\(\int \frac {(e+f x^2)^{3/2}}{\sqrt {a+b x^2} (c+d x^2)^{5/2}} \, dx\) [444]

Optimal result

Integrand size = 34, antiderivative size = 427 \[ \int \frac {\left (e+f x^2\right )^{3/2}}{\sqrt {a+b x^2} \left (c+d x^2\right )^{5/2}} \, dx=-\frac {(d e-c f) x \sqrt {a+b x^2} \sqrt {e+f x^2}}{3 c (b c-a d) \left (c+d x^2\right )^{3/2}}-\frac {2 \sqrt {e} \sqrt {d e-c f} (2 b c e-a d e-a c f) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{3 c^2 (b c-a d)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}+\frac {\sqrt {e} (b e-a f) (3 b c e-a d e-2 a c f) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{3 a c (b c-a d)^2 \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}} \] Output:

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

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

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

Output:

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

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

\(\Big \downarrow \) 434

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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