Integrand size = 27, antiderivative size = 90 \[ \int \frac {\sqrt {-c+d x^2}}{\sqrt {-a+b x^2}} \, dx=\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {-c+d x^2} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {-a+b x^2} \sqrt {1-\frac {d x^2}{c}}} \] Output:
a^(1/2)*(1-b*x^2/a)^(1/2)*(d*x^2-c)^(1/2)*EllipticE(b^(1/2)*x/a^(1/2),(a*d /b/c)^(1/2))/b^(1/2)/(b*x^2-a)^(1/2)/(1-d*x^2/c)^(1/2)
Time = 0.54 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {-c+d x^2}}{\sqrt {-a+b x^2}} \, dx=\frac {\sqrt {\frac {a-b x^2}{a}} \sqrt {-c+d x^2} E\left (\arcsin \left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )}{\sqrt {\frac {b}{a}} \sqrt {-a+b x^2} \sqrt {\frac {c-d x^2}{c}}} \] Input:
Integrate[Sqrt[-c + d*x^2]/Sqrt[-a + b*x^2],x]
Output:
(Sqrt[(a - b*x^2)/a]*Sqrt[-c + d*x^2]*EllipticE[ArcSin[Sqrt[b/a]*x], (a*d) /(b*c)])/(Sqrt[b/a]*Sqrt[-a + b*x^2]*Sqrt[(c - d*x^2)/c])
Time = 0.20 (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.
\(\displaystyle \int \frac {\sqrt {d x^2-c}}{\sqrt {b x^2-a}} \, dx\) |
\(\Big \downarrow \) 331 |
\(\displaystyle \frac {\sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {d x^2-c}}{\sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {b x^2-a}}\) |
\(\Big \downarrow \) 330 |
\(\displaystyle \frac {\sqrt {1-\frac {b x^2}{a}} \sqrt {d x^2-c} \int \frac {\sqrt {1-\frac {d x^2}{c}}}{\sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {b x^2-a} \sqrt {1-\frac {d x^2}{c}}}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle \frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {d x^2-c} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {b x^2-a} \sqrt {1-\frac {d x^2}{c}}}\) |
Input:
Int[Sqrt[-c + d*x^2]/Sqrt[-a + b*x^2],x]
Output:
(Sqrt[a]*Sqrt[1 - (b*x^2)/a]*Sqrt[-c + d*x^2]*EllipticE[ArcSin[(Sqrt[b]*x) /Sqrt[a]], (a*d)/(b*c)])/(Sqrt[b]*Sqrt[-a + b*x^2]*Sqrt[1 - (d*x^2)/c])
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. \(162\) vs. \(2(75)=150\).
Time = 1.37 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.81
method | result | size |
default | \(\frac {\sqrt {x^{2} d -c}\, \sqrt {b \,x^{2}-a}\, \sqrt {\frac {-x^{2} d +c}{c}}\, \sqrt {\frac {-b \,x^{2}+a}{a}}\, \left (a d \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {\frac {b c}{a d}}\right )-c \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {\frac {b c}{a d}}\right ) b -a d \operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {\frac {b c}{a d}}\right )\right )}{\left (b d \,x^{4}-a d \,x^{2}-x^{2} b c +a c \right ) \sqrt {\frac {d}{c}}\, b}\) | \(163\) |
