\(\int \frac {1}{\sqrt {\text {a1}+\text {a2}+b x^2+c x^4}} \, dx\) [167]

Optimal result

Integrand size = 17, antiderivative size = 133 \[ \int \frac {1}{\sqrt {\text {a1}+\text {a2}+b x^2+c x^4}} \, dx=\frac {\sqrt [4]{\text {a1}+\text {a2}} \left (1+\frac {\sqrt {c} x^2}{\sqrt {\text {a1}+\text {a2}}}\right ) \sqrt {\frac {\text {a1}+\text {a2}+b x^2+c x^4}{(\text {a1}+\text {a2}) \left (1+\frac {\sqrt {c} x^2}{\sqrt {\text {a1}+\text {a2}}}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{\text {a1}+\text {a2}}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {\text {a1}+\text {a2}} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \sqrt {\text {a1}+\text {a2}+b x^2+c x^4}} \] Output:

1/2*(a1+a2)^(1/4)*(1+c^(1/2)*x^2/(a1+a2)^(1/2))*((c*x^4+b*x^2+a1+a2)/(a1+a 
2)/(1+c^(1/2)*x^2/(a1+a2)^(1/2))^2)^(1/2)*InverseJacobiAM(2*arctan(c^(1/4) 
*x/(a1+a2)^(1/4)),1/2*(2-b/(a1+a2)^(1/2)/c^(1/2))^(1/2))/c^(1/4)/(c*x^4+b* 
x^2+a1+a2)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.20 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.41 \[ \int \frac {1}{\sqrt {\text {a1}+\text {a2}+b x^2+c x^4}} \, dx=-\frac {i \sqrt {1-\frac {2 c x^2}{-b+\sqrt {b^2-4 (\text {a1}+\text {a2}) c}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 (\text {a1}+\text {a2}) c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 (\text {a1}+\text {a2}) c}}} x\right ),-\frac {b+\sqrt {b^2-4 (\text {a1}+\text {a2}) c}}{-b+\sqrt {b^2-4 (\text {a1}+\text {a2}) c}}\right )}{\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 (\text {a1}+\text {a2}) c}}} \sqrt {\text {a1}+\text {a2}+b x^2+c x^4}} \] Input:

Integrate[1/Sqrt[a1 + a2 + b*x^2 + c*x^4],x]
 

Output:

((-I)*Sqrt[1 - (2*c*x^2)/(-b + Sqrt[b^2 - 4*(a1 + a2)*c])]*Sqrt[1 + (2*c*x 
^2)/(b + Sqrt[b^2 - 4*(a1 + a2)*c])]*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b 
 + Sqrt[b^2 - 4*(a1 + a2)*c])]*x], -((b + Sqrt[b^2 - 4*(a1 + a2)*c])/(-b + 
 Sqrt[b^2 - 4*(a1 + a2)*c]))])/(Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*(a1 + a2) 
*c])]*Sqrt[a1 + a2 + b*x^2 + c*x^4])
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {1416}

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.

\(\displaystyle \int \frac {1}{\sqrt {\text {a1}+\text {a2}+b x^2+c x^4}} \, dx\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {\sqrt [4]{\text {a1}+\text {a2}} \left (\frac {\sqrt {c} x^2}{\sqrt {\text {a1}+\text {a2}}}+1\right ) \sqrt {\frac {\text {a1}+\text {a2}+b x^2+c x^4}{(\text {a1}+\text {a2}) \left (\frac {\sqrt {c} x^2}{\sqrt {\text {a1}+\text {a2}}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{\text {a1}+\text {a2}}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {\text {a1}+\text {a2}} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \sqrt {\text {a1}+\text {a2}+b x^2+c x^4}}\)

Input:

Int[1/Sqrt[a1 + a2 + b*x^2 + c*x^4],x]
 

Output:

((a1 + a2)^(1/4)*(1 + (Sqrt[c]*x^2)/Sqrt[a1 + a2])*Sqrt[(a1 + a2 + b*x^2 + 
 c*x^4)/((a1 + a2)*(1 + (Sqrt[c]*x^2)/Sqrt[a1 + a2])^2)]*EllipticF[2*ArcTa 
n[(c^(1/4)*x)/(a1 + a2)^(1/4)], (2 - b/(Sqrt[a1 + a2]*Sqrt[c]))/4])/(2*c^( 
1/4)*Sqrt[a1 + a2 + b*x^2 + c*x^4])
 

