3.15.53 \(\int \frac {-b+a x^4}{\sqrt [4]{-b x^2+a x^4} (-b+2 a x^4+2 x^8)} \, dx\)
Optimal. Leaf size=102 \[ -\frac {1}{8} \text {RootSum}\left [\text {$\#$1}^{16}-4 \text {$\#$1}^{12} a+6 \text {$\#$1}^8 a^2-2 \text {$\#$1}^8 a b-4 \text {$\#$1}^4 a^3+4 \text {$\#$1}^4 a^2 b+a^4-2 a^3 b-2 b^3\& ,\frac {\log \left (\sqrt [4]{a x^4-b x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ] \]
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Rubi [B] time = 4.88, antiderivative size = 1425, normalized size of antiderivative = 13.97,
number of steps used = 22, number of rules used = 8, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used
= {2056, 6715, 6728, 1429, 377, 212, 208, 205} \begin {gather*} \frac {\sqrt [8]{-a-\sqrt {a^2+2 b}} \sqrt {x} \sqrt [4]{a x^2-b} \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt {2} a \sqrt {-a-\sqrt {a^2+2 b}}-2 b} \sqrt {x}}{\sqrt [8]{2} \sqrt [8]{-a-\sqrt {a^2+2 b}} \sqrt [4]{a x^2-b}}\right )}{2\ 2^{7/8} \sqrt [4]{\sqrt {2} a \sqrt {-a-\sqrt {a^2+2 b}}-2 b} \sqrt [4]{a x^4-b x^2}}+\frac {\sqrt [8]{-a-\sqrt {a^2+2 b}} \sqrt {x} \sqrt [4]{a x^2-b} \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt {2} \sqrt {-a-\sqrt {a^2+2 b}} a+2 b} \sqrt {x}}{\sqrt [8]{2} \sqrt [8]{-a-\sqrt {a^2+2 b}} \sqrt [4]{a x^2-b}}\right )}{2\ 2^{7/8} \sqrt [4]{\sqrt {2} \sqrt {-a-\sqrt {a^2+2 b}} a+2 b} \sqrt [4]{a x^4-b x^2}}+\frac {\sqrt [8]{\sqrt {a^2+2 b}-a} \sqrt {x} \sqrt [4]{a x^2-b} \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt {2} a \sqrt {\sqrt {a^2+2 b}-a}-2 b} \sqrt {x}}{\sqrt [8]{2} \sqrt [8]{\sqrt {a^2+2 b}-a} \sqrt [4]{a x^2-b}}\right )}{2\ 2^{7/8} \sqrt [4]{\sqrt {2} a \sqrt {\sqrt {a^2+2 b}-a}-2 b} \sqrt [4]{a x^4-b x^2}}+\frac {\sqrt [8]{\sqrt {a^2+2 b}-a} \sqrt {x} \sqrt [4]{a x^2-b} \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt {2} \sqrt {\sqrt {a^2+2 b}-a} a+2 b} \sqrt {x}}{\sqrt [8]{2} \sqrt [8]{\sqrt {a^2+2 b}-a} \sqrt [4]{a x^2-b}}\right )}{2\ 2^{7/8} \sqrt [4]{\sqrt {2} \sqrt {\sqrt {a^2+2 b}-a} a+2 b} \sqrt [4]{a x^4-b x^2}}+\frac {\sqrt [8]{-a-\sqrt {a^2+2 b}} \sqrt {x} \sqrt [4]{a x^2-b} \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {2} a \sqrt {-a-\sqrt {a^2+2 b}}-2 b} \sqrt {x}}{\sqrt [8]{2} \sqrt [8]{-a-\sqrt {a^2+2 b}} \sqrt [4]{a x^2-b}}\right )}{2\ 2^{7/8} \sqrt [4]{\sqrt {2} a \sqrt {-a-\sqrt {a^2+2 b}}-2 b} \sqrt [4]{a x^4-b x^2}}+\frac {\sqrt [8]{-a-\sqrt {a^2+2 b}} \sqrt {x} \sqrt [4]{a x^2-b} \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {2} \sqrt {-a-\sqrt {a^2+2 b}} a+2 b} \sqrt {x}}{\sqrt [8]{2} \sqrt [8]{-a-\sqrt {a^2+2 b}} \sqrt [4]{a x^2-b}}\right )}{2\ 2^{7/8} \sqrt [4]{\sqrt {2} \sqrt {-a-\sqrt {a^2+2 b}} a+2 b} \sqrt [4]{a x^4-b x^2}}+\frac {\sqrt [8]{\sqrt {a^2+2 b}-a} \sqrt {x} \sqrt [4]{a x^2-b} \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {2} a \sqrt {\sqrt {a^2+2 b}-a}-2 b} \sqrt {x}}{\sqrt [8]{2} \sqrt [8]{\sqrt {a^2+2 b}-a} \sqrt [4]{a x^2-b}}\right )}{2\ 2^{7/8} \sqrt [4]{\sqrt {2} a \sqrt {\sqrt {a^2+2 b}-a}-2 b} \sqrt [4]{a x^4-b x^2}}+\frac {\sqrt [8]{\sqrt {a^2+2 b}-a} \sqrt {x} \sqrt [4]{a x^2-b} \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {2} \sqrt {\sqrt {a^2+2 b}-a} a+2 b} \sqrt {x}}{\sqrt [8]{2} \sqrt [8]{\sqrt {a^2+2 b}-a} \sqrt [4]{a x^2-b}}\right )}{2\ 2^{7/8} \sqrt [4]{\sqrt {2} \sqrt {\sqrt {a^2+2 b}-a} a+2 b} \sqrt [4]{a x^4-b x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(-b + a*x^4)/((-(b*x^2) + a*x^4)^(1/4)*(-b + 2*a*x^4 + 2*x^8)),x]
[Out]
((-a - Sqrt[a^2 + 2*b])^(1/8)*Sqrt[x]*(-b + a*x^2)^(1/4)*ArcTan[((-2*b + Sqrt[2]*a*Sqrt[-a - Sqrt[a^2 + 2*b]])
^(1/4)*Sqrt[x])/(2^(1/8)*(-a - Sqrt[a^2 + 2*b])^(1/8)*(-b + a*x^2)^(1/4))])/(2*2^(7/8)*(-2*b + Sqrt[2]*a*Sqrt[
-a - Sqrt[a^2 + 2*b]])^(1/4)*(-(b*x^2) + a*x^4)^(1/4)) + ((-a - Sqrt[a^2 + 2*b])^(1/8)*Sqrt[x]*(-b + a*x^2)^(1
/4)*ArcTan[((2*b + Sqrt[2]*a*Sqrt[-a - Sqrt[a^2 + 2*b]])^(1/4)*Sqrt[x])/(2^(1/8)*(-a - Sqrt[a^2 + 2*b])^(1/8)*
(-b + a*x^2)^(1/4))])/(2*2^(7/8)*(2*b + Sqrt[2]*a*Sqrt[-a - Sqrt[a^2 + 2*b]])^(1/4)*(-(b*x^2) + a*x^4)^(1/4))
+ ((-a + Sqrt[a^2 + 2*b])^(1/8)*Sqrt[x]*(-b + a*x^2)^(1/4)*ArcTan[((-2*b + Sqrt[2]*a*Sqrt[-a + Sqrt[a^2 + 2*b]
])^(1/4)*Sqrt[x])/(2^(1/8)*(-a + Sqrt[a^2 + 2*b])^(1/8)*(-b + a*x^2)^(1/4))])/(2*2^(7/8)*(-2*b + Sqrt[2]*a*Sqr
