\(\int \frac {\sqrt {a+b x^2}}{(c+d x^2)^{7/2} \sqrt {e+f x^2}} \, dx\) [463]

Optimal result

Integrand size = 34, antiderivative size = 588 \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{7/2} \sqrt {e+f x^2}} \, dx=\frac {d x \sqrt {a+b x^2} \sqrt {e+f x^2}}{5 c (d e-c f) \left (c+d x^2\right )^{5/2}}+\frac {d (b c (3 d e-7 c f)-4 a d (d e-2 c f)) x \sqrt {a+b x^2} \sqrt {e+f x^2}}{15 c^2 (b c-a d) (d e-c f)^2 \left (c+d x^2\right )^{3/2}}+\frac {\sqrt {e} \left (b^2 c^2 \left (3 d^2 e^2-10 c d e f+15 c^2 f^2\right )+a^2 d^2 \left (8 d^2 e^2-23 c d e f+23 c^2 f^2\right )-a b c d \left (13 d^2 e^2-37 c d e f+40 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{15 c^3 (b c-a d)^2 (d e-c f)^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}+\frac {2 d \sqrt {e} (b e-a f) (b c (3 d e-5 c f)-2 a d (d e-2 c f)) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{15 c^2 (b c-a d)^2 (d e-c f)^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}} \] Output:

1/5*d*x*(b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/c/(-c*f+d*e)/(d*x^2+c)^(5/2)+1/15* 
d*(b*c*(-7*c*f+3*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/(-a*d+b*c)/(-c*f+d*e)^2/(d*x^2+c)^(3/2)+1/15*e^(1/2)*(b^2*c^2*(15*c^ 
2*f^2-10*c*d*e*f+3*d^2*e^2)+a^2*d^2*(23*c^2*f^2-23*c*d*e*f+8*d^2*e^2)-a*b* 
c*d*(40*c^2*f^2-37*c*d*e*f+13*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/(-a*d+b*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)+2/15*d*e^(1/2)*(-a*f+b*e)*(b*c*(-5*c* 
f+3*d*e)-2*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))/c^2/(-a*d+b*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]

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

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

Output:

Integrate[Sqrt[a + b*x^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.

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

\(\Big \downarrow \) 434

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

Input:

Int[Sqrt[a + b*x^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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(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]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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