Integrand size = 27, antiderivative size = 91 \[ \int \frac {1}{\sqrt {a c+(b c-a d) x^2-b d x^4}} \, dx=\frac {\sqrt {c} \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {a c+(b c-a d) x^2-b d x^4}} \] Output:
c^(1/2)*(1+b*x^2/a)^(1/2)*(1-d*x^2/c)^(1/2)*EllipticF(d^(1/2)*x/c^(1/2),(- b*c/a/d)^(1/2))/d^(1/2)/(a*c+(-a*d+b*c)*x^2-b*d*x^4)^(1/2)
Time = 0.05 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.95 \[ \int \frac {1}{\sqrt {a c+(b c-a d) x^2-b d x^4}} \, dx=\frac {\sqrt {\frac {a+b x^2}{a}} \sqrt {\frac {c-d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )}{\sqrt {-\frac {b}{a}} \sqrt {\left (a+b x^2\right ) \left (c-d x^2\right )}} \] Input:
Integrate[1/Sqrt[a*c + (b*c - a*d)*x^2 - b*d*x^4],x]
Output:
(Sqrt[(a + b*x^2)/a]*Sqrt[(c - d*x^2)/c]*EllipticF[ArcSin[Sqrt[-(b/a)]*x], -((a*d)/(b*c))])/(Sqrt[-(b/a)]*Sqrt[(a + b*x^2)*(c - d*x^2)])
Time = 0.34 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1417, 321}
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.
\(\displaystyle \int \frac {1}{\sqrt {x^2 (b c-a d)+a c-b d x^4}} \, dx\) |
\(\Big \downarrow \) 1417 |
\(\displaystyle \frac {\sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} \int \frac {1}{\sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}}}dx}{\sqrt {x^2 (b c-a d)+a c-b d x^4}}\) |
\(\Big \downarrow \) 321 |
\(\displaystyle \frac {\sqrt {c} \sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {x^2 (b c-a d)+a c-b d x^4}}\) |
Input:
Int[1/Sqrt[a*c + (b*c - a*d)*x^2 - b*d*x^4],x]
Output:
(Sqrt[c]*Sqrt[1 + (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(Sqrt[d] *x)/Sqrt[c]], -((b*c)/(a*d))])/(Sqrt[d]*Sqrt[a*c + (b*c - a*d)*x^2 - b*d*x ^4])
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b ^2 - 4*a*c, 2]}, Simp[Sqrt[1 + 2*c*(x^2/(b - q))]*(Sqrt[1 + 2*c*(x^2/(b + q ))]/Sqrt[a + b*x^2 + c*x^4]) Int[1/(Sqrt[1 + 2*c*(x^2/(b - q))]*Sqrt[1 + 2*c*(x^2/(b + q))]), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[c/a]
Time = 0.54 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.99
method | result | size |
default | \(\frac {\sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}}\) | \(90\) |
elliptic | \(\frac {\sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}}\) | \(90\) |
Input:
int(1/(a*c+(-a*d+b*c)*x^2-b*d*x^4)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/(d/c)^(1/2)*(1-d*x^2/c)^(1/2)*(1+b*x^2/a)^(1/2)/(-b*d*x^4-a*d*x^2+b*c*x^ 2+a*c)^(1/2)*EllipticF(x*(d/c)^(1/2),(-1-(-a*d+b*c)/a/d)^(1/2))
Time = 0.08 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.44 \[ \int \frac {1}{\sqrt {a c+(b c-a d) x^2-b d x^4}} \, dx=\frac {\sqrt {a c} \sqrt {\frac {d}{c}} F(\arcsin \left (x \sqrt {\frac {d}{c}}\right )\,|\,-\frac {b c}{a d})}{a d} \] Input:
integrate(1/(a*c+(-a*d+b*c)*x^2-b*d*x^4)^(1/2),x, algorithm="fricas")
Output:
sqrt(a*c)*sqrt(d/c)*elliptic_f(arcsin(x*sqrt(d/c)), -b*c/(a*d))/(a*d)
\[ \int \frac {1}{\sqrt {a c+(b c-a d) x^2-b d x^4}} \, dx=\int \frac {1}{\sqrt {a c - b d x^{4} + x^{2} \left (- a d + b c\right )}}\, dx \] Input:
integrate(1/(a*c+(-a*d+b*c)*x**2-b*d*x**4)**(1/2),x)
Output:
Integral(1/sqrt(a*c - b*d*x**4 + x**2*(-a*d + b*c)), x)
\[ \int \frac {1}{\sqrt {a c+(b c-a d) x^2-b d x^4}} \, dx=\int { \frac {1}{\sqrt {-b d x^{4} + {\left (b c - a d\right )} x^{2} + a c}} \,d x } \] Input:
integrate(1/(a*c+(-a*d+b*c)*x^2-b*d*x^4)^(1/2),x, algorithm="maxima")
Output:
integrate(1/sqrt(-b*d*x^4 + (b*c - a*d)*x^2 + a*c), x)
\[ \int \frac {1}{\sqrt {a c+(b c-a d) x^2-b d x^4}} \, dx=\int { \frac {1}{\sqrt {-b d x^{4} + {\left (b c - a d\right )} x^{2} + a c}} \,d x } \] Input:
integrate(1/(a*c+(-a*d+b*c)*x^2-b*d*x^4)^(1/2),x, algorithm="giac")
Output:
integrate(1/sqrt(-b*d*x^4 + (b*c - a*d)*x^2 + a*c), x)
Timed out. \[ \int \frac {1}{\sqrt {a c+(b c-a d) x^2-b d x^4}} \, dx=\int \frac {1}{\sqrt {-b\,d\,x^4+\left (b\,c-a\,d\right )\,x^2+a\,c}} \,d x \] Input:
int(1/(a*c - x^2*(a*d - b*c) - b*d*x^4)^(1/2),x)
Output:
int(1/(a*c - x^2*(a*d - b*c) - b*d*x^4)^(1/2), x)
\[ \int \frac {1}{\sqrt {a c+(b c-a d) x^2-b d x^4}} \, dx=\int \frac {\sqrt {-d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{-b d \,x^{4}-a d \,x^{2}+b c \,x^{2}+a c}d x \] Input:
int(1/(a*c+(-a*d+b*c)*x^2-b*d*x^4)^(1/2),x)
Output:
int((sqrt(c - d*x**2)*sqrt(a + b*x**2))/(a*c - a*d*x**2 + b*c*x**2 - b*d*x **4),x)