\(\int \frac {(e+f x^2)^{3/2}}{(a+b x^2)^{3/2} (c+d x^2)^{5/2}} \, dx\) [451]

Optimal result

Integrand size = 34, antiderivative size = 530 \[ \int \frac {\left (e+f x^2\right )^{3/2}}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2}} \, dx=-\frac {(d e-c f) x \sqrt {e+f x^2}}{3 c (b c-a d) \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}+\frac {b (3 b c e+a d e-4 a c f) x \sqrt {e+f x^2}}{3 a c (b c-a d)^2 \sqrt {a+b x^2} \sqrt {c+d x^2}}+\frac {\sqrt {e} \sqrt {d e-c f} \left (3 b^2 c^2 e+a b c (7 d e-6 c f)-2 a^2 d (d e+c f)\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{3 a c^2 (b c-a d)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}-\frac {\sqrt {e} (b e-a f) (3 b c (3 d e-2 c f)-a d (d e+2 c f)) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{3 a c (b c-a d)^3 \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}} \] Output:

-1/3*(-c*f+d*e)*x*(f*x^2+e)^(1/2)/c/(-a*d+b*c)/(b*x^2+a)^(1/2)/(d*x^2+c)^( 
3/2)+1/3*b*(-4*a*c*f+a*d*e+3*b*c*e)*x*(f*x^2+e)^(1/2)/a/c/(-a*d+b*c)^2/(b* 
x^2+a)^(1/2)/(d*x^2+c)^(1/2)+1/3*e^(1/2)*(-c*f+d*e)^(1/2)*(3*b^2*c^2*e+a*b 
*c*(-6*c*f+7*d*e)-2*a^2*d*(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))/a/c^2/(-a*d+b*c)^3/(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)*(3*b*c*(-2*c*f+3*d*e)-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)*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)^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]

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

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

Output:

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

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

\(\Big \downarrow \) 434

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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