\(\int \frac {1}{(-2 b+a x^4) \sqrt [4]{b x^2+a x^4}} \, dx\) [857]

Optimal result

Integrand size = 27, antiderivative size = 65 \[ \int \frac {1}{\left (-2 b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\frac {\text {RootSum}\left [2 a^2-a b-4 a \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{8 b} \] Output:

Unintegrable
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.54 \[ \int \frac {1}{\left (-2 b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\frac {\sqrt {x} \sqrt [4]{b+a x^2} \text {RootSum}\left [2 a^2-a b-4 a \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right )+\log \left (\sqrt [4]{b+a x^2}-\sqrt {x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{8 b \sqrt [4]{x^2 \left (b+a x^2\right )}} \] Input:

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

Output:

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(428\) vs. \(2(65)=130\).

Time = 0.57 (sec) , antiderivative size = 428, normalized size of antiderivative = 6.58, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2467, 25, 1593, 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.

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

\(\Big \downarrow \) 2467

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1593

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

\(\Big \downarrow \) 1759

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 902

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

\(\Big \downarrow \) 756

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

\(\Big \downarrow \) 216

\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {\frac {\int \frac {1}{\sqrt [4]{2}-\sqrt [4]{a} \sqrt {\sqrt {2} \sqrt {a}-\sqrt {b}} x}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}}{2 \sqrt [4]{2} \sqrt {b}}+\frac {\arctan \left (\frac {\sqrt [8]{a} \sqrt {x} \sqrt [4]{\sqrt {2} \sqrt {a}-\sqrt {b}}}{\sqrt [8]{2} \sqrt [4]{a x^2+b}}\right )}{2\ 2^{3/8} \sqrt [8]{a} \sqrt {b} \sqrt [4]{\sqrt {2} \sqrt {a}-\sqrt {b}}}}{2 \sqrt {2} \sqrt {b}}+\frac {\frac {\int \frac {1}{\sqrt [4]{2}-\sqrt [4]{a} \sqrt {\sqrt {2} \sqrt {a}+\sqrt {b}} x}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}}{2 \sqrt [4]{2} \sqrt {b}}+\frac {\arctan \left (\frac {\sqrt [8]{a} \sqrt {x} \sqrt [4]{\sqrt {2} \sqrt {a}+\sqrt {b}}}{\sqrt [8]{2} \sqrt [4]{a x^2+b}}\right )}{2\ 2^{3/8} \sqrt [8]{a} \sqrt {b} \sqrt [4]{\sqrt {2} \sqrt {a}+\sqrt {b}}}}{2 \sqrt {2} \sqrt {b}}\right )}{\sqrt [4]{a x^4+b x^2}}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {\frac {\arctan \left (\frac {\sqrt [8]{a} \sqrt {x} \sqrt [4]{\sqrt {2} \sqrt {a}-\sqrt {b}}}{\sqrt [8]{2} \sqrt [4]{a x^2+b}}\right )}{2\ 2^{3/8} \sqrt [8]{a} \sqrt {b} \sqrt [4]{\sqrt {2} \sqrt {a}-\sqrt {b}}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt {x} \sqrt [4]{\sqrt {2} \sqrt {a}-\sqrt {b}}}{\sqrt [8]{2} \sqrt [4]{a x^2+b}}\right )}{2\ 2^{3/8} \sqrt [8]{a} \sqrt {b} \sqrt [4]{\sqrt {2} \sqrt {a}-\sqrt {b}}}}{2 \sqrt {2} \sqrt {b}}+\frac {\frac {\arctan \left (\frac {\sqrt [8]{a} \sqrt {x} \sqrt [4]{\sqrt {2} \sqrt {a}+\sqrt {b}}}{\sqrt [8]{2} \sqrt [4]{a x^2+b}}\right )}{2\ 2^{3/8} \sqrt [8]{a} \sqrt {b} \sqrt [4]{\sqrt {2} \sqrt {a}+\sqrt {b}}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt {x} \sqrt [4]{\sqrt {2} \sqrt {a}+\sqrt {b}}}{\sqrt [8]{2} \sqrt [4]{a x^2+b}}\right )}{2\ 2^{3/8} \sqrt [8]{a} \sqrt {b} \sqrt [4]{\sqrt {2} \sqrt {a}+\sqrt {b}}}}{2 \sqrt {2} \sqrt {b}}\right )}{\sqrt [4]{a x^4+b x^2}}\)

Input:

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

Output:

(-2*Sqrt[x]*(b + a*x^2)^(1/4)*((ArcTan[(a^(1/8)*(Sqrt[2]*Sqrt[a] - Sqrt[b] 
)^(1/4)*Sqrt[x])/(2^(1/8)*(b + a*x^2)^(1/4))]/(2*2^(3/8)*a^(1/8)*(Sqrt[2]* 
Sqrt[a] - Sqrt[b])^(1/4)*Sqrt[b]) + ArcTanh[(a^(1/8)*(Sqrt[2]*Sqrt[a] - Sq 
rt[b])^(1/4)*Sqrt[x])/(2^(1/8)*(b + a*x^2)^(1/4))]/(2*2^(3/8)*a^(1/8)*(Sqr 
t[2]*Sqrt[a] - Sqrt[b])^(1/4)*Sqrt[b]))/(2*Sqrt[2]*Sqrt[b]) + (ArcTan[(a^( 
1/8)*(Sqrt[2]*Sqrt[a] + Sqrt[b])^(1/4)*Sqrt[x])/(2^(1/8)*(b + a*x^2)^(1/4) 
)]/(2*2^(3/8)*a^(1/8)*(Sqrt[2]*Sqrt[a] + Sqrt[b])^(1/4)*Sqrt[b]) + ArcTanh 
[(a^(1/8)*(Sqrt[2]*Sqrt[a] + Sqrt[b])^(1/4)*Sqrt[x])/(2^(1/8)*(b + a*x^2)^ 
(1/4))]/(2*2^(3/8)*a^(1/8)*(Sqrt[2]*Sqrt[a] + Sqrt[b])^(1/4)*Sqrt[b]))/(2* 
Sqrt[2]*Sqrt[b])))/(b*x^2 + a*x^4)^(1/4)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 1593

