3.8.41 \(\int \frac {1}{\sqrt [4]{b+a x^4} (-b+a x^8)} \, dx\) [741]

3.8.41.1 Optimal result

Integrand size = 23, antiderivative size = 57 \[ \int \frac {1}{\sqrt [4]{b+a x^4} \left (-b+a x^8\right )} \, dx=\frac {\text {RootSum}\left [a^2-a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{8 b} \]

output

Unintegrable
 

3.8.41.2 Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt [4]{b+a x^4} \left (-b+a x^8\right )} \, dx=\frac {\text {RootSum}\left [a^2-a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{8 b} \]

input

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

output

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

3.8.41.3 Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(281\) vs. \(2(57)=114\).

Time = 0.39 (sec) , antiderivative size = 281, normalized size of antiderivative = 4.93, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {1759, 27, 902, 756, 216, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {1}{\sqrt [4]{a x^4+b} \left (a x^8-b\right )} \, dx\)

\(\Big \downarrow \) 1759

\(\displaystyle -\frac {\sqrt {a} \int \frac {1}{\sqrt {a} \left (\sqrt {b}-\sqrt {a} x^4\right ) \sqrt [4]{a x^4+b}}dx}{2 \sqrt {b}}-\frac {\sqrt {a} \int \frac {1}{\sqrt {a} \left (\sqrt {a} x^4+\sqrt {b}\right ) \sqrt [4]{a x^4+b}}dx}{2 \sqrt {b}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {1}{\left (\sqrt {b}-\sqrt {a} x^4\right ) \sqrt [4]{a x^4+b}}dx}{2 \sqrt {b}}-\frac {\int \frac {1}{\left (\sqrt {a} x^4+\sqrt {b}\right ) \sqrt [4]{a x^4+b}}dx}{2 \sqrt {b}}\)

\(\Big \downarrow \) 902

\(\displaystyle -\frac {\int \frac {1}{\sqrt {b}-\frac {\left (a \sqrt {b}-\sqrt {a} b\right ) x^4}{a x^4+b}}d\frac {x}{\sqrt [4]{a x^4+b}}}{2 \sqrt {b}}-\frac {\int \frac {1}{\sqrt {b}-\frac {\left (\sqrt {b} a+b \sqrt {a}\right ) x^4}{a x^4+b}}d\frac {x}{\sqrt [4]{a x^4+b}}}{2 \sqrt {b}}\)

\(\Big \downarrow \) 756

\(\displaystyle -\frac {\frac {\int \frac {1}{1-\frac {\sqrt [4]{a} \sqrt {\sqrt {a}-\sqrt {b}} x^2}{\sqrt {a x^4+b}}}d\frac {x}{\sqrt [4]{a x^4+b}}}{2 \sqrt {b}}+\frac {\int \frac {1}{\frac {\sqrt [4]{a} \sqrt {\sqrt {a}-\sqrt {b}} x^2}{\sqrt {a x^4+b}}+1}d\frac {x}{\sqrt [4]{a x^4+b}}}{2 \sqrt {b}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {1}{1-\frac {\sqrt [4]{a} \sqrt {\sqrt {a}+\sqrt {b}} x^2}{\sqrt {a x^4+b}}}d\frac {x}{\sqrt [4]{a x^4+b}}}{2 \sqrt {b}}+\frac {\int \frac {1}{\frac {\sqrt [4]{a} \sqrt {\sqrt {a}+\sqrt {b}} x^2}{\sqrt {a x^4+b}}+1}d\frac {x}{\sqrt [4]{a x^4+b}}}{2 \sqrt {b}}}{2 \sqrt {b}}\)

