Integrand size = 32, antiderivative size = 105 \[ \int \frac {1}{\sqrt {a+\frac {\left (c d^2+a e^2\right ) x^2}{d e}+c x^4}} \, dx=\frac {\sqrt {\frac {d \left (a e+c d x^2\right )}{a e \left (d+e x^2\right )}} \left (d+e x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),1-\frac {c d^2}{a e^2}\right )}{\sqrt {d} \sqrt {e} \sqrt {a+\left (\frac {c d}{e}+\frac {a e}{d}\right ) x^2+c x^4}} \] Output:
(d*(c*d*x^2+a*e)/a/e/(e*x^2+d))^(1/2)*(e*x^2+d)*InverseJacobiAM(arctan(e^( 1/2)*x/d^(1/2)),(1-c*d^2/a/e^2)^(1/2))/d^(1/2)/e^(1/2)/(a+(c*d/e+a*e/d)*x^ 2+c*x^4)^(1/2)
Time = 1.73 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.99 \[ \int \frac {1}{\sqrt {a+\frac {\left (c d^2+a e^2\right ) x^2}{d e}+c x^4}} \, dx=\frac {\sqrt {1+\frac {c d x^2}{a e}} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {-\frac {c d}{a e}} x\right ),\frac {a e^2}{c d^2}\right )}{\sqrt {-\frac {c d}{a e}} \sqrt {\frac {\left (a e+c d x^2\right ) \left (d+e x^2\right )}{d e}}} \] Input:
Integrate[1/Sqrt[a + ((c*d^2 + a*e^2)*x^2)/(d*e) + c*x^4],x]
Output:
(Sqrt[1 + (c*d*x^2)/(a*e)]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[Sqrt[-((c* d)/(a*e))]*x], (a*e^2)/(c*d^2)])/(Sqrt[-((c*d)/(a*e))]*Sqrt[((a*e + c*d*x^ 2)*(d + e*x^2))/(d*e)])
Time = 0.37 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.47, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, 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.
\(\displaystyle \int \frac {1}{\sqrt {\frac {x^2 \left (a e^2+c d^2\right )}{d e}+a+c x^4}} \, dx\) |
\(\Big \downarrow \) 1416 |
\(\displaystyle \frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {x^2 \left (\frac {a e}{d}+\frac {c d}{e}\right )+a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {c d^2+a e^2}{\sqrt {a} \sqrt {c} d e}\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {x^2 \left (\frac {a e}{d}+\frac {c d}{e}\right )+a+c x^4}}\) |
Input:
Int[1/Sqrt[a + ((c*d^2 + a*e^2)*x^2)/(d*e) + c*x^4],x]
Output:
((Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + ((c*d)/e + (a*e)/d)*x^2 + c*x^4)/(Sqrt[ a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - (c*d^2 + a*e^2)/(Sqrt[a]*Sqrt[c]*d*e))/4])/(2*a^(1/4)*c^(1/4)*Sqrt[a + ((c*d)/e + (a*e)/d)*x^2 + c*x^4])
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]
Time = 0.88 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.04
method | result | size |
default | \(\frac {\sqrt {1+\frac {x^{2} d c}{a e}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {c d}{a e}}, \sqrt {-1+\frac {\left (\frac {c d}{e}+\frac {a e}{d}\right ) e}{d c}}\right )}{\sqrt {-\frac {c d}{a e}}\, \sqrt {a +\frac {x^{2} e a}{d}+\frac {x^{2} d c}{e}+c \,x^{4}}}\) | \(109\) |
