\(\int \frac {-b+2 a x^4}{\sqrt [4]{-b+a x^4} (-b-a x^4+x^8)} \, dx\) [1467]

Optimal result

Integrand size = 39, antiderivative size = 103 \[ \int \frac {-b+2 a x^4}{\sqrt [4]{-b+a x^4} \left (-b-a x^4+x^8\right )} \, dx=\frac {1}{4} \text {RootSum}\left [2 a^2-b-3 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-a \log (x)+a \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}^4+\log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{3 a \text {$\#$1}-2 \text {$\#$1}^5}\&\right ] \]

[Out]

Unintegrable

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(499\) vs. \(2(103)=206\).

Time = 0.62 (sec) , antiderivative size = 499, normalized size of antiderivative = 4.84, 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} \[ \int \frac {-b+2 a x^4}{\sqrt [4]{-b+a x^4} \left (-b-a x^4+x^8\right )} \, dx=-\frac {\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \arctan \left (\frac {x \sqrt [4]{-a \sqrt {a^2+4 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{a x^4-b}}\right )}{\left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+4 b}+a^2-2 b}}-\frac {\left (\frac {a^2-b}{\sqrt {a^2+4 b}}+a\right ) \arctan \left (\frac {x \sqrt [4]{a \sqrt {a^2+4 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2+4 b}+a} \sqrt [4]{a x^4-b}}\right )}{\left (\sqrt {a^2+4 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+4 b}+a^2-2 b}}-\frac {\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \text {arctanh}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+4 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{a x^4-b}}\right )}{\left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+4 b}+a^2-2 b}}-\frac {\left (\frac {a^2-b}{\sqrt {a^2+4 b}}+a\right ) \text {arctanh}\left (\frac {x \sqrt [4]{a \sqrt {a^2+4 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2+4 b}+a} \sqrt [4]{a x^4-b}}\right )}{\left (\sqrt {a^2+4 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+4 b}+a^2-2 b}} \]

[In]

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

[Out]

