\(\int \frac {-2 b+a x^4}{\sqrt [4]{-b+a x^4} (-b+a x^4+2 x^8)} \, dx\) [685]

Optimal result

Integrand size = 39, antiderivative size = 53 \[ \int \frac {-2 b+a x^4}{\sqrt [4]{-b+a x^4} \left (-b+a x^4+2 x^8\right )} \, dx=-\frac {1}{4} \text {RootSum}\left [2 b+a \text {$\#$1}^4-\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]

[Out]

Unintegrable

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(421\) vs. \(2(53)=106\).

Time = 0.67 (sec) , antiderivative size = 421, normalized size of antiderivative = 7.94, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {6860, 385, 218, 214, 211} \[ \int \frac {-2 b+a x^4}{\sqrt [4]{-b+a x^4} \left (-b+a x^4+2 x^8\right )} \, dx=\frac {\sqrt [4]{a-\sqrt {a^2+8 b}} \arctan \left (\frac {x \sqrt [4]{-a \sqrt {a^2+8 b}+a^2+4 b}}{\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{-a \sqrt {a^2+8 b}+a^2+4 b}}+\frac {\sqrt [4]{\sqrt {a^2+8 b}+a} \arctan \left (\frac {x \sqrt [4]{a \sqrt {a^2+8 b}+a^2+4 b}}{\sqrt [4]{\sqrt {a^2+8 b}+a} \sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a \sqrt {a^2+8 b}+a^2+4 b}}+\frac {\sqrt [4]{a-\sqrt {a^2+8 b}} \text {arctanh}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+8 b}+a^2+4 b}}{\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{-a \sqrt {a^2+8 b}+a^2+4 b}}+\frac {\sqrt [4]{\sqrt {a^2+8 b}+a} \text {arctanh}\left (\frac {x \sqrt [4]{a \sqrt {a^2+8 b}+a^2+4 b}}{\sqrt [4]{\sqrt {a^2+8 b}+a} \sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a \sqrt {a^2+8 b}+a^2+4 b}} \]

[In]

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

[Out]

((a - Sqrt[a^2 + 8*b])^(1/4)*ArcTan[((a^2 + 4*b - a*Sqrt[a^2 + 8*b])^(1/4)*x)/((a - Sqrt[a^2 + 8*b])^(1/4)*(-b
+ a*x^4)^(1/4))])/(2*(a^2 + 4*b - a*Sqrt[a^2 + 8*b])^(1/4)) + ((a + Sqrt[a^2 + 8*b])^(1/4)*ArcTan[((a^2 + 4*b
+ a*Sqrt[a^2 + 8*b])^(1/4)*x)/((a + Sqrt[a^2 + 8*b])^(1/4)*(-b + a*x^4)^(1/4))])/(2*(a^2 + 4*b + a*Sqrt[a^2 +
8*b])^(1/4)) + ((a - Sqrt[a^2 + 8*b])^(1/4)*ArcTanh[((a^2 + 4*b - a*Sqrt[a^2 + 8*b])^(1/4)*x)/((a - Sqrt[a^2
+ 8*b])^(1/4)*(-b + a*x^4)^(1/4))])/(2*(a^2 + 4*b - a*Sqrt[a^2 + 8*b])^(1/4)) + ((a + Sqrt[a^2 + 8*b])^(1/4)*A
rcTanh[((a^2 + 4*b + a*Sqrt[a^2 + 8*b])^(1/4)*x)/((a + Sqrt[a^2 + 8*b])^(1/4)*(-b + a*x^4)^(1/4))])/(2*(a^2 +
4*b + a*Sqrt[a^2 + 8*b])^(1/4))

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

Mathematica [A] (verified)

Time = 0.70 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00 \[ \int \frac {-2 b+a x^4}{\sqrt [4]{-b+a x^4} \left (-b+a x^4+2 x^8\right )} \, dx=-\frac {1}{4} \text {RootSum}\left [2 b+a \text {$\#$1}^4-\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Time = 1.44 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.85

method	result	size
pseudoelliptic	\(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-a \,\textit {\_Z}^{4}-2 b \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (a \,x^{4}-b \right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}}\right )}{4}\)	\(45\)

[In]

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

[Out]

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

Fricas [F(-1)]

Timed out. \[ \int \frac {-2 b+a x^4}{\sqrt [4]{-b+a x^4} \left (-b+a x^4+2 x^8\right )} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [N/A]

Not integrable

Time = 65.04 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.58 \[ \int \frac {-2 b+a x^4}{\sqrt [4]{-b+a x^4} \left (-b+a x^4+2 x^8\right )} \, dx=\int \frac {a x^{4} - 2 b}{\sqrt [4]{a x^{4} - b} \left (a x^{4} - b + 2 x^{8}\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.74 \[ \int \frac {-2 b+a x^4}{\sqrt [4]{-b+a x^4} \left (-b+a x^4+2 x^8\right )} \, dx=\int { \frac {a x^{4} - 2 \, b}{{\left (2 \, x^{8} + a x^{4} - b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 1.64 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.74 \[ \int \frac {-2 b+a x^4}{\sqrt [4]{-b+a x^4} \left (-b+a x^4+2 x^8\right )} \, dx=\int { \frac {a x^{4} - 2 \, b}{{\left (2 \, x^{8} + a x^{4} - b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 5.72 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.77 \[ \int \frac {-2 b+a x^4}{\sqrt [4]{-b+a x^4} \left (-b+a x^4+2 x^8\right )} \, dx=\int -\frac {2\,b-a\,x^4}{{\left (a\,x^4-b\right )}^{1/4}\,\left (2\,x^8+a\,x^4-b\right )} \,d x \]

[In]

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

[Out]

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