\(\int \frac {(a+b x^2)^{3/2}}{(c+d x^2)^{9/2} \sqrt {e+f x^2}} \, dx\) [469]

Optimal result

Integrand size = 34, antiderivative size = 877 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{9/2} \sqrt {e+f x^2}} \, dx=-\frac {(b c-a d) x \sqrt {a+b x^2} \sqrt {e+f x^2}}{7 c (d e-c f) \left (c+d x^2\right )^{7/2}}+\frac {2 (3 a d (d e-2 c f)+b c (d e+2 c f)) x \sqrt {a+b x^2} \sqrt {e+f x^2}}{35 c^2 (d e-c f)^2 \left (c+d x^2\right )^{5/2}}+\frac {\left (2 b^2 c^2 \left (3 d^2 e^2-11 c d e f-4 c^2 f^2\right )-a^2 d^2 \left (24 d^2 e^2-71 c d e f+71 c^2 f^2\right )+a b c d \left (15 d^2 e^2-43 c d e f+76 c^2 f^2\right )\right ) x \sqrt {a+b x^2} \sqrt {e+f x^2}}{105 c^3 (b c-a d) (d e-c f)^3 \left (c+d x^2\right )^{3/2}}+\frac {2 \sqrt {e} \left (b^3 c^3 e \left (3 d^2 e^2-14 c d e f+35 c^2 f^2\right )-a^2 b c d \left (36 d^3 e^3-137 c d^2 e^2 f+190 c^2 d e f^2-161 c^3 f^3\right )+2 a b^2 c^2 \left (3 d^3 e^3-11 c d^2 e^2 f+7 c^2 d e f^2-35 c^3 f^3\right )+4 a^3 d^2 \left (6 d^3 e^3-23 c d^2 e^2 f+33 c^2 d e f^2-22 c^3 f^3\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{105 c^4 (b c-a d)^2 (d e-c f)^{7/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}-\frac {\sqrt {e} (b e-a f) \left (b^2 c^2 \left (3 d^2 e^2-14 c d e f+35 c^2 f^2\right )+a^2 d^2 \left (24 d^2 e^2-71 c d e f+71 c^2 f^2\right )-a b c d \left (33 d^2 e^2-97 c d e f+112 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{105 c^3 (b c-a d)^2 (d e-c f)^{7/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}} \] Output:

-1/7*(-a*d+b*c)*x*(b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/c/(-c*f+d*e)/(d*x^2+c)^( 
7/2)+2/35*(3*a*d*(-2*c*f+d*e)+b*c*(2*c*f+d*e))*x*(b*x^2+a)^(1/2)*(f*x^2+e) 
^(1/2)/c^2/(-c*f+d*e)^2/(d*x^2+c)^(5/2)+1/105*(2*b^2*c^2*(-4*c^2*f^2-11*c* 
d*e*f+3*d^2*e^2)-a^2*d^2*(71*c^2*f^2-71*c*d*e*f+24*d^2*e^2)+a*b*c*d*(76*c^ 
2*f^2-43*c*d*e*f+15*d^2*e^2))*x*(b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/c^3/(-a*d+ 
b*c)/(-c*f+d*e)^3/(d*x^2+c)^(3/2)+2/105*e^(1/2)*(b^3*c^3*e*(35*c^2*f^2-14* 
c*d*e*f+3*d^2*e^2)-a^2*b*c*d*(-161*c^3*f^3+190*c^2*d*e*f^2-137*c*d^2*e^2*f 
+36*d^3*e^3)+2*a*b^2*c^2*(-35*c^3*f^3+7*c^2*d*e*f^2-11*c*d^2*e^2*f+3*d^3*e 
^3)+4*a^3*d^2*(-22*c^3*f^3+33*c^2*d*e*f^2-23*c*d^2*e^2*f+6*d^3*e^3))*(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^4/(-a*d+b*c)^2 
/(-c*f+d*e)^(7/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(f*x^2+e)^(1/2)-1/105*e^ 
(1/2)*(-a*f+b*e)*(b^2*c^2*(35*c^2*f^2-14*c*d*e*f+3*d^2*e^2)+a^2*d^2*(71*c^ 
2*f^2-71*c*d*e*f+24*d^2*e^2)-a*b*c*d*(112*c^2*f^2-97*c*d*e*f+33*d^2*e^2))* 
(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^3/(-a*d+b 
*c)^2/(-c*f+d*e)^(7/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(f*x^2+e)^(1/2)
 

