\(\int \frac {b-2 a x^4+2 x^8}{\sqrt [4]{b+a x^4} (-2 b-a x^4+x^8)} \, dx\) [2079]

Optimal result

Integrand size = 40, antiderivative size = 150 \[ \int \frac {b-2 a x^4+2 x^8}{\sqrt [4]{b+a x^4} \left (-2 b-a x^4+x^8\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}-\frac {5}{4} \text {RootSum}\left [a^2-b-3 a \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {a \log (x)-a \log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}^4+\log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{3 a \text {$\#$1}-4 \text {$\#$1}^5}\&\right ] \]

[Out]

Unintegrable

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(503\) vs. \(2(150)=300\).

Time = 0.70 (sec) , antiderivative size = 503, normalized size of antiderivative = 3.35, number of steps used = 15, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.225, Rules used = {6860, 246, 218, 212, 209, 1442, 385, 214, 211} \[ \int \frac {b-2 a x^4+2 x^8}{\sqrt [4]{b+a x^4} \left (-2 b-a x^4+x^8\right )} \, dx=\frac {5 b \arctan \left (\frac {x \sqrt [4]{-a \sqrt {a^2+8 b}+a^2+2 b}}{\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt [4]{a x^4+b}}\right )}{\sqrt {a^2+8 b} \left (a-\sqrt {a^2+8 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+8 b}+a^2+2 b}}-\frac {5 b \arctan \left (\frac {x \sqrt [4]{a \sqrt {a^2+8 b}+a^2+2 b}}{\sqrt [4]{\sqrt {a^2+8 b}+a} \sqrt [4]{a x^4+b}}\right )}{\sqrt {a^2+8 b} \left (\sqrt {a^2+8 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+8 b}+a^2+2 b}}+\frac {5 b \text {arctanh}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+8 b}+a^2+2 b}}{\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt [4]{a x^4+b}}\right )}{\sqrt {a^2+8 b} \left (a-\sqrt {a^2+8 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+8 b}+a^2+2 b}}-\frac {5 b \text {arctanh}\left (\frac {x \sqrt [4]{a \sqrt {a^2+8 b}+a^2+2 b}}{\sqrt [4]{\sqrt {a^2+8 b}+a} \sqrt [4]{a x^4+b}}\right )}{\sqrt {a^2+8 b} \left (\sqrt {a^2+8 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+8 b}+a^2+2 b}}+\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{\sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{\sqrt [4]{a}} \]

[In]

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

[Out]

ArcTan[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/a^(1/4) + (5*b*ArcTan[((a^2 + 2*b - a*Sqrt[a^2 + 8*b])^(1/4)*x)/((a - Sq
rt[a^2 + 8*b])^(1/4)*(b + a*x^4)^(1/4))])/(Sqrt[a^2 + 8*b]*(a - Sqrt[a^2 + 8*b])^(3/4)*(a^2 + 2*b - a*Sqrt[a^2
+ 8*b])^(1/4)) - (5*b*ArcTan[((a^2 + 2*b + a*Sqrt[a^2 + 8*b])^(1/4)*x)/((a + Sqrt[a^2 + 8*b])^(1/4)*(b + a*x^
4)^(1/4))])/(Sqrt[a^2 + 8*b]*(a + Sqrt[a^2 + 8*b])^(3/4)*(a^2 + 2*b + a*Sqrt[a^2 + 8*b])^(1/4)) + ArcTanh[(a^(
1/4)*x)/(b + a*x^4)^(1/4)]/a^(1/4) + (5*b*ArcTanh[((a^2 + 2*b - a*Sqrt[a^2 + 8*b])^(1/4)*x)/((a - Sqrt[a^2 + 8
*b])^(1/4)*(b + a*x^4)^(1/4))])/(Sqrt[a^2 + 8*b]*(a - Sqrt[a^2 + 8*b])^(3/4)*(a^2 + 2*b - a*Sqrt[a^2 + 8*b])^(
1/4)) - (5*b*ArcTanh[((a^2 + 2*b + a*Sqrt[a^2 + 8*b])^(1/4)*x)/((a + Sqrt[a^2 + 8*b])^(1/4)*(b + a*x^4)^(1/4))
])/(Sqrt[a^2 + 8*b]*(a + Sqrt[a^2 + 8*b])^(3/4)*(a^2 + 2*b + a*Sqrt[a^2 + 8*b])^(1/4))

