3.15.43 \(\int \frac {-2 b+a x^2}{\sqrt [4]{-b+a x^2} (-b+a x^2+x^4)} \, dx\) [1443]

3.15.43.1 Optimal result

Integrand size = 37, antiderivative size = 102 \[ \int \frac {-2 b+a x^2}{\sqrt [4]{-b+a x^2} \left (-b+a x^2+x^4\right )} \, dx=-\frac {\arctan \left (\frac {-\frac {x^2}{\sqrt {2}}+\frac {\sqrt {-b+a x^2}}{\sqrt {2}}}{x \sqrt [4]{-b+a x^2}}\right )}{\sqrt {2}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-b+a x^2}}{x^2+\sqrt {-b+a x^2}}\right )}{\sqrt {2}} \]

output

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

3.15.43.2 Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.89 \[ \int \frac {-2 b+a x^2}{\sqrt [4]{-b+a x^2} \left (-b+a x^2+x^4\right )} \, dx=\frac {-\arctan \left (\frac {-x^2+\sqrt {-b+a x^2}}{\sqrt {2} x \sqrt [4]{-b+a x^2}}\right )+\text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-b+a x^2}}{x^2+\sqrt {-b+a x^2}}\right )}{\sqrt {2}} \]

input

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

output

(-ArcTan[(-x^2 + Sqrt[-b + a*x^2])/(Sqrt[2]*x*(-b + a*x^2)^(1/4))] + ArcTa 
nh[(Sqrt[2]*x*(-b + a*x^2)^(1/4))/(x^2 + Sqrt[-b + a*x^2])])/Sqrt[2]
 

3.15.43.3 Rubi [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 9.54 (sec) , antiderivative size = 2422, normalized size of antiderivative = 23.75, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {2256, 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.