elliptic | \(\frac {\sqrt {1+\frac {x^{2} d c}{a e}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {c d}{a e}}, \sqrt {-1+\frac {\left (\frac {c d}{e}+\frac {a e}{d}\right ) e}{d c}}\right )}{\sqrt {-\frac {c d}{a e}}\, \sqrt {a +\frac {x^{2} e a}{d}+\frac {x^{2} d c}{e}+c \,x^{4}}}\) | \(109\) |
Input:
int(1/(a+(a*e^2+c*d^2)*x^2/d/e+c*x^4)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/(-c*d/a/e)^(1/2)*(1+1/a*x^2*d/e*c)^(1/2)*(1+e*x^2/d)^(1/2)/(a+x^2/d*e*a+ x^2*d/e*c+c*x^4)^(1/2)*EllipticF(x*(-c*d/a/e)^(1/2),(-1+(c*d/e+1/d*a*e)*e/ d/c)^(1/2))
Time = 0.08 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.49 \[ \int \frac {1}{\sqrt {a+\frac {\left (c d^2+a e^2\right ) x^2}{d e}+c x^4}} \, dx=-\frac {\sqrt {a} e \sqrt {-\frac {c d}{a e}} F(\arcsin \left (x \sqrt {-\frac {c d}{a e}}\right )\,|\,\frac {a e^{2}}{c d^{2}})}{c d} \] Input:
integrate(1/(a+(a*e^2+c*d^2)*x^2/d/e+c*x^4)^(1/2),x, algorithm="fricas")
Output:
-sqrt(a)*e*sqrt(-c*d/(a*e))*elliptic_f(arcsin(x*sqrt(-c*d/(a*e))), a*e^2/( c*d^2))/(c*d)
\[ \int \frac {1}{\sqrt {a+\frac {\left (c d^2+a e^2\right ) x^2}{d e}+c x^4}} \, dx=\int \frac {1}{\sqrt {a + c x^{4} + \frac {x^{2} \left (a e^{2} + c d^{2}\right )}{d e}}}\, dx \] Input:
integrate(1/(a+(a*e**2+c*d**2)*x**2/d/e+c*x**4)**(1/2),x)
Output:
Integral(1/sqrt(a + c*x**4 + x**2*(a*e**2 + c*d**2)/(d*e)), x)
\[ \int \frac {1}{\sqrt {a+\frac {\left (c d^2+a e^2\right ) x^2}{d e}+c x^4}} \, dx=\int { \frac {1}{\sqrt {c x^{4} + a + \frac {{\left (c d^{2} + a e^{2}\right )} x^{2}}{d e}}} \,d x } \] Input:
integrate(1/(a+(a*e^2+c*d^2)*x^2/d/e+c*x^4)^(1/2),x, algorithm="maxima")
Output:
integrate(1/sqrt(c*x^4 + a + (c*d^2 + a*e^2)*x^2/(d*e)), x)
\[ \int \frac {1}{\sqrt {a+\frac {\left (c d^2+a e^2\right ) x^2}{d e}+c x^4}} \, dx=\int { \frac {1}{\sqrt {c x^{4} + a + \frac {{\left (c d^{2} + a e^{2}\right )} x^{2}}{d e}}} \,d x } \] Input:
integrate(1/(a+(a*e^2+c*d^2)*x^2/d/e+c*x^4)^(1/2),x, algorithm="giac")
Output:
integrate(1/sqrt(c*x^4 + a + (c*d^2 + a*e^2)*x^2/(d*e)), x)
Timed out. \[ \int \frac {1}{\sqrt {a+\frac {\left (c d^2+a e^2\right ) x^2}{d e}+c x^4}} \, dx=\int \frac {1}{\sqrt {a+c\,x^4+\frac {x^2\,\left (c\,d^2+a\,e^2\right )}{d\,e}}} \,d x \] Input:
int(1/(a + c*x^4 + (x^2*(a*e^2 + c*d^2))/(d*e))^(1/2),x)
Output:
int(1/(a + c*x^4 + (x^2*(a*e^2 + c*d^2))/(d*e))^(1/2), x)
\[ \int \frac {1}{\sqrt {a+\frac {\left (c d^2+a e^2\right ) x^2}{d e}+c x^4}} \, dx=\sqrt {e}\, \sqrt {d}\, \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {c d \,x^{2}+a e}}{c d e \,x^{4}+a \,e^{2} x^{2}+c \,d^{2} x^{2}+a d e}d x \right ) \] Input:
int(1/(a+(a*e^2+c*d^2)*x^2/d/e+c*x^4)^(1/2),x)
Output:
sqrt(e)*sqrt(d)*int((sqrt(d + e*x**2)*sqrt(a*e + c*d*x**2))/(a*d*e + a*e** 2*x**2 + c*d**2*x**2 + c*d*e*x**4),x)