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 211

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 214

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

Rule 218

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

Rule 246

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 385

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

Rule 1442

Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{r = Rt[b^2 -
4*a*c, 2]}, Dist[2*(c/r), Int[(d + e*x^n)^q/(b - r + 2*c*x^n), x], x] - Dist[2*(c/r), Int[(d + e*x^n)^q/(b +
r + 2*c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] &&  !IntegerQ[q]

Rule 6860

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*} \text {integral}& = \int \left (\frac {2}{\sqrt [4]{b+a x^4}}+\frac {5 b}{\sqrt [4]{b+a x^4} \left (-2 b-a x^4+x^8\right )}\right ) \, dx \\ & = 2 \int \frac {1}{\sqrt [4]{b+a x^4}} \, dx+(5 b) \int \frac {1}{\sqrt [4]{b+a x^4} \left (-2 b-a x^4+x^8\right )} \, dx \\ & = 2 \text {Subst}\left (\int \frac {1}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )+\frac {(10 b) \int \frac {1}{\left (-a-\sqrt {a^2+8 b}+2 x^4\right ) \sqrt [4]{b+a x^4}} \, dx}{\sqrt {a^2+8 b}}-\frac {(10 b) \int \frac {1}{\left (-a+\sqrt {a^2+8 b}+2 x^4\right ) \sqrt [4]{b+a x^4}} \, dx}{\sqrt {a^2+8 b}} \\ & = \frac {(10 b) \text {Subst}\left (\int \frac {1}{-a-\sqrt {a^2+8 b}-\left (-2 b+a \left (-a-\sqrt {a^2+8 b}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt {a^2+8 b}}-\frac {(10 b) \text {Subst}\left (\int \frac {1}{-a+\sqrt {a^2+8 b}-\left (-2 b+a \left (-a+\sqrt {a^2+8 b}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt {a^2+8 b}}+\text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )+\text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right ) \\ & = \frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {(5 b) \text {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+8 b}}-\sqrt {a^2+2 b-a \sqrt {a^2+8 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt {a^2+8 b} \sqrt {a-\sqrt {a^2+8 b}}}+\frac {(5 b) \text {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+8 b}}+\sqrt {a^2+2 b-a \sqrt {a^2+8 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt {a^2+8 b} \sqrt {a-\sqrt {a^2+8 b}}}-\frac {(5 b) \text {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+8 b}}-\sqrt {a^2+2 b+a \sqrt {a^2+8 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt {a^2+8 b} \sqrt {a+\sqrt {a^2+8 b}}}-\frac {(5 b) \text {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+8 b}}+\sqrt {a^2+2 b+a \sqrt {a^2+8 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt {a^2+8 b} \sqrt {a+\sqrt {a^2+8 b}}} \\ & = \frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {5 b \arctan \left (\frac {\sqrt [4]{a^2+2 b-a \sqrt {a^2+8 b}} x}{\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt [4]{b+a x^4}}\right )}{\sqrt {a^2+8 b} \left (a-\sqrt {a^2+8 b}\right )^{3/4} \sqrt [4]{a^2+2 b-a \sqrt {a^2+8 b}}}-\frac {5 b \arctan \left (\frac {\sqrt [4]{a^2+2 b+a \sqrt {a^2+8 b}} x}{\sqrt [4]{a+\sqrt {a^2+8 b}} \sqrt [4]{b+a x^4}}\right )}{\sqrt {a^2+8 b} \left (a+\sqrt {a^2+8 b}\right )^{3/4} \sqrt [4]{a^2+2 b+a \sqrt {a^2+8 b}}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {5 b \text {arctanh}\left (\frac {\sqrt [4]{a^2+2 b-a \sqrt {a^2+8 b}} x}{\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt [4]{b+a x^4}}\right )}{\sqrt {a^2+8 b} \left (a-\sqrt {a^2+8 b}\right )^{3/4} \sqrt [4]{a^2+2 b-a \sqrt {a^2+8 b}}}-\frac {5 b \text {arctanh}\left (\frac {\sqrt [4]{a^2+2 b+a \sqrt {a^2+8 b}} x}{\sqrt [4]{a+\sqrt {a^2+8 b}} \sqrt [4]{b+a x^4}}\right )}{\sqrt {a^2+8 b} \left (a+\sqrt {a^2+8 b}\right )^{3/4} \sqrt [4]{a^2+2 b+a \sqrt {a^2+8 b}}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.04 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.97 \[ \int \frac {b-2 a x^4+2 x^8}{\sqrt [4]{b+a x^4} \left (-2 b-a x^4+x^8\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}-\frac {5}{4} \text {RootSum}\left [a^2-b-3 a \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-a \log (x)+a \log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-3 a \text {$\#$1}+4 \text {$\#$1}^5}\&\right ] \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Time = 0.38 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.90

