\(\int \frac {b+a x^4}{(-b+x^2+a x^4) \sqrt [4]{-b x^2+a x^6}} \, dx\) [1721]

Optimal result

Integrand size = 38, antiderivative size = 116 \[ \int \frac {b+a x^4}{\left (-b+x^2+a x^4\right ) \sqrt [4]{-b x^2+a x^6}} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} x \sqrt [4]{-b x^2+a x^6}}{-x^2+\sqrt {-b x^2+a x^6}}\right )}{\sqrt {2}}-\frac {\text {arctanh}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {-b x^2+a x^6}}{\sqrt {2}}}{x \sqrt [4]{-b x^2+a x^6}}\right )}{\sqrt {2}} \] Output:

-1/2*arctan(2^(1/2)*x*(a*x^6-b*x^2)^(1/4)/(-x^2+(a*x^6-b*x^2)^(1/2)))*2^(1 
/2)-1/2*arctanh((1/2*2^(1/2)*x^2+1/2*(a*x^6-b*x^2)^(1/2)*2^(1/2))/x/(a*x^6 
-b*x^2)^(1/4))*2^(1/2)
                                                                                    
                                                                                    
 

Mathematica [F]

\[ \int \frac {b+a x^4}{\left (-b+x^2+a x^4\right ) \sqrt [4]{-b x^2+a x^6}} \, dx=\int \frac {b+a x^4}{\left (-b+x^2+a x^4\right ) \sqrt [4]{-b x^2+a x^6}} \, dx \] Input:

Integrate[(b + a*x^4)/((-b + x^2 + a*x^4)*(-(b*x^2) + a*x^6)^(1/4)),x]
 

Output:

Integrate[(b + a*x^4)/((-b + x^2 + a*x^4)*(-(b*x^2) + a*x^6)^(1/4)), x]
 

Rubi [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 1.57 (sec) , antiderivative size = 482, normalized size of antiderivative = 4.16, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {2467, 25, 2035, 7279, 2009}

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.

\(\displaystyle \int \frac {a x^4+b}{\left (a x^4-b+x^2\right ) \sqrt [4]{a x^6-b x^2}} \, dx\)

\(\Big \downarrow \) 2467

\(\displaystyle \frac {\sqrt {x} \sqrt [4]{a x^4-b} \int -\frac {a x^4+b}{\sqrt {x} \left (-a x^4-x^2+b\right ) \sqrt [4]{a x^4-b}}dx}{\sqrt [4]{a x^6-b x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\sqrt {x} \sqrt [4]{a x^4-b} \int \frac {a x^4+b}{\sqrt {x} \left (-a x^4-x^2+b\right ) \sqrt [4]{a x^4-b}}dx}{\sqrt [4]{a x^6-b x^2}}\)

\(\Big \downarrow \) 2035

\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{a x^4-b} \int \frac {a x^4+b}{\left (-a x^4-x^2+b\right ) \sqrt [4]{a x^4-b}}d\sqrt {x}}{\sqrt [4]{a x^6-b x^2}}\)

