Integrand size = 29, antiderivative size = 104 \[ \int \frac {1}{\sqrt {a+b x^2+\frac {\left (b d e-a e^2\right ) x^4}{d^2}}} \, dx=\frac {\left (d+e x^2\right ) \sqrt {\frac {a d+(b d-a e) x^2}{a \left (d+e x^2\right )}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),2-\frac {b d}{a e}\right )}{\sqrt {d} \sqrt {e} \sqrt {a+b x^2+\frac {e (b d-a e) x^4}{d^2}}} \] Output:
(e*x^2+d)*((a*d+(-a*e+b*d)*x^2)/a/(e*x^2+d))^(1/2)*InverseJacobiAM(arctan( e^(1/2)*x/d^(1/2)),(2-b*d/a/e)^(1/2))/d^(1/2)/e^(1/2)/(a+b*x^2+e*(-a*e+b*d )*x^4/d^2)^(1/2)
Time = 2.09 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.01 \[ \int \frac {1}{\sqrt {a+b x^2+\frac {\left (b d e-a e^2\right ) x^4}{d^2}}} \, dx=\frac {\sqrt {1+\frac {b x^2}{a}-\frac {e x^2}{d}} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {-\frac {e}{d}} x\right ),-1+\frac {b d}{a e}\right )}{\sqrt {-\frac {e}{d}} \sqrt {\frac {\left (d+e x^2\right ) \left (b d x^2+a \left (d-e x^2\right )\right )}{d^2}}} \] Input:
Integrate[1/Sqrt[a + b*x^2 + ((b*d*e - a*e^2)*x^4)/d^2],x]
Output:
(Sqrt[1 + (b*x^2)/a - (e*x^2)/d]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[Sqrt [-(e/d)]*x], -1 + (b*d)/(a*e)])/(Sqrt[-(e/d)]*Sqrt[((d + e*x^2)*(b*d*x^2 + a*(d - e*x^2)))/d^2])
Leaf count is larger than twice the leaf count of optimal. \(214\) vs. \(2(104)=208\).
Time = 0.45 (sec) , antiderivative size = 214, normalized size of antiderivative = 2.06, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, 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.
\(\displaystyle \int \frac {1}{\sqrt {\frac {x^4 \left (b d e-a e^2\right )}{d^2}+a+b x^2}} \, dx\) |
\(\Big \downarrow \) 1416 |
\(\displaystyle \frac {\left (\sqrt {e} x^2 \sqrt {b d-a e}+\sqrt {a} d\right ) \sqrt {\frac {d^2 \left (\frac {e x^4 (b d-a e)}{d^2}+a+b x^2\right )}{\left (\sqrt {e} x^2 \sqrt {b d-a e}+\sqrt {a} d\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{e} \sqrt [4]{b d-a e} x}{\sqrt [4]{a} \sqrt {d}}\right ),\frac {1}{4} \left (2-\frac {b d}{\sqrt {a} \sqrt {e} \sqrt {b d-a e}}\right )\right )}{2 \sqrt [4]{a} \sqrt {d} \sqrt [4]{e} \sqrt [4]{b d-a e} \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}\) |
Input:
Int[1/Sqrt[a + b*x^2 + ((b*d*e - a*e^2)*x^4)/d^2],x]
Output:
((Sqrt[a]*d + Sqrt[e]*Sqrt[b*d - a*e]*x^2)*Sqrt[(d^2*(a + b*x^2 + (e*(b*d - a*e)*x^4)/d^2))/(Sqrt[a]*d + Sqrt[e]*Sqrt[b*d - a*e]*x^2)^2]*EllipticF[2 *ArcTan[(e^(1/4)*(b*d - a*e)^(1/4)*x)/(a^(1/4)*Sqrt[d])], (2 - (b*d)/(Sqrt [a]*Sqrt[e]*Sqrt[b*d - a*e]))/4])/(2*a^(1/4)*Sqrt[d]*e^(1/4)*(b*d - a*e)^( 1/4)*Sqrt[a + b*x^2 + (e*(b*d - a*e)*x^4)/d^2])
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 = 0.78 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.27
method | result | size |
default | \(\frac {\sqrt {1-\frac {\left (a e -b d \right ) x^{2}}{a d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {a e -b d}{a d}}, \sqrt {-1+\frac {b e}{d \left (-\frac {a \,e^{2}}{d^{2}}+\frac {e b}{d}\right )}}\right )}{\sqrt {\frac {a e -b d}{a d}}\, \sqrt {-\frac {x^{4} a \,e^{2}}{d^{2}}+\frac {e \,x^{4} b}{d}+b \,x^{2}+a}}\) | \(132\) |
