ArcTan[(a^(1/4)*Sqrt[x])/(-b + a*x^2)^(1/4)])/(-(b*x^2) + a*x^4)^(1/4) - (
(a^2 - (a^3 + 2*b + 2*a*b)/Sqrt[a^2 + 4*b])*Sqrt[x]*(-b + a*x^2)^(1/4)*ArcTan[((a^2 - 2*b - a*Sqrt[a^2 + 4*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])^(3/4)*(a^2 - 2*b - a
*Sqrt[a^2 + 4*b])^(1/4)*(-(b*x^2) + a*x^4)^(1/4)) - ((a^2 + (a^3 + 2*b + 2*a*b)/Sqrt[a^2 + 4*b])*Sqrt[x]*(-b +
a*x^2)^(1/4)*ArcTan[((a^2 - 2*b + a*Sqrt[a^2 + 4*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])^(3/4)*(a^2 - 2*b + a*Sqrt[a^2 + 4*b])^(1/4)*(-(b*x^2) + a*x^4)^(1/4)) + (a^(3
/4)*Sqrt[x]*(-b + a*x^2)^(1/4)*ArcTanh[(a^(1/4)*Sqrt[x])/(-b + a*x^2)^(1/4)])/(-(b*x^2) + a*x^4)^(1/4) - ((a^2
- (a^3 + 2*b + 2*a*b)/Sqrt[a^2 + 4*b])*Sqrt[x]*(-b + a*x^2)^(1/4)*ArcTanh[((a^2 - 2*b - a*Sqrt[a^2 + 4*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])^(3/4)*(a^2 - 2*b - a*Sq
rt[a^2 + 4*b])^(1/4)*(-(b*x^2) + a*x^4)^(1/4)) - ((a^2 + (a^3 + 2*b + 2*a*b)/Sqrt[a^2 + 4*b])*Sqrt[x]*(-b + a*
x^2)^(1/4)*ArcTanh[((a^2 - 2*b + a*Sqrt[a^2 + 4*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])^(3/4)*(a^2 - 2*b + a*Sqrt[a^2 + 4*b])^(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 205

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

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 208

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

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 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 6728

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

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Mathematica [B]  time = 2.00, size = 656, normalized size = 2.70 \begin {gather*} \frac {\sqrt {x} \sqrt [4]{a x^2-b} \left (a^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )+a^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )-\frac {\left (a^2-\frac {a^3+2 a b+2 b}{\sqrt {a^2+4 b}}\right ) \tan ^{-1}\left (\frac {\sqrt {x} \sqrt [4]{-a \sqrt {a^2+4 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{a x^2-b}}\right )}{\left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+4 b}+a^2-2 b}}-\frac {\left (a^3+a^2 \sqrt {a^2+4 b}+2 a b+2 b\right ) \tan ^{-1}\left (\frac {\sqrt {x} \sqrt [4]{a \sqrt {a^2+4 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2+4 b}+a} \sqrt [4]{a x^2-b}}\right )}{\sqrt {a^2+4 b} \left (\sqrt {a^2+4 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+4 b}+a^2-2 b}}-\frac {\left (a^2-\frac {a^3+2 a b+2 b}{\sqrt {a^2+4 b}}\right ) \tanh ^{-1}\left (\frac {\sqrt {x} \sqrt [4]{-a \sqrt {a^2+4 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{a x^2-b}}\right )}{\left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+4 b}+a^2-2 b}}-\frac {\left (a^3+a^2 \sqrt {a^2+4 b}+2 a b+2 b\right ) \tanh ^{-1}\left (\frac {\sqrt {x} \sqrt [4]{a \sqrt {a^2+4 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2+4 b}+a} \sqrt [4]{a x^2-b}}\right )}{\sqrt {a^2+4 b} \left (\sqrt {a^2+4 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+4 b}+a^2-2 b}}\right )}{\sqrt [4]{a x^4-b x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x]*(-b + a*x^2)^(1/4)*(a^(3/4)*ArcTan[(a^(1/4)*Sqrt[x])/(-b + a*x^2)^(1/4)] - ((a^2 - (a^3 + 2*b + 2*a*b
)/Sqrt[a^2 + 4*b])*ArcTan[((a^2 - 2*b - a*Sqrt[a^2 + 4*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])^(3/4)*(a^2 - 2*b - a*Sqrt[a^2 + 4*b])^(1/4)) - ((a^3 + 2*b + 2*a*b + a^2
*Sqrt[a^2 + 4*b])*ArcTan[((a^2 - 2*b + a*Sqrt[a^2 + 4*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 + Sqrt[a^2 + 4*b])^(3/4)*(a^2 - 2*b + a*Sqrt[a^2 + 4*b])^(1/4)) + a^(3/4)*Ar
cTanh[(a^(1/4)*Sqrt[x])/(-b + a*x^2)^(1/4)] - ((a^2 - (a^3 + 2*b + 2*a*b)/Sqrt[a^2 + 4*b])*ArcTanh[((a^2 - 2*b
- a*Sqrt[a^2 + 4*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])
^(3/4)*(a^2 - 2*b - a*Sqrt[a^2 + 4*b])^(1/4)) - ((a^3 + 2*b + 2*a*b + a^2*Sqrt[a^2 + 4*b])*ArcTanh[((a^2 - 2*b
+ a*Sqrt[a^2 + 4*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 +
Sqrt[a^2 + 4*b])^(3/4)*(a^2 - 2*b + a*Sqrt[a^2 + 4*b])^(1/4))))/(-(b*x^2) + a*x^4)^(1/4)

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IntegrateAlgebraic [A]  time = 3.15, size = 243, normalized size = 1.00 \begin {gather*} a^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )+a^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )+\frac {1}{2} \text {RootSum}\left [2 a^2-b-3 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {a \log (x)+2 a^2 \log (x)-a \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )-2 a^2 \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}^4-a \log (x) \text {$\#$1}^4+\log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4+a \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-3 a \text {$\#$1}+2 \text {$\#$1}^5}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a^(3/4)*ArcTan[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)] + a^(3/4)*ArcTanh[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)] +
RootSum[2*a^2 - b - 3*a*#1^4 + #1^8 & , (a*Log[x] + 2*a^2*Log[x] - a*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1] - 2
*a^2*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1] - Log[x]*#1^4 - a*Log[x]*#1^4 + Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]
*#1^4 + a*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1^4)/(-3*a*#1 + 2*#1^5) & ]/2

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was
done assuming [a,b]=[-86,67]Warning, need to choose a branch for the root of a polynomial with parameters. Thi
s might be wrong.The choice was done assuming [a,b]=[89,-63]Evaluation time: 5.96Unable to convert to real 1/4
Error: Bad Argument Value

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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