\(\Big \downarrow \) 7279

\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{a x^4-b} \int \left (\frac {2 b-x^2}{\left (-a x^4-x^2+b\right ) \sqrt [4]{a x^4-b}}-\frac {1}{\sqrt [4]{a x^4-b}}\right )d\sqrt {x}}{\sqrt [4]{a x^6-b x^2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{a x^4-b} \left (\frac {\sqrt {x} \sqrt [4]{1-\frac {a x^4}{b}} \operatorname {AppellF1}\left (\frac {1}{8},1,\frac {1}{4},\frac {9}{8},\frac {2 a^2 x^4}{2 a b-\sqrt {4 a b+1}+1},\frac {a x^4}{b}\right )}{\sqrt [4]{a x^4-b}}+\frac {\sqrt {x} \sqrt [4]{1-\frac {a x^4}{b}} \operatorname {AppellF1}\left (\frac {1}{8},1,\frac {1}{4},\frac {9}{8},\frac {2 a^2 x^4}{2 a b+\sqrt {4 a b+1}+1},\frac {a x^4}{b}\right )}{\sqrt [4]{a x^4-b}}-\frac {a x^{5/2} \left (1-\sqrt {4 a b+1}\right ) \sqrt [4]{1-\frac {a x^4}{b}} \operatorname {AppellF1}\left (\frac {5}{8},1,\frac {1}{4},\frac {13}{8},\frac {2 a^2 x^4}{2 a b-\sqrt {4 a b+1}+1},\frac {a x^4}{b}\right )}{5 \left (2 a b-\sqrt {4 a b+1}+1\right ) \sqrt [4]{a x^4-b}}-\frac {a x^{5/2} \left (\sqrt {4 a b+1}+1\right ) \sqrt [4]{1-\frac {a x^4}{b}} \operatorname {AppellF1}\left (\frac {5}{8},1,\frac {1}{4},\frac {13}{8},\frac {2 a^2 x^4}{2 a b+\sqrt {4 a b+1}+1},\frac {a x^4}{b}\right )}{5 \left (2 a b+\sqrt {4 a b+1}+1\right ) \sqrt [4]{a x^4-b}}-\frac {\sqrt {x} \sqrt [4]{1-\frac {a x^4}{b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{8},\frac {1}{4},\frac {9}{8},\frac {a x^4}{b}\right )}{\sqrt [4]{a x^4-b}}\right )}{\sqrt [4]{a x^6-b x^2}}\)

Input:

Int[(b + a*x^4)/((-b + x^2 + a*x^4)*(-(b*x^2) + a*x^6)^(1/4)),x]
 

Output:

(-2*Sqrt[x]*(-b + a*x^4)^(1/4)*((Sqrt[x]*(1 - (a*x^4)/b)^(1/4)*AppellF1[1/ 
8, 1, 1/4, 9/8, (2*a^2*x^4)/(1 + 2*a*b - Sqrt[1 + 4*a*b]), (a*x^4)/b])/(-b 
 + a*x^4)^(1/4) + (Sqrt[x]*(1 - (a*x^4)/b)^(1/4)*AppellF1[1/8, 1, 1/4, 9/8 
, (2*a^2*x^4)/(1 + 2*a*b + Sqrt[1 + 4*a*b]), (a*x^4)/b])/(-b + a*x^4)^(1/4 
) - (a*(1 - Sqrt[1 + 4*a*b])*x^(5/2)*(1 - (a*x^4)/b)^(1/4)*AppellF1[5/8, 1 
, 1/4, 13/8, (2*a^2*x^4)/(1 + 2*a*b - Sqrt[1 + 4*a*b]), (a*x^4)/b])/(5*(1 
+ 2*a*b - Sqrt[1 + 4*a*b])*(-b + a*x^4)^(1/4)) - (a*(1 + Sqrt[1 + 4*a*b])* 
x^(5/2)*(1 - (a*x^4)/b)^(1/4)*AppellF1[5/8, 1, 1/4, 13/8, (2*a^2*x^4)/(1 + 
 2*a*b + Sqrt[1 + 4*a*b]), (a*x^4)/b])/(5*(1 + 2*a*b + Sqrt[1 + 4*a*b])*(- 
b + a*x^4)^(1/4)) - (Sqrt[x]*(1 - (a*x^4)/b)^(1/4)*Hypergeometric2F1[1/8, 
1/4, 9/8, (a*x^4)/b])/(-b + a*x^4)^(1/4)))/(-(b*x^2) + a*x^6)^(1/4)
 

Defintions of rubi rules used

rule 25

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

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2035

Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k   Subst 
[Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti 
onQ[m] && AlgebraicFunctionQ[Fx, x]
 

