\(\int \frac {-b+2 a x^8}{\sqrt [4]{b+a x^4} (-b+a x^8)} \, dx\) [1521]

Optimal result

Integrand size = 33, antiderivative size = 105 \[ \int \frac {-b+2 a x^8}{\sqrt [4]{b+a x^4} \left (-b+a x^8\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {1}{8} \text {RootSum}\left [a^2-a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]

[Out]

Unintegrable

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(291\) vs. \(2(105)=210\).

Time = 0.35 (sec) , antiderivative size = 291, normalized size of antiderivative = 2.77, number of steps used = 15, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {6857, 246, 218, 212, 209, 1443, 385} \[ \int \frac {-b+2 a x^8}{\sqrt [4]{b+a x^4} \left (-b+a x^8\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{\sqrt [4]{a}}-\frac {\arctan \left (\frac {\sqrt [8]{a} x \sqrt [4]{\sqrt {a}-\sqrt {b}}}{\sqrt [4]{a x^4+b}}\right )}{4 \sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}}}-\frac {\arctan \left (\frac {\sqrt [8]{a} x \sqrt [4]{\sqrt {a}+\sqrt {b}}}{\sqrt [4]{a x^4+b}}\right )}{4 \sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{\sqrt [4]{a}}-\frac {\text {arctanh}\left (\frac {\sqrt [8]{a} x \sqrt [4]{\sqrt {a}-\sqrt {b}}}{\sqrt [4]{a x^4+b}}\right )}{4 \sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}}}-\frac {\text {arctanh}\left (\frac {\sqrt [8]{a} x \sqrt [4]{\sqrt {a}+\sqrt {b}}}{\sqrt [4]{a x^4+b}}\right )}{4 \sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}}} \]

[In]

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

[Out]

ArcTan[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/a^(1/4) - ArcTan[(a^(1/8)*(Sqrt[a] - Sqrt[b])^(1/4)*x)/(b + a*x^4)^(1/4)
]/(4*a^(1/8)*(Sqrt[a] - Sqrt[b])^(1/4)) - ArcTan[(a^(1/8)*(Sqrt[a] + Sqrt[b])^(1/4)*x)/(b + a*x^4)^(1/4)]/(4*a
^(1/8)*(Sqrt[a] + Sqrt[b])^(1/4)) + ArcTanh[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/a^(1/4) - ArcTanh[(a^(1/8)*(Sqrt[a]
- Sqrt[b])^(1/4)*x)/(b + a*x^4)^(1/4)]/(4*a^(1/8)*(Sqrt[a] - Sqrt[b])^(1/4)) - ArcTanh[(a^(1/8)*(Sqrt[a] + Sq
rt[b])^(1/4)*x)/(b + a*x^4)^(1/4)]/(4*a^(1/8)*(Sqrt[a] + Sqrt[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 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 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 1443

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 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

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

Mathematica [A] (verified)

Time = 0.98 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.95 \[ \int \frac {-b+2 a x^8}{\sqrt [4]{b+a x^4} \left (-b+a x^8\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {1}{8} \text {RootSum}\left [a^2-a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]

[In]

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

Maple [N/A] (verified)

Time = 0.58 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.10

method	result	size
pseudoelliptic	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2 a \,\textit {\_Z}^{4}+a^{2}-a b \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}}\right ) a^{\frac {1}{4}}-8 \arctan \left (\frac {\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )+4 \ln \left (\frac {-a^{\frac {1}{4}} x -\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (a \,x^{4}+b \right )^{\frac {1}{4}}}\right )}{8 a^{\frac {1}{4}}}\)	\(115\)

[In]

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

[Out]