-(((a - (a^2 - b)/Sqrt[a^2 + 4*b])*ArcTan[((a^2 - 2*b - a*Sqrt[a^2 + 4*b])^(1/4)*x)/((a - Sqrt[a^2 + 4*b])^(1/
4)*(-b + a*x^4)^(1/4))])/((a - Sqrt[a^2 + 4*b])^(3/4)*(a^2 - 2*b - a*Sqrt[a^2 + 4*b])^(1/4))) - ((a + (a^2 - b
)/Sqrt[a^2 + 4*b])*ArcTan[((a^2 - 2*b + a*Sqrt[a^2 + 4*b])^(1/4)*x)/((a + Sqrt[a^2 + 4*b])^(1/4)*(-b + a*x^4)^
(1/4))])/((a + Sqrt[a^2 + 4*b])^(3/4)*(a^2 - 2*b + a*Sqrt[a^2 + 4*b])^(1/4)) - ((a - (a^2 - b)/Sqrt[a^2 + 4*b]
)*ArcTanh[((a^2 - 2*b - a*Sqrt[a^2 + 4*b])^(1/4)*x)/((a - Sqrt[a^2 + 4*b])^(1/4)*(-b + a*x^4)^(1/4))])/((a - S
qrt[a^2 + 4*b])^(3/4)*(a^2 - 2*b - a*Sqrt[a^2 + 4*b])^(1/4)) - ((a + (a^2 - b)/Sqrt[a^2 + 4*b])*ArcTanh[((a^2
- 2*b + a*Sqrt[a^2 + 4*b])^(1/4)*x)/((a + Sqrt[a^2 + 4*b])^(1/4)*(-b + a*x^4)^(1/4))])/((a + Sqrt[a^2 + 4*b])^
(3/4)*(a^2 - 2*b + a*Sqrt[a^2 + 4*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 {2 a+\frac {2 \left (a^2-b\right )}{\sqrt {a^2+4 b}}}{\left (-a-\sqrt {a^2+4 b}+2 x^4\right ) \sqrt [4]{-b+a x^4}}+\frac {2 a-\frac {2 \left (a^2-b\right )}{\sqrt {a^2+4 b}}}{\left (-a+\sqrt {a^2+4 b}+2 x^4\right ) \sqrt [4]{-b+a x^4}}\right ) \, dx \\ & = \left (2 \left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right )\right ) \int \frac {1}{\left (-a+\sqrt {a^2+4 b}+2 x^4\right ) \sqrt [4]{-b+a x^4}} \, dx+\left (2 \left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right )\right ) \int \frac {1}{\left (-a-\sqrt {a^2+4 b}+2 x^4\right ) \sqrt [4]{-b+a x^4}} \, dx \\ & = \left (2 \left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right )\right ) \text {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 {x}{\sqrt [4]{-b+a x^4}}\right )+\left (2 \left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right )\right ) \text {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 {x}{\sqrt [4]{-b+a x^4}}\right ) \\ & = -\frac {\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \text {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 {x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt {a-\sqrt {a^2+4 b}}}-\frac {\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \text {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 {x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt {a-\sqrt {a^2+4 b}}}-\frac {\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \text {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 {x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt {a+\sqrt {a^2+4 b}}}-\frac {\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \text {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 {x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt {a+\sqrt {a^2+4 b}}} \\ & = -\frac {\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \arctan \left (\frac {\sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}} x}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{-b+a x^4}}\right )}{\left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}}}-\frac {\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \arctan \left (\frac {\sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}} x}{\sqrt [4]{a+\sqrt {a^2+4 b}} \sqrt [4]{-b+a x^4}}\right )}{\left (a+\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}}}-\frac {\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}} x}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{-b+a x^4}}\right )}{\left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}}}-\frac {\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}} x}{\sqrt [4]{a+\sqrt {a^2+4 b}} \sqrt [4]{-b+a x^4}}\right )}{\left (a+\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.84 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00 \[ \int \frac {-b+2 a x^4}{\sqrt [4]{-b+a x^4} \left (-b-a x^4+x^8\right )} \, dx=\frac {1}{4} \text {RootSum}\left [2 a^2-b-3 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {a \log (x)-a \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-3 a \text {$\#$1}+2 \text {$\#$1}^5}\&\right ] \]

[In]

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

[Out]

RootSum[2*a^2 - b - 3*a*#1^4 + #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 + 2*#1^5) & ]/4

Maple [N/A] (verified)

Time = 0.28 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.64

method	result	size
pseudoelliptic	\(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-3 a \,\textit {\_Z}^{4}+2 a^{2}-b \right )}{\sum }\frac {\left (\textit {\_R}^{4}+a \right ) \ln \left (\frac {-\textit {\_R} x +\left (a \,x^{4}-b \right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R} \left (2 \textit {\_R}^{4}-3 a \right )}\right )}{4}\)	\(66\)

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [N/A]

Not integrable

Time = 58.40 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.28 \[ \int \frac {-b+2 a x^4}{\sqrt [4]{-b+a x^4} \left (-b-a x^4+x^8\right )} \, dx=\int \frac {2 a x^{4} - b}{\sqrt [4]{a x^{4} - b} \left (- a x^{4} - b + x^{8}\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.38 \[ \int \frac {-b+2 a x^4}{\sqrt [4]{-b+a x^4} \left (-b-a x^4+x^8\right )} \, dx=\int { \frac {2 \, a x^{4} - b}{{\left (x^{8} - a x^{4} - b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 1.62 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.38 \[ \int \frac {-b+2 a x^4}{\sqrt [4]{-b+a x^4} \left (-b-a x^4+x^8\right )} \, dx=\int { \frac {2 \, a x^{4} - b}{{\left (x^{8} - a x^{4} - b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 6.41 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.35 \[ \int \frac {-b+2 a x^4}{\sqrt [4]{-b+a x^4} \left (-b-a x^4+x^8\right )} \, dx=\int \frac {b-2\,a\,x^4}{{\left (a\,x^4-b\right )}^{1/4}\,\left (-x^8+a\,x^4+b\right )} \,d x \]

[In]

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

[Out]

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