\(\int \frac {\sqrt {e+f x^2}}{(a+b x^2)^{3/2} \sqrt {c+d x^2}} \, dx\) [410]

Optimal result

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

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

Mathematica [A] (verified)

Time = 5.31 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {c} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2} E\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )|\frac {c (b e-a f)}{(b c-a d) e}\right )}{a \sqrt {b c-a d} \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {429, 327}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 429

\(\displaystyle \frac {\sqrt {e+f x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \int \frac {\sqrt {1-\frac {(b e-a f) x^2}{e \left (b x^2+a\right )}}}{\sqrt {1-\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}}}d\frac {x}{\sqrt {b x^2+a}}}{a \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\sqrt {c} \sqrt {e+f x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )|\frac {c (b e-a f)}{(b c-a d) e}\right )}{a \sqrt {c+d x^2} \sqrt {b c-a d} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}}}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[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 429

Int[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)^(3/2)*Sqrt[(e_) + (f_. 
)*(x_)^2]), x_Symbol] :> Simp[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)))]))   Subs 
t[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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\int \frac {\sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}+2 a b c \,x^{2}+a^{2} c}d x \] Input:

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

Output:

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