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]
 

rule 7279

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] /; Su 
mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
 

Maple [A] (verified)

Time = 1.49 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.28

method	result	size
pseudoelliptic	\(\frac {\sqrt {2}\, \left (\ln \left (\frac {-\left (x^{2} \left (a \,x^{4}-b \right )\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{2} \left (a \,x^{4}-b \right )}}{\left (x^{2} \left (a \,x^{4}-b \right )\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{2} \left (a \,x^{4}-b \right )}}\right )+2 \arctan \left (\frac {\left (x^{2} \left (a \,x^{4}-b \right )\right )^{\frac {1}{4}} \sqrt {2}+x}{x}\right )+2 \arctan \left (\frac {\left (x^{2} \left (a \,x^{4}-b \right )\right )^{\frac {1}{4}} \sqrt {2}-x}{x}\right )\right )}{4}\)	\(148\)

Input:

int((a*x^4+b)/(a*x^4+x^2-b)/(a*x^6-b*x^2)^(1/4),x,method=_RETURNVERBOSE)
 

Output:

1/4*2^(1/2)*(ln((-(x^2*(a*x^4-b))^(1/4)*2^(1/2)*x+x^2+(x^2*(a*x^4-b))^(1/2 
))/((x^2*(a*x^4-b))^(1/4)*2^(1/2)*x+x^2+(x^2*(a*x^4-b))^(1/2)))+2*arctan(( 
(x^2*(a*x^4-b))^(1/4)*2^(1/2)+x)/x)+2*arctan(((x^2*(a*x^4-b))^(1/4)*2^(1/2 
)-x)/x))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {b+a x^4}{\left (-b+x^2+a x^4\right ) \sqrt [4]{-b x^2+a x^6}} \, dx=\text {Timed out} \] Input:

integrate((a*x^4+b)/(a*x^4+x^2-b)/(a*x^6-b*x^2)^(1/4),x, algorithm="fricas 
")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {b+a x^4}{\left (-b+x^2+a x^4\right ) \sqrt [4]{-b x^2+a x^6}} \, dx=\int \frac {a x^{4} + b}{\sqrt [4]{x^{2} \left (a x^{4} - b\right )} \left (a x^{4} - b + x^{2}\right )}\, dx \] Input:

integrate((a*x**4+b)/(a*x**4+x**2-b)/(a*x**6-b*x**2)**(1/4),x)
 

Output:

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

Maxima [F]

\[ \int \frac {b+a x^4}{\left (-b+x^2+a x^4\right ) \sqrt [4]{-b x^2+a x^6}} \, dx=\int { \frac {a x^{4} + b}{{\left (a x^{6} - b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} + x^{2} - b\right )}} \,d x } \] Input:

integrate((a*x^4+b)/(a*x^4+x^2-b)/(a*x^6-b*x^2)^(1/4),x, algorithm="maxima 
")
 

Output:

integrate((a*x^4 + b)/((a*x^6 - b*x^2)^(1/4)*(a*x^4 + x^2 - b)), x)
 

Giac [F]

\[ \int \frac {b+a x^4}{\left (-b+x^2+a x^4\right ) \sqrt [4]{-b x^2+a x^6}} \, dx=\int { \frac {a x^{4} + b}{{\left (a x^{6} - b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} + x^{2} - b\right )}} \,d x } \] Input:

integrate((a*x^4+b)/(a*x^4+x^2-b)/(a*x^6-b*x^2)^(1/4),x, algorithm="giac")
 

Output:

integrate((a*x^4 + b)/((a*x^6 - b*x^2)^(1/4)*(a*x^4 + x^2 - b)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {b+a x^4}{\left (-b+x^2+a x^4\right ) \sqrt [4]{-b x^2+a x^6}} \, dx=\int \frac {a\,x^4+b}{{\left (a\,x^6-b\,x^2\right )}^{1/4}\,\left (a\,x^4+x^2-b\right )} \,d x \] Input:

int((b + a*x^4)/((a*x^6 - b*x^2)^(1/4)*(a*x^4 - b + x^2)),x)
 

