Integrand size = 30, antiderivative size = 126 \[ \int \frac {1}{\sqrt {\frac {-c d^2+b d e}{e^2}+b x^2+c x^4}} \, dx=\frac {\sqrt {c d-b e} \sqrt {1+\frac {e x^2}{d}} \sqrt {1-\frac {c e x^2}{c d-b e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right ),-1+\frac {b e}{c d}\right )}{\sqrt {c} \sqrt {e} \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}} \] Output:
(-b*e+c*d)^(1/2)*(1+e*x^2/d)^(1/2)*(1-c*e*x^2/(-b*e+c*d))^(1/2)*EllipticF( c^(1/2)*e^(1/2)*x/(-b*e+c*d)^(1/2),(-1+b*e/c/d)^(1/2))/c^(1/2)/e^(1/2)/(-d *(-b*e+c*d)/e^2+b*x^2+c*x^4)^(1/2)
Time = 1.93 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\sqrt {\frac {-c d^2+b d e}{e^2}+b x^2+c x^4}} \, dx=\frac {\sqrt {\frac {-c d+b e+c e x^2}{-c d+b e}} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {-\frac {e}{d}} x\right ),\frac {c d}{-c d+b e}\right )}{\sqrt {-\frac {e}{d}} \sqrt {\frac {\left (d+e x^2\right ) \left (-c d+b e+c e x^2\right )}{e^2}}} \] Input:
Integrate[1/Sqrt[(-(c*d^2) + b*d*e)/e^2 + b*x^2 + c*x^4],x]
Output:
(Sqrt[(-(c*d) + b*e + c*e*x^2)/(-(c*d) + b*e)]*Sqrt[1 + (e*x^2)/d]*Ellipti cF[ArcSin[Sqrt[-(e/d)]*x], (c*d)/(-(c*d) + b*e)])/(Sqrt[-(e/d)]*Sqrt[((d + e*x^2)*(-(c*d) + b*e + c*e*x^2))/e^2])
Time = 0.41 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, 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.
\(\displaystyle \int \frac {1}{\sqrt {\frac {b d e-c d^2}{e^2}+b x^2+c x^4}} \, dx\) |
\(\Big \downarrow \) 1417 |
\(\displaystyle \frac {\sqrt {\frac {e x^2}{d}+1} \sqrt {1-\frac {c e x^2}{c d-b e}} \int \frac {1}{\sqrt {\frac {e x^2}{d}+1} \sqrt {1-\frac {c e x^2}{c d-b e}}}dx}{\sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}\) |
\(\Big \downarrow \) 321 |
\(\displaystyle \frac {\sqrt {\frac {e x^2}{d}+1} \sqrt {c d-b e} \sqrt {1-\frac {c e x^2}{c d-b e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right ),\frac {b e}{c d}-1\right )}{\sqrt {c} \sqrt {e} \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}\) |
Input:
Int[1/Sqrt[(-(c*d^2) + b*d*e)/e^2 + b*x^2 + c*x^4],x]
Output:
(Sqrt[c*d - b*e]*Sqrt[1 + (e*x^2)/d]*Sqrt[1 - (c*e*x^2)/(c*d - b*e)]*Ellip ticF[ArcSin[(Sqrt[c]*Sqrt[e]*x)/Sqrt[c*d - b*e]], -1 + (b*e)/(c*d)])/(Sqrt [c]*Sqrt[e]*Sqrt[-((d*(c*d - b*e))/e^2) + b*x^2 + c*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 = 1.86 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.87
method | result | size |
default | \(\frac {\sqrt {1+\frac {c e \,x^{2}}{b e -c d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {c e}{b e -c d}}, \sqrt {-1+\frac {b e}{c d}}\right )}{\sqrt {-\frac {c e}{b e -c d}}\, \sqrt {c \,x^{4}+b \,x^{2}+\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}}}\) | \(110\) |
