\(\int \frac {\sqrt {a+b x^2}}{(c+d x^2)^{5/2} (e+f x^2)^{5/2}} \, dx\) [540]

Optimal result

Integrand size = 34, antiderivative size = 798 \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^{5/2}} \, dx=\frac {d x \sqrt {a+b x^2}}{3 c (d e-c f) \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}+\frac {d (b c (d e-7 c f)-2 a d (d e-4 c f)) x \sqrt {a+b x^2}}{3 c^2 (b c-a d) (d e-c f)^2 \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}-\frac {f \left (a d \left (2 d^2 e^2-9 c d e f-c^2 f^2\right )-b c \left (d^2 e^2-8 c d e f-c^2 f^2\right )\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c^2 (b c-a d) e (d e-c f)^3 \left (e+f x^2\right )^{3/2}}-\frac {\left (b^2 c e \left (d^3 e^3-9 c d^2 e^2 f-9 c^2 d e f^2+c^3 f^3\right )+2 a^2 d f \left (d^3 e^3-5 c d^2 e^2 f-5 c^2 d e f^2+c^3 f^3\right )-a b \left (2 d^4 e^4-9 c d^3 e^3 f-18 c^2 d^2 e^2 f^2-9 c^3 d e f^3+2 c^4 f^4\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )|\frac {a (d e-c f)}{c (b e-a f)}\right )}{3 \sqrt {a} c (b c-a d) e^2 \sqrt {-b e+a f} (d e-c f)^4 \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}-\frac {\left (a d f \left (d^2 e^2-18 c d e f+c^2 f^2\right )-b \left (d^3 e^3-9 c d^2 e^2 f-9 c^2 d e f^2+c^3 f^3\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{3 \sqrt {a} c e \sqrt {-b e+a f} (d e-c f)^4 \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}} \] Output:

1/3*d*x*(b*x^2+a)^(1/2)/c/(-c*f+d*e)/(d*x^2+c)^(3/2)/(f*x^2+e)^(3/2)+1/3*d 
*(b*c*(-7*c*f+d*e)-2*a*d*(-4*c*f+d*e))*x*(b*x^2+a)^(1/2)/c^2/(-a*d+b*c)/(- 
c*f+d*e)^2/(d*x^2+c)^(1/2)/(f*x^2+e)^(3/2)-1/3*f*(a*d*(-c^2*f^2-9*c*d*e*f+ 
2*d^2*e^2)-b*c*(-c^2*f^2-8*c*d*e*f+d^2*e^2))*x*(b*x^2+a)^(1/2)*(d*x^2+c)^( 
1/2)/c^2/(-a*d+b*c)/e/(-c*f+d*e)^3/(f*x^2+e)^(3/2)-1/3*(b^2*c*e*(c^3*f^3-9 
*c^2*d*e*f^2-9*c*d^2*e^2*f+d^3*e^3)+2*a^2*d*f*(c^3*f^3-5*c^2*d*e*f^2-5*c*d 
^2*e^2*f+d^3*e^3)-a*b*(2*c^4*f^4-9*c^3*d*e*f^3-18*c^2*d^2*e^2*f^2-9*c*d^3* 
e^3*f+2*d^4*e^4))*(b*x^2+a)^(1/2)*(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)*Elliptic 
E((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)/c/(-a*d+b*c)/e^2/(a*f-b*e)^(1/2)/(-c*f+d*e)^4/(d*x^2+c)^(1/2) 
/(e*(b*x^2+a)/a/(f*x^2+e))^(1/2)-1/3*(a*d*f*(c^2*f^2-18*c*d*e*f+d^2*e^2)-b 
*(c^3*f^3-9*c^2*d*e*f^2-9*c*d^2*e^2*f+d^3*e^3))*(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)/c/e/(a*f-b*e)^(1/2)/(-c*f+d*e)^ 
4/(d*x^2+c)^(1/2)/(e*(b*x^2+a)/a/(f*x^2+e))^(1/2)
 

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 434

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)/(d^3*f^3*x^12 + 3 
*(d^3*e*f^2 + c*d^2*f^3)*x^10 + 3*(d^3*e^2*f + 3*c*d^2*e*f^2 + c^2*d*f^3)* 
x^8 + (d^3*e^3 + 9*c*d^2*e^2*f + 9*c^2*d*e*f^2 + c^3*f^3)*x^6 + c^3*e^3 + 
3*(c*d^2*e^3 + 3*c^2*d*e^2*f + c^3*e*f^2)*x^4 + 3*(c^2*d*e^3 + c^3*e^2*f)* 
x^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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