Int[((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_ 
), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/f   Subst[Int[x^(k*(m + 1 
) - 1)*(d + e*(x^(2*k)/f))^q*(a + c*(x^(4*k)/f))^p, x], x, (f*x)^(1/k)], x] 
] /; FreeQ[{a, c, d, e, f, p, q}, x] && FractionQ[m] && IntegerQ[p]
 

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]
 

rule 2467

Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F 
racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p])   Int[x^( 
p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P 
olyQ[Px, x] &&  !IntegerQ[p] &&  !MonomialQ[Px, x] &&  !PolyQ[Fx, x]
 

Maple [N/A] (verified)

Time = 0.34 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.89

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

Input:

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

Output:

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

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\left (-2 b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [N/A]

Not integrable

Time = 2.93 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.37 \[ \int \frac {1}{\left (-2 b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int \frac {1}{\sqrt [4]{x^{2} \left (a x^{2} + b\right )} \left (a x^{4} - 2 b\right )}\, dx \] Input:

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

Output:

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

Maxima [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.42 \[ \int \frac {1}{\left (-2 b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int { \frac {1}{{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} - 2 \, b\right )}} \,d x } \] Input:

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

Output:

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

Giac [N/A]

Not integrable

Time = 4.14 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.42 \[ \int \frac {1}{\left (-2 b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int { \frac {1}{{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} - 2 \, b\right )}} \,d x } \] Input:

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

Output:

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

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.46 \[ \int \frac {1}{\left (-2 b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=-\int \frac {1}{{\left (a\,x^4+b\,x^2\right )}^{1/4}\,\left (2\,b-a\,x^4\right )} \,d x \] Input:

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

Output:

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

Reduce [N/A]

Not integrable

Time = 2.02 (sec) , antiderivative size = 903, normalized size of antiderivative = 13.89 \[ \int \frac {1}{\left (-2 b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 2*sqrt(x)*(a*x**2 + b)**(5/4)*x**2 - 16*sqrt(x)*(a*x**2 + b)**(1/4)*a* 
x**2 - 16*sqrt(x)*(a*x**2 + b)**(1/4)*b + 16*sqrt(a*x**2 + b)*int((a*x**2 
+ b)**(3/4)/(sqrt(x)*a**3*x**8 + 2*sqrt(x)*a**2*b*x**6 - 2*sqrt(x)*a**2*b* 
x**4 + sqrt(x)*a*b**2*x**4 - 4*sqrt(x)*a*b**2*x**2 - 2*sqrt(x)*b**3),x)*a* 
b**2*x**2 + 16*sqrt(a*x**2 + b)*int((a*x**2 + b)**(3/4)/(sqrt(x)*a**3*x**8 
 + 2*sqrt(x)*a**2*b*x**6 - 2*sqrt(x)*a**2*b*x**4 + sqrt(x)*a*b**2*x**4 - 4 
*sqrt(x)*a*b**2*x**2 - 2*sqrt(x)*b**3),x)*b**3 + 4*sqrt(a*x**2 + b)*int((s 
qrt(x)*(a*x**2 + b)**(3/4)*x**7)/(a**3*x**8 + 2*a**2*b*x**6 - 2*a**2*b*x** 
4 + a*b**2*x**4 - 4*a*b**2*x**2 - 2*b**3),x)*a**3*x**2 + 4*sqrt(a*x**2 + b 
)*int((sqrt(x)*(a*x**2 + b)**(3/4)*x**7)/(a**3*x**8 + 2*a**2*b*x**6 - 2*a* 
*2*b*x**4 + a*b**2*x**4 - 4*a*b**2*x**2 - 2*b**3),x)*a**2*b + 16*sqrt(a*x* 
*2 + b)*int((sqrt(x)*(a*x**2 + b)**(3/4)*x**5)/(a**3*x**8 + 2*a**2*b*x**6 
- 2*a**2*b*x**4 + a*b**2*x**4 - 4*a*b**2*x**2 - 2*b**3),x)*a**3*x**2 + 5*s 
qrt(a*x**2 + b)*int((sqrt(x)*(a*x**2 + b)**(3/4)*x**5)/(a**3*x**8 + 2*a**2 
*b*x**6 - 2*a**2*b*x**4 + a*b**2*x**4 - 4*a*b**2*x**2 - 2*b**3),x)*a**2*b* 
x**2 + 16*sqrt(a*x**2 + b)*int((sqrt(x)*(a*x**2 + b)**(3/4)*x**5)/(a**3*x* 
*8 + 2*a**2*b*x**6 - 2*a**2*b*x**4 + a*b**2*x**4 - 4*a*b**2*x**2 - 2*b**3) 
,x)*a**2*b + 5*sqrt(a*x**2 + b)*int((sqrt(x)*(a*x**2 + b)**(3/4)*x**5)/(a* 
*3*x**8 + 2*a**2*b*x**6 - 2*a**2*b*x**4 + a*b**2*x**4 - 4*a*b**2*x**2 - 2* 
b**3),x)*a*b**2 - 10*sqrt(a*x**2 + b)*int((sqrt(x)*(a*x**2 + b)**(3/4)*...