\(\int \frac {\sqrt {c+d x^2}}{(a+b x^2)^{3/2} \sqrt {e+f x^2}} \, dx\) [106]

Optimal result

Integrand size = 34, antiderivative size = 148 \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}} \, dx=\frac {\sqrt {e} \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )|\frac {(b c-a d) e}{c (b e-a f)}\right )}{a \sqrt {b e-a f} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {568, 435} \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}} \, dx=\frac {\sqrt {e} \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )|\frac {(b c-a d) e}{c (b e-a f)}\right )}{a \sqrt {e+f x^2} \sqrt {b e-a f} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \]

[In]

Int[Sqrt[c + d*x^2]/((a + b*x^2)^(3/2)*Sqrt[e + f*x^2]),x]

[Out]

(Sqrt[e]*Sqrt[c + d*x^2]*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))]*EllipticE[ArcSin[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*S
qrt[a + b*x^2])], ((b*c - a*d)*e)/(c*(b*e - a*f))])/(a*Sqrt[b*e - a*f]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]*S
qrt[e + f*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 568

Int[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)^(3/2)*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[
c + d*x^2]*(Sqrt[a*((e + f*x^2)/(e*(a + b*x^2)))]/(a*Sqrt[e + f*x^2]*Sqrt[a*((c + d*x^2)/(c*(a + b*x^2)))])),
Subst[Int[Sqrt[1 - (b*c - a*d)*(x^2/c)]/Sqrt[1 - (b*e - a*f)*(x^2/e)], x], x, x/Sqrt[a + b*x^2]], x] /; FreeQ[
{a, b, c, d, e, f}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}}\right ) \text {Subst}\left (\int \frac {\sqrt {1-\frac {(b c-a d) x^2}{c}}}{\sqrt {1-\frac {(b e-a f) x^2}{e}}} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{a \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}} \\ & = \frac {\sqrt {e} \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} E\left (\sin ^{-1}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )|\frac {(b c-a d) e}{c (b e-a f)}\right )}{a \sqrt {b e-a f} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.65 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}} \, dx=\frac {\sqrt {e} \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )|\frac {(b c-a d) e}{c (b e-a f)}\right )}{a \sqrt {b e-a f} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}} \]

[In]

Integrate[Sqrt[c + d*x^2]/((a + b*x^2)^(3/2)*Sqrt[e + f*x^2]),x]

[Out]

(Sqrt[e]*Sqrt[c + d*x^2]*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))]*EllipticE[ArcSin[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*S
qrt[a + b*x^2])], ((b*c - a*d)*e)/(c*(b*e - a*f))])/(a*Sqrt[b*e - a*f]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]*S
qrt[e + f*x^2])

Maple [F]

\[\int \frac {\sqrt {d \,x^{2}+c}}{\left (b \,x^{2}+a \right )^{\frac {3}{2}} \sqrt {f \,x^{2}+e}}d x\]

[In]

int((d*x^2+c)^(1/2)/(b*x^2+a)^(3/2)/(f*x^2+e)^(1/2),x)

[Out]

int((d*x^2+c)^(1/2)/(b*x^2+a)^(3/2)/(f*x^2+e)^(1/2),x)

Fricas [F]

\[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}} \, dx=\int { \frac {\sqrt {d x^{2} + c}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {f x^{2} + e}} \,d x } \]

[In]

integrate((d*x^2+c)^(1/2)/(b*x^2+a)^(3/2)/(f*x^2+e)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)/(b^2*f*x^6 + (b^2*e + 2*a*b*f)*x^4 + a^2*e + (2*a*b*e
+ a^2*f)*x^2), x)

Sympy [F]

\[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}} \, dx=\int \frac {\sqrt {c + d x^{2}}}{\left (a + b x^{2}\right )^{\frac {3}{2}} \sqrt {e + f x^{2}}}\, dx \]

[In]

integrate((d*x**2+c)**(1/2)/(b*x**2+a)**(3/2)/(f*x**2+e)**(1/2),x)

[Out]

Integral(sqrt(c + d*x**2)/((a + b*x**2)**(3/2)*sqrt(e + f*x**2)), x)

Maxima [F]

\[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}} \, dx=\int { \frac {\sqrt {d x^{2} + c}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {f x^{2} + e}} \,d x } \]

[In]

integrate((d*x^2+c)^(1/2)/(b*x^2+a)^(3/2)/(f*x^2+e)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(d*x^2 + c)/((b*x^2 + a)^(3/2)*sqrt(f*x^2 + e)), x)

Giac [F]

\[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}} \, dx=\int { \frac {\sqrt {d x^{2} + c}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {f x^{2} + e}} \,d x } \]

[In]

integrate((d*x^2+c)^(1/2)/(b*x^2+a)^(3/2)/(f*x^2+e)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(d*x^2 + c)/((b*x^2 + a)^(3/2)*sqrt(f*x^2 + e)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}} \, dx=\int \frac {\sqrt {d\,x^2+c}}{{\left (b\,x^2+a\right )}^{3/2}\,\sqrt {f\,x^2+e}} \,d x \]

[In]

int((c + d*x^2)^(1/2)/((a + b*x^2)^(3/2)*(e + f*x^2)^(1/2)),x)

[Out]

int((c + d*x^2)^(1/2)/((a + b*x^2)^(3/2)*(e + f*x^2)^(1/2)), x)