\(\int \frac {(a+b x^2)^{5/2}}{(c+d x^2)^{7/2} \sqrt {e+f x^2}} \, dx\) [474]

Optimal result

Integrand size = 34, antiderivative size = 531 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{7/2} \sqrt {e+f x^2}} \, dx=\frac {(b c-a d)^2 x \sqrt {a+b x^2} \sqrt {e+f x^2}}{5 c d (d e-c f) \left (c+d x^2\right )^{5/2}}-\frac {(b c-a d) (b c (7 d e-3 c f)+4 a d (d e-2 c f)) x \sqrt {a+b x^2} \sqrt {e+f x^2}}{15 c^2 d (d e-c f)^2 \left (c+d x^2\right )^{3/2}}+\frac {\sqrt {e} \left (8 b^2 c^2 e^2+a b c e (7 d e-23 c f)+a^2 \left (8 d^2 e^2-23 c d e f+23 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{15 c^3 (d e-c f)^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}-\frac {4 \sqrt {e} (b e-a f) (b c e+a d e-2 a c f) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{15 c^2 (d e-c f)^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}} \] Output:

1/5*(-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 
)^(5/2)-1/15*(-a*d+b*c)*(b*c*(-3*c*f+7*d*e)+4*a*d*(-2*c*f+d*e))*x*(b*x^2+a 
)^(1/2)*(f*x^2+e)^(1/2)/c^2/d/(-c*f+d*e)^2/(d*x^2+c)^(3/2)+1/15*e^(1/2)*(8 
*b^2*c^2*e^2+a*b*c*e*(-23*c*f+7*d*e)+a^2*(23*c^2*f^2-23*c*d*e*f+8*d^2*e^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^3/(-c*f 
+d*e)^(5/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(f*x^2+e)^(1/2)-4/15*e^(1/2)*( 
-a*f+b*e)*(-2*a*c*f+a*d*e+b*c*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))/c^2/(-c*f+d*e)^(5/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2) 
/(f*x^2+e)^(1/2)
 

Mathematica [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{7/2} \sqrt {e+f x^2}} \, dx=\int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{7/2} \sqrt {e+f x^2}} \, dx \] Input:

Integrate[(a + b*x^2)^(5/2)/((c + d*x^2)^(7/2)*Sqrt[e + f*x^2]),x]
 

Output:

Integrate[(a + b*x^2)^(5/2)/((c + d*x^2)^(7/2)*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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{7/2} \sqrt {e+f x^2}} \, dx\)

\(\Big \downarrow \) 434

\(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{7/2} \sqrt {e+f x^2}}dx\)

Input:

Int[(a + b*x^2)^(5/2)/((c + d*x^2)^(7/2)*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]

\[\int \frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{\left (x^{2} d +c \right )^{\frac {7}{2}} \sqrt {f \,x^{2}+e}}d x\]

Input:

int((b*x^2+a)^(5/2)/(d*x^2+c)^(7/2)/(f*x^2+e)^(1/2),x)
 

Output:

int((b*x^2+a)^(5/2)/(d*x^2+c)^(7/2)/(f*x^2+e)^(1/2),x)
 

Fricas [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{7/2} \sqrt {e+f x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{{\left (d x^{2} + c\right )}^{\frac {7}{2}} \sqrt {f x^{2} + e}} \,d x } \] Input:

integrate((b*x^2+a)^(5/2)/(d*x^2+c)^(7/2)/(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^4*f*x^10 + (d^4*e + 4*c*d^3*f)*x^8 + 2*(2*c*d^3*e + 3*c^2*d^ 
2*f)*x^6 + c^4*e + 2*(3*c^2*d^2*e + 2*c^3*d*f)*x^4 + (4*c^3*d*e + c^4*f)*x 
^2), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{7/2} \sqrt {e+f x^2}} \, dx=\text {Timed out} \] Input:

integrate((b*x**2+a)**(5/2)/(d*x**2+c)**(7/2)/(f*x**2+e)**(1/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{7/2} \sqrt {e+f x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{{\left (d x^{2} + c\right )}^{\frac {7}{2}} \sqrt {f x^{2} + e}} \,d x } \] Input:

integrate((b*x^2+a)^(5/2)/(d*x^2+c)^(7/2)/(f*x^2+e)^(1/2),x, algorithm="ma 
xima")
 

Output:

integrate((b*x^2 + a)^(5/2)/((d*x^2 + c)^(7/2)*sqrt(f*x^2 + e)), x)
 

Giac [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{7/2} \sqrt {e+f x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{{\left (d x^{2} + c\right )}^{\frac {7}{2}} \sqrt {f x^{2} + e}} \,d x } \] Input:

integrate((b*x^2+a)^(5/2)/(d*x^2+c)^(7/2)/(f*x^2+e)^(1/2),x, algorithm="gi 
ac")
 

Output:

integrate((b*x^2 + a)^(5/2)/((d*x^2 + c)^(7/2)*sqrt(f*x^2 + e)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{7/2} \sqrt {e+f x^2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{5/2}}{{\left (d\,x^2+c\right )}^{7/2}\,\sqrt {f\,x^2+e}} \,d x \] Input:

int((a + b*x^2)^(5/2)/((c + d*x^2)^(7/2)*(e + f*x^2)^(1/2)),x)
 

Output:

int((a + b*x^2)^(5/2)/((c + d*x^2)^(7/2)*(e + f*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{7/2} \sqrt {e+f x^2}} \, dx=\int \frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{\left (d \,x^{2}+c \right )^{\frac {7}{2}} \sqrt {f \,x^{2}+e}}d x \] Input:

int((b*x^2+a)^(5/2)/(d*x^2+c)^(7/2)/(f*x^2+e)^(1/2),x)
 

Output:

int((b*x^2+a)^(5/2)/(d*x^2+c)^(7/2)/(f*x^2+e)^(1/2),x)