t[-a + Sqrt[a^2 + 2*b]])^(1/4)*(-(b*x^2) + a*x^4)^(1/4)) + ((-a + Sqrt[a^2 + 2*b])^(1/8)*Sqrt[x]*(-b + a*x^2)^
(1/4)*ArcTan[((2*b + Sqrt[2]*a*Sqrt[-a + Sqrt[a^2 + 2*b]])^(1/4)*Sqrt[x])/(2^(1/8)*(-a + Sqrt[a^2 + 2*b])^(1/8
)*(-b + a*x^2)^(1/4))])/(2*2^(7/8)*(2*b + Sqrt[2]*a*Sqrt[-a + Sqrt[a^2 + 2*b]])^(1/4)*(-(b*x^2) + a*x^4)^(1/4)
) + ((-a - Sqrt[a^2 + 2*b])^(1/8)*Sqrt[x]*(-b + a*x^2)^(1/4)*ArcTanh[((-2*b + Sqrt[2]*a*Sqrt[-a - Sqrt[a^2 + 2
*b]])^(1/4)*Sqrt[x])/(2^(1/8)*(-a - Sqrt[a^2 + 2*b])^(1/8)*(-b + a*x^2)^(1/4))])/(2*2^(7/8)*(-2*b + Sqrt[2]*a*
Sqrt[-a - Sqrt[a^2 + 2*b]])^(1/4)*(-(b*x^2) + a*x^4)^(1/4)) + ((-a - Sqrt[a^2 + 2*b])^(1/8)*Sqrt[x]*(-b + a*x^
2)^(1/4)*ArcTanh[((2*b + Sqrt[2]*a*Sqrt[-a - Sqrt[a^2 + 2*b]])^(1/4)*Sqrt[x])/(2^(1/8)*(-a - Sqrt[a^2 + 2*b])^
(1/8)*(-b + a*x^2)^(1/4))])/(2*2^(7/8)*(2*b + Sqrt[2]*a*Sqrt[-a - Sqrt[a^2 + 2*b]])^(1/4)*(-(b*x^2) + a*x^4)^(
1/4)) + ((-a + Sqrt[a^2 + 2*b])^(1/8)*Sqrt[x]*(-b + a*x^2)^(1/4)*ArcTanh[((-2*b + Sqrt[2]*a*Sqrt[-a + Sqrt[a^2
+ 2*b]])^(1/4)*Sqrt[x])/(2^(1/8)*(-a + Sqrt[a^2 + 2*b])^(1/8)*(-b + a*x^2)^(1/4))])/(2*2^(7/8)*(-2*b + Sqrt[2
]*a*Sqrt[-a + Sqrt[a^2 + 2*b]])^(1/4)*(-(b*x^2) + a*x^4)^(1/4)) + ((-a + Sqrt[a^2 + 2*b])^(1/8)*Sqrt[x]*(-b +
a*x^2)^(1/4)*ArcTanh[((2*b + Sqrt[2]*a*Sqrt[-a + Sqrt[a^2 + 2*b]])^(1/4)*Sqrt[x])/(2^(1/8)*(-a + Sqrt[a^2 + 2*
b])^(1/8)*(-b + a*x^2)^(1/4))])/(2*2^(7/8)*(2*b + Sqrt[2]*a*Sqrt[-a + Sqrt[a^2 + 2*b]])^(1/4)*(-(b*x^2) + a*x^
4)^(1/4))
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 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 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 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}{\sqrt [4]{-b x^2+a x^4} \left (-b+2 a x^4+2 x^8\right )} \, 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+2 a x^4+2 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^8}{\sqrt [4]{-b+a x^4} \left (-b+2 a x^8+2 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 {a-\sqrt {a^2+2 b}}{\sqrt [4]{-b+a x^4} \left (2 a-2 \sqrt {a^2+2 b}+4 x^8\right )}+\frac {a+\sqrt {a^2+2 b}}{\sqrt [4]{-b+a x^4} \left (2 a+2 \sqrt {a^2+2 b}+4 x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^4}}\\ &=\frac {\left (2 \left (a-\sqrt {a^2+2 b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-b+a x^4} \left (2 a-2 \sqrt {a^2+2 b}+4 x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^4}}+\frac {\left (2 \left (a+\sqrt {a^2+2 b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-b+a x^4} \left (2 a+2 \sqrt {a^2+2 b}+4 x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^4}}\\ &=-\frac {\left (\sqrt {2} \left (a-\sqrt {a^2+2 b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (2 \sqrt {2} \sqrt {-a+\sqrt {a^2+2 b}}-4 x^4\right ) \sqrt [4]{-b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-a+\sqrt {a^2+2 b}} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (\sqrt {2} \left (a-\sqrt {a^2+2 b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (2 \sqrt {2} \sqrt {-a+\sqrt {a^2+2 b}}+4 x^4\right ) \sqrt [4]{-b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-a+\sqrt {a^2+2 b}} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (\sqrt {2} \left (a+\sqrt {a^2+2 b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (2 \sqrt {2} \sqrt {-a-\sqrt {a^2+2 b}}-4 x^4\right ) \sqrt [4]{-b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-a-\sqrt {a^2+2 b}} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (\sqrt {2} \left (a+\sqrt {a^2+2 b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (2 \sqrt {2} \sqrt {-a-\sqrt {a^2+2 b}}+4 x^4\right ) \sqrt [4]{-b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-a-\sqrt {a^2+2 b}} \sqrt [4]{-b x^2+a x^4}}\\ &=-\frac {\left (\sqrt {2} \left (a-\sqrt {a^2+2 b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{2 \sqrt {2} \sqrt {-a+\sqrt {a^2+2 b}}-\left (-4 b+2 \sqrt {2} a \sqrt {-a+\sqrt {a^2+2 b}}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt {-a+\sqrt {a^2+2 b}} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (\sqrt {2} \left (a-\sqrt {a^2+2 b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{2 \sqrt {2} \sqrt {-a+\sqrt {a^2+2 b}}-\left (4 b+2 \sqrt {2} a \sqrt {-a+\sqrt {a^2+2 