\(\int \frac {(a+b x^2)^{3/2} \sqrt {e+f x^2}}{(c+d x^2)^{3/2}} \, dx\) [391]

Optimal result

Integrand size = 34, antiderivative size = 584 \[ \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {b x \sqrt {a+b x^2} \sqrt {e+f x^2}}{2 d \sqrt {c+d x^2}}-\frac {(3 b c-2 a d) \sqrt {e} \sqrt {d e-c f} \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{2 c d^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}+\frac {\sqrt {e} \left (2 a^2 d^2 f-b^2 c (d e-3 c f)+2 a b d (d e-3 c f)\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{2 a d^3 \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}+\frac {b c \sqrt {e} (b d e-3 b c f+3 a d f) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticPi}\left (\frac {d e}{d e-c f},\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{2 a d^3 \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}} \] Output:

1/2*b*x*(b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/d/(d*x^2+c)^(1/2)-1/2*(-2*a*d+3*b* 
c)*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 
)*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/d^2/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(f*x^2+e)^(1/2)+1/2 
*e^(1/2)*(2*a^2*d^2*f-b^2*c*(-3*c*f+d*e)+2*a*b*d*(-3*c*f+d*e))*(b*x^2+a)^( 
1/2)*(c*(f*x^2+e)/e/(d*x^2+c))^(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))/a/d^3/(-c*f+d*e)^(1/2) 
/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(f*x^2+e)^(1/2)+1/2*b*c*e^(1/2)*(3*a*d*f- 
3*b*c*f+b*d*e)*(b*x^2+a)^(1/2)*(c*(f*x^2+e)/e/(d*x^2+c))^(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))/a/d^3/(-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 [F]

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

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

Output:

Integrate[((a + b*x^2)^(3/2)*Sqrt[e + f*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.

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

\(\Big \downarrow \) 434

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)/(d^2*x^4 + 2*c* 
d*x^2 + c^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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