Mathematica [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{9/2} \sqrt {e+f x^2}} \, dx=\int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{9/2} \sqrt {e+f x^2}} \, dx \] Input:

Integrate[(a + b*x^2)^(3/2)/((c + d*x^2)^(9/2)*Sqrt[e + f*x^2]),x]
 

Output:

Integrate[(a + b*x^2)^(3/2)/((c + d*x^2)^(9/2)*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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{9/2} \sqrt {e+f x^2}} \, dx\)

\(\Big \downarrow \) 434

\(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{9/2} \sqrt {e+f x^2}}dx\)

Input:

Int[(a + b*x^2)^(3/2)/((c + d*x^2)^(9/2)*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]

\[\int \frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{\left (x^{2} d +c \right )^{\frac {9}{2}} \sqrt {f \,x^{2}+e}}d x\]

Input:

int((b*x^2+a)^(3/2)/(d*x^2+c)^(9/2)/(f*x^2+e)^(1/2),x)
 

Output:

int((b*x^2+a)^(3/2)/(d*x^2+c)^(9/2)/(f*x^2+e)^(1/2),x)
 

Fricas [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{9/2} \sqrt {e+f x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (d x^{2} + c\right )}^{\frac {9}{2}} \sqrt {f x^{2} + e}} \,d x } \] Input:

integrate((b*x^2+a)^(3/2)/(d*x^2+c)^(9/2)/(f*x^2+e)^(1/2),x, algorithm="fr 
icas")
 

Output:

integral((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)/(d^5*f*x^12 + ( 
d^5*e + 5*c*d^4*f)*x^10 + 5*(c*d^4*e + 2*c^2*d^3*f)*x^8 + 10*(c^2*d^3*e + 
c^3*d^2*f)*x^6 + c^5*e + 5*(2*c^3*d^2*e + c^4*d*f)*x^4 + (5*c^4*d*e + c^5* 
f)*x^2), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{9/2} \sqrt {e+f x^2}} \, dx=\text {Timed out} \] Input:

integrate((b*x**2+a)**(3/2)/(d*x**2+c)**(9/2)/(f*x**2+e)**(1/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{9/2} \sqrt {e+f x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (d x^{2} + c\right )}^{\frac {9}{2}} \sqrt {f x^{2} + e}} \,d x } \] Input:

integrate((b*x^2+a)^(3/2)/(d*x^2+c)^(9/2)/(f*x^2+e)^(1/2),x, algorithm="ma 
xima")
 

Output:

integrate((b*x^2 + a)^(3/2)/((d*x^2 + c)^(9/2)*sqrt(f*x^2 + e)), x)
 

Giac [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{9/2} \sqrt {e+f x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (d x^{2} + c\right )}^{\frac {9}{2}} \sqrt {f x^{2} + e}} \,d x } \] Input:

integrate((b*x^2+a)^(3/2)/(d*x^2+c)^(9/2)/(f*x^2+e)^(1/2),x, algorithm="gi 
ac")
 

Output:

integrate((b*x^2 + a)^(3/2)/((d*x^2 + c)^(9/2)*sqrt(f*x^2 + e)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{9/2} \sqrt {e+f x^2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}}{{\left (d\,x^2+c\right )}^{9/2}\,\sqrt {f\,x^2+e}} \,d x \] Input:

int((a + b*x^2)^(3/2)/((c + d*x^2)^(9/2)*(e + f*x^2)^(1/2)),x)
 

Output:

int((a + b*x^2)^(3/2)/((c + d*x^2)^(9/2)*(e + f*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{9/2} \sqrt {e+f x^2}} \, dx=\int \frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{\left (d \,x^{2}+c \right )^{\frac {9}{2}} \sqrt {f \,x^{2}+e}}d x \] Input:

int((b*x^2+a)^(3/2)/(d*x^2+c)^(9/2)/(f*x^2+e)^(1/2),x)
 

Output:

int((b*x^2+a)^(3/2)/(d*x^2+c)^(9/2)/(f*x^2+e)^(1/2),x)