elliptic | \(\frac {\sqrt {\left (-x^{2} d +c \right ) \left (-b \,x^{2}+a \right )}\, \left (-\frac {c \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1-\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d -b c}{a d}}\right )}{\sqrt {\frac {d}{c}}\, \sqrt {b d \,x^{4}-a d \,x^{2}-x^{2} b c +a c}}+\frac {d a \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1-\frac {b \,x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d -b c}{a d}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d -b c}{a d}}\right )\right )}{\sqrt {\frac {d}{c}}\, \sqrt {b d \,x^{4}-a d \,x^{2}-x^{2} b c +a c}\, b}\right )}{\sqrt {x^{2} d -c}\, \sqrt {b \,x^{2}-a}}\) | \(267\) |
Input:
int((d*x^2-c)^(1/2)/(b*x^2-a)^(1/2),x,method=_RETURNVERBOSE)
Output:
(d*x^2-c)^(1/2)*(b*x^2-a)^(1/2)*((-d*x^2+c)/c)^(1/2)*((-b*x^2+a)/a)^(1/2)* (a*d*EllipticF(x*(1/c*d)^(1/2),(b*c/a/d)^(1/2))-c*EllipticF(x*(1/c*d)^(1/2 ),(b*c/a/d)^(1/2))*b-a*d*EllipticE(x*(1/c*d)^(1/2),(b*c/a/d)^(1/2)))/(b*d* x^4-a*d*x^2-b*c*x^2+a*c)/(1/c*d)^(1/2)/b
Time = 0.09 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.42 \[ \int \frac {\sqrt {-c+d x^2}}{\sqrt {-a+b x^2}} \, dx=\frac {\sqrt {b d} a^{2} d x \sqrt {\frac {a}{b}} E(\arcsin \left (\frac {\sqrt {\frac {a}{b}}}{x}\right )\,|\,\frac {b c}{a d}) + \sqrt {b x^{2} - a} \sqrt {d x^{2} - c} a b d + {\left (b^{2} c - a^{2} d\right )} \sqrt {b d} x \sqrt {\frac {a}{b}} F(\arcsin \left (\frac {\sqrt {\frac {a}{b}}}{x}\right )\,|\,\frac {b c}{a d})}{a b^{2} d x} \] Input:
integrate((d*x^2-c)^(1/2)/(b*x^2-a)^(1/2),x, algorithm="fricas")
Output:
(sqrt(b*d)*a^2*d*x*sqrt(a/b)*elliptic_e(arcsin(sqrt(a/b)/x), b*c/(a*d)) + sqrt(b*x^2 - a)*sqrt(d*x^2 - c)*a*b*d + (b^2*c - a^2*d)*sqrt(b*d)*x*sqrt(a /b)*elliptic_f(arcsin(sqrt(a/b)/x), b*c/(a*d)))/(a*b^2*d*x)
\[ \int \frac {\sqrt {-c+d x^2}}{\sqrt {-a+b x^2}} \, dx=\int \frac {\sqrt {- c + d x^{2}}}{\sqrt {- a + b x^{2}}}\, dx \] Input:
integrate((d*x**2-c)**(1/2)/(b*x**2-a)**(1/2),x)
Output:
Integral(sqrt(-c + d*x**2)/sqrt(-a + b*x**2), x)
\[ \int \frac {\sqrt {-c+d x^2}}{\sqrt {-a+b x^2}} \, dx=\int { \frac {\sqrt {d x^{2} - c}}{\sqrt {b x^{2} - a}} \,d x } \] Input:
integrate((d*x^2-c)^(1/2)/(b*x^2-a)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(d*x^2 - c)/sqrt(b*x^2 - a), x)
\[ \int \frac {\sqrt {-c+d x^2}}{\sqrt {-a+b x^2}} \, dx=\int { \frac {\sqrt {d x^{2} - c}}{\sqrt {b x^{2} - a}} \,d x } \] Input:
integrate((d*x^2-c)^(1/2)/(b*x^2-a)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(d*x^2 - c)/sqrt(b*x^2 - a), x)
Timed out. \[ \int \frac {\sqrt {-c+d x^2}}{\sqrt {-a+b x^2}} \, dx=\int \frac {\sqrt {d\,x^2-c}}{\sqrt {b\,x^2-a}} \,d x \] Input:
int((d*x^2 - c)^(1/2)/(b*x^2 - a)^(1/2),x)
Output:
int((d*x^2 - c)^(1/2)/(b*x^2 - a)^(1/2), x)
\[ \int \frac {\sqrt {-c+d x^2}}{\sqrt {-a+b x^2}} \, dx=-\left (\int \frac {\sqrt {d \,x^{2}-c}\, \sqrt {b \,x^{2}-a}}{-b \,x^{2}+a}d x \right ) \] Input:
int((d*x^2-c)^(1/2)/(b*x^2-a)^(1/2),x)
Output:
- int((sqrt( - c + d*x**2)*sqrt( - a + b*x**2))/(a - b*x**2),x)