1/8*(sum(ln((-_R*x+(a*x^4+b)^(1/4))/x)/_R,_R=RootOf(_Z^8-2*_Z^4*a+a^2-a*b))*a^(1/4)-8*arctan(1/a^(1/4)/x*(a*x^
4+b)^(1/4))+4*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 = 0.29 (sec) , antiderivative size = 1368, normalized size of antiderivative = 13.03 \[ \int \frac {-b+2 a x^8}{\sqrt [4]{b+a x^4} \left (-b+a x^8\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/8*sqrt(-sqrt(((a - b)*sqrt(b/(a^3 - 2*a^2*b + a*b^2)) + 1)/(a - b)))*log((((a^2 - a*b)*x*sqrt(b/(a^3 - 2*a^
2*b + a*b^2)) - a*x)*sqrt(-sqrt(((a - b)*sqrt(b/(a^3 - 2*a^2*b + a*b^2)) + 1)/(a - b)))*sqrt(((a - b)*sqrt(b/(
a^3 - 2*a^2*b + a*b^2)) + 1)/(a - b)) + (a*x^4 + b)^(1/4))/x) + 1/8*sqrt(-sqrt(((a - b)*sqrt(b/(a^3 - 2*a^2*b
+ a*b^2)) + 1)/(a - b)))*log(-(((a^2 - a*b)*x*sqrt(b/(a^3 - 2*a^2*b + a*b^2)) - a*x)*sqrt(-sqrt(((a - b)*sqrt(
b/(a^3 - 2*a^2*b + a*b^2)) + 1)/(a - b)))*sqrt(((a - b)*sqrt(b/(a^3 - 2*a^2*b + a*b^2)) + 1)/(a - b)) - (a*x^4
+ b)^(1/4))/x) + 1/8*sqrt(-sqrt(-((a - b)*sqrt(b/(a^3 - 2*a^2*b + a*b^2)) - 1)/(a - b)))*log((((a^2 - a*b)*x*
sqrt(b/(a^3 - 2*a^2*b + a*b^2)) + a*x)*sqrt(-sqrt(-((a - b)*sqrt(b/(a^3 - 2*a^2*b + a*b^2)) - 1)/(a - b)))*sqr
t(-((a - b)*sqrt(b/(a^3 - 2*a^2*b + a*b^2)) - 1)/(a - b)) + (a*x^4 + b)^(1/4))/x) - 1/8*sqrt(-sqrt(-((a - b)*s
qrt(b/(a^3 - 2*a^2*b + a*b^2)) - 1)/(a - b)))*log(-(((a^2 - a*b)*x*sqrt(b/(a^3 - 2*a^2*b + a*b^2)) + a*x)*sqrt
(-sqrt(-((a - b)*sqrt(b/(a^3 - 2*a^2*b + a*b^2)) - 1)/(a - b)))*sqrt(-((a - b)*sqrt(b/(a^3 - 2*a^2*b + a*b^2))
- 1)/(a - b)) - (a*x^4 + b)^(1/4))/x) + 1/8*(((a - b)*sqrt(b/(a^3 - 2*a^2*b + a*b^2)) + 1)/(a - b))^(1/4)*log
((((a^2 - a*b)*x*sqrt(b/(a^3 - 2*a^2*b + a*b^2)) - a*x)*(((a - b)*sqrt(b/(a^3 - 2*a^2*b + a*b^2)) + 1)/(a - b)
)^(3/4) + (a*x^4 + b)^(1/4))/x) - 1/8*(((a - b)*sqrt(b/(a^3 - 2*a^2*b + a*b^2)) + 1)/(a - b))^(1/4)*log(-(((a^
2 - a*b)*x*sqrt(b/(a^3 - 2*a^2*b + a*b^2)) - a*x)*(((a - b)*sqrt(b/(a^3 - 2*a^2*b + a*b^2)) + 1)/(a - b))^(3/4
) - (a*x^4 + b)^(1/4))/x) - 1/8*(-((a - b)*sqrt(b/(a^3 - 2*a^2*b + a*b^2)) - 1)/(a - b))^(1/4)*log((((a^2 - a*
b)*x*sqrt(b/(a^3 - 2*a^2*b + a*b^2)) + a*x)*(-((a - b)*sqrt(b/(a^3 - 2*a^2*b + a*b^2)) - 1)/(a - b))^(3/4) + (
a*x^4 + b)^(1/4))/x) + 1/8*(-((a - b)*sqrt(b/(a^3 - 2*a^2*b + a*b^2)) - 1)/(a - b))^(1/4)*log(-(((a^2 - a*b)*x
*sqrt(b/(a^3 - 2*a^2*b + a*b^2)) + a*x)*(-((a - b)*sqrt(b/(a^3 - 2*a^2*b + a*b^2)) - 1)/(a - b))^(3/4) - (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 = 15.83 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.25 \[ \int \frac {-b+2 a x^8}{\sqrt [4]{b+a x^4} \left (-b+a x^8\right )} \, dx=\int \frac {2 a x^{8} - b}{\sqrt [4]{a x^{4} + b} \left (a x^{8} - b\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.31 \[ \int \frac {-b+2 a x^8}{\sqrt [4]{b+a x^4} \left (-b+a x^8\right )} \, dx=\int { \frac {2 \, a x^{8} - b}{{\left (a x^{8} - b\right )} {\left (a x^{4} + b\right )}^{\frac {1}{4}}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.31 \[ \int \frac {-b+2 a x^8}{\sqrt [4]{b+a x^4} \left (-b+a x^8\right )} \, dx=\int { \frac {2 \, a x^{8} - b}{{\left (a x^{8} - b\right )} {\left (a x^{4} + b\right )}^{\frac {1}{4}}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 6.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.29 \[ \int \frac {-b+2 a x^8}{\sqrt [4]{b+a x^4} \left (-b+a x^8\right )} \, dx=\int \frac {b-2\,a\,x^8}{{\left (a\,x^4+b\right )}^{1/4}\,\left (b-a\,x^8\right )} \,d x \]

[In]

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

[Out]

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