Output:

int((b + a*x^4)/((a*x^6 - b*x^2)^(1/4)*(a*x^4 - b + x^2)), x)
 

Reduce [F]

\[ \int \frac {b+a x^4}{\left (-b+x^2+a x^4\right ) \sqrt [4]{-b x^2+a x^6}} \, dx =\text {Too large to display} \] Input:

int((a*x^4+b)/(a*x^4+x^2-b)/(a*x^6-b*x^2)^(1/4),x)
 

Output:

(6*sqrt(x)*(a*x**4 - b)**(5/4) + 2*sqrt(x)*(a*x**4 - b)**(1/4)*a*x**4 - 2* 
sqrt(x)*(a*x**4 - b)**(1/4)*b - 7*sqrt(a*x**4 - b)*int((a*x**4 - b)**(3/4) 
/(sqrt(x)*a**3*x**12 - 3*sqrt(x)*a**2*b*x**8 + sqrt(x)*a**2*x**10 + 3*sqrt 
(x)*a*b**2*x**4 - 2*sqrt(x)*a*b*x**6 - sqrt(x)*b**3 + sqrt(x)*b**2*x**2),x 
)*a*b**2*x**4 + 7*sqrt(a*x**4 - b)*int((a*x**4 - b)**(3/4)/(sqrt(x)*a**3*x 
**12 - 3*sqrt(x)*a**2*b*x**8 + sqrt(x)*a**2*x**10 + 3*sqrt(x)*a*b**2*x**4 
- 2*sqrt(x)*a*b*x**6 - sqrt(x)*b**3 + sqrt(x)*b**2*x**2),x)*b**3 + 3*sqrt( 
a*x**4 - b)*int((sqrt(x)*(a*x**4 - b)**(3/4)*x**7)/(a**3*x**12 - 3*a**2*b* 
x**8 + a**2*x**10 + 3*a*b**2*x**4 - 2*a*b*x**6 - b**3 + b**2*x**2),x)*a**3 
*x**4 - 3*sqrt(a*x**4 - b)*int((sqrt(x)*(a*x**4 - b)**(3/4)*x**7)/(a**3*x* 
*12 - 3*a**2*b*x**8 + a**2*x**10 + 3*a*b**2*x**4 - 2*a*b*x**6 - b**3 + b** 
2*x**2),x)*a**2*b + 4*sqrt(a*x**4 - b)*int((sqrt(x)*(a*x**4 - b)**(3/4)*x* 
*3)/(a**3*x**12 - 3*a**2*b*x**8 + a**2*x**10 + 3*a*b**2*x**4 - 2*a*b*x**6 
- b**3 + b**2*x**2),x)*a**2*b*x**4 - 4*sqrt(a*x**4 - b)*int((sqrt(x)*(a*x* 
*4 - b)**(3/4)*x**3)/(a**3*x**12 - 3*a**2*b*x**8 + a**2*x**10 + 3*a*b**2*x 
**4 - 2*a*b*x**6 - b**3 + b**2*x**2),x)*a*b**2 + 4*sqrt(a*x**4 - b)*int((s 
qrt(x)*(a*x**4 - b)**(3/4)*x)/(a**3*x**12 - 3*a**2*b*x**8 + a**2*x**10 + 3 
*a*b**2*x**4 - 2*a*b*x**6 - b**3 + b**2*x**2),x)*a*b*x**4 - 4*sqrt(a*x**4 
- b)*int((sqrt(x)*(a*x**4 - b)**(3/4)*x)/(a**3*x**12 - 3*a**2*b*x**8 + a** 
2*x**10 + 3*a*b**2*x**4 - 2*a*b*x**6 - b**3 + b**2*x**2),x)*b**2)/(3*sq...