b}}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt {-a+\sqrt {a^2+2 b}} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (\sqrt {2} \left (a+\sqrt {a^2+2 b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{2 \sqrt {2} \sqrt {-a-\sqrt {a^2+2 b}}-\left (-4 b+2 \sqrt {2} a \sqrt {-a-\sqrt {a^2+2 b}}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt {-a-\sqrt {a^2+2 b}} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (\sqrt {2} \left (a+\sqrt {a^2+2 b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{2 \sqrt {2} \sqrt {-a-\sqrt {a^2+2 b}}-\left (4 b+2 \sqrt {2} a \sqrt {-a-\sqrt {a^2+2 b}}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt {-a-\sqrt {a^2+2 b}} \sqrt [4]{-b x^2+a x^4}}\\ &=-\frac {\left (\left (a-\sqrt {a^2+2 b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2} \sqrt [4]{-a+\sqrt {a^2+2 b}}-\sqrt {-2 b+\sqrt {2} a \sqrt {-a+\sqrt {a^2+2 b}}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2\ 2^{3/4} \left (-a+\sqrt {a^2+2 b}\right )^{3/4} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (\left (a-\sqrt {a^2+2 b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2} \sqrt [4]{-a+\sqrt {a^2+2 b}}+\sqrt {-2 b+\sqrt {2} a \sqrt {-a+\sqrt {a^2+2 b}}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2\ 2^{3/4} \left (-a+\sqrt {a^2+2 b}\right )^{3/4} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (\left (a-\sqrt {a^2+2 b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2} \sqrt [4]{-a+\sqrt {a^2+2 b}}-\sqrt {2 b+\sqrt {2} a \sqrt {-a+\sqrt {a^2+2 b}}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2\ 2^{3/4} \left (-a+\sqrt {a^2+2 b}\right )^{3/4} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (\left (a-\sqrt {a^2+2 b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2} \sqrt [4]{-a+\sqrt {a^2+2 b}}+\sqrt {2 b+\sqrt {2} a \sqrt {-a+\sqrt {a^2+2 b}}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2\ 2^{3/4} \left (-a+\sqrt {a^2+2 b}\right )^{3/4} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (\left (a+\sqrt {a^2+2 b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2} \sqrt [4]{-a-\sqrt {a^2+2 b}}-\sqrt {-2 b+\sqrt {2} a \sqrt {-a-\sqrt {a^2+2 b}}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2\ 2^{3/4} \left (-a-\sqrt {a^2+2 b}\right )^{3/4} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (\left (a+\sqrt {a^2+2 b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2} \sqrt [4]{-a-\sqrt {a^2+2 b}}+\sqrt {-2 b+\sqrt {2} a \sqrt {-a-\sqrt {a^2+2 b}}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2\ 2^{3/4} \left (-a-\sqrt {a^2+2 b}\right )^{3/4} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (\left (a+\sqrt {a^2+2 b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2} \sqrt [4]{-a-\sqrt {a^2+2 b}}-\sqrt {2 b+\sqrt {2} a \sqrt {-a-\sqrt {a^2+2 b}}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2\ 2^{3/4} \left (-a-\sqrt {a^2+2 b}\right )^{3/4} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (\left (a+\sqrt {a^2+2 b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2} \sqrt [4]{-a-\sqrt {a^2+2 b}}+\sqrt {2 b+\sqrt {2} a \sqrt {-a-\sqrt {a^2+2 b}}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2\ 2^{3/4} \left (-a-\sqrt {a^2+2 b}\right )^{3/4} \sqrt [4]{-b x^2+a x^4}}\\ &=\frac {\sqrt [8]{-a-\sqrt {a^2+2 b}} \sqrt {x} \sqrt [4]{-b+a x^2} \tan ^{-1}\left (\frac {\sqrt [4]{-2 b+\sqrt {2} a \sqrt {-a-\sqrt {a^2+2 b}}} \sqrt {x}}{\sqrt [8]{2} \sqrt [8]{-a-\sqrt {a^2+2 b}} \sqrt [4]{-b+a x^2}}\right )}{2\ 2^{7/8} \sqrt [4]{-2 b+\sqrt {2} a \sqrt {-a-\sqrt {a^2+2 b}}} \sqrt [4]{-b x^2+a x^4}}+\frac {\sqrt [8]{-a-\sqrt {a^2+2 b}} \sqrt {x} \sqrt [4]{-b+a x^2} \tan ^{-1}\left (\frac {\sqrt [4]{2 b+\sqrt {2} a \sqrt {-a-\sqrt {a^2+2 b}}} \sqrt {x}}{\sqrt [8]{2} \sqrt [8]{-a-\sqrt {a^2+2 b}} \sqrt [4]{-b+a x^2}}\right )}{2\ 2^{7/8} \sqrt [4]{2 b+\sqrt {2} a \sqrt {-a-\sqrt {a^2+2 b}}} \sqrt [4]{-b x^2+a x^4}}+\frac {\sqrt [8]{-a+\sqrt {a^2+2 b}} \sqrt {x} \sqrt [4]{-b+a x^2} \tan ^{-1}\left (\frac {\sqrt [4]{-2 b+\sqrt {2} a \sqrt {-a+\sqrt {a^2+2 b}}} \sqrt {x}}{\sqrt [8]{2} \sqrt [8]{-a+\sqrt {a^2+2 b}} \sqrt [4]{-b+a x^2}}\right )}{2\ 2^{7/8} \sqrt [4]{-2 b+\sqrt {2} a \sqrt {-a+\sqrt {a^2+2 b}}} \sqrt [4]{-b x^2+a x^4}}+\frac {\sqrt [8]{-a+\sqrt {a^2+2 b}} \sqrt {x} \sqrt [4]{-b+a x^2} \tan ^{-1}\left (\frac {\sqrt [4]{2 b+\sqrt {2} a \sqrt {-a+\sqrt {a^2+2 b}}} \sqrt {x}}{\sqrt [8]{2} \sqrt [8]{-a+\sqrt {a^2+2 b}} \sqrt [4]{-b+a x^2}}\right )}{2\ 2^{7/8} \sqrt [4]{2 b+\sqrt {2} a \sqrt {-a+\sqrt {a^2+2 b}}} \sqrt [4]{-b x^2+a x^4}}+\frac {\sqrt [8]{-a-\sqrt {a^2+2 b}} \sqrt {x} \sqrt [4]{-b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{-2 b+\sqrt {2} a \sqrt {-a-\sqrt {a^2+2 b}}} \sqrt {x}}{\sqrt [8]{2} \sqrt [8]{-a-\sqrt {a^2+2 b}} \sqrt [4]{-b+a x^2}}\right )}{2\ 2^{7/8} \sqrt [4]{-2 b+\sqrt {2} a \sqrt {-a-\sqrt {a^2+2 b}}} \sqrt [4]{-b x^2+a x^4}}+\frac {\sqrt [8]{-a-\sqrt {a^2+2 b}} \sqrt {x} \sqrt [4]{-b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{2 b+\sqrt {2} a \sqrt {-a-\sqrt {a^2+2 b}}} \sqrt {x}}{\sqrt [8]{2} \sqrt [8]{-a-\sqrt {a^2+2 b}} \sqrt [4]{-b+a x^2}}\right )}{2\ 2^{7/8} \sqrt [4]{2 b+\sqrt {2} a \sqrt {-a-\sqrt {a^2+2 b}}} \sqrt [4]{-b x^2+a x^4}}+\frac {\sqrt [8]{-a+\sqrt {a^2+2 b}} \sqrt {x} \sqrt [4]{-b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{-2 b+\sqrt {2} a \sqrt {-a+\sqrt {a^2+2 b}}} \sqrt {x}}{\sqrt [8]{2} \sqrt [8]{-a+\sqrt {a^2+2 b}} \sqrt [4]{-b+a x^2}}\right )}{2\ 2^{7/8} \sqrt [4]{-2 b+\sqrt {2} a \sqrt {-a+\sqrt {a^2+2 b}}} \sqrt [4]{-b x^2+a x^4}}+\frac {\sqrt [8]{-a+\sqrt {a^2+2 b}} \sqrt {x} \sqrt [4]{-b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{2 b+\sqrt {2} a \sqrt {-a+\sqrt {a^2+2 b}}} \sqrt {x}}{\sqrt [8]{2} \sqrt [8]{-a+\sqrt {a^2+2 b}} \sqrt [4]{-b+a x^2}}\right )}{2\ 2^{7/8} \sqrt [4]{2 b+\sqrt {2} a \sqrt {-a+\sqrt {a^2+2 b}}} \sqrt [4]{-b x^2+a x^4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 3.62, size = 804, normalized size = 7.88 \begin {gather*} -\frac {\sqrt [4]{\frac {b}{x^2}-a} x \left ((a-b) \text {RootSum}\left [a^4-2 b a^3+4 \text {$\#$1} a^3+6 \text {$\#$1}^2 a^2-4 b \text {$\#$1} a^2+4 \text {$\#$1}^3 a-2 b \text {$\#$1}^2 a+\text {$\#$1}^4-2 b^3\&,\frac {\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [4]{\text {$\#$1}}}\right )}{\sqrt [4]{\text {$\#$1}}}+\frac {\log \left (\sqrt [4]{\text {$\#$1}}-\sqrt [4]{\frac {b}{x^2}-a}\right )}{\sqrt [4]{\text {$\#$1}}}-\frac {\log \left (\sqrt [4]{\frac {b}{x^2}-a}+\sqrt [4]{\text {$\#$1}}\right )}{\sqrt [4]{\text {$\#$1}}}}{a^3-b a^2+3 \text {$\#$1} a^2+3 \text {$\#$1}^2 a-b \text {$\#$1} a+\text {$\#$1}^3}\&\right ] a^2+(3 a-b) \text {RootSum}\left [a^4-2 b a^3+4 \text {$\#$1} a^3+6 \text {$\#$1}^2 a^2-4 b \text {$\#$1} a^2+4 \text {$\#$1}^3 a-2 b \text {$\#$1}^2 a+\text {$\#$1}^4-2 b^3\&,\frac {2 \text {$\#$1}^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [4]{\text {$\#$1}}}\right )+\log \left (\sqrt [4]{\text {$\#$1}}-\sqrt [4]{\frac {b}{x^2}-a}\right ) \text {$\#$1}^{3/4}-\log \left (\sqrt [4]{\frac {b}{x^2}-a}+\sqrt [4]{\text {$\#$1}}\right ) \text {$\#$1}^{3/4}}{a^3-b a^2+3 \text {$\#$1} a^2+3 \text {$\#$1}^2 a-b \text {$\#$1} a+\text {$\#$1}^3}\&\right ] a+3 \text {RootSum}\left [a^4-2 b a^3+4 \text {$\#$1} a^3+6 \text {$\#$1}^2 a^2-4 b \text {$\#$1} a^2+4 \text {$\#$1}^3 a-2 b \text {$\#$1}^2 a+\text {$\#$1}^4-2 b^3\&,\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [4]{\text {$\#$1}}}\right ) \text {$\#$1}^{7/4}+\log \left (\sqrt [4]{\text {$\#$1}}-\sqrt [4]{\frac {b}{x^2}-a}\right ) \text {$\#$1}^{7/4}-\log \left (\sqrt [4]{\frac {b}{x^2}-a}+\sqrt [4]{\text {$\#$1}}\right ) \text {$\#$1}^{7/4}}{a^3-b a^2+3 \text {$\#$1} a^2+3 \text {$\#$1}^2 a-b \text {$\#$1} a+\text {$\#$1}^3}\&\right ] a+\text {RootSum}\left [a^4-2 b a^3+4 \text {$\#$1} a^3+6 \text {$\#$1}^2 a^2-4 b \text {$\#$1} a^2+4 \text {$\#$1}^3 a-2 b \text {$\#$1}^2 a+\text {$\#$1}^4-2 b^3\&,\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [4]{\text {$\#$1}}}\right ) \text {$\#$1}^{11/4}+\log \left (\sqrt [4]{\text {$\#$1}}-\sqrt [4]{\frac {b}{x^2}-a}\right ) \text {$\#$1}^{11/4}-\log \left (\sqrt [4]{\frac {b}{x^2}-a}+\sqrt [4]{\text {$\#$1}}\right ) \text {$\#$1}^{11/4}}{a^3-b a^2+3 \text {$\#$1} a^2+3 \text {$\#$1}^2 a-b \text {$\#$1} a+\text {$\#$1}^3}\&\right ]\right )}{8 \sqrt [4]{a x^4-b x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(-b + a*x^4)/((-(b*x^2) + a*x^4)^(1/4)*(-b + 2*a*x^4 + 2*x^8)),x]
