\(\int \frac {(e+f x^2)^{3/2}}{\sqrt {a+b x^2} (c+d x^2)^{3/2}} \, dx\) [443]

Optimal result

Integrand size = 34, antiderivative size = 504 \[ \int \frac {\left (e+f x^2\right )^{3/2}}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}} \, dx=-\frac {a (d e-c f)^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2} E\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{c d (b c-a d) \sqrt {e} \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {\sqrt {d e-c f} (b d e+b c f-2 a d f) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{d^2 (b c-a d) \sqrt {e} \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {c f^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2} \operatorname {EllipticPi}\left (\frac {d e}{d e-c f},\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{d^2 \sqrt {e} \sqrt {d e-c f} \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}} \] Output:

-a*(-c*f+d*e)^(3/2)*(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)*(f*x^2+e)^(1/2)*Ellipt 
icE((-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/d/(-a*d+b*c)/e^(1/2)/(b*x^2+a)^(1/2)/(c*(f*x^2+e)/e/(d*x^2+c))^ 
(1/2)+(-c*f+d*e)^(1/2)*(-2*a*d*f+b*c*f+b*d*e)*(c*(b*x^2+a)/a/(d*x^2+c))^(1 
/2)*(f*x^2+e)^(1/2)*EllipticF((-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))/d^2/(-a*d+b*c)/e^(1/2)/(b*x^2+a)^(1/2)/ 
(c*(f*x^2+e)/e/(d*x^2+c))^(1/2)+c*f^2*(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)*(f*x 
^2+e)^(1/2)*EllipticPi((-c*f+d*e)^(1/2)*x/e^(1/2)/(d*x^2+c)^(1/2),d*e/(-c* 
f+d*e),(-(-a*d+b*c)*e/a/(-c*f+d*e))^(1/2))/d^2/e^(1/2)/(-c*f+d*e)^(1/2)/(b 
*x^2+a)^(1/2)/(c*(f*x^2+e)/e/(d*x^2+c))^(1/2)
 

Mathematica [F]

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

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

Output:

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

Rubi [F]

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.

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

\(\Big \downarrow \) 434

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 434

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2)^(r_.), x_Symbol] :> Unintegrable[(a + b*x^2)^p*(c + d*x^2)^q*(e + f* 
x^2)^r, x] /; FreeQ[{a, b, c, d, e, f, p, q, r}, x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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