\(\int \frac {\sqrt {-a-b x^2}}{\sqrt {c-d x^2}} \, dx\) [259]

Optimal result

Integrand size = 27, antiderivative size = 90 \[ \int \frac {\sqrt {-a-b x^2}}{\sqrt {c-d x^2}} \, dx=\frac {\sqrt {c} \sqrt {-a-b x^2} \sqrt {1-\frac {d x^2}{c}} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {1+\frac {b x^2}{a}} \sqrt {c-d x^2}} \]

[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 = 0.03 (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 = {438, 437, 435} \[ \int \frac {\sqrt {-a-b x^2}}{\sqrt {c-d x^2}} \, dx=\frac {\sqrt {c} \sqrt {-a-b x^2} \sqrt {1-\frac {d x^2}{c}} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {\frac {b x^2}{a}+1} \sqrt {c-d x^2}} \]

[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]
*Sqrt[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 = 0.76 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {-a-b x^2}}{\sqrt {c-d x^2}} \, dx=\frac {\sqrt {-a-b x^2} \sqrt {\frac {c-d x^2}{c}} E\left (\arcsin \left (\sqrt {\frac {d}{c}} x\right )|-\frac {b c}{a d}\right )}{\sqrt {\frac {d}{c}} \sqrt {\frac {a+b x^2}{a}} \sqrt {c-d x^2}} \]

[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. \(167\) vs. \(2(75)=150\).

Time = 2.48 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.87

method	result	size
default	\(\frac {\sqrt {-b \,x^{2}-a}\, \sqrt {-d \,x^{2}+c}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {-d \,x^{2}+c}{c}}\, \left (a F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) d +b c F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right )-b c E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right )\right )}{\left (-b d \,x^{4}-a d \,x^{2}+c b \,x^{2}+a c \right ) \sqrt {-\frac {b}{a}}\, d}\)	\(168\)
elliptic	\(\frac {\sqrt {-\left (b \,x^{2}+a \right ) \left (-d \,x^{2}+c \right )}\, \left (-\frac {a \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1-\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}-c b \,x^{2}-a c}}-\frac {b c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1-\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}-c b \,x^{2}-a c}\, d}\right )}{\sqrt {-b \,x^{2}-a}\, \sqrt {-d \,x^{2}+c}}\)	\(268\)

[In]

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

[Out]

(-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

Fricas [A] (verification not implemented)

none

Time = 0.08 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.46 \[ \int \frac {\sqrt {-a-b x^2}}{\sqrt {c-d x^2}} \, dx=-\frac {\sqrt {b d} b c^{2} x \sqrt {\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {\frac {c}{d}}}{x}\right )\,|\,-\frac {a d}{b c}) + \sqrt {-b x^{2} - a} \sqrt {-d x^{2} + c} b c d - {\left (b c^{2} + a d^{2}\right )} \sqrt {b d} x \sqrt {\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {\frac {c}{d}}}{x}\right )\,|\,-\frac {a d}{b c})}{b c d^{2} x} \]

[In]

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

[Out]

-(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*sqrt(c/d)*elliptic_f(arcsin(sqrt(c/d)/x), -a*d/(b*c)))/(b*c*d^2*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. \[ \int \frac {\sqrt {-a-b x^2}}{\sqrt {c-d x^2}} \, dx=\int \frac {\sqrt {-b\,x^2-a}}{\sqrt {c-d\,x^2}} \,d x \]

[In]

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

[Out]

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