method	result	size
pseudoelliptic	\(\frac {5 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{8}-3 a \,\textit {\_Z}^{4}+a^{2}-b \right )}{\sum }\frac {\left (\textit {\_R}^{4}-a \right ) \ln \left (\frac {-\textit {\_R} x +\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R} \left (4 \textit {\_R}^{4}-3 a \right )}\right ) a^{\frac {1}{4}}-4 \arctan \left (\frac {\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )+2 \ln \left (\frac {-a^{\frac {1}{4}} x -\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (a \,x^{4}+b \right )^{\frac {1}{4}}}\right )}{4 a^{\frac {1}{4}}}\)	\(135\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.67 (sec) , antiderivative size = 4684, normalized size of antiderivative = 31.23 \[ \int \frac {b-2 a x^4+2 x^8}{\sqrt [4]{b+a x^4} \left (-2 b-a x^4+x^8\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-5/8*sqrt(-sqrt((a^5 + 9*a^3*b + 14*a*b^2 + (a^6 + 15*a^4*b + 48*a^2*b^2 - 64*b^3)*sqrt((a^8 + 10*a^6*b + 29*a
^4*b^2 + 20*a^2*b^3 + 4*b^4)/(a^10 + 22*a^8*b + 145*a^6*b^2 + 152*a^4*b^3 - 832*a^2*b^4 + 512*b^5)))/(a^6 + 15
*a^4*b + 48*a^2*b^2 - 64*b^3)))*log(78125/16*(((a^12 + 30*a^10*b + 333*a^8*b^2 + 1588*a^6*b^3 + 2400*a^4*b^4 -
2304*a^2*b^5 - 2048*b^6)*x*sqrt((a^8 + 10*a^6*b + 29*a^4*b^2 + 20*a^2*b^3 + 4*b^4)/(a^10 + 22*a^8*b + 145*a^6
*b^2 + 152*a^4*b^3 - 832*a^2*b^4 + 512*b^5)) - (a^11 + 24*a^9*b + 209*a^7*b^2 + 790*a^5*b^3 + 1184*a^3*b^4 + 3
84*a*b^5)*x)*sqrt(-sqrt((a^5 + 9*a^3*b + 14*a*b^2 + (a^6 + 15*a^4*b + 48*a^2*b^2 - 64*b^3)*sqrt((a^8 + 10*a^6*
b + 29*a^4*b^2 + 20*a^2*b^3 + 4*b^4)/(a^10 + 22*a^8*b + 145*a^6*b^2 + 152*a^4*b^3 - 832*a^2*b^4 + 512*b^5)))/(
a^6 + 15*a^4*b + 48*a^2*b^2 - 64*b^3)))*sqrt((a^5 + 9*a^3*b + 14*a*b^2 + (a^6 + 15*a^4*b + 48*a^2*b^2 - 64*b^3
)*sqrt((a^8 + 10*a^6*b + 29*a^4*b^2 + 20*a^2*b^3 + 4*b^4)/(a^10 + 22*a^8*b + 145*a^6*b^2 + 152*a^4*b^3 - 832*a
^2*b^4 + 512*b^5)))/(a^6 + 15*a^4*b + 48*a^2*b^2 - 64*b^3)) + 16*(a^4*b^3 + 5*a^2*b^4 + 2*b^5)*(a*x^4 + b)^(1/
4))/x) + 5/8*sqrt(-sqrt((a^5 + 9*a^3*b + 14*a*b^2 + (a^6 + 15*a^4*b + 48*a^2*b^2 - 64*b^3)*sqrt((a^8 + 10*a^6*
b + 29*a^4*b^2 + 20*a^2*b^3 + 4*b^4)/(a^10 + 22*a^8*b + 145*a^6*b^2 + 152*a^4*b^3 - 832*a^2*b^4 + 512*b^5)))/(
a^6 + 15*a^4*b + 48*a^2*b^2 - 64*b^3)))*log(-78125/16*(((a^12 + 30*a^10*b + 333*a^8*b^2 + 1588*a^6*b^3 + 2400*
