3.20.15 \(\int \frac {-2 b+a x^2}{\sqrt [4]{-b+a x^2} (-b+a x^2+c x^4)} \, dx\) [1915]

3.20.15.1 Optimal result

Integrand size = 39, antiderivative size = 133 \[ \int \frac {-2 b+a x^2}{\sqrt [4]{-b+a x^2} \left (-b+a x^2+c x^4\right )} \, dx=-\frac {\arctan \left (\frac {-\frac {\sqrt [4]{c} x^2}{\sqrt {2}}+\frac {\sqrt {-b+a x^2}}{\sqrt {2} \sqrt [4]{c}}}{x \sqrt [4]{-b+a x^2}}\right )}{\sqrt {2} \sqrt [4]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{-b+a x^2}}{\sqrt {c} x^2+\sqrt {-b+a x^2}}\right )}{\sqrt {2} \sqrt [4]{c}} \]

output

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

3.20.15.2 Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.88 \[ \int \frac {-2 b+a x^2}{\sqrt [4]{-b+a x^2} \left (-b+a x^2+c x^4\right )} \, dx=\frac {-\arctan \left (\frac {-\sqrt {c} x^2+\sqrt {-b+a x^2}}{\sqrt {2} \sqrt [4]{c} x \sqrt [4]{-b+a x^2}}\right )+\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{-b+a x^2}}{\sqrt {c} x^2+\sqrt {-b+a x^2}}\right )}{\sqrt {2} \sqrt [4]{c}} \]

input

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

output

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

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

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

Time = 21.61 (sec) , antiderivative size = 2667, normalized size of antiderivative = 20.05, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2256, 7239, 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.

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

\(\Big \downarrow \) 2256

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2256

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

\(\Big \downarrow \) 2009

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

input

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

output

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

3.20.15.3.1 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]
 

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

3.20.15.4 Maple [F]

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

input

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

output

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

3.20.15.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.20.15.6 Sympy [F]

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

input

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

output

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

3.20.15.7 Maxima [F]

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

input

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

output

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

3.20.15.8 Giac [F]

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

input

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

output

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

3.20.15.9 Mupad [F(-1)]

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

input

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

output

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