elliptic | \(\frac {\sqrt {1-\frac {\left (a e -b d \right ) x^{2}}{a d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {a e -b d}{a d}}, \sqrt {-1+\frac {b e}{d \left (-\frac {a \,e^{2}}{d^{2}}+\frac {e b}{d}\right )}}\right )}{\sqrt {\frac {a e -b d}{a d}}\, \sqrt {-\frac {x^{4} a \,e^{2}}{d^{2}}+\frac {e \,x^{4} b}{d}+b \,x^{2}+a}}\) | \(132\) |
Input:
int(1/(a+b*x^2+(-a*e^2+b*d*e)*x^4/d^2)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/((a*e-b*d)/a/d)^(1/2)*(1-(a*e-b*d)/a/d*x^2)^(1/2)*(1+e*x^2/d)^(1/2)/(-x^ 4*a/d^2*e^2+e/d*x^4*b+b*x^2+a)^(1/2)*EllipticF(x*((a*e-b*d)/a/d)^(1/2),(-1 +b*e/d/(-a/d^2*e^2+e/d*b))^(1/2))
Time = 0.09 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.69 \[ \int \frac {1}{\sqrt {a+b x^2+\frac {\left (b d e-a e^2\right ) x^4}{d^2}}} \, dx=-\frac {\sqrt {a d^{2}} \sqrt {-\frac {b d - a e}{a d}} F(\arcsin \left (x \sqrt {-\frac {b d - a e}{a d}}\right )\,|\,\frac {a e}{b d - a e})}{b d - a e} \] Input:
integrate(1/(a+b*x^2+(-a*e^2+b*d*e)*x^4/d^2)^(1/2),x, algorithm="fricas")
Output:
-sqrt(a*d^2)*sqrt(-(b*d - a*e)/(a*d))*elliptic_f(arcsin(x*sqrt(-(b*d - a*e )/(a*d))), a*e/(b*d - a*e))/(b*d - a*e)
\[ \int \frac {1}{\sqrt {a+b x^2+\frac {\left (b d e-a e^2\right ) x^4}{d^2}}} \, dx=\int \frac {1}{\sqrt {a + b x^{2} + \frac {x^{4} \left (- a e^{2} + b d e\right )}{d^{2}}}}\, dx \] Input:
integrate(1/(a+b*x**2+(-a*e**2+b*d*e)*x**4/d**2)**(1/2),x)
Output:
Integral(1/sqrt(a + b*x**2 + x**4*(-a*e**2 + b*d*e)/d**2), x)
\[ \int \frac {1}{\sqrt {a+b x^2+\frac {\left (b d e-a e^2\right ) x^4}{d^2}}} \, dx=\int { \frac {1}{\sqrt {b x^{2} + \frac {{\left (b d e - a e^{2}\right )} x^{4}}{d^{2}} + a}} \,d x } \] Input:
integrate(1/(a+b*x^2+(-a*e^2+b*d*e)*x^4/d^2)^(1/2),x, algorithm="maxima")
Output:
integrate(1/sqrt(b*x^2 + (b*d*e - a*e^2)*x^4/d^2 + a), x)
\[ \int \frac {1}{\sqrt {a+b x^2+\frac {\left (b d e-a e^2\right ) x^4}{d^2}}} \, dx=\int { \frac {1}{\sqrt {b x^{2} + \frac {{\left (b d e - a e^{2}\right )} x^{4}}{d^{2}} + a}} \,d x } \] Input:
integrate(1/(a+b*x^2+(-a*e^2+b*d*e)*x^4/d^2)^(1/2),x, algorithm="giac")
Output:
integrate(1/sqrt(b*x^2 + (b*d*e - a*e^2)*x^4/d^2 + a), x)
Timed out. \[ \int \frac {1}{\sqrt {a+b x^2+\frac {\left (b d e-a e^2\right ) x^4}{d^2}}} \, dx=\int \frac {1}{\sqrt {a+b\,x^2-\frac {x^4\,\left (a\,e^2-b\,d\,e\right )}{d^2}}} \,d x \] Input:
int(1/(a + b*x^2 - (x^4*(a*e^2 - b*d*e))/d^2)^(1/2),x)
Output:
int(1/(a + b*x^2 - (x^4*(a*e^2 - b*d*e))/d^2)^(1/2), x)
\[ \int \frac {1}{\sqrt {a+b x^2+\frac {\left (b d e-a e^2\right ) x^4}{d^2}}} \, dx=\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-a e \,x^{2}+b d \,x^{2}+a d}}{-a \,e^{2} x^{4}+b d e \,x^{4}+b \,d^{2} x^{2}+a \,d^{2}}d x \right ) d \] Input:
int(1/(a+b*x^2+(-a*e^2+b*d*e)*x^4/d^2)^(1/2),x)
Output:
int((sqrt(d + e*x**2)*sqrt(a*d - a*e*x**2 + b*d*x**2))/(a*d**2 - a*e**2*x* *4 + b*d**2*x**2 + b*d*e*x**4),x)*d