[Out]
-1/8*((-a + b/x^2)^(1/4)*x*(a^2*(a - b)*RootSum[a^4 - 2*a^3*b - 2*b^3 + 4*a^3*#1 - 4*a^2*b*#1 + 6*a^2*#1^2 - 2
*a*b*#1^2 + 4*a*#1^3 + #1^4 & , ((2*ArcTan[(-a + b/x^2)^(1/4)/#1^(1/4)])/#1^(1/4) + Log[-(-a + b/x^2)^(1/4) +
#1^(1/4)]/#1^(1/4) - Log[(-a + b/x^2)^(1/4) + #1^(1/4)]/#1^(1/4))/(a^3 - a^2*b + 3*a^2*#1 - a*b*#1 + 3*a*#1^2
+ #1^3) & ] + a*(3*a - b)*RootSum[a^4 - 2*a^3*b - 2*b^3 + 4*a^3*#1 - 4*a^2*b*#1 + 6*a^2*#1^2 - 2*a*b*#1^2 + 4*
a*#1^3 + #1^4 & , (2*ArcTan[(-a + b/x^2)^(1/4)/#1^(1/4)]*#1^(3/4) + Log[-(-a + b/x^2)^(1/4) + #1^(1/4)]*#1^(3/
4) - Log[(-a + b/x^2)^(1/4) + #1^(1/4)]*#1^(3/4))/(a^3 - a^2*b + 3*a^2*#1 - a*b*#1 + 3*a*#1^2 + #1^3) & ] + 3*
a*RootSum[a^4 - 2*a^3*b - 2*b^3 + 4*a^3*#1 - 4*a^2*b*#1 + 6*a^2*#1^2 - 2*a*b*#1^2 + 4*a*#1^3 + #1^4 & , (2*Arc
Tan[(-a + b/x^2)^(1/4)/#1^(1/4)]*#1^(7/4) + Log[-(-a + b/x^2)^(1/4) + #1^(1/4)]*#1^(7/4) - Log[(-a + b/x^2)^(1
/4) + #1^(1/4)]*#1^(7/4))/(a^3 - a^2*b + 3*a^2*#1 - a*b*#1 + 3*a*#1^2 + #1^3) & ] + RootSum[a^4 - 2*a^3*b - 2*
b^3 + 4*a^3*#1 - 4*a^2*b*#1 + 6*a^2*#1^2 - 2*a*b*#1^2 + 4*a*#1^3 + #1^4 & , (2*ArcTan[(-a + b/x^2)^(1/4)/#1^(1
/4)]*#1^(11/4) + Log[-(-a + b/x^2)^(1/4) + #1^(1/4)]*#1^(11/4) - Log[(-a + b/x^2)^(1/4) + #1^(1/4)]*#1^(11/4))
/(a^3 - a^2*b + 3*a^2*#1 - a*b*#1 + 3*a*#1^2 + #1^3) & ]))/(-(b*x^2) + a*x^4)^(1/4)
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 11.65, size = 102, normalized size = 1.00 \begin {gather*} -\frac {1}{8} \text {RootSum}\left [a^4-2 a^3 b-2 b^3-4 a^3 \text {$\#$1}^4+4 a^2 b \text {$\#$1}^4+6 a^2 \text {$\#$1}^8-2 a b \text {$\#$1}^8-4 a \text {$\#$1}^{12}+\text {$\#$1}^{16}\&,\frac {-\log (x)+\log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(-b + a*x^4)/((-(b*x^2) + a*x^4)^(1/4)*(-b + 2*a*x^4 + 2*x^8)),x]
[Out]
-1/8*RootSum[a^4 - 2*a^3*b - 2*b^3 - 4*a^3*#1^4 + 4*a^2*b*#1^4 + 6*a^2*#1^8 - 2*a*b*#1^8 - 4*a*#1^12 + #1^16 &
, (-Log[x] + Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1])/#1 & ]
________________________________________________________________________________________
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)/(a*x^4-b*x^2)^(1/4)/(2*x^8+2*a*x^4-b),x, algorithm="fricas")
[Out]
Timed out
________________________________________________________________________________________
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)/(a*x^4-b*x^2)^(1/4)/(2*x^8+2*a*x^4-b),x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Eval
uation time: 2.76Unable to divide, perhaps due to rounding error%%%{36028797018963968,[2,35,24,18]%%%}+%%%{-21
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+%%%{504403158265495552,[2,33,25,18]%%%}+%%%{193514046488576,[2,33,25,14]%%%}+%%%{12884901888,[2,33,24,10]%%%}
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2,31,26,10]%%%}+%%%{180388626432,[2,31,25,10]%%%}+%%%{11534336,[2,31,25,6]%%%}+%%%{128,[2,31,24,2]%%%}+%%%{-11
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0,27,10]%%%}+%%%{-2955487255461888,[2,30,26,14]%%%}+%%%{-987842478080,[2,30,26,10]%%%}+%%%{-22020096,[2,30,26,
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%}+%%%{8690539905941504,[2,29,28,14]%%%}+%%%{94489280512,[2,29,28,10]%%%}+%%%{10088063165309911040,[2,29,27,18
]%%%}+%%%{16747761114284032,[2,29,27,14]%%%}+%%%{1954210119680,[2,29,27,10]%%%}+%%%{16777216,[2,29,27,6]%%%}+%
%%{1082331758592,[2,29,26,10]%%%}+%%%{167772160,[2,29,26,6]%%%}+%%%{1664,[2,29,26,2]%%%}+%%%{1792,[2,29,25,2]%
%%}+%%%{-18446744073709551616,[2,28,30,18]%%%}+%%%{-104339396166919651328,[2,28,29,18]%%%}+%%%{-75998243711877
12,[2,28,29,14]%%%}+%%%{-34359738368,[2,28,29,10]%%%}+%%%{-60528378991859466240,[2,28,28,18]%%%}+%%%{-37717646
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960,[2,28,27,14]%%%}+%%%{-6133213298688,[2,28,27,10]%%%}+%%%{-327155712,[2,28,27,6]%%%}+%%%{-1536,[2,28,27,2]%
%%}+%%%{-176160768,[2,28,26,6]%%%}+%%%{-11264,[2,28,26,2]%%%}+%%%{10376293541461622784,[2,27,31,18]%%%}+%%%{14
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8,6]%%%}+%%%{512,[2,27,28,2]%%%}+%%%{3607772528640,[2,27,27,10]%%%}+%%%{1056964608,[2,27,27,6]%%%}+%%%{25344,[
2,27,27,2]%%%}+%%%{10752,[2,27,26,2]%%%}+%%%{-129127208515966861312,[2,26,31,18]%%%}+%%%{-36317027395115679744
