\(\int \frac {1}{\sqrt {4+x^2} \sqrt {c+d x^2}} \, dx\) [292]

Optimal result

Integrand size = 21, antiderivative size = 61 \[ \int \frac {1}{\sqrt {4+x^2} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {x}{2}\right ),1-\frac {4 d}{c}\right )}{c \sqrt {4+x^2} \sqrt {\frac {c+d x^2}{c \left (4+x^2\right )}}} \]

[Out]

(1/(x^2+4))^(1/2)*EllipticF(x/(x^2+4)^(1/2),(1-4*d/c)^(1/2))*(d*x^2+c)^(1/2)/c/((d*x^2+c)/c/(x^2+4))^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {429} \[ \int \frac {1}{\sqrt {4+x^2} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {x}{2}\right ),1-\frac {4 d}{c}\right )}{c \sqrt {x^2+4} \sqrt {\frac {c+d x^2}{c \left (x^2+4\right )}}} \]

[In]

Int[1/(Sqrt[4 + x^2]*Sqrt[c + d*x^2]),x]

[Out]

(Sqrt[c + d*x^2]*EllipticF[ArcTan[x/2], 1 - (4*d)/c])/(c*Sqrt[4 + x^2]*Sqrt[(c + d*x^2)/(c*(4 + x^2))])

Rule 429

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*
Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /
; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {x}{2}\right )|1-\frac {4 d}{c}\right )}{c \sqrt {4+x^2} \sqrt {\frac {c+d x^2}{c \left (4+x^2\right )}}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.46 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.77 \[ \int \frac {1}{\sqrt {4+x^2} \sqrt {c+d x^2}} \, dx=-\frac {i \sqrt {\frac {c+d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {x}{2}\right ),\frac {4 d}{c}\right )}{\sqrt {c+d x^2}} \]

[In]

Integrate[1/(Sqrt[4 + x^2]*Sqrt[c + d*x^2]),x]

[Out]

((-I)*Sqrt[(c + d*x^2)/c]*EllipticF[I*ArcSinh[x/2], (4*d)/c])/Sqrt[c + d*x^2]

Maple [A] (verified)

Time = 2.42 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.87

method	result	size
default	\(\frac {\sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {d}{c}}, \frac {\sqrt {\frac {c}{d}}}{2}\right )}{2 \sqrt {d \,x^{2}+c}\, \sqrt {-\frac {d}{c}}}\)	\(53\)
elliptic	\(\frac {\sqrt {\left (d \,x^{2}+c \right ) \left (x^{2}+4\right )}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {d}{c}}, \frac {\sqrt {-4+\frac {c +4 d}{d}}}{2}\right )}{2 \sqrt {d \,x^{2}+c}\, \sqrt {-\frac {d}{c}}\, \sqrt {d \,x^{4}+c \,x^{2}+4 d \,x^{2}+4 c}}\)	\(95\)

[In]

int(1/(x^2+4)^(1/2)/(d*x^2+c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/2/(d*x^2+c)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-d/c)^(1/2),1/2*(c/d)^(1/2))/(-d/c)^(1/2)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.26 \[ \int \frac {1}{\sqrt {4+x^2} \sqrt {c+d x^2}} \, dx=-\frac {i \, F(\arcsin \left (\frac {1}{2} i \, x\right )\,|\,\frac {4 \, d}{c})}{\sqrt {c}} \]

[In]

integrate(1/(x^2+4)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

-I*elliptic_f(arcsin(1/2*I*x), 4*d/c)/sqrt(c)

Sympy [F]

\[ \int \frac {1}{\sqrt {4+x^2} \sqrt {c+d x^2}} \, dx=\int \frac {1}{\sqrt {c + d x^{2}} \sqrt {x^{2} + 4}}\, dx \]

[In]

integrate(1/(x**2+4)**(1/2)/(d*x**2+c)**(1/2),x)

[Out]

Integral(1/(sqrt(c + d*x**2)*sqrt(x**2 + 4)), x)

Maxima [F]

\[ \int \frac {1}{\sqrt {4+x^2} \sqrt {c+d x^2}} \, dx=\int { \frac {1}{\sqrt {d x^{2} + c} \sqrt {x^{2} + 4}} \,d x } \]

[In]

integrate(1/(x^2+4)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(d*x^2 + c)*sqrt(x^2 + 4)), x)

Giac [F]

\[ \int \frac {1}{\sqrt {4+x^2} \sqrt {c+d x^2}} \, dx=\int { \frac {1}{\sqrt {d x^{2} + c} \sqrt {x^{2} + 4}} \,d x } \]

[In]

integrate(1/(x^2+4)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(d*x^2 + c)*sqrt(x^2 + 4)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {4+x^2} \sqrt {c+d x^2}} \, dx=\int \frac {1}{\sqrt {x^2+4}\,\sqrt {d\,x^2+c}} \,d x \]

[In]

int(1/((x^2 + 4)^(1/2)*(c + d*x^2)^(1/2)),x)

[Out]

int(1/((x^2 + 4)^(1/2)*(c + d*x^2)^(1/2)), x)