\(\int \frac {(a+b x^2)^{3/2} \sqrt {e+f x^2}}{(c+d x^2)^{7/2}} \, dx\) [393]

Optimal result

Integrand size = 34, antiderivative size = 531 \[ \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{7/2}} \, dx=\frac {\left (\frac {a}{c}-\frac {b}{d}\right ) x \sqrt {a+b x^2} \sqrt {e+f x^2}}{5 \left (c+d x^2\right )^{5/2}}+\frac {(b c (2 d e-3 c f)+a d (4 d e-3 c f)) x \sqrt {a+b x^2} \sqrt {e+f x^2}}{15 c^2 d (d e-c f) \left (c+d x^2\right )^{3/2}}+\frac {\sqrt {e} \left (2 b^2 c^2 e^2+a b c e (3 d e-7 c f)-a^2 \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{15 c^3 (b c-a d) (d e-c f)^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}-\frac {\sqrt {e} (b e-a f) (b c e-4 a d e+3 a c f) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{15 c^2 (b c-a d) (d e-c f)^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}} \] Output:

1/5*(a/c-b/d)*x*(b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(5/2)+1/15*(b*c* 
(-3*c*f+2*d*e)+a*d*(-3*c*f+4*d*e))*x*(b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/c^2/d 
/(-c*f+d*e)/(d*x^2+c)^(3/2)+1/15*e^(1/2)*(2*b^2*c^2*e^2+a*b*c*e*(-7*c*f+3* 
d*e)-a^2*(3*c^2*f^2-13*c*d*e*f+8*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)/(-c*f+d*e)^(3/2)/(c*(b*x^2+ 
a)/a/(d*x^2+c))^(1/2)/(f*x^2+e)^(1/2)-1/15*e^(1/2)*(-a*f+b*e)*(3*a*c*f-4*a 
*d*e+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)) 
/c^2/(-a*d+b*c)/(-c*f+d*e)^(3/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(f*x^2+e) 
^(1/2)
 

Mathematica [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{7/2}} \, dx=\int \frac {\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{7/2}} \, dx \] Input:

Integrate[((a + b*x^2)^(3/2)*Sqrt[e + f*x^2])/(c + d*x^2)^(7/2),x]
 

Output:

Integrate[((a + b*x^2)^(3/2)*Sqrt[e + f*x^2])/(c + d*x^2)^(7/2), 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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 434

\(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{7/2}}dx\)

Input:

Int[((a + b*x^2)^(3/2)*Sqrt[e + f*x^2])/(c + d*x^2)^(7/2),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]

\[\int \frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} \sqrt {f \,x^{2}+e}}{\left (x^{2} d +c \right )^{\frac {7}{2}}}d x\]

Input:

int((b*x^2+a)^(3/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(7/2),x)
 

Output:

int((b*x^2+a)^(3/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(7/2),x)
 

Fricas [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{7/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {f x^{2} + e}}{{\left (d x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(3/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(7/2),x, algorithm="fr 
icas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{7/2}} \, dx=\text {Timed out} \] Input:

integrate((b*x**2+a)**(3/2)*(f*x**2+e)**(1/2)/(d*x**2+c)**(7/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{7/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {f x^{2} + e}}{{\left (d x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(3/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(7/2),x, algorithm="ma 
xima")
 

Output:

integrate((b*x^2 + a)^(3/2)*sqrt(f*x^2 + e)/(d*x^2 + c)^(7/2), x)
 

Giac [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{7/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {f x^{2} + e}}{{\left (d x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(3/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(7/2),x, algorithm="gi 
ac")
 

Output:

integrate((b*x^2 + a)^(3/2)*sqrt(f*x^2 + e)/(d*x^2 + c)^(7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{7/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}\,\sqrt {f\,x^2+e}}{{\left (d\,x^2+c\right )}^{7/2}} \,d x \] Input:

int(((a + b*x^2)^(3/2)*(e + f*x^2)^(1/2))/(c + d*x^2)^(7/2),x)
 

Output:

int(((a + b*x^2)^(3/2)*(e + f*x^2)^(1/2))/(c + d*x^2)^(7/2), x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{7/2}} \, dx=\int \frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} \sqrt {f \,x^{2}+e}}{\left (d \,x^{2}+c \right )^{\frac {7}{2}}}d x \] Input:

int((b*x^2+a)^(3/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(7/2),x)
 

Output:

int((b*x^2+a)^(3/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(7/2),x)