0,[2,26,30,18]%%%}+%%%{-51439551993872384,[2,26,30,14]%%%}+%%%{-584115552256,[2,26,30,10]%%%}+%%%{-12105675798
3718932480,[2,26,29,18]%%%}+%%%{-126100789566373888,[2,26,29,14]%%%}+%%%{-12369505812480,[2,26,29,10]%%%}+%%%{
-58720256,[2,26,29,6]%%%}+%%%{-19703248369745920,[2,26,28,14]%%%}+%%%{-21406117003264,[2,26,28,10]%%%}+%%%{-21
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%%%{84163269836299829248,[2,25,32,18]%%%}+%%%{544755410926735196160,[2,25,31,18]%%%}+%%%{13370061393756160,[2,
25,31,14]%%%}+%%%{318782796023793188864,[2,25,30,18]%%%}+%%%{188025284442718208,[2,25,30,14]%%%}+%%%{101618926
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,28,10]%%%}+%%%{3758096384,[2,25,28,6]%%%}+%%%{168960,[2,25,28,2]%%%}+%%%{35840,[2,25,27,2]%%%}+%%%{-514202991
054653751296,[2,24,32,18]%%%}+%%%{-758622350031305310208,[2,24,31,18]%%%}+%%%{-200269445929631744,[2,24,31,14]
%%%}+%%%{-4432406249472,[2,24,31,10]%%%}+%%%{-145268109580462718976,[2,24,30,18]%%%}+%%%{-248260929458798592,[
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24,29,14]%%%}+%%%{-45698452029440,[2,24,29,10]%%%}+%%%{-7751073792,[2,24,29,6]%%%}+%%%{-165888,[2,24,29,2]%%%}
+%%%{-1174405120,[2,24,28,6]%%%}+%%%{-258048,[2,24,28,2]%%%}+%%%{405828369621610135552,[2,23,33,18]%%%}+%%%{12
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,[2,23,30,18]%%%}+%%%{149744687610068992,[2,23,30,14]%%%}+%%%{88991722373120,[2,23,30,10]%%%}+%%%{5570035712,[
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{645120,[2,23,29,2]%%%}+%%%{71680,[2,23,28,2]%%%}+%%%{-1257837361526070050816,[2,22,33,18]%%%}+%%%{-9361722617
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{-96845406386975145984,[2,22,31,18]%%%}+%%%{-275845477176442880,[2,22,31,14]%%%}+%%%{-83700322664448,[2,22,31,
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]%%%}+%%%{-17616076800,[2,22,30,6]%%%}+%%%{-659456,[2,22,30,2]%%%}+%%%{-1409286144,[2,22,29,6]%%%}+%%%{-573440
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%%{16064,[1,24,29,1]%%%}+%%%{601295421440,[1,24,28,9]%%%}+%%%{1078984704,[1,24,28,5]%%%}+%%%{123648,[1,24,28,1
]%%%}+%%%{5376,[1,24,27,1]%%%}+%%%{-644898172816885924495360,[1,23,33,21]%%%}+%%%{50728546202701266944,[1,23,3
3,17]%%%}+%%%{-1144583576285530258669568,[1,23,32,21]%%%}+%%%{-391921253972290043904,[1,23,32,17]%%%}+%%%{2301
9375439118336,[1,23,32,13]%%%}+%%%{-147573952589676412928000,[1,23,31,21]%%%}+%%%{-497341514049778614272,[1,23
,31,17]%%%}+%%%{-115017712358391808,[1,23,31,13]%%%}+%%%{-1434519076864,[1,23,31,9]%%%}+%%%{-29687728743626309
632,[1,23,30,17]%%%}+%%%{-114771421753769984,[1,23,30,13]%%%}+%%%{-30726196035584,[1,23,30,9]%%%}+%%%{-8441036
8,[1,23,30,5]%%%}+%%%{7232,[1,23,30,1]%%%}+%%%{-703687441776640,[1,23,29,13]%%%}+%%%{-7215545057280,[1,23,29,9
]%%%}+%%%{-3317694464,[1,23,29,5]%%%}+%%%{-141568,[1,23,29,1]%%%}+%%%{-73400320,[1,23,28,5]%%%}+%%%{-55552,[1,
23,28,1]%%%}+%%%{-661426455506929682743296,[1,22,34,21]%%%}+%%%{1913148721372565017198592,[1,22,33,21]%%%}+%%%
{18807032043899191296,[1,22,33,17]%%%}+%%%{258549564937113075449856,[1,22,32,21]%%%}+%%%{434795522424857165824
,[1,22,32,17]%%%}+%%%{13299692649578496,[1,22,32,13]%%%}+%%%{-2469606195200,[1,22,32,9]%%%}+%%%{47223664828696
45213696,[1,22,31,21]%%%}+%%%{144403418452007583744,[1,22,31,17]%%%}+%%%{155163080911749120,[1,22,31,13]%%%}+%
%%{8598524526592,[1,22,31,9]%%%}+%%%{-425197568,[1,22,31,5]%%%}+%%%{13510798882111488,[1,22,30,13]%%%}+%%%{248
93630447616,[1,22,30,9]%%%}+%%%{3126853632,[1,22,30,5]%%%}+%%%{24192,[1,22,30,1]%%%}+%%%{171798691840,[1,22,29
,9]%%%}+%%%{895483904,[1,22,29,5]%%%}+%%%{186368,[1,22,29,1]%%%}+%%%{3584,[1,22,28,1]%%%}+%%%{-830546205174698
851958784,[1,21,34,21]%%%}+%%%{161481068238996504576,[1,21,34,17]%%%}+%%%{-435638308044724770963456,[1,21,33,2
1]%%%}+%%%{-354091017102377877504,[1,21,33,17]%%%}+%%%{63155947899453440,[1,21,33,13]%%%}+%%%{-306953821386526
93889024,[1,21,32,21]%%%}+%%%{-195708425407012274176,[1,21,32,17]%%%}+%%%{-66005882038648832,[1,21,32,13]%%%}+
%%%{-240518168576,[1,21,32,9]%%%}+%%%{-5188146770730811392,[1,21,31,17]%%%}+%%%{-58969007620882432,[1,21,31,13
]%%%}+%%%{-25134148616192,[1,21,31,9]%%%}+%%%{726663168,[1,21,31,5]%%%}+%%%{29568,[1,21,31,1]%%%}+%%%{-3229815