\(\Big \downarrow \) 216

\(\displaystyle -\frac {\frac {\int \frac {1}{1-\frac {\sqrt [4]{a} \sqrt {\sqrt {a}-\sqrt {b}} x^2}{\sqrt {a x^4+b}}}d\frac {x}{\sqrt [4]{a x^4+b}}}{2 \sqrt {b}}+\frac {\arctan \left (\frac {\sqrt [8]{a} x \sqrt [4]{\sqrt {a}-\sqrt {b}}}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [8]{a} \sqrt {b} \sqrt [4]{\sqrt {a}-\sqrt {b}}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {1}{1-\frac {\sqrt [4]{a} \sqrt {\sqrt {a}+\sqrt {b}} x^2}{\sqrt {a x^4+b}}}d\frac {x}{\sqrt [4]{a x^4+b}}}{2 \sqrt {b}}+\frac {\arctan \left (\frac {\sqrt [8]{a} x \sqrt [4]{\sqrt {a}+\sqrt {b}}}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [8]{a} \sqrt {b} \sqrt [4]{\sqrt {a}+\sqrt {b}}}}{2 \sqrt {b}}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {\frac {\arctan \left (\frac {\sqrt [8]{a} x \sqrt [4]{\sqrt {a}-\sqrt {b}}}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [8]{a} \sqrt {b} \sqrt [4]{\sqrt {a}-\sqrt {b}}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{a} x \sqrt [4]{\sqrt {a}-\sqrt {b}}}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [8]{a} \sqrt {b} \sqrt [4]{\sqrt {a}-\sqrt {b}}}}{2 \sqrt {b}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{a} x \sqrt [4]{\sqrt {a}+\sqrt {b}}}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [8]{a} \sqrt {b} \sqrt [4]{\sqrt {a}+\sqrt {b}}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{a} x \sqrt [4]{\sqrt {a}+\sqrt {b}}}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [8]{a} \sqrt {b} \sqrt [4]{\sqrt {a}+\sqrt {b}}}}{2 \sqrt {b}}\)

input

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

output

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

3.8.41.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 216

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

rule 219

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] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 756

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

rule 902

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Su 
bst[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 1759

Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (c_.)*(x_)^(n2_)), x_Symbol] :> W 
ith[{r = Rt[(-a)*c, 2]}, Simp[-c/(2*r)   Int[(d + e*x^n)^q/(r - c*x^n), x], 
 x] - Simp[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]
 

3.8.41.4 Maple [N/A] (verified)

Time = 1.22 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.88

method	result	size
pseudoelliptic	\(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2 a \,\textit {\_Z}^{4}+a^{2}-a b \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}}}{8 b}\)	\(50\)

input

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

output

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)) 
/b
 

3.8.41.5 Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt [4]{b+a x^4} \left (-b+a x^8\right )} \, dx=\text {Timed out} \]

input

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

output

Timed out
 

3.8.41.6 Sympy [N/A]

Not integrable

Time = 5.43 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.33 \[ \int \frac {1}{\sqrt [4]{b+a x^4} \left (-b+a x^8\right )} \, dx=\int \frac {1}{\sqrt [4]{a x^{4} + b} \left (a x^{8} - b\right )}\, dx \]

input

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

output

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

3.8.41.7 Maxima [N/A]

Not integrable

Time = 0.22 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.40 \[ \int \frac {1}{\sqrt [4]{b+a x^4} \left (-b+a x^8\right )} \, dx=\int { \frac {1}{{\left (a x^{8} - b\right )} {\left (a x^{4} + b\right )}^{\frac {1}{4}}} \,d x } \]

input

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

output

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

3.8.41.8 Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.40 \[ \int \frac {1}{\sqrt [4]{b+a x^4} \left (-b+a x^8\right )} \, dx=\int { \frac {1}{{\left (a x^{8} - b\right )} {\left (a x^{4} + b\right )}^{\frac {1}{4}}} \,d x } \]

input

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

output

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

3.8.41.9 Mupad [N/A]

Not integrable

Time = 5.50 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.42 \[ \int \frac {1}{\sqrt [4]{b+a x^4} \left (-b+a x^8\right )} \, dx=-\int \frac {1}{{\left (a\,x^4+b\right )}^{1/4}\,\left (b-a\,x^8\right )} \,d x \]

input

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

output

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