Integrand size = 27, antiderivative size = 90 \[ \int \frac {\sqrt {-a-b x^2}}{\sqrt {c-d x^2}} \, dx=\frac {\sqrt {c} \sqrt {-a-b x^2} \sqrt {1-\frac {d x^2}{c}} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {1+\frac {b x^2}{a}} \sqrt {c-d x^2}} \] Output:
c^(1/2)*(-b*x^2-a)^(1/2)*(1-d*x^2/c)^(1/2)*EllipticE(d^(1/2)*x/c^(1/2),(-b *c/a/d)^(1/2))/d^(1/2)/(1+b*x^2/a)^(1/2)/(-d*x^2+c)^(1/2)
Time = 0.75 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {-a-b x^2}}{\sqrt {c-d x^2}} \, dx=\frac {\sqrt {-a-b x^2} \sqrt {\frac {c-d x^2}{c}} E\left (\arcsin \left (\sqrt {\frac {d}{c}} x\right )|-\frac {b c}{a d}\right )}{\sqrt {\frac {d}{c}} \sqrt {\frac {a+b x^2}{a}} \sqrt {c-d x^2}} \] Input:
Integrate[Sqrt[-a - b*x^2]/Sqrt[c - d*x^2],x]
Output:
(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 = 0.21 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {331, 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.
\(\displaystyle \int \frac {\sqrt {-a-b x^2}}{\sqrt {c-d x^2}} \, dx\) |
\(\Big \downarrow \) 331 |
\(\displaystyle \frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {\sqrt {-b x^2-a}}{\sqrt {1-\frac {d x^2}{c}}}dx}{\sqrt {c-d x^2}}\) |
\(\Big \downarrow \) 330 |
\(\displaystyle \frac {\sqrt {-a-b x^2} \sqrt {1-\frac {d x^2}{c}} \int \frac {\sqrt {\frac {b x^2}{a}+1}}{\sqrt {1-\frac {d x^2}{c}}}dx}{\sqrt {\frac {b x^2}{a}+1} \sqrt {c-d x^2}}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle \frac {\sqrt {c} \sqrt {-a-b x^2} \sqrt {1-\frac {d x^2}{c}} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {\frac {b x^2}{a}+1} \sqrt {c-d x^2}}\) |
Input:
Int[Sqrt[-a - b*x^2]/Sqrt[c - d*x^2],x]
Output:
(Sqrt[c]*Sqrt[-a - b*x^2]*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(Sqrt[d]*x) /Sqrt[c]], -((b*c)/(a*d))])/(Sqrt[d]*Sqrt[1 + (b*x^2)/a]*Sqrt[c - d*x^2])
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]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[Sqrt[a + b*x^2]/Sqrt[1 + (d/c)*x^ 2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && !GtQ[c, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(167\) vs. \(2(75)=150\).
Time = 0.96 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.87
method | result | size |
default | \(\frac {\sqrt {-b \,x^{2}-a}\, \sqrt {-x^{2} d +c}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {-x^{2} d +c}{c}}\, \left (a \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) d +b c \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right )-b c \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right )\right )}{\left (-b d \,x^{4}-a d \,x^{2}+x^{2} b c +a c \right ) \sqrt {-\frac {b}{a}}\, d}\) | \(168\) |
elliptic | \(\frac {\sqrt {-\left (-x^{2} d +c \right ) \left (b \,x^{2}+a \right )}\, \left (-\frac {a \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1-\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}-x^{2} b c -a c}}-\frac {b c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1-\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}-x^{2} b c -a c}\, d}\right )}{\sqrt {-b \,x^{2}-a}\, \sqrt {-x^{2} d +c}}\) | \(268\) |
Input:
int((-b*x^2-a)^(1/2)/(-d*x^2+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
(-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) *(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* d*x^4-a*d*x^2+b*c*x^2+a*c)/(-b/a)^(1/2)/d
Time = 0.09 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.46 \[ \int \frac {\sqrt {-a-b x^2}}{\sqrt {c-d x^2}} \, dx=-\frac {\sqrt {b d} b c^{2} x \sqrt {\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {\frac {c}{d}}}{x}\right )\,|\,-\frac {a d}{b c}) + \sqrt {-b x^{2} - a} \sqrt {-d x^{2} + c} b c d - {\left (b c^{2} + a d^{2}\right )} \sqrt {b d} x \sqrt {\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {\frac {c}{d}}}{x}\right )\,|\,-\frac {a d}{b c})}{b c d^{2} x} \] Input:
integrate((-b*x^2-a)^(1/2)/(-d*x^2+c)^(1/2),x, algorithm="fricas")
Output:
-(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)
\[ \int \frac {\sqrt {-a-b x^2}}{\sqrt {c-d x^2}} \, dx=\int \frac {\sqrt {- a - b x^{2}}}{\sqrt {c - d x^{2}}}\, dx \] Input:
integrate((-b*x**2-a)**(1/2)/(-d*x**2+c)**(1/2),x)
Output:
Integral(sqrt(-a - b*x**2)/sqrt(c - d*x**2), x)
\[ \int \frac {\sqrt {-a-b x^2}}{\sqrt {c-d x^2}} \, dx=\int { \frac {\sqrt {-b x^{2} - a}}{\sqrt {-d x^{2} + c}} \,d x } \] Input:
integrate((-b*x^2-a)^(1/2)/(-d*x^2+c)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(-b*x^2 - a)/sqrt(-d*x^2 + c), x)
\[ \int \frac {\sqrt {-a-b x^2}}{\sqrt {c-d x^2}} \, dx=\int { \frac {\sqrt {-b x^{2} - a}}{\sqrt {-d x^{2} + c}} \,d x } \] Input:
integrate((-b*x^2-a)^(1/2)/(-d*x^2+c)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(-b*x^2 - a)/sqrt(-d*x^2 + c), x)
Timed out. \[ \int \frac {\sqrt {-a-b x^2}}{\sqrt {c-d x^2}} \, dx=\int \frac {\sqrt {-b\,x^2-a}}{\sqrt {c-d\,x^2}} \,d x \] Input:
int((- a - b*x^2)^(1/2)/(c - d*x^2)^(1/2),x)
Output:
int((- a - b*x^2)^(1/2)/(c - d*x^2)^(1/2), x)
\[ \int \frac {\sqrt {-a-b x^2}}{\sqrt {c-d x^2}} \, dx=\left (\int \frac {\sqrt {-d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{-d \,x^{2}+c}d x \right ) i \] Input:
int((-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))/(c - d*x**2),x)*i