406592,[1,21,30,9]%%%}+%%%{-3082813440,[1,21,30,5]%%%}+%%%{-215040,[1,21,30,1]%%%}+%%%{-20971520,[1,21,29,5]%%
%}+%%%{-46592,[1,21,29,1]%%%}+%%%{-1458620947396361665380352,[1,20,35,21]%%%}+%%%{1557200347726265509216256,[1
,20,34,21]%%%}+%%%{-31272995812460724224,[1,20,34,17]%%%}+%%%{-44862481587261629530112,[1,20,33,21]%%%}+%%%{34
299414762053697536,[1,20,33,17]%%%}+%%%{-19703248369745920,[1,20,33,13]%%%}+%%%{-7155415515136,[1,20,33,9]%%%}
+%%%{32858262881295138816,[1,20,32,17]%%%}+%%%{62205969853054976,[1,20,32,13]%%%}+%%%{-1082331758592,[1,20,32,
9]%%%}+%%%{-1020264448,[1,20,32,5]%%%}+%%%{2392537302040576,[1,20,31,13]%%%}+%%%{13159779794944,[1,20,31,9]%%%
}+%%%{2583691264,[1,20,31,5]%%%}+%%%{1792,[1,20,31,1]%%%}+%%%{406847488,[1,20,30,5]%%%}+%%%{179200,[1,20,30,1]
%%%}+%%%{1024,[1,20,29,1]%%%}+%%%{-855928925020123194982400,[1,19,35,21]%%%}+%%%{356108629735439859712,[1,19,3
5,17]%%%}+%%%{186533476073350985940992,[1,19,34,21]%%%}+%%%{-172073534562571911168,[1,19,34,17]%%%}+%%%{114771
421753769984,[1,19,34,13]%%%}+%%%{22481969339833516032,[1,19,33,17]%%%}+%%%{23643898043695104,[1,19,33,13]%%%}
+%%%{2164663517184,[1,19,33,9]%%%}+%%%{-14355223812243456,[1,19,32,13]%%%}+%%%{-9414568312832,[1,19,32,9]%%%}+
%%%{2187329536,[1,19,32,5]%%%}+%%%{77056,[1,19,32,1]%%%}+%%%{-584115552256,[1,19,31,9]%%%}+%%%{-1702887424,[1,
19,31,5]%%%}+%%%{-207872,[1,19,31,1]%%%}+%%%{-21504,[1,19,30,1]%%%}+%%%{-2272638869881016759091200,[1,18,36,21
]%%%}+%%%{866554249606579896713216,[1,18,35,21]%%%}+%%%{-33146493257446850560,[1,18,35,17]%%%}+%%%{-7555786372
5914323419136,[1,18,34,21]%%%}+%%%{-230007840169065971712,[1,18,34,17]%%%}+%%%{-46584108645613568,[1,18,34,13]
%%%}+%%%{-14070312861696,[1,18,34,9]%%%}+%%%{1152921504606846976,[1,18,33,17]%%%}+%%%{-3940649673949184,[1,18,
33,13]%%%}+%%%{-12128987643904,[1,18,33,9]%%%}+%%%{-1541406720,[1,18,33,5]%%%}+%%%{3435973836800,[1,18,32,9]%%
%}+%%%{1015021568,[1,18,32,5]%%%}+%%%{-60928,[1,18,32,1]%%%}+%%%{75497472,[1,18,31,5]%%%}+%%%{104448,[1,18,31,
1]%%%}+%%%{-819330584777883444576256,[1,17,36,21]%%%}+%%%{554843474092045107200,[1,17,36,17]%%%}+%%%{288064355
455048358035456,[1,17,35,21]%%%}+%%%{-36317027395115679744,[1,17,35,17]%%%}+%%%{136937576169734144,[1,17,35,13
]%%%}+%%%{47269781688880726016,[1,17,34,17]%%%}+%%%{65583669573582848,[1,17,34,13]%%%}+%%%{2095944040448,[1,17
,34,9]%%%}+%%%{-844424930131968,[1,17,33,13]%%%}+%%%{1099511627776,[1,17,33,9]%%%}+%%%{3066036224,[1,17,33,5]%
%%}+%%%{132608,[1,17,33,1]%%%}+%%%{-486539264,[1,17,32,5]%%%}+%%%{-120832,[1,17,32,1]%%%}+%%%{-4096,[1,17,31,1
]%%%}+%%%{-2500493052679477140652032,[1,16,37,21]%%%}+%%%{344732753249484100599808,[1,16,36,21]%%%}+%%%{812809
66074782711808,[1,16,36,17]%%%}+%%%{-18889465931478580854784,[1,16,35,21]%%%}+%%%{-197149577287770832896,[1,16
,35,17]%%%}+%%%{-33776997205278720,[1,16,35,13]%%%}+%%%{-19000935317504,[1,16,35,9]%%%}+%%%{-12103423998558208
,[1,16,34,13]%%%}+%%%{-13125420056576,[1,16,34,9]%%%}+%%%{-1379926016,[1,16,34,5]%%%}+%%%{274877906944,[1,16,3
3,9]%%%}+%%%{-50331648,[1,16,33,5]%%%}+%%%{-113664,[1,16,33,1]%%%}+%%%{32768,[1,16,32,1]%%%}+%%%{-731966804844
795008122880,[1,15,37,21]%%%}+%%%{610471936689325473792,[1,15,37,17]%%%}+%%%{122781528554610775556096,[1,15,36
,21]%%%}+%%%{-19599665578316398592,[1,15,36,17]%%%}+%%%{100205091708993536,[1,15,36,13]%%%}+%%%{11529215046068
469760,[1,15,35,17]%%%}+%%%{42784196460019712,[1,15,35,13]%%%}+%%%{-1786706395136,[1,15,35,9]%%%}+%%%{19928648
25344,[1,15,34,9]%%%}+%%%{2390753280,[1,15,34,5]%%%}+%%%{150528,[1,15,34,1]%%%}+%%%{-50331648,[1,15,33,5]%%%}+
%%%{-36864,[1,15,33,1]%%%}+%%%{-1903113692596467021119488,[1,14,38,21]%%%}+%%%{103892062623132194701312,[1,14,
37,21]%%%}+%%%{210984635343052996608,[1,14,37,17]%%%}+%%%{-69175290276410818560,[1,14,36,17]%%%}+%%%{-22517998
13685248,[1,14,36,13]%%%}+%%%{-17386027614208,[1,14,36,9]%%%}+%%%{-2814749767106560,[1,14,35,13]%%%}+%%%{-6184
752906240,[1,14,35,9]%%%}+%%%{-545259520,[1,14,35,5]%%%}+%%%{-167772160,[1,14,34,5]%%%}+%%%{-100352,[1,14,34,1
]%%%}+%%%{4096,[1,14,33,1]%%%}+%%%{-500570847184182392651776,[1,13,38,21]%%%}+%%%{464627366356559331328,[1,13,
38,17]%%%}+%%%{18889465931478580854784,[1,13,37,21]%%%}+%%%{-25364273101350633472,[1,13,37,17]%%%}+%%%{3433994