a^4*b^4 - 2304*a^2*b^5 - 2048*b^6)*x*sqrt((a^8 + 10*a^6*b + 29*a^4*b^2 + 20*a^2*b^3 + 4*b^4)/(a^10 + 22*a^8*b
+ 145*a^6*b^2 + 152*a^4*b^3 - 832*a^2*b^4 + 512*b^5)) - (a^11 + 24*a^9*b + 209*a^7*b^2 + 790*a^5*b^3 + 1184*a^
3*b^4 + 384*a*b^5)*x)*sqrt(-sqrt((a^5 + 9*a^3*b + 14*a*b^2 + (a^6 + 15*a^4*b + 48*a^2*b^2 - 64*b^3)*sqrt((a^8
+ 10*a^6*b + 29*a^4*b^2 + 20*a^2*b^3 + 4*b^4)/(a^10 + 22*a^8*b + 145*a^6*b^2 + 152*a^4*b^3 - 832*a^2*b^4 + 512
*b^5)))/(a^6 + 15*a^4*b + 48*a^2*b^2 - 64*b^3)))*sqrt((a^5 + 9*a^3*b + 14*a*b^2 + (a^6 + 15*a^4*b + 48*a^2*b^2
- 64*b^3)*sqrt((a^8 + 10*a^6*b + 29*a^4*b^2 + 20*a^2*b^3 + 4*b^4)/(a^10 + 22*a^8*b + 145*a^6*b^2 + 152*a^4*b^
3 - 832*a^2*b^4 + 512*b^5)))/(a^6 + 15*a^4*b + 48*a^2*b^2 - 64*b^3)) - 16*(a^4*b^3 + 5*a^2*b^4 + 2*b^5)*(a*x^4
+ b)^(1/4))/x) + 5/8*sqrt(-sqrt((a^5 + 9*a^3*b + 14*a*b^2 - (a^6 + 15*a^4*b + 48*a^2*b^2 - 64*b^3)*sqrt((a^8
+ 10*a^6*b + 29*a^4*b^2 + 20*a^2*b^3 + 4*b^4)/(a^10 + 22*a^8*b + 145*a^6*b^2 + 152*a^4*b^3 - 832*a^2*b^4 + 512
*b^5)))/(a^6 + 15*a^4*b + 48*a^2*b^2 - 64*b^3)))*log(78125/16*(((a^12 + 30*a^10*b + 333*a^8*b^2 + 1588*a^6*b^3
+ 2400*a^4*b^4 - 2304*a^2*b^5 - 2048*b^6)*x*sqrt((a^8 + 10*a^6*b + 29*a^4*b^2 + 20*a^2*b^3 + 4*b^4)/(a^10 + 2
2*a^8*b + 145*a^6*b^2 + 152*a^4*b^3 - 832*a^2*b^4 + 512*b^5)) + (a^11 + 24*a^9*b + 209*a^7*b^2 + 790*a^5*b^3 +
1184*a^3*b^4 + 384*a*b^5)*x)*sqrt(-sqrt((a^5 + 9*a^3*b + 14*a*b^2 - (a^6 + 15*a^4*b + 48*a^2*b^2 - 64*b^3)*sq
rt((a^8 + 10*a^6*b + 29*a^4*b^2 + 20*a^2*b^3 + 4*b^4)/(a^10 + 22*a^8*b + 145*a^6*b^2 + 152*a^4*b^3 - 832*a^2*b
^4 + 512*b^5)))/(a^6 + 15*a^4*b + 48*a^2*b^2 - 64*b^3)))*sqrt((a^5 + 9*a^3*b + 14*a*b^2 - (a^6 + 15*a^4*b + 48
*a^2*b^2 - 64*b^3)*sqrt((a^8 + 10*a^6*b + 29*a^4*b^2 + 20*a^2*b^3 + 4*b^4)/(a^10 + 22*a^8*b + 145*a^6*b^2 + 15
2*a^4*b^3 - 832*a^2*b^4 + 512*b^5)))/(a^6 + 15*a^4*b + 48*a^2*b^2 - 64*b^3)) + 16*(a^4*b^3 + 5*a^2*b^4 + 2*b^5
)*(a*x^4 + b)^(1/4))/x) - 5/8*sqrt(-sqrt((a^5 + 9*a^3*b + 14*a*b^2 - (a^6 + 15*a^4*b + 48*a^2*b^2 - 64*b^3)*sq
rt((a^8 + 10*a^6*b + 29*a^4*b^2 + 20*a^2*b^3 + 4*b^4)/(a^10 + 22*a^8*b + 145*a^6*b^2 + 152*a^4*b^3 - 832*a^2*b
^4 + 512*b^5)))/(a^6 + 15*a^4*b + 48*a^2*b^2 - 64*b^3)))*log(-78125/16*(((a^12 + 30*a^10*b + 333*a^8*b^2 + 158
