\(\int \frac {\sqrt {a+b x^2}}{\sqrt {c-d x^2}} \, dx\) [258]

Optimal result

Integrand size = 24, antiderivative size = 87 \[ \int \frac {\sqrt {a+b x^2}}{\sqrt {c-d x^2}} \, dx=\frac {\sqrt {c} \sqrt {a+b x^2} \sqrt {1-\frac {d x^2}{c}} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {1+\frac {b x^2}{a}} \sqrt {c-d x^2}} \]

[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 = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, 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 = 0.74 (sec) , antiderivative size = 87, 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 [A] (verified)

Time = 2.44 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.20

method	result	size
default	\(\frac {\sqrt {b \,x^{2}+a}\, \sqrt {-d \,x^{2}+c}\, a \sqrt {\frac {-d \,x^{2}+c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, E\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right )}{\left (-b d \,x^{4}-a d \,x^{2}+c b \,x^{2}+a c \right ) \sqrt {\frac {d}{c}}}\)	\(104\)
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (-d \,x^{2}+c \right )}\, \left (\frac {a \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, F\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+c b \,x^{2}+a c}}-\frac {a \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \left (F\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )-E\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+c b \,x^{2}+a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {-d \,x^{2}+c}}\)	\(254\)

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

Fricas [A] (verification not implemented)

none

Time = 0.09 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.49 \[ \int \frac {\sqrt {a+b x^2}}{\sqrt {c-d x^2}} \, dx=-\frac {\sqrt {-b d} b c^{2} x \sqrt {\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {\frac {c}{d}}}{x}\right )\,|\,-\frac {a d}{b c}) + \sqrt {b x^{2} + a} \sqrt {-d x^{2} + c} b c d - {\left (b c^{2} + a d^{2}\right )} \sqrt {-b d} x \sqrt {\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {\frac {c}{d}}}{x}\right )\,|\,-\frac {a d}{b c})}{b c d^{2} x} \]

[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)