Integrand size = 29, antiderivative size = 212 \[ \int \frac {\sqrt {-a-b x^2}}{\sqrt {-c-d x^2}} \, dx=\frac {x \sqrt {-a-b x^2}}{\sqrt {-c-d x^2}}-\frac {\sqrt {c} \sqrt {-a-b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {-c-d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {\sqrt {c} \sqrt {-a-b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {-c-d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]
x*(-b*x^2-a)^(1/2)/(-d*x^2-c)^(1/2)-(1/(1+d*x^2/c))^(1/2)*(1+d*x^2/c)^(1/2 )*EllipticE(x*d^(1/2)/c^(1/2)/(1+d*x^2/c)^(1/2),(1-b*c/a/d)^(1/2))*c^(1/2) *(-b*x^2-a)^(1/2)/d^(1/2)/(-d*x^2-c)^(1/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2) +(1/(1+d*x^2/c))^(1/2)*(1+d*x^2/c)^(1/2)*EllipticF(x*d^(1/2)/c^(1/2)/(1+d* x^2/c)^(1/2),(1-b*c/a/d)^(1/2))*c^(1/2)*(-b*x^2-a)^(1/2)/d^(1/2)/(-d*x^2-c )^(1/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)
Time = 0.59 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.43 \[ \int \frac {\sqrt {-a-b x^2}}{\sqrt {-c-d x^2}} \, dx=\frac {\sqrt {-a-b x^2} \sqrt {\frac {c+d x^2}{c}} E\left (\arcsin \left (\sqrt {-\frac {d}{c}} x\right )|\frac {b c}{a d}\right )}{\sqrt {-\frac {d}{c}} \sqrt {\frac {a+b x^2}{a}} \sqrt {-c-d x^2}} \]
(Sqrt[-a - b*x^2]*Sqrt[(c + d*x^2)/c]*EllipticE[ArcSin[Sqrt[-(d/c)]*x], (b *c)/(a*d)])/(Sqrt[-(d/c)]*Sqrt[(a + b*x^2)/a]*Sqrt[-c - d*x^2])
Time = 0.31 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {324, 320, 388, 313}
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.
| \(\displaystyle \int \frac {\sqrt {-a-b x^2}}{\sqrt {-c-d x^2}} \, dx\) |
| \(\Big \downarrow \) 324 |
| \(\displaystyle -a \int \frac {1}{\sqrt {-b x^2-a} \sqrt {-d x^2-c}}dx-b \int \frac {x^2}{\sqrt {-b x^2-a} \sqrt {-d x^2-c}}dx\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\sqrt {c} \sqrt {-a-b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {-c-d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-b \int \frac {x^2}{\sqrt {-b x^2-a} \sqrt {-d x^2-c}}dx\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {\sqrt {c} \sqrt {-a-b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {-c-d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-b \left (-\frac {c \int \frac {\sqrt {-b x^2-a}}{\left (-d x^2-c\right )^{3/2}}dx}{b}-\frac {x \sqrt {-a-b x^2}}{b \sqrt {-c-d x^2}}\right )\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\sqrt {c} \sqrt {-a-b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {-c-d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-b \left (\frac {\sqrt {c} \sqrt {-a-b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {-c-d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {x \sqrt {-a-b x^2}}{b \sqrt {-c-d x^2}}\right )\) |
-(b*(-((x*Sqrt[-a - b*x^2])/(b*Sqrt[-c - d*x^2])) + (Sqrt[c]*Sqrt[-a - b*x ^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b*Sqrt[d]*Sq rt[-c - d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]))) + (Sqrt[c]*Sqrt[-a - b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d ]*Sqrt[-c - d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))])
3.3.69.3.1 Defintions of rubi rules used
Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ [{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] && !SimplerSqrtQ[b/a, d/c]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ a Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] + Simp[b Int[x^2/(Sqr t[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && PosQ[d/c ] && PosQ[b/a]
Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b Int[Sqrt[ a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] && !SimplerSqrtQ[b/a, d/c]
Time = 2.57 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.78
| method | result | size |
| default | \(\frac {\left (-a F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) d +b c F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right )-b c E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right )\right ) \sqrt {-b \,x^{2}-a}\, \sqrt {-d \,x^{2}-c}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}}{\left (b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c \right ) \sqrt {-\frac {b}{a}}\, d}\) | \(165\) |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (-\frac {a \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}+\frac {b c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}\, d}\right )}{\sqrt {-b \,x^{2}-a}\, \sqrt {-d \,x^{2}-c}}\) | \(255\) |
(-a*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*d+b*c*EllipticF(x*(-b/a)^(1/ 2),(a*d/b/c)^(1/2))-b*c*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2)))*(-b*x^2 -a)^(1/2)*(-d*x^2-c)^(1/2)*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)/(b*d*x^ 4+a*d*x^2+b*c*x^2+a*c)/(-b/a)^(1/2)/d
Time = 0.08 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.64 \[ \int \frac {\sqrt {-a-b x^2}}{\sqrt {-c-d x^2}} \, dx=\frac {\sqrt {b d} b c^{2} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - \sqrt {-b x^{2} - a} \sqrt {-d x^{2} - c} b c d - {\left (b c^{2} + a d^{2}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c})}{b c d^{2} x} \]
(sqrt(b*d)*b*c^2*x*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - sqrt(-b*x^2 - a)*sqrt(-d*x^2 - c)*b*c*d - (b*c^2 + a*d^2)*sqrt(b*d)*x*sq rt(-c/d)*elliptic_f(arcsin(sqrt(-c/d)/x), a*d/(b*c)))/(b*c*d^2*x)
\[ \int \frac {\sqrt {-a-b x^2}}{\sqrt {-c-d x^2}} \, dx=\int \frac {\sqrt {- a - b x^{2}}}{\sqrt {- c - d x^{2}}}\, dx \]
\[ \int \frac {\sqrt {-a-b x^2}}{\sqrt {-c-d x^2}} \, dx=\int { \frac {\sqrt {-b x^{2} - a}}{\sqrt {-d x^{2} - c}} \,d x } \]
\[ \int \frac {\sqrt {-a-b x^2}}{\sqrt {-c-d x^2}} \, dx=\int { \frac {\sqrt {-b x^{2} - a}}{\sqrt {-d x^{2} - c}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {-a-b x^2}}{\sqrt {-c-d x^2}} \, dx=\int \frac {\sqrt {-b\,x^2-a}}{\sqrt {-d\,x^2-c}} \,d x \]