\(\int \frac {\sqrt {-c-d x^2}}{\sqrt {-a+b x^2}} \, dx\) [277]

Optimal result

Integrand size = 28, antiderivative size = 91 \[ \int \frac {\sqrt {-c-d x^2}}{\sqrt {-a+b x^2}} \, dx=\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {-c-d x^2} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {-a+b x^2} \sqrt {1+\frac {d x^2}{c}}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {438, 437, 435} \[ \int \frac {\sqrt {-c-d x^2}}{\sqrt {-a+b x^2}} \, dx=\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {-c-d x^2} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {b x^2-a} \sqrt {\frac {d x^2}{c}+1}} \]

[In]

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

[Out]

(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])

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

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {-c-d x^2}}{\sqrt {-a+b x^2}} \, dx=\frac {\sqrt {\frac {a-b x^2}{a}} \sqrt {-c-d x^2} E\left (\arcsin \left (\sqrt {\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )}{\sqrt {\frac {b}{a}} \sqrt {-a+b x^2} \sqrt {\frac {c+d x^2}{c}}} \]

[In]

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

[Out]

(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])

Maple [A] (verified)

Time = 2.41 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.19

method	result	size
default	\(\frac {\sqrt {-d \,x^{2}-c}\, \sqrt {b \,x^{2}-a}\, c \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right )}{\left (b d \,x^{4}-a d \,x^{2}+c b \,x^{2}-a c \right ) \sqrt {\frac {b}{a}}}\)	\(108\)
elliptic	\(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (-\frac {c \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-c b \,x^{2}+a c}}+\frac {c \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )-E\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-c b \,x^{2}+a c}}\right )}{\sqrt {b \,x^{2}-a}\, \sqrt {-d \,x^{2}-c}}\)	\(258\)

[In]

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

[Out]

1/(b*d*x^4-a*d*x^2+b*c*x^2-a*c)/(b/a)^(1/2)*(-d*x^2-c)^(1/2)*(b*x^2-a)^(1/2)*c*((-b*x^2+a)/a)^(1/2)*((d*x^2+c)
/c)^(1/2)*EllipticE(x*(b/a)^(1/2),(-a*d/b/c)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.08 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.46 \[ \int \frac {\sqrt {-c-d x^2}}{\sqrt {-a+b x^2}} \, dx=\frac {\sqrt {-b d} a^{2} d x \sqrt {\frac {a}{b}} E(\arcsin \left (\frac {\sqrt {\frac {a}{b}}}{x}\right )\,|\,-\frac {b c}{a d}) + \sqrt {b x^{2} - a} \sqrt {-d x^{2} - c} a b d - {\left (b^{2} c + a^{2} d\right )} \sqrt {-b d} x \sqrt {\frac {a}{b}} F(\arcsin \left (\frac {\sqrt {\frac {a}{b}}}{x}\right )\,|\,-\frac {b c}{a d})}{a b^{2} d x} \]

[In]

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

[Out]

(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)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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