Integrand size = 23, antiderivative size = 91 \[ \int \frac {\sqrt {4-x^2}}{\sqrt {c+d x^2}} \, dx=-\frac {\sqrt {c+d x^2} E\left (\arcsin \left (\frac {x}{2}\right )|-\frac {4 d}{c}\right )}{d \sqrt {1+\frac {d x^2}{c}}}+\frac {(c+4 d) \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{2}\right ),-\frac {4 d}{c}\right )}{d \sqrt {c+d x^2}} \]
-EllipticE(1/2*x,2*(-d/c)^(1/2))*(d*x^2+c)^(1/2)/d/(1+d*x^2/c)^(1/2)+(c+4* d)*EllipticF(1/2*x,2*(-d/c)^(1/2))*(1+d*x^2/c)^(1/2)/d/(d*x^2+c)^(1/2)
Time = 0.58 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.66 \[ \int \frac {\sqrt {4-x^2}}{\sqrt {c+d x^2}} \, dx=\frac {2 \sqrt {\frac {c+d x^2}{c}} E\left (\arcsin \left (\sqrt {-\frac {d}{c}} x\right )|-\frac {c}{4 d}\right )}{\sqrt {-\frac {d}{c}} \sqrt {c+d x^2}} \]
(2*Sqrt[(c + d*x^2)/c]*EllipticE[ArcSin[Sqrt[-(d/c)]*x], -1/4*c/d])/(Sqrt[ -(d/c)]*Sqrt[c + d*x^2])
Time = 0.23 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {326, 323, 321, 330, 327}
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.
\(\displaystyle \int \frac {\sqrt {4-x^2}}{\sqrt {c+d x^2}} \, dx\) |
\(\Big \downarrow \) 326 |
\(\displaystyle \frac {(c+4 d) \int \frac {1}{\sqrt {4-x^2} \sqrt {d x^2+c}}dx}{d}-\frac {\int \frac {\sqrt {d x^2+c}}{\sqrt {4-x^2}}dx}{d}\) |
\(\Big \downarrow \) 323 |
\(\displaystyle \frac {(c+4 d) \sqrt {\frac {d x^2}{c}+1} \int \frac {1}{\sqrt {4-x^2} \sqrt {\frac {d x^2}{c}+1}}dx}{d \sqrt {c+d x^2}}-\frac {\int \frac {\sqrt {d x^2+c}}{\sqrt {4-x^2}}dx}{d}\) |
\(\Big \downarrow \) 321 |
\(\displaystyle \frac {(c+4 d) \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{2}\right ),-\frac {4 d}{c}\right )}{d \sqrt {c+d x^2}}-\frac {\int \frac {\sqrt {d x^2+c}}{\sqrt {4-x^2}}dx}{d}\) |
\(\Big \downarrow \) 330 |
\(\displaystyle \frac {(c+4 d) \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{2}\right ),-\frac {4 d}{c}\right )}{d \sqrt {c+d x^2}}-\frac {\sqrt {c+d x^2} \int \frac {\sqrt {\frac {d x^2}{c}+1}}{\sqrt {4-x^2}}dx}{d \sqrt {\frac {d x^2}{c}+1}}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle \frac {(c+4 d) \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{2}\right ),-\frac {4 d}{c}\right )}{d \sqrt {c+d x^2}}-\frac {\sqrt {c+d x^2} E\left (\arcsin \left (\frac {x}{2}\right )|-\frac {4 d}{c}\right )}{d \sqrt {\frac {d x^2}{c}+1}}\) |
-((Sqrt[c + d*x^2]*EllipticE[ArcSin[x/2], (-4*d)/c])/(d*Sqrt[1 + (d*x^2)/c ])) + ((c + 4*d)*Sqrt[1 + (d*x^2)/c]*EllipticF[ArcSin[x/2], (-4*d)/c])/(d* Sqrt[c + d*x^2])
3.2.82.3.1 Defintions of rubi rules used
