\(\int \frac {\sqrt {a-b x^2}}{\sqrt {-c+d x^2}} \, dx\) [264]

Optimal result

Integrand size = 26, antiderivative size = 89 \[ \int \frac {\sqrt {a-b x^2}}{\sqrt {-c+d x^2}} \, dx=\frac {\sqrt {c} \sqrt {a-b x^2} \sqrt {1-\frac {d x^2}{c}} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {1-\frac {b x^2}{a}} \sqrt {-c+d x^2}} \]

[Out]

EllipticE(x*d^(1/2)/c^(1/2),(b*c/a/d)^(1/2))*c^(1/2)*(-b*x^2+a)^(1/2)*(1-d*x^2/c)^(1/2)/d^(1/2)/(1-b*x^2/a)^(1
/2)/(d*x^2-c)^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {438, 437, 435} \[ \int \frac {\sqrt {a-b x^2}}{\sqrt {-c+d x^2}} \, dx=\frac {\sqrt {c} \sqrt {a-b x^2} \sqrt {1-\frac {d x^2}{c}} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {1-\frac {b x^2}{a}} \sqrt {d x^2-c}} \]

[In]

Int[Sqrt[a - b*x^2]/Sqrt[-c + d*x^2],x]

[Out]

(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]*Sqr
t[1 - (b*x^2)/a]*Sqrt[-c + d*x^2])

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[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
]

Rule 437

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[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]

Rule 438

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {\sqrt {a-b x^2}}{\sqrt {1-\frac {d x^2}{c}}} \, dx}{\sqrt {-c+d x^2}} \\ & = \frac {\left (\sqrt {a-b x^2} \sqrt {1-\frac {d x^2}{c}}\right ) \int \frac {\sqrt {1-\frac {b x^2}{a}}}{\sqrt {1-\frac {d x^2}{c}}} \, dx}{\sqrt {1-\frac {b x^2}{a}} \sqrt {-c+d x^2}} \\ & = \frac {\sqrt {c} \sqrt {a-b x^2} \sqrt {1-\frac {d x^2}{c}} E\left (\sin ^{-1}\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}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a-b x^2}}{\sqrt {-c+d x^2}} \, dx=\frac {\sqrt {a-b x^2} \sqrt {\frac {c-d x^2}{c}} E\left (\arcsin \left (\sqrt {\frac {d}{c}} x\right )|\frac {b c}{a d}\right )}{\sqrt {\frac {d}{c}} \sqrt {\frac {a-b x^2}{a}} \sqrt {-c+d x^2}} \]

[In]

Integrate[Sqrt[a - b*x^2]/Sqrt[-c + d*x^2],x]

[Out]

(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])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(161\) vs. \(2(74)=148\).

Time = 2.56 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.82

method	result	size
default	\(\frac {\left (-a F\left (x \sqrt {\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) d +b c F\left (x \sqrt {\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right )-b c E\left (x \sqrt {\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right )\right ) \sqrt {-b \,x^{2}+a}\, \sqrt {d \,x^{2}-c}\, \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {-d \,x^{2}+c}{c}}}{\left (b d \,x^{4}-a d \,x^{2}-c b \,x^{2}+a c \right ) \sqrt {\frac {b}{a}}\, d}\)	\(162\)
elliptic	\(\frac {\sqrt {-\left (-b \,x^{2}+a \right ) \left (-d \,x^{2}+c \right )}\, \left (\frac {a \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1-\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}+c b \,x^{2}-a c}}-\frac {b c \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1-\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}+c b \,x^{2}-a c}\, d}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {d \,x^{2}-c}}\)	\(258\)

[In]

int((-b*x^2+a)^(1/2)/(d*x^2-c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

(-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

Fricas [A] (verification not implemented)

none

Time = 0.09 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.29 \[ \int \frac {\sqrt {a-b x^2}}{\sqrt {-c+d x^2}} \, dx=\frac {\sqrt {-b d} a x \sqrt {\frac {a}{b}} E(\arcsin \left (\frac {\sqrt {\frac {a}{b}}}{x}\right )\,|\,\frac {b c}{a d}) - \sqrt {-b d} {\left (a - b\right )} x \sqrt {\frac {a}{b}} F(\arcsin \left (\frac {\sqrt {\frac {a}{b}}}{x}\right )\,|\,\frac {b c}{a d}) + \sqrt {-b x^{2} + a} \sqrt {d x^{2} - c} b}{b d x} \]

[In]

integrate((-b*x^2+a)^(1/2)/(d*x^2-c)^(1/2),x, algorithm="fricas")

[Out]

(sqrt(-b*d)*a*x*sqrt(a/b)*elliptic_e(arcsin(sqrt(a/b)/x), b*c/(a*d)) - sqrt(-b*d)*(a - b)*x*sqrt(a/b)*elliptic
_f(arcsin(sqrt(a/b)/x), b*c/(a*d)) + sqrt(-b*x^2 + a)*sqrt(d*x^2 - c)*b)/(b*d*x)

Sympy [F]

\[ \int \frac {\sqrt {a-b x^2}}{\sqrt {-c+d x^2}} \, dx=\int \frac {\sqrt {a - b x^{2}}}{\sqrt {- c + d x^{2}}}\, dx \]

[In]

integrate((-b*x**2+a)**(1/2)/(d*x**2-c)**(1/2),x)

[Out]

Integral(sqrt(a - b*x**2)/sqrt(-c + d*x**2), x)

Maxima [F]

\[ \int \frac {\sqrt {a-b x^2}}{\sqrt {-c+d x^2}} \, dx=\int { \frac {\sqrt {-b x^{2} + a}}{\sqrt {d x^{2} - c}} \,d x } \]

[In]

integrate((-b*x^2+a)^(1/2)/(d*x^2-c)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(-b*x^2 + a)/sqrt(d*x^2 - c), x)

Giac [F]

\[ \int \frac {\sqrt {a-b x^2}}{\sqrt {-c+d x^2}} \, dx=\int { \frac {\sqrt {-b x^{2} + a}}{\sqrt {d x^{2} - c}} \,d x } \]

[In]

integrate((-b*x^2+a)^(1/2)/(d*x^2-c)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(-b*x^2 + a)/sqrt(d*x^2 - c), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a-b x^2}}{\sqrt {-c+d x^2}} \, dx=\int \frac {\sqrt {a-b\,x^2}}{\sqrt {d\,x^2-c}} \,d x \]

[In]

int((a - b*x^2)^(1/2)/(d*x^2 - c)^(1/2),x)

[Out]

int((a - b*x^2)^(1/2)/(d*x^2 - c)^(1/2), x)