8*a^6*b^3 + 2400*a^4*b^4 - 2304*a^2*b^5 - 2048*b^6)*x*sqrt((a^8 + 10*a^6*b + 29*a^4*b^2 + 20*a^2*b^3 + 4*b^4)/
(a^10 + 22*a^8*b + 145*a^6*b^2 + 152*a^4*b^3 - 832*a^2*b^4 + 512*b^5)) + (a^11 + 24*a^9*b + 209*a^7*b^2 + 790*
a^5*b^3 + 1184*a^3*b^4 + 384*a*b^5)*x)*sqrt(-sqrt((a^5 + 9*a^3*b + 14*a*b^2 - (a^6 + 15*a^4*b + 48*a^2*b^2 - 6
4*b^3)*sqrt((a^8 + 10*a^6*b + 29*a^4*b^2 + 20*a^2*b^3 + 4*b^4)/(a^10 + 22*a^8*b + 145*a^6*b^2 + 152*a^4*b^3 -
832*a^2*b^4 + 512*b^5)))/(a^6 + 15*a^4*b + 48*a^2*b^2 - 64*b^3)))*sqrt((a^5 + 9*a^3*b + 14*a*b^2 - (a^6 + 15*a
^4*b + 48*a^2*b^2 - 64*b^3)*sqrt((a^8 + 10*a^6*b + 29*a^4*b^2 + 20*a^2*b^3 + 4*b^4)/(a^10 + 22*a^8*b + 145*a^6
*b^2 + 152*a^4*b^3 - 832*a^2*b^4 + 512*b^5)))/(a^6 + 15*a^4*b + 48*a^2*b^2 - 64*b^3)) - 16*(a^4*b^3 + 5*a^2*b^
4 + 2*b^5)*(a*x^4 + b)^(1/4))/x) + 5/8*((a^5 + 9*a^3*b + 14*a*b^2 + (a^6 + 15*a^4*b + 48*a^2*b^2 - 64*b^3)*sqr
t((a^8 + 10*a^6*b + 29*a^4*b^2 + 20*a^2*b^3 + 4*b^4)/(a^10 + 22*a^8*b + 145*a^6*b^2 + 152*a^4*b^3 - 832*a^2*b^
4 + 512*b^5)))/(a^6 + 15*a^4*b + 48*a^2*b^2 - 64*b^3))^(1/4)*log(78125/16*(((a^12 + 30*a^10*b + 333*a^8*b^2 +
1588*a^6*b^3 + 2400*a^4*b^4 - 2304*a^2*b^5 - 2048*b^6)*x*sqrt((a^8 + 10*a^6*b + 29*a^4*b^2 + 20*a^2*b^3 + 4*b^
4)/(a^10 + 22*a^8*b + 145*a^6*b^2 + 152*a^4*b^3 - 832*a^2*b^4 + 512*b^5)) - (a^11 + 24*a^9*b + 209*a^7*b^2 + 7
90*a^5*b^3 + 1184*a^3*b^4 + 384*a*b^5)*x)*((a^5 + 9*a^3*b + 14*a*b^2 + (a^6 + 15*a^4*b + 48*a^2*b^2 - 64*b^3)*
sqrt((a^8 + 10*a^6*b + 29*a^4*b^2 + 20*a^2*b^3 + 4*b^4)/(a^10 + 22*a^8*b + 145*a^6*b^2 + 152*a^4*b^3 - 832*a^2
*b^4 + 512*b^5)))/(a^6 + 15*a^4*b + 48*a^2*b^2 - 64*b^3))^(3/4) + 16*(a^4*b^3 + 5*a^2*b^4 + 2*b^5)*(a*x^4 + b)
^(1/4))/x) - 5/8*((a^5 + 9*a^3*b + 14*a*b^2 + (a^6 + 15*a^4*b + 48*a^2*b^2 - 64*b^3)*sqrt((a^8 + 10*a^6*b + 29
*a^4*b^2 + 20*a^2*b^3 + 4*b^4)/(a^10 + 22*a^8*b + 145*a^6*b^2 + 152*a^4*b^3 - 832*a^2*b^4 + 512*b^5)))/(a^6 +
15*a^4*b + 48*a^2*b^2 - 64*b^3))^(1/4)*log(-78125/16*(((a^12 + 30*a^10*b + 333*a^8*b^2 + 1588*a^6*b^3 + 2400*a
^4*b^4 - 2304*a^2*b^5 - 2048*b^6)*x*sqrt((a^8 + 10*a^6*b + 29*a^4*b^2 + 20*a^2*b^3 + 4*b^4)/(a^10 + 22*a^8*b +
145*a^6*b^2 + 152*a^4*b^3 - 832*a^2*b^4 + 512*b^5)) - (a^11 + 24*a^9*b + 209*a^7*b^2 + 790*a^5*b^3 + 1184*a^3