7158700032,[1,13,37,13]%%%}+%%%{11821949021847552,[1,13,36,13]%%%}+%%%{-4672924418048,[1,13,36,9]%%%}+%%%{4123
16860416,[1,13,35,9]%%%}+%%%{989855744,[1,13,35,5]%%%}+%%%{108544,[1,13,35,1]%%%}+%%%{-4096,[1,13,34,1]%%%}+%%
%{-953918029539668333166592,[1,12,39,21]%%%}+%%%{18889465931478580854784,[1,12,38,21]%%%}+%%%{2052200278200187
61728,[1,12,38,17]%%%}+%%%{-9223372036854775808,[1,12,37,17]%%%}+%%%{9007199254740992,[1,12,37,13]%%%}+%%%{-10
307921510400,[1,12,37,9]%%%}+%%%{-1099511627776,[1,12,36,9]%%%}+%%%{150994944,[1,12,36,5]%%%}+%%%{-33554432,[1
,12,35,5]%%%}+%%%{-45056,[1,12,35,1]%%%}+%%%{-207784125246264389402624,[1,11,39,21]%%%}+%%%{232890143930583089
152,[1,11,39,17]%%%}+%%%{-9223372036854775808,[1,11,38,17]%%%}+%%%{-5629499534213120,[1,11,38,13]%%%}+%%%{1125
899906842624,[1,11,37,13]%%%}+%%%{-3298534883328,[1,11,37,9]%%%}+%%%{167772160,[1,11,36,5]%%%}+%%%{45056,[1,11
,36,1]%%%}+%%%{-283341988972178712821760,[1,10,40,21]%%%}+%%%{96845406386975145984,[1,10,39,17]%%%}+%%%{337769
9720527872,[1,10,38,13]%%%}+%%%{-3573412790272,[1,10,38,9]%%%}+%%%{234881024,[1,10,37,5]%%%}+%%%{-8192,[1,10,3
6,1]%%%}+%%%{-37778931862957161709568,[1,9,40,21]%%%}+%%%{69175290276410818560,[1,9,40,17]%%%}+%%%{-9007199254
740992,[1,9,39,13]%%%}+%%%{-824633720832,[1,9,38,9]%%%}+%%%{8192,[1,9,37,1]%%%}+%%%{-37778931862957161709568,[
1,8,41,21]%%%}+%%%{18446744073709551616,[1,8,40,17]%%%}+%%%{-549755813888,[1,8,39,9]%%%}+%%%{67108864,[1,8,38,
5]%%%}+%%%{9223372036854775808,[1,7,41,17]%%%}+%%%{-2251799813685248,[1,7,40,13]%%%} / %%%{-134217728,[0,17,10
,9]%%%}+%%%{402653184,[0,16,11,9]%%%}+%%%{65536,[0,16,10,5]%%%}+%%%{-536870912,[0,15,12,9]%%%}+%%%{-805306368,
[0,15,11,9]%%%}+%%%{-163840,[0,15,11,5]%%%}+%%%{-8,[0,15,10,1]%%%}+%%%{536870912,[0,14,13,9]%%%}+%%%{241591910
4,[0,14,12,9]%%%}+%%%{65536,[0,14,12,5]%%%}+%%%{393216,[0,14,11,5]%%%}+%%%{24,[0,14,11,1]%%%}+%%%{-3221225472,
[0,13,13,9]%%%}+%%%{-1610612736,[0,13,12,9]%%%}+%%%{-1048576,[0,13,12,5]%%%}+%%%{-16,[0,13,12,1]%%%}+%%%{-48,[
0,13,11,1]%%%}+%%%{4294967296,[0,12,14,9]%%%}+%%%{4831838208,[0,12,13,9]%%%}+%%%{327680,[0,12,13,5]%%%}+%%%{78
6432,[0,12,12,5]%%%}+%%%{160,[0,12,12,1]%%%}+%%%{-5905580032,[0,11,14,9]%%%}+%%%{-1073741824,[0,11,13,9]%%%}+%
%%{-2359296,[0,11,13,5]%%%}+%%%{-112,[0,11,13,1]%%%}+%%%{-96,[0,11,12,1]%%%}+%%%{13421772800,[0,10,15,9]%%%}+%
%%{3221225472,[0,10,14,9]%%%}+%%%{262144,[0,10,14,5]%%%}+%%%{524288,[0,10,13,5]%%%}+%%%{384,[0,10,13,1]%%%}+%%
%{-1073741824,[0,9,15,9]%%%}+%%%{-2097152,[0,9,14,5]%%%}+%%%{-288,[0,9,14,1]%%%}+%%%{-64,[0,9,13,1]%%%}+%%%{20
401094656,[0,8,16,9]%%%}+%%%{-1048576,[0,8,15,5]%%%}+%%%{384,[0,8,14,1]%%%}+%%%{6442450944,[0,7,16,9]%%%}+%%%{
-524288,[0,7,15,5]%%%}+%%%{-320,[0,7,15,1]%%%}+%%%{15032385536,[0,6,17,9]%%%}+%%%{-2097152,[0,6,16,5]%%%}+%%%{
128,[0,6,15,1]%%%}+%%%{4294967296,[0,5,17,9]%%%}+%%%{-128,[0,5,16,1]%%%}+%%%{4294967296,[0,4,18,9]%%%}+%%%{-10
48576,[0,4,17,5]%%%} Error: Bad Argument Value
________________________________________________________________________________________
maple [F] time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{4}-b}{\left (a \,x^{4}-b \,x^{2}\right )^{\frac {1}{4}} \left (2 x^{8}+2 a \,x^{4}-b \right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a*x^4-b)/(a*x^4-b*x^2)^(1/4)/(2*x^8+2*a*x^4-b),x)
[Out]
int((a*x^4-b)/(a*x^4-b*x^2)^(1/4)/(2*x^8+2*a*x^4-b),x)
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{4} - b}{{\left (2 \, x^{8} + 2 \, a x^{4} - 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^4-b)/(a*x^4-b*x^2)^(1/4)/(2*x^8+2*a*x^4-b),x, algorithm="maxima")
[Out]
integrate((a*x^4 - b)/((2*x^8 + 2*a*x^4 - b)*(a*x^4 - b*x^2)^(1/4)), x)
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {b-a\,x^4}{{\left (a\,x^4-b\,x^2\right )}^{1/4}\,\left (2\,x^8+2\,a\,x^4-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)*(2*a*x^4 - b + 2*x^8)),x)
[Out]
int(-(b - a*x^4)/((a*x^4 - b*x^2)^(1/4)*(2*a*x^4 - b + 2*x^8)), x)
________________________________________________________________________________________
sympy [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)/(a*x**4-b*x**2)**(1/4)/(2*x**8+2*a*x**4-b),x)
[Out]
Timed out
________________________________________________________________________________________