elliptic | \(\frac {\sqrt {1+\frac {c e \,x^{2}}{b e -c d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {c e}{b e -c d}}, \sqrt {-1+\frac {b e}{c d}}\right )}{\sqrt {-\frac {c e}{b e -c d}}\, \sqrt {c \,x^{4}+b \,x^{2}+\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}}}\) | \(110\) |
Input:
int(1/((b*d*e-c*d^2)/e^2+b*x^2+c*x^4)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/(-c*e/(b*e-c*d))^(1/2)*(1+c*e/(b*e-c*d)*x^2)^(1/2)*(1+e*x^2/d)^(1/2)/(c* x^4+b*x^2+b*d/e-c*d^2/e^2)^(1/2)*EllipticF(x*(-c*e/(b*e-c*d))^(1/2),(-1+b* e/c/d)^(1/2))
Time = 0.08 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.56 \[ \int \frac {1}{\sqrt {\frac {-c d^2+b d e}{e^2}+b x^2+c x^4}} \, dx=-\frac {\sqrt {-c d^{2} + b d e} \sqrt {\frac {c e}{c d - b e}} F(\arcsin \left (\sqrt {\frac {c e}{c d - b e}} x\right )\,|\,-\frac {c d - b e}{c d})}{c d} \] Input:
integrate(1/((b*d*e-c*d^2)/e^2+b*x^2+c*x^4)^(1/2),x, algorithm="fricas")
Output:
-sqrt(-c*d^2 + b*d*e)*sqrt(c*e/(c*d - b*e))*elliptic_f(arcsin(sqrt(c*e/(c* d - b*e))*x), -(c*d - b*e)/(c*d))/(c*d)
\[ \int \frac {1}{\sqrt {\frac {-c d^2+b d e}{e^2}+b x^2+c x^4}} \, dx=\int \frac {1}{\sqrt {b x^{2} + c x^{4} + \frac {b d e - c d^{2}}{e^{2}}}}\, dx \] Input:
integrate(1/((b*d*e-c*d**2)/e**2+b*x**2+c*x**4)**(1/2),x)
Output:
Integral(1/sqrt(b*x**2 + c*x**4 + (b*d*e - c*d**2)/e**2), x)
\[ \int \frac {1}{\sqrt {\frac {-c d^2+b d e}{e^2}+b x^2+c x^4}} \, dx=\int { \frac {1}{\sqrt {c x^{4} + b x^{2} - \frac {c d^{2} - b d e}{e^{2}}}} \,d x } \] Input:
integrate(1/((b*d*e-c*d^2)/e^2+b*x^2+c*x^4)^(1/2),x, algorithm="maxima")
Output:
integrate(1/sqrt(c*x^4 + b*x^2 - (c*d^2 - b*d*e)/e^2), x)
\[ \int \frac {1}{\sqrt {\frac {-c d^2+b d e}{e^2}+b x^2+c x^4}} \, dx=\int { \frac {1}{\sqrt {c x^{4} + b x^{2} - \frac {c d^{2} - b d e}{e^{2}}}} \,d x } \] Input:
integrate(1/((b*d*e-c*d^2)/e^2+b*x^2+c*x^4)^(1/2),x, algorithm="giac")
Output:
integrate(1/sqrt(c*x^4 + b*x^2 - (c*d^2 - b*d*e)/e^2), x)
Timed out. \[ \int \frac {1}{\sqrt {\frac {-c d^2+b d e}{e^2}+b x^2+c x^4}} \, dx=\int \frac {1}{\sqrt {b\,x^2-\frac {c\,d^2-b\,d\,e}{e^2}+c\,x^4}} \,d x \] Input:
int(1/(b*x^2 - (c*d^2 - b*d*e)/e^2 + c*x^4)^(1/2),x)
Output:
int(1/(b*x^2 - (c*d^2 - b*d*e)/e^2 + c*x^4)^(1/2), x)
\[ \int \frac {1}{\sqrt {\frac {-c d^2+b d e}{e^2}+b x^2+c x^4}} \, dx=\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {c e \,x^{2}+b e -c d}}{c \,e^{2} x^{4}+b \,e^{2} x^{2}+b d e -c \,d^{2}}d x \right ) e \] Input:
int(1/((b*d*e-c*d^2)/e^2+b*x^2+c*x^4)^(1/2),x)
Output:
int((sqrt(d + e*x**2)*sqrt(b*e - c*d + c*e*x**2))/(b*d*e + b*e**2*x**2 - c *d**2 + c*e**2*x**4),x)*e