*b^4 + 384*a*b^5)*x)*((a^5 + 9*a^3*b + 14*a*b^2 + (a^6 + 15*a^4*b + 48*a^2*b^2 - 64*b^3)*sqrt((a^8 + 10*a^6*b
+ 29*a^4*b^2 + 20*a^2*b^3 + 4*b^4)/(a^10 + 22*a^8*b + 145*a^6*b^2 + 152*a^4*b^3 - 832*a^2*b^4 + 512*b^5)))/(a^
6 + 15*a^4*b + 48*a^2*b^2 - 64*b^3))^(3/4) - 16*(a^4*b^3 + 5*a^2*b^4 + 2*b^5)*(a*x^4 + b)^(1/4))/x) - 5/8*((a^
5 + 9*a^3*b + 14*a*b^2 - (a^6 + 15*a^4*b + 48*a^2*b^2 - 64*b^3)*sqrt((a^8 + 10*a^6*b + 29*a^4*b^2 + 20*a^2*b^3
+ 4*b^4)/(a^10 + 22*a^8*b + 145*a^6*b^2 + 152*a^4*b^3 - 832*a^2*b^4 + 512*b^5)))/(a^6 + 15*a^4*b + 48*a^2*b^2
- 64*b^3))^(1/4)*log(78125/16*(((a^12 + 30*a^10*b + 333*a^8*b^2 + 1588*a^6*b^3 + 2400*a^4*b^4 - 2304*a^2*b^5
- 2048*b^6)*x*sqrt((a^8 + 10*a^6*b + 29*a^4*b^2 + 20*a^2*b^3 + 4*b^4)/(a^10 + 22*a^8*b + 145*a^6*b^2 + 152*a^4
*b^3 - 832*a^2*b^4 + 512*b^5)) + (a^11 + 24*a^9*b + 209*a^7*b^2 + 790*a^5*b^3 + 1184*a^3*b^4 + 384*a*b^5)*x)*(
(a^5 + 9*a^3*b + 14*a*b^2 - (a^6 + 15*a^4*b + 48*a^2*b^2 - 64*b^3)*sqrt((a^8 + 10*a^6*b + 29*a^4*b^2 + 20*a^2*
b^3 + 4*b^4)/(a^10 + 22*a^8*b + 145*a^6*b^2 + 152*a^4*b^3 - 832*a^2*b^4 + 512*b^5)))/(a^6 + 15*a^4*b + 48*a^2*
b^2 - 64*b^3))^(3/4) + 16*(a^4*b^3 + 5*a^2*b^4 + 2*b^5)*(a*x^4 + b)^(1/4))/x) + 5/8*((a^5 + 9*a^3*b + 14*a*b^2
- (a^6 + 15*a^4*b + 48*a^2*b^2 - 64*b^3)*sqrt((a^8 + 10*a^6*b + 29*a^4*b^2 + 20*a^2*b^3 + 4*b^4)/(a^10 + 22*a
^8*b + 145*a^6*b^2 + 152*a^4*b^3 - 832*a^2*b^4 + 512*b^5)))/(a^6 + 15*a^4*b + 48*a^2*b^2 - 64*b^3))^(1/4)*log(
-78125/16*(((a^12 + 30*a^10*b + 333*a^8*b^2 + 1588*a^6*b^3 + 2400*a^4*b^4 - 2304*a^2*b^5 - 2048*b^6)*x*sqrt((a
^8 + 10*a^6*b + 29*a^4*b^2 + 20*a^2*b^3 + 4*b^4)/(a^10 + 22*a^8*b + 145*a^6*b^2 + 152*a^4*b^3 - 832*a^2*b^4 +
512*b^5)) + (a^11 + 24*a^9*b + 209*a^7*b^2 + 790*a^5*b^3 + 1184*a^3*b^4 + 384*a*b^5)*x)*((a^5 + 9*a^3*b + 14*a
*b^2 - (a^6 + 15*a^4*b + 48*a^2*b^2 - 64*b^3)*sqrt((a^8 + 10*a^6*b + 29*a^4*b^2 + 20*a^2*b^3 + 4*b^4)/(a^10 +
22*a^8*b + 145*a^6*b^2 + 152*a^4*b^3 - 832*a^2*b^4 + 512*b^5)))/(a^6 + 15*a^4*b + 48*a^2*b^2 - 64*b^3))^(3/4)
- 16*(a^4*b^3 + 5*a^2*b^4 + 2*b^5)*(a*x^4 + b)^(1/4))/x) + 1/2*log((a^(1/4)*x + (a*x^4 + b)^(1/4))/x)/a^(1/4)
- 1/2*log(-(a^(1/4)*x - (a*x^4 + b)^(1/4))/x)/a^(1/4) - 1/2*I*log((I*a^(1/4)*x + (a*x^4 + b)^(1/4))/x)/a^(1/4)
+ 1/2*I*log((-I*a^(1/4)*x + (a*x^4 + b)^(1/4))/x)/a^(1/4)

