\(\int \frac {x^4}{\sqrt [4]{-b+a x^4} (-b+a x^8)} \, dx\) [863]

Optimal result

Integrand size = 28, antiderivative size = 65 \[ \int \frac {x^4}{\sqrt [4]{-b+a x^4} \left (-b+a x^8\right )} \, dx=\frac {1}{8} \text {RootSum}\left [a^2-a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )}{a \text {$\#$1}-\text {$\#$1}^5}\&\right ] \]

[Out]

Unintegrable

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(269\) vs. \(2(65)=130\).

Time = 0.18 (sec) , antiderivative size = 269, normalized size of antiderivative = 4.14, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {1543, 385, 218, 212, 209} \[ \int \frac {x^4}{\sqrt [4]{-b+a x^4} \left (-b+a x^8\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt [8]{a} x \sqrt [4]{\sqrt {a}-\sqrt {b}}}{\sqrt [4]{a x^4-b}}\right )}{4 a^{5/8} \sqrt {b} \sqrt [4]{\sqrt {a}-\sqrt {b}}}+\frac {\arctan \left (\frac {\sqrt [8]{a} x \sqrt [4]{\sqrt {a}+\sqrt {b}}}{\sqrt [4]{a x^4-b}}\right )}{4 a^{5/8} \sqrt {b} \sqrt [4]{\sqrt {a}+\sqrt {b}}}-\frac {\text {arctanh}\left (\frac {\sqrt [8]{a} x \sqrt [4]{\sqrt {a}-\sqrt {b}}}{\sqrt [4]{a x^4-b}}\right )}{4 a^{5/8} \sqrt {b} \sqrt [4]{\sqrt {a}-\sqrt {b}}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{a} x \sqrt [4]{\sqrt {a}+\sqrt {b}}}{\sqrt [4]{a x^4-b}}\right )}{4 a^{5/8} \sqrt {b} \sqrt [4]{\sqrt {a}+\sqrt {b}}} \]

[In]

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

[Out]

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

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 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 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 1543

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))^(q_))/((a_) + (c_.)*(x_)^(n2_.)), x_Symbol] :> Int[ExpandInte
grand[(d + e*x^n)^q, (f*x)^m/(a + c*x^(2*n)), x], x] /; FreeQ[{a, c, d, e, f, q, n}, x] && EqQ[n2, 2*n] && IGt
Q[n, 0] &&  !IntegerQ[q] && IntegerQ[m]

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

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.98 \[ \int \frac {x^4}{\sqrt [4]{-b+a x^4} \left (-b+a x^8\right )} \, dx=\frac {1}{8} \text {RootSum}\left [a^2-a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x)-\log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )}{-a \text {$\#$1}+\text {$\#$1}^5}\&\right ] \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Time = 1.28 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.89

method	result	size
pseudoelliptic	\(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2 \textit {\_Z}^{4} a +a^{2}-a b \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (a \,x^{4}-b \right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R} \left (\textit {\_R}^{4}-a \right )}\right )}{8}\)	\(58\)

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [N/A]

Not integrable

Time = 9.44 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.31 \[ \int \frac {x^4}{\sqrt [4]{-b+a x^4} \left (-b+a x^8\right )} \, dx=\int \frac {x^{4}}{\sqrt [4]{a x^{4} - b} \left (a x^{8} - b\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.22 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.43 \[ \int \frac {x^4}{\sqrt [4]{-b+a x^4} \left (-b+a x^8\right )} \, dx=\int { \frac {x^{4}}{{\left (a x^{8} - b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.43 \[ \int \frac {x^4}{\sqrt [4]{-b+a x^4} \left (-b+a x^8\right )} \, dx=\int { \frac {x^{4}}{{\left (a x^{8} - b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 5.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.45 \[ \int \frac {x^4}{\sqrt [4]{-b+a x^4} \left (-b+a x^8\right )} \, dx=-\int \frac {x^4}{{\left (a\,x^4-b\right )}^{1/4}\,\left (b-a\,x^8\right )} \,d x \]

[In]

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

[Out]

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