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

\(\Big \downarrow \) 2256

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\left (a+\sqrt {a^2+4 b}\right ) \sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {a x^2-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x^2-b}\right ) \operatorname {EllipticPi}\left (\frac {\left (\sqrt {2} \sqrt {b}+\sqrt {-a^2-\sqrt {a^2+4 b} a-2 b}\right )^2}{4 \sqrt {2} \sqrt {b} \sqrt {-a^2-\sqrt {a^2+4 b} a-2 b}},2 \arctan \left (\frac {\sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right ) \left (\sqrt {2} \sqrt {b}-\sqrt {-a^2-\sqrt {a^2+4 b} a-2 b}\right )^2}{4 \sqrt {2} \sqrt [4]{b} \sqrt {-a^2-\sqrt {a^2+4 b} a-2 b} \left (a^2+\sqrt {a^2+4 b} a+4 b\right ) x}-\frac {\sqrt {b} \sqrt {a+\sqrt {a^2+4 b}} \sqrt {\frac {a x^2}{b}} \arctan \left (\frac {\sqrt {a} \sqrt {a+\sqrt {a^2+4 b}} \sqrt [4]{a x^2-b}}{\sqrt [4]{2} \sqrt {b} \sqrt [4]{-a^2-\sqrt {a^2+4 b} a-2 b} \sqrt {\frac {a x^2}{b}}}\right )}{2 \sqrt [4]{2} \sqrt {a} \sqrt [4]{-a^2-\sqrt {a^2+4 b} a-2 b} x}-\frac {\sqrt {b} \sqrt {a-\sqrt {a^2+4 b}} \sqrt {\frac {a x^2}{b}} \arctan \left (\frac {\sqrt {a} \sqrt {a-\sqrt {a^2+4 b}} \sqrt [4]{a x^2-b}}{\sqrt [4]{2} \sqrt {b} \sqrt [4]{-a^2+\sqrt {a^2+4 b} a-2 b} \sqrt {\frac {a x^2}{b}}}\right )}{2 \sqrt [4]{2} \sqrt {a} \sqrt [4]{-a^2+\sqrt {a^2+4 b} a-2 b} x}+\frac {\sqrt {b} \sqrt {a+\sqrt {a^2+4 b}} \sqrt {\frac {a x^2}{b}} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {a+\sqrt {a^2+4 b}} \sqrt [4]{a x^2-b}}{\sqrt [4]{2} \sqrt {b} \sqrt [4]{-a^2-\sqrt {a^2+4 b} a-2 b} \sqrt {\frac {a x^2}{b}}}\right )}{2 \sqrt [4]{2} \sqrt {a} \sqrt [4]{-a^2-\sqrt {a^2+4 b} a-2 b} x}+\frac {\sqrt {b} \sqrt {a-\sqrt {a^2+4 b}} \sqrt {\frac {a x^2}{b}} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {a-\sqrt {a^2+4 b}} \sqrt [4]{a x^2-b}}{\sqrt [4]{2} \sqrt {b} \sqrt [4]{-a^2+\sqrt {a^2+4 b} a-2 b} \sqrt {\frac {a x^2}{b}}}\right )}{2 \sqrt [4]{2} \sqrt {a} \sqrt [4]{-a^2+\sqrt {a^2+4 b} a-2 b} x}-\frac {\left (a+\sqrt {a^2+4 b}\right ) \left (2 \sqrt {b}-\sqrt {2} \sqrt {-a^2-\sqrt {a^2+4 b} a-2 b}\right ) \sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {a x^2-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x^2-b}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{b} \left (a^2+\sqrt {a^2+4 b} a+4 b\right ) x}-\frac {\left (a+\sqrt {a^2+4 b}\right ) \left (2 \sqrt {b}+\sqrt {2} \sqrt {-a^2-\sqrt {a^2+4 b} a-2 b}\right ) \sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {a x^2-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x^2-b}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{b} \left (a^2+\sqrt {a^2+4 b} a+4 b\right ) x}-\frac {\left (a-\sqrt {a^2+4 b}\right ) \left (2 \sqrt {b}-\sqrt {-2 a^2+2 \sqrt {a^2+4 b} a-4 b}\right ) \sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {a x^2-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x^2-b}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{b} \left (a^2-\sqrt {a^2+4 b} a+4 b\right ) x}-\frac {\left (a-\sqrt {a^2+4 b}\right ) \left (2 \sqrt {b}+\sqrt {-2 a^2+2 \sqrt {a^2+4 b} a-4 b}\right ) \sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {a x^2-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x^2-b}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{b} \left (a^2-\sqrt {a^2+4 b} a+4 b\right ) x}+\frac {\left (a+\sqrt {a^2+4 b}\right ) \left (\sqrt {2} \sqrt {b}+\sqrt {-a^2-\sqrt {a^2+4 b} a-2 b}\right )^2 \sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {a x^2-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x^2-b}\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \left (\sqrt {2}-\frac {\sqrt {-a^2-\sqrt {a^2+4 b} a-2 b}}{\sqrt {b}}\right )^2}{4 \sqrt {2} \sqrt {-a^2-\sqrt {a^2+4 b} a-2 b}},2 \arctan \left (\frac {\sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} \sqrt [4]{b} \sqrt {-a^2-\sqrt {a^2+4 b} a-2 b} \left (a^2+\sqrt {a^2+4 b} a+4 b\right ) x}-\frac {\left (a-\sqrt {a^2+4 b}\right ) \left (\sqrt {2} \sqrt {b}-\sqrt {-a^2+\sqrt {a^2+4 b} a-2 b}\right )^2 \sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {a x^2-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x^2-b}\right ) \operatorname {EllipticPi}\left (\frac {\left (\sqrt {2} \sqrt {b}+\sqrt {-a^2+\sqrt {a^2+4 b} a-2 b}\right )^2}{4 \sqrt {2} \sqrt {b} \sqrt {-a^2+\sqrt {a^2+4 b} a-2 b}},2 \arctan \left (\frac {\sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} \sqrt [4]{b} \left (a^2-\sqrt {a^2+4 b} a+4 b\right ) \sqrt {-a^2+\sqrt {a^2+4 b} a-2 b} x}+\frac {\left (a-\sqrt {a^2+4 b}\right ) \left (\sqrt {2} \sqrt {b}+\sqrt {-a^2+\sqrt {a^2+4 b} a-2 b}\right )^2 \sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {a x^2-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x^2-b}\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \left (\sqrt {2}-\frac {\sqrt {-a^2+\sqrt {a^2+4 b} a-2 b}}{\sqrt {b}}\right )^2}{4 \sqrt {2} \sqrt {-a^2+\sqrt {a^2+4 b} a-2 b}},2 \arctan \left (\frac {\sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} \sqrt [4]{b} \left (a^2-\sqrt {a^2+4 b} a+4 b\right ) \sqrt {-a^2+\sqrt {a^2+4 b} a-2 b} x}\)

