Integrand size = 24, antiderivative size = 87 \[ \int \frac {1}{\sqrt {a+b x^2} \sqrt {c-d x^2}} \, dx=\frac {\sqrt {c} \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {a+b x^2} \sqrt {c-d x^2}} \] Output:
c^(1/2)*(1+b*x^2/a)^(1/2)*(1-d*x^2/c)^(1/2)*EllipticF(d^(1/2)*x/c^(1/2),(- b*c/a/d)^(1/2))/d^(1/2)/(b*x^2+a)^(1/2)/(-d*x^2+c)^(1/2)
Time = 0.58 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.02 \[ \int \frac {1}{\sqrt {a+b x^2} \sqrt {c-d x^2}} \, dx=\frac {\sqrt {\frac {a+b x^2}{a}} \sqrt {\frac {c-d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )}{\sqrt {-\frac {b}{a}} \sqrt {a+b x^2} \sqrt {c-d x^2}} \] Input:
Integrate[1/(Sqrt[a + b*x^2]*Sqrt[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]*Sqrt[c - d*x^2])
Time = 0.21 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {323, 323, 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.
\(\displaystyle \int \frac {1}{\sqrt {a+b x^2} \sqrt {c-d x^2}} \, dx\) |
\(\Big \downarrow \) 323 |
\(\displaystyle \frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {1}{\sqrt {b x^2+a} \sqrt {1-\frac {d x^2}{c}}}dx}{\sqrt {c-d x^2}}\) |
\(\Big \downarrow \) 323 |
\(\displaystyle \frac {\sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} \int \frac {1}{\sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}}}dx}{\sqrt {a+b x^2} \sqrt {c-d x^2}}\) |
\(\Big \downarrow \) 321 |
\(\displaystyle \frac {\sqrt {c} \sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {a+b x^2} \sqrt {c-d x^2}}\) |
Input:
Int[1/(Sqrt[a + b*x^2]*Sqrt[c - d*x^2]),x]
Output:
(Sqrt[c]*Sqrt[1 + (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(Sqrt[d] *x)/Sqrt[c]], -((b*c)/(a*d))])/(Sqrt[d]*Sqrt[a + b*x^2]*Sqrt[c - d*x^2])
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]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + ( d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && !GtQ[c, 0]
Time = 3.77 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.18
method | result | size |
default | \(\frac {\operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {-x^{2} d +c}{c}}\, \sqrt {b \,x^{2}+a}\, \sqrt {-x^{2} d +c}}{\sqrt {\frac {d}{c}}\, \left (-b d \,x^{4}-a d \,x^{2}+x^{2} b c +a c \right )}\) | \(103\) |
elliptic | \(\frac {\sqrt {\left (-x^{2} d +c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )}{\sqrt {-x^{2} d +c}\, \sqrt {b \,x^{2}+a}\, \sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+x^{2} b c +a c}}\) | \(127\) |
Input:
int(1/(b*x^2+a)^(1/2)/(-d*x^2+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
EllipticF(x*(1/c*d)^(1/2),(-b*c/a/d)^(1/2))*((b*x^2+a)/a)^(1/2)*((-d*x^2+c )/c)^(1/2)*(b*x^2+a)^(1/2)*(-d*x^2+c)^(1/2)/(1/c*d)^(1/2)/(-b*d*x^4-a*d*x^ 2+b*c*x^2+a*c)
Time = 0.08 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.46 \[ \int \frac {1}{\sqrt {a+b x^2} \sqrt {c-d x^2}} \, dx=\frac {\sqrt {a c} \sqrt {\frac {d}{c}} F(\arcsin \left (x \sqrt {\frac {d}{c}}\right )\,|\,-\frac {b c}{a d})}{a d} \] Input:
integrate(1/(b*x^2+a)^(1/2)/(-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)
\[ \int \frac {1}{\sqrt {a+b x^2} \sqrt {c-d x^2}} \, dx=\int \frac {1}{\sqrt {a + b x^{2}} \sqrt {c - d x^{2}}}\, dx \] Input:
integrate(1/(b*x**2+a)**(1/2)/(-d*x**2+c)**(1/2),x)
Output:
Integral(1/(sqrt(a + b*x**2)*sqrt(c - d*x**2)), x)
\[ \int \frac {1}{\sqrt {a+b x^2} \sqrt {c-d x^2}} \, dx=\int { \frac {1}{\sqrt {b x^{2} + a} \sqrt {-d x^{2} + c}} \,d x } \] Input:
integrate(1/(b*x^2+a)^(1/2)/(-d*x^2+c)^(1/2),x, algorithm="maxima")
Output:
integrate(1/(sqrt(b*x^2 + a)*sqrt(-d*x^2 + c)), x)
\[ \int \frac {1}{\sqrt {a+b x^2} \sqrt {c-d x^2}} \, dx=\int { \frac {1}{\sqrt {b x^{2} + a} \sqrt {-d x^{2} + c}} \,d x } \] Input:
integrate(1/(b*x^2+a)^(1/2)/(-d*x^2+c)^(1/2),x, algorithm="giac")
Output:
integrate(1/(sqrt(b*x^2 + a)*sqrt(-d*x^2 + c)), x)
Timed out. \[ \int \frac {1}{\sqrt {a+b x^2} \sqrt {c-d x^2}} \, dx=\int \frac {1}{\sqrt {b\,x^2+a}\,\sqrt {c-d\,x^2}} \,d x \] Input:
int(1/((a + b*x^2)^(1/2)*(c - d*x^2)^(1/2)),x)
Output:
int(1/((a + b*x^2)^(1/2)*(c - d*x^2)^(1/2)), x)
\[ \int \frac {1}{\sqrt {a+b x^2} \sqrt {c-d x^2}} \, dx=\int \frac {\sqrt {-d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{-b d \,x^{4}-a d \,x^{2}+b c \,x^{2}+a c}d x \] Input:
int(1/(b*x^2+a)^(1/2)/(-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)