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]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ b/d Int[Sqrt[c + d*x^2]/Sqrt[a + b*x^2], x], x] - Simp[(b*c - a*d)/d In t[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && NegQ[b/a]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[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]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ Sqrt[a + b*x^2]/Sqrt[1 + (b/a)*x^2] Int[Sqrt[1 + (b/a)*x^2]/Sqrt[c + d*x^ 2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && !GtQ[a, 0]
Time = 2.40 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.86
method | result | size |
default | \(\frac {\left (c F\left (\frac {x}{2}, 2 \sqrt {-\frac {d}{c}}\right )+4 F\left (\frac {x}{2}, 2 \sqrt {-\frac {d}{c}}\right ) d -c E\left (\frac {x}{2}, 2 \sqrt {-\frac {d}{c}}\right )\right ) \sqrt {\frac {d \,x^{2}+c}{c}}}{\sqrt {d \,x^{2}+c}\, d}\) | \(78\) |
elliptic | \(\frac {\sqrt {-\left (d \,x^{2}+c \right ) \left (x^{2}-4\right )}\, \left (\frac {4 \sqrt {-x^{2}+4}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (\frac {x}{2}, \sqrt {-1-\frac {-c +4 d}{c}}\right )}{\sqrt {-d \,x^{4}-c \,x^{2}+4 d \,x^{2}+4 c}}+\frac {c \sqrt {-x^{2}+4}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (\frac {x}{2}, \sqrt {-1-\frac {-c +4 d}{c}}\right )-E\left (\frac {x}{2}, \sqrt {-1-\frac {-c +4 d}{c}}\right )\right )}{\sqrt {-d \,x^{4}-c \,x^{2}+4 d \,x^{2}+4 c}\, d}\right )}{\sqrt {-x^{2}+4}\, \sqrt {d \,x^{2}+c}}\) | \(197\) |
(c*EllipticF(1/2*x,2*(-d/c)^(1/2))+4*EllipticF(1/2*x,2*(-d/c)^(1/2))*d-c*E llipticE(1/2*x,2*(-d/c)^(1/2)))*((d*x^2+c)/c)^(1/2)/(d*x^2+c)^(1/2)/d
Time = 0.08 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {4-x^2}}{\sqrt {c+d x^2}} \, dx=\frac {2 \, {\left (4 \, x E(\arcsin \left (\frac {2}{x}\right )\,|\,-\frac {c}{4 \, d}) - 3 \, x F(\arcsin \left (\frac {2}{x}\right )\,|\,-\frac {c}{4 \, d})\right )} \sqrt {-d} + \sqrt {d x^{2} + c} \sqrt {-x^{2} + 4}}{d x} \]
(2*(4*x*elliptic_e(arcsin(2/x), -1/4*c/d) - 3*x*elliptic_f(arcsin(2/x), -1 /4*c/d))*sqrt(-d) + sqrt(d*x^2 + c)*sqrt(-x^2 + 4))/(d*x)
\[ \int \frac {\sqrt {4-x^2}}{\sqrt {c+d x^2}} \, dx=\int \frac {\sqrt {- \left (x - 2\right ) \left (x + 2\right )}}{\sqrt {c + d x^{2}}}\, dx \]
\[ \int \frac {\sqrt {4-x^2}}{\sqrt {c+d x^2}} \, dx=\int { \frac {\sqrt {-x^{2} + 4}}{\sqrt {d x^{2} + c}} \,d x } \]
\[ \int \frac {\sqrt {4-x^2}}{\sqrt {c+d x^2}} \, dx=\int { \frac {\sqrt {-x^{2} + 4}}{\sqrt {d x^{2} + c}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {4-x^2}}{\sqrt {c+d x^2}} \, dx=\int \frac {\sqrt {4-x^2}}{\sqrt {d\,x^2+c}} \,d x \]