Sympy [N/A]

Not integrable

Time = 118.19 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.24 \[ \int \frac {b-2 a x^4+2 x^8}{\sqrt [4]{b+a x^4} \left (-2 b-a x^4+x^8\right )} \, dx=\int \frac {- 2 a x^{4} + b + 2 x^{8}}{\sqrt [4]{a x^{4} + b} \left (- a x^{4} - 2 b + x^{8}\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.27 \[ \int \frac {b-2 a x^4+2 x^8}{\sqrt [4]{b+a x^4} \left (-2 b-a x^4+x^8\right )} \, dx=\int { \frac {2 \, x^{8} - 2 \, a x^{4} + b}{{\left (x^{8} - a x^{4} - 2 \, b\right )} {\left (a x^{4} + b\right )}^{\frac {1}{4}}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.94 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.27 \[ \int \frac {b-2 a x^4+2 x^8}{\sqrt [4]{b+a x^4} \left (-2 b-a x^4+x^8\right )} \, dx=\int { \frac {2 \, x^{8} - 2 \, a x^{4} + b}{{\left (x^{8} - a x^{4} - 2 \, b\right )} {\left (a x^{4} + b\right )}^{\frac {1}{4}}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 6.64 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.28 \[ \int \frac {b-2 a x^4+2 x^8}{\sqrt [4]{b+a x^4} \left (-2 b-a x^4+x^8\right )} \, dx=\int -\frac {2\,x^8-2\,a\,x^4+b}{{\left (a\,x^4+b\right )}^{1/4}\,\left (-x^8+a\,x^4+2\,b\right )} \,d x \]

[In]

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

[Out]

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