\(\int \frac {\sqrt {a+b x^2}}{(c+d x^2)^{3/2} \sqrt {e+f x^2}} \, dx\) [461]

Optimal result

Integrand size = 34, antiderivative size = 149 \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}} \, dx=\frac {\sqrt {e} \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{c \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 5.37 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}} \, dx=\frac {\sqrt {e} \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|\frac {(-b c+a d) e}{a (d e-c f)}\right )}{c \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 149, 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.

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

\(\Big \downarrow \) 429

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

\(\Big \downarrow \) 327

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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