\(\int \frac {(a+b x^2)^{5/2}}{(c+d x^2)^{5/2} \sqrt {e+f x^2}} \, dx\) [473]

Optimal result

Integrand size = 34, antiderivative size = 619 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}} \, dx=\frac {(b c-a d)^2 x \sqrt {a+b x^2} \sqrt {e+f x^2}}{3 c d (d e-c f) \left (c+d x^2\right )^{3/2}}-\frac {(b c-a d) \sqrt {e} (b c (5 d e-3 c f)+2 a d (d e-2 c f)) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{3 c^2 d^2 (d e-c f)^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}+\frac {(b c-a d) \sqrt {e} \left (a b d^2 e-a^2 d^2 f-3 b^2 c (d e-c f)\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{3 a c d^3 (d e-c f)^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}+\frac {b^3 c \sqrt {e} \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticPi}\left (\frac {d e}{d e-c f},\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{a d^3 \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}} \] Output:

1/3*(-a*d+b*c)^2*x*(b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/c/d/(-c*f+d*e)/(d*x^2+c 
)^(3/2)-1/3*(-a*d+b*c)*e^(1/2)*(b*c*(-3*c*f+5*d*e)+2*a*d*(-2*c*f+d*e))*(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^2/d^2/(-c*f+ 
d*e)^(3/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(f*x^2+e)^(1/2)+1/3*(-a*d+b*c)* 
e^(1/2)*(a*b*d^2*e-a^2*d^2*f-3*b^2*c*(-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/c/d^3/(-c*f+d*e)^(3/2)/(c*(b*x^2+a 
)/a/(d*x^2+c))^(1/2)/(f*x^2+e)^(1/2)+b^3*c*e^(1/2)*(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]

\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}} \, dx=\int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}} \, dx \] Input:

Integrate[(a + b*x^2)^(5/2)/((c + d*x^2)^(5/2)*Sqrt[e + f*x^2]),x]
 

Output:

Integrate[(a + b*x^2)^(5/2)/((c + d*x^2)^(5/2)*Sqrt[e + f*x^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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}} \, dx\)

\(\Big \downarrow \) 434

\(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}}dx\)

Input:

Int[(a + b*x^2)^(5/2)/((c + d*x^2)^(5/2)*Sqrt[e + f*x^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]

\[\int \frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{\left (x^{2} d +c \right )^{\frac {5}{2}} \sqrt {f \,x^{2}+e}}d x\]

Input:

int((b*x^2+a)^(5/2)/(d*x^2+c)^(5/2)/(f*x^2+e)^(1/2),x)
 

Output:

int((b*x^2+a)^(5/2)/(d*x^2+c)^(5/2)/(f*x^2+e)^(1/2),x)
 

Fricas [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}} \sqrt {f x^{2} + e}} \,d x } \] Input:

integrate((b*x^2+a)^(5/2)/(d*x^2+c)^(5/2)/(f*x^2+e)^(1/2),x, algorithm="fr 
icas")
 

Output:

integral((b^2*x^4 + 2*a*b*x^2 + a^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt( 
f*x^2 + e)/(d^3*f*x^8 + (d^3*e + 3*c*d^2*f)*x^6 + 3*(c*d^2*e + c^2*d*f)*x^ 
4 + c^3*e + (3*c^2*d*e + c^3*f)*x^2), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}} \, dx=\text {Timed out} \] Input:

integrate((b*x**2+a)**(5/2)/(d*x**2+c)**(5/2)/(f*x**2+e)**(1/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}} \sqrt {f x^{2} + e}} \,d x } \] Input:

integrate((b*x^2+a)^(5/2)/(d*x^2+c)^(5/2)/(f*x^2+e)^(1/2),x, algorithm="ma 
xima")
 

Output:

integrate((b*x^2 + a)^(5/2)/((d*x^2 + c)^(5/2)*sqrt(f*x^2 + e)), x)
 

Giac [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}} \sqrt {f x^{2} + e}} \,d x } \] Input:

integrate((b*x^2+a)^(5/2)/(d*x^2+c)^(5/2)/(f*x^2+e)^(1/2),x, algorithm="gi 
ac")
 

Output:

integrate((b*x^2 + a)^(5/2)/((d*x^2 + c)^(5/2)*sqrt(f*x^2 + e)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{5/2}}{{\left (d\,x^2+c\right )}^{5/2}\,\sqrt {f\,x^2+e}} \,d x \] Input:

int((a + b*x^2)^(5/2)/((c + d*x^2)^(5/2)*(e + f*x^2)^(1/2)),x)
 

Output:

int((a + b*x^2)^(5/2)/((c + d*x^2)^(5/2)*(e + f*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}} \, dx=\int \frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{\left (d \,x^{2}+c \right )^{\frac {5}{2}} \sqrt {f \,x^{2}+e}}d x \] Input:

int((b*x^2+a)^(5/2)/(d*x^2+c)^(5/2)/(f*x^2+e)^(1/2),x)
 

Output:

int((b*x^2+a)^(5/2)/(d*x^2+c)^(5/2)/(f*x^2+e)^(1/2),x)