Defintions of rubi rules used

rule 1416

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c 
/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ 
(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) 
], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.32

method	result	size
default	\(\frac {\sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 \operatorname {a1} c -4 c \operatorname {a2} +b^{2}}\right ) x^{2}}{\operatorname {a1} +\operatorname {a2}}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 \operatorname {a1} c -4 c \operatorname {a2} +b^{2}}\right ) x^{2}}{\operatorname {a1} +\operatorname {a2}}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 \operatorname {a1} c -4 c \operatorname {a2} +b^{2}}}{\operatorname {a1} +\operatorname {a2}}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 \operatorname {a1} c -4 c \operatorname {a2} +b^{2}}\right )}{\left (\operatorname {a1} +\operatorname {a2} \right ) c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 \operatorname {a1} c -4 c \operatorname {a2} +b^{2}}}{\operatorname {a1} +\operatorname {a2}}}\, \sqrt {c \,x^{4}+b \,x^{2}+\operatorname {a1} +\operatorname {a2}}}\)	\(175\)
elliptic	\(\frac {\sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 \operatorname {a1} c -4 c \operatorname {a2} +b^{2}}\right ) x^{2}}{\operatorname {a1} +\operatorname {a2}}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 \operatorname {a1} c -4 c \operatorname {a2} +b^{2}}\right ) x^{2}}{\operatorname {a1} +\operatorname {a2}}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 \operatorname {a1} c -4 c \operatorname {a2} +b^{2}}}{\operatorname {a1} +\operatorname {a2}}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 \operatorname {a1} c -4 c \operatorname {a2} +b^{2}}\right )}{\left (\operatorname {a1} +\operatorname {a2} \right ) c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 \operatorname {a1} c -4 c \operatorname {a2} +b^{2}}}{\operatorname {a1} +\operatorname {a2}}}\, \sqrt {c \,x^{4}+b \,x^{2}+\operatorname {a1} +\operatorname {a2}}}\)	\(175\)

Input:

int(1/(c*x^4+b*x^2+a1+a2)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/4*2^(1/2)/((-b+(-4*a1*c-4*a2*c+b^2)^(1/2))/(a1+a2))^(1/2)*(4-2*(-b+(-4*a 
1*c-4*a2*c+b^2)^(1/2))/(a1+a2)*x^2)^(1/2)*(4+2*(b+(-4*a1*c-4*a2*c+b^2)^(1/ 
2))/(a1+a2)*x^2)^(1/2)/(c*x^4+b*x^2+a1+a2)^(1/2)*EllipticF(1/2*x*2^(1/2)*( 
(-b+(-4*a1*c-4*a2*c+b^2)^(1/2))/(a1+a2))^(1/2),1/2*(-4+2*b*(b+(-4*a1*c-4*a 
2*c+b^2)^(1/2))/(a1+a2)/c)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.41 \[ \int \frac {1}{\sqrt {\text {a1}+\text {a2}+b x^2+c x^4}} \, dx=-\frac {\sqrt {\frac {1}{2}} {\left ({\left (a_{1} + a_{2}\right )} \sqrt {\frac {b^{2} - 4 \, {\left (a_{1} + a_{2}\right )} c}{a_{1}^{2} + 2 \, a_{1} a_{2} + a_{2}^{2}}} + b\right )} \sqrt {\frac {{\left (a_{1} + a_{2}\right )} \sqrt {\frac {b^{2} - 4 \, {\left (a_{1} + a_{2}\right )} c}{a_{1}^{2} + 2 \, a_{1} a_{2} + a_{2}^{2}}} - b}{a_{1} + a_{2}}} F(\arcsin \left (\sqrt {\frac {1}{2}} x \sqrt {\frac {{\left (a_{1} + a_{2}\right )} \sqrt {\frac {b^{2} - 4 \, {\left (a_{1} + a_{2}\right )} c}{a_{1}^{2} + 2 \, a_{1} a_{2} + a_{2}^{2}}} - b}{a_{1} + a_{2}}}\right )\,|\,\frac {{\left (a_{1} + a_{2}\right )} b \sqrt {\frac {b^{2} - 4 \, {\left (a_{1} + a_{2}\right )} c}{a_{1}^{2} + 2 \, a_{1} a_{2} + a_{2}^{2}}} + b^{2} - 2 \, {\left (a_{1} + a_{2}\right )} c}{2 \, {\left (a_{1} + a_{2}\right )} c})}{2 \, \sqrt {a_{1} + a_{2}} c} \] Input:

integrate(1/(c*x^4+b*x^2+a1+a2)^(1/2),x, algorithm="fricas")
 

Output:

-1/2*sqrt(1/2)*((a1 + a2)*sqrt((b^2 - 4*(a1 + a2)*c)/(a1^2 + 2*a1*a2 + a2^ 
2)) + b)*sqrt(((a1 + a2)*sqrt((b^2 - 4*(a1 + a2)*c)/(a1^2 + 2*a1*a2 + a2^2 
)) - b)/(a1 + a2))*elliptic_f(arcsin(sqrt(1/2)*x*sqrt(((a1 + a2)*sqrt((b^2 
 - 4*(a1 + a2)*c)/(a1^2 + 2*a1*a2 + a2^2)) - b)/(a1 + a2))), 1/2*((a1 + a2 
)*b*sqrt((b^2 - 4*(a1 + a2)*c)/(a1^2 + 2*a1*a2 + a2^2)) + b^2 - 2*(a1 + a2 
)*c)/((a1 + a2)*c))/(sqrt(a1 + a2)*c)
 

Sympy [F]

\[ \int \frac {1}{\sqrt {\text {a1}+\text {a2}+b x^2+c x^4}} \, dx=\int \frac {1}{\sqrt {a_{1} + a_{2} + b x^{2} + c x^{4}}}\, dx \] Input:

integrate(1/(c*x**4+b*x**2+a1+a2)**(1/2),x)
 

Output:

Integral(1/sqrt(a1 + a2 + b*x**2 + c*x**4), x)
 

Maxima [F]

\[ \int \frac {1}{\sqrt {\text {a1}+\text {a2}+b x^2+c x^4}} \, dx=\int { \frac {1}{\sqrt {c x^{4} + b x^{2} + a_{1} + a_{2}}} \,d x } \] Input:

integrate(1/(c*x^4+b*x^2+a1+a2)^(1/2),x, algorithm="maxima")
 

Output:

integrate(1/sqrt(c*x^4 + b*x^2 + a1 + a2), x)
 

Giac [F]

\[ \int \frac {1}{\sqrt {\text {a1}+\text {a2}+b x^2+c x^4}} \, dx=\int { \frac {1}{\sqrt {c x^{4} + b x^{2} + a_{1} + a_{2}}} \,d x } \] Input:

integrate(1/(c*x^4+b*x^2+a1+a2)^(1/2),x, algorithm="giac")
 

Output:

integrate(1/sqrt(c*x^4 + b*x^2 + a1 + a2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {\text {a1}+\text {a2}+b x^2+c x^4}} \, dx=\int \frac {1}{\sqrt {c\,x^4+b\,x^2+a_{1}+a_{2}}} \,d x \] Input:

int(1/(a1 + a2 + b*x^2 + c*x^4)^(1/2),x)
 

Output:

int(1/(a1 + a2 + b*x^2 + c*x^4)^(1/2), x)
 

Reduce [F]

\[ \int \frac {1}{\sqrt {\text {a1}+\text {a2}+b x^2+c x^4}} \, dx=\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+\mathit {a1} +\mathit {a2}}}{c \,x^{4}+b \,x^{2}+\mathit {a1} +\mathit {a2}}d x \] Input:

int(1/(c*x^4+b*x^2+a1+a2)^(1/2),x)
 

Output:

int(sqrt(a1 + a2 + b*x**2 + c*x**4)/(a1 + a2 + b*x**2 + c*x**4),x)