Integrand size = 20, antiderivative size = 92 \[ \int \frac {1}{\sqrt {\left (a-b x^2\right ) \left (c+d x^2\right )}} \, dx=\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a c-(b c-a d) x^2-b d x^4}} \] 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 = 2.00 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\sqrt {\left (a-b x^2\right ) \left (c+d x^2\right )}} \, dx=\frac {\sqrt {\frac {a-b x^2}{a}} \sqrt {\frac {c+d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )}{\sqrt {\frac {b}{a}} \sqrt {\left (a-b x^2\right ) \left (c+d x^2\right )}} \] 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)])
Time = 0.39 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2048, 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.
\(\displaystyle \int \frac {1}{\sqrt {\left (a-b x^2\right ) \left (c+d x^2\right )}} \, dx\) |
\(\Big \downarrow \) 2048 |
\(\displaystyle \int \frac {1}{\sqrt {x^2 (a d-b c)+a c-b d x^4}}dx\) |
\(\Big \downarrow \) 1417 |
\(\displaystyle \frac {\sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \int \frac {1}{\sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1}}dx}{\sqrt {-x^2 (b c-a d)+a c-b d x^4}}\) |
\(\Big \downarrow \) 321 |
\(\displaystyle \frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {-x^2 (b c-a d)+a c-b d x^4}}\) |
Input:
Int[1/Sqrt[(a - b*x^2)*(c + d*x^2)],x]
Output:
(Sqrt[a]*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[ArcSin[(Sqrt[b] *x)/Sqrt[a]], -((a*d)/(b*c))])/(Sqrt[b]*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]
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 = 90, normalized size of antiderivative = 0.98
method | result | size |
default | \(\frac {\sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+x^{2} d a -b c \,x^{2}+a c}}\) | \(90\) |
elliptic | \(\frac {\sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+x^{2} d a -b c \,x^{2}+a c}}\) | \(90\) |
Input:
int(1/((-b*x^2+a)*(d*x^2+c))^(1/2),x,method=_RETURNVERBOSE)
Output:
1/(b/a)^(1/2)*(1-b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(-b*d*x^4+a*d*x^2-b*c*x^ 2+a*c)^(1/2)*EllipticF(x*(b/a)^(1/2),(-1-(a*d-b*c)/c/b)^(1/2))
Time = 0.08 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.43 \[ \int \frac {1}{\sqrt {\left (a-b x^2\right ) \left (c+d x^2\right )}} \, dx=\frac {\sqrt {a c} \sqrt {\frac {b}{a}} F(\arcsin \left (x \sqrt {\frac {b}{a}}\right )\,|\,-\frac {a d}{b c})}{b c} \] Input:
integrate(1/((-b*x^2+a)*(d*x^2+c))^(1/2),x, algorithm="fricas")
Output:
sqrt(a*c)*sqrt(b/a)*elliptic_f(arcsin(x*sqrt(b/a)), -a*d/(b*c))/(b*c)
\[ \int \frac {1}{\sqrt {\left (a-b x^2\right ) \left (c+d x^2\right )}} \, dx=\int \frac {1}{\sqrt {\left (a - b x^{2}\right ) \left (c + d x^{2}\right )}}\, dx \] 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. \[ \int \frac {1}{\sqrt {\left (a-b x^2\right ) \left (c+d x^2\right )}} \, dx=\int \frac {1}{\sqrt {\left (a-b\,x^2\right )\,\left (d\,x^2+c\right )}} \,d x \] 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)