input

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

output

-1/2*(Sqrt[b]*Sqrt[a + Sqrt[a^2 + 4*b]]*Sqrt[(a*x^2)/b]*ArcTan[(Sqrt[a]*Sq 
rt[a + Sqrt[a^2 + 4*b]]*(-b + a*x^2)^(1/4))/(2^(1/4)*Sqrt[b]*(-a^2 - 2*b - 
 a*Sqrt[a^2 + 4*b])^(1/4)*Sqrt[(a*x^2)/b])])/(2^(1/4)*Sqrt[a]*(-a^2 - 2*b 
- a*Sqrt[a^2 + 4*b])^(1/4)*x) - (Sqrt[b]*Sqrt[a - Sqrt[a^2 + 4*b]]*Sqrt[(a 
*x^2)/b]*ArcTan[(Sqrt[a]*Sqrt[a - Sqrt[a^2 + 4*b]]*(-b + a*x^2)^(1/4))/(2^ 
(1/4)*Sqrt[b]*(-a^2 - 2*b + a*Sqrt[a^2 + 4*b])^(1/4)*Sqrt[(a*x^2)/b])])/(2 
*2^(1/4)*Sqrt[a]*(-a^2 - 2*b + a*Sqrt[a^2 + 4*b])^(1/4)*x) + (Sqrt[b]*Sqrt 
[a + Sqrt[a^2 + 4*b]]*Sqrt[(a*x^2)/b]*ArcTanh[(Sqrt[a]*Sqrt[a + Sqrt[a^2 + 
 4*b]]*(-b + a*x^2)^(1/4))/(2^(1/4)*Sqrt[b]*(-a^2 - 2*b - a*Sqrt[a^2 + 4*b 
])^(1/4)*Sqrt[(a*x^2)/b])])/(2*2^(1/4)*Sqrt[a]*(-a^2 - 2*b - a*Sqrt[a^2 + 
4*b])^(1/4)*x) + (Sqrt[b]*Sqrt[a - Sqrt[a^2 + 4*b]]*Sqrt[(a*x^2)/b]*ArcTan 
h[(Sqrt[a]*Sqrt[a - Sqrt[a^2 + 4*b]]*(-b + a*x^2)^(1/4))/(2^(1/4)*Sqrt[b]* 
(-a^2 - 2*b + a*Sqrt[a^2 + 4*b])^(1/4)*Sqrt[(a*x^2)/b])])/(2*2^(1/4)*Sqrt[ 
a]*(-a^2 - 2*b + a*Sqrt[a^2 + 4*b])^(1/4)*x) - ((a + Sqrt[a^2 + 4*b])*(2*S 
qrt[b] - Sqrt[2]*Sqrt[-a^2 - 2*b - a*Sqrt[a^2 + 4*b]])*Sqrt[(a*x^2)/(Sqrt[ 
b] + Sqrt[-b + a*x^2])^2]*(Sqrt[b] + Sqrt[-b + a*x^2])*EllipticF[2*ArcTan[ 
(-b + a*x^2)^(1/4)/b^(1/4)], 1/2])/(4*b^(1/4)*(a^2 + 4*b + a*Sqrt[a^2 + 4* 
b])*x) - ((a + Sqrt[a^2 + 4*b])*(2*Sqrt[b] + Sqrt[2]*Sqrt[-a^2 - 2*b - a*S 
qrt[a^2 + 4*b]])*Sqrt[(a*x^2)/(Sqrt[b] + Sqrt[-b + a*x^2])^2]*(Sqrt[b] + S 
qrt[-b + a*x^2])*EllipticF[2*ArcTan[(-b + a*x^2)^(1/4)/b^(1/4)], 1/2])/...
 

3.15.43.3.1 Defintions of rubi rules used

rule 2009

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

rule 2256

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^ 
(p_.), x_Symbol] :> Int[ExpandIntegrand[Px*(d + e*x^2)^q*(a + b*x^2 + c*x^4 
)^p, x], x] /; FreeQ[{a, b, c, d, e, q}, x] && PolyQ[Px, x] && IntegerQ[p]
 

3.15.43.4 Maple [F]

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

input

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

output

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

3.15.43.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.15.43.6 Sympy [F]

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

input

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

output

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

3.15.43.7 Maxima [F]

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

input

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

output

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

3.15.43.8 Giac [F]

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

input

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

output

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

3.15.43.9 Mupad [F(-1)]

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

input

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

output

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