\(\int \frac {(a+b x^2)^{3/2}}{\sqrt {c+d x^2} (e+f x^2)^{5/2}} \, dx\) [543]

Optimal result

Integrand size = 34, antiderivative size = 420 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx=\frac {(b e-a f) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 e (d e-c f) \left (e+f x^2\right )^{3/2}}+\frac {2 c \sqrt {-b e+a f} (b c e-2 a d e+a c f) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )|\frac {a (d e-c f)}{c (b e-a f)}\right )}{3 \sqrt {a} e^2 (d e-c f)^2 \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}+\frac {(b c-a d) (2 b c e-3 a d e+a c f) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{3 \sqrt {a} e \sqrt {-b e+a f} (d e-c f)^2 \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}} \] Output:

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

Mathematica [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx=\int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx \] Input:

Integrate[(a + b*x^2)^(3/2)/(Sqrt[c + d*x^2]*(e + f*x^2)^(5/2)),x]
 

Output:

Integrate[(a + b*x^2)^(3/2)/(Sqrt[c + d*x^2]*(e + f*x^2)^(5/2)), 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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 434

\(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}}dx\)

Input:

Int[(a + b*x^2)^(3/2)/(Sqrt[c + d*x^2]*(e + f*x^2)^(5/2)),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]

\[\int \frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{\sqrt {x^{2} d +c}\, \left (f \,x^{2}+e \right )^{\frac {5}{2}}}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{\sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(5/2),x, algorithm="fr 
icas")
 

Output:

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

Sympy [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {3}{2}}}{\sqrt {c + d x^{2}} \left (e + f x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:

integrate((b*x**2+a)**(3/2)/(d*x**2+c)**(1/2)/(f*x**2+e)**(5/2),x)
 

Output:

Integral((a + b*x**2)**(3/2)/(sqrt(c + d*x**2)*(e + f*x**2)**(5/2)), x)
 

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{\sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(5/2),x, algorithm="ma 
xima")
 

Output:

integrate((b*x^2 + a)^(3/2)/(sqrt(d*x^2 + c)*(f*x^2 + e)^(5/2)), x)
 

Giac [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{\sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(5/2),x, algorithm="gi 
ac")
 

Output:

integrate((b*x^2 + a)^(3/2)/(sqrt(d*x^2 + c)*(f*x^2 + e)^(5/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}}{\sqrt {d\,x^2+c}\,{\left (f\,x^2+e\right )}^{5/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx=\int \frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{\sqrt {d \,x^{2}+c}\, \left (f \,x^{2}+e \right )^{\frac {5}{2}}}d x \] Input:

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

Output:

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