\(\int \frac {\sqrt {c+d x^2}}{\sqrt {a-b x^2}} \, dx\) [274]

Optimal result

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

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

[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 = 0.77 (sec) , antiderivative size = 87, 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 = 2.47 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.20

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.08 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.49 \[ \int \frac {\sqrt {c+d x^2}}{\sqrt {a-b x^2}} \, dx=-\frac {\sqrt {-b d} a^{2} d x \sqrt {\frac {a}{b}} E(\arcsin \left (\frac {\sqrt {\frac {a}{b}}}{x}\right )\,|\,-\frac {b c}{a d}) + \sqrt {-b x^{2} + a} \sqrt {d x^{2} + c} a b d - {\left (b^{2} c + a^{2} d\right )} \sqrt {-b d} x \sqrt {\frac {a}{b}} F(\arcsin \left (\frac {\sqrt {\frac {a}{b}}}{x}\right )\,|\,-\frac {b c}{a d})}{a b^{2} d x} \]

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

[In]

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

[Out]

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