Integrand size = 21, antiderivative size = 90 \[ \int \frac {1}{\sqrt {\left (a-b x^2\right ) \left (c-d x^2\right )}} \, dx=\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a c-(b c+a d) x^2+b d x^4}} \] Output:
a^(1/2)*(1-b*x^2/a)^(1/2)*(1-d*x^2/c)^(1/2)*EllipticF(b^(1/2)*x/a^(1/2),(a *d/b/c)^(1/2))/b^(1/2)/(a*c-(a*d+b*c)*x^2+b*d*x^4)^(1/2)
Time = 2.02 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\sqrt {\left (a-b x^2\right ) \left (c-d x^2\right )}} \, dx=\frac {\sqrt {\frac {a-b x^2}{a}} \sqrt {\frac {c-d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{\sqrt {\frac {b}{a}} \sqrt {\left (a-b x^2\right ) \left (c-d x^2\right )}} \] Input:
Integrate[1/Sqrt[(a - b*x^2)*(c - d*x^2)],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)])
Leaf count is larger than twice the leaf count of optimal. \(191\) vs. \(2(90)=180\).
Time = 0.44 (sec) , antiderivative size = 191, normalized size of antiderivative = 2.12, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2048, 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.
\(\displaystyle \int \frac {1}{\sqrt {\left (a-b x^2\right ) \left (c-d x^2\right )}} \, dx\) |
\(\Big \downarrow \) 2048 |
\(\displaystyle \int \frac {1}{\sqrt {x^2 (-a d-b c)+a c+b d x^4}}dx\) |
\(\Big \downarrow \) 1416 |
\(\displaystyle \frac {\left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right ) \sqrt {\frac {-x^2 (a d+b c)+a c+b d x^4}{\left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{d} x}{\sqrt [4]{a} \sqrt [4]{c}}\right ),\frac {1}{4} \left (\frac {b c+a d}{\sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}}+2\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{c} \sqrt [4]{d} \sqrt {-x^2 (a d+b c)+a c+b d x^4}}\) |
Input:
Int[1/Sqrt[(a - b*x^2)*(c - d*x^2)],x]
Output:
((Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt[d]*x^2)*Sqrt[(a*c - (b*c + a*d)*x^2 + b*d *x^4)/(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt[d]*x^2)^2]*EllipticF[2*ArcTan[(b^(1/ 4)*d^(1/4)*x)/(a^(1/4)*c^(1/4))], (2 + (b*c + a*d)/(Sqrt[a]*Sqrt[b]*Sqrt[c ]*Sqrt[d]))/4])/(2*a^(1/4)*b^(1/4)*c^(1/4)*d^(1/4)*Sqrt[a*c - (b*c + a*d)* x^2 + b*d*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]
Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))*((c_) + (d_.)*(x_)^(n_.)))^(p_) , x_Symbol] :> Int[u*(a*c*e + (b*c + a*d)*e*x^n + b*d*e*x^(2*n))^p, x] /; F reeQ[{a, b, c, d, e, n, p}, x]
Time = 0.57 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.02
method | result | size |
default | \(\frac {\sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1-\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d -b c}{a d}}\right )}{\sqrt {\frac {d}{c}}\, \sqrt {b d \,x^{4}-x^{2} d a -b c \,x^{2}+a c}}\) | \(92\) |
elliptic | \(\frac {\sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1-\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d -b c}{a d}}\right )}{\sqrt {\frac {d}{c}}\, \sqrt {b d \,x^{4}-x^{2} d a -b c \,x^{2}+a c}}\) | \(92\) |
Input:
int(1/((-b*x^2+a)*(-d*x^2+c))^(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 = 0.09 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.43 \[ \int \frac {1}{\sqrt {\left (a-b x^2\right ) \left (c-d x^2\right )}} \, dx=\frac {\sqrt {a c} \sqrt {\frac {d}{c}} F(\arcsin \left (x \sqrt {\frac {d}{c}}\right )\,|\,\frac {b c}{a d})}{a d} \] Input:
integrate(1/((-b*x^2+a)*(-d*x^2+c))^(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)
\[ \int \frac {1}{\sqrt {\left (a-b x^2\right ) \left (c-d x^2\right )}} \, dx=\int \frac {1}{\sqrt {\left (a - b x^{2}\right ) \left (c - d x^{2}\right )}}\, dx \] Input:
integrate(1/((-b*x**2+a)*(-d*x**2+c))**(1/2),x)
Output:
Integral(1/sqrt((a - b*x**2)*(c - d*x**2)), x)
\[ \int \frac {1}{\sqrt {\left (a-b x^2\right ) \left (c-d x^2\right )}} \, dx=\int { \frac {1}{\sqrt {{\left (b x^{2} - a\right )} {\left (d x^{2} - c\right )}}} \,d x } \] Input:
integrate(1/((-b*x^2+a)*(-d*x^2+c))^(1/2),x, algorithm="maxima")
Output:
integrate(1/sqrt((b*x^2 - a)*(d*x^2 - c)), x)
\[ \int \frac {1}{\sqrt {\left (a-b x^2\right ) \left (c-d x^2\right )}} \, dx=\int { \frac {1}{\sqrt {{\left (b x^{2} - a\right )} {\left (d x^{2} - c\right )}}} \,d x } \] Input:
integrate(1/((-b*x^2+a)*(-d*x^2+c))^(1/2),x, algorithm="giac")
Output:
integrate(1/sqrt((b*x^2 - a)*(d*x^2 - c)), x)
Timed out. \[ \int \frac {1}{\sqrt {\left (a-b x^2\right ) \left (c-d x^2\right )}} \, dx=\int \frac {1}{\sqrt {\left (a-b\,x^2\right )\,\left (c-d\,x^2\right )}} \,d x \] Input:
int(1/((a - b*x^2)*(c - d*x^2))^(1/2),x)
Output:
int(1/((a - b*x^2)*(c - d*x^2))^(1/2), x)
\[ \int \frac {1}{\sqrt {\left (a-b x^2\right ) \left (c-d x^2\right )}} \, dx=\int \frac {\sqrt {-d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}}{b d \,x^{4}-a d \,x^{2}-b c \,x^{2}+a c}d x \] Input:
int(1/((-b*x^2+a)*(-d*x^2+c))^(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)