\(\int \frac {(a+b x^2)^{5/2}}{(c+d x^2)^{9/2} \sqrt {e+f x^2}} \, dx\) [475]

Optimal result

Integrand size = 34, antiderivative size = 840 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{9/2} \sqrt {e+f x^2}} \, dx=\frac {(b c-a d)^2 x \sqrt {a+b x^2} \sqrt {e+f x^2}}{7 c d (d e-c f) \left (c+d x^2\right )^{7/2}}-\frac {3 (b c-a d) (2 a d (d e-2 c f)+b c (3 d e-c f)) x \sqrt {a+b x^2} \sqrt {e+f x^2}}{35 c^2 d (d e-c f)^2 \left (c+d x^2\right )^{5/2}}+\frac {\left (a b c d \left (13 d^2 e^2-41 c d e f-20 c^2 f^2\right )+2 b^2 c^2 \left (4 d^2 e^2+11 c d e f-3 c^2 f^2\right )+a^2 d^2 \left (24 d^2 e^2-71 c d e f+71 c^2 f^2\right )\right ) x \sqrt {a+b x^2} \sqrt {e+f x^2}}{105 c^3 d (d e-c f)^3 \left (c+d x^2\right )^{3/2}}+\frac {\sqrt {e} \left (8 b^3 c^3 e^2 (d e-7 c f)+a b^2 c^2 e \left (9 d^2 e^2-26 c d e f+161 c^2 f^2\right )+a^2 b c \left (16 d^3 e^3-57 c d^2 e^2 f+58 c^2 d e f^2-161 c^3 f^3\right )-8 a^3 d \left (6 d^3 e^3-23 c d^2 e^2 f+33 c^2 d e f^2-22 c^3 f^3\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{105 c^4 (b c-a d) (d e-c f)^{7/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}-\frac {\sqrt {e} (b e-a f) \left (4 b^2 c^2 e (d e-7 c f)+a b c \left (5 d^2 e^2-13 c d e f+56 c^2 f^2\right )-a^2 d \left (24 d^2 e^2-71 c d e f+71 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{105 c^3 (b c-a d) (d e-c f)^{7/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}} \] Output:

1/7*(-a*d+b*c)^2*x*(b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/c/d/(-c*f+d*e)/(d*x^2+c 
)^(7/2)-3/35*(-a*d+b*c)*(2*a*d*(-2*c*f+d*e)+b*c*(-c*f+3*d*e))*x*(b*x^2+a)^ 
(1/2)*(f*x^2+e)^(1/2)/c^2/d/(-c*f+d*e)^2/(d*x^2+c)^(5/2)+1/105*(a*b*c*d*(- 
20*c^2*f^2-41*c*d*e*f+13*d^2*e^2)+2*b^2*c^2*(-3*c^2*f^2+11*c*d*e*f+4*d^2*e 
^2)+a^2*d^2*(71*c^2*f^2-71*c*d*e*f+24*d^2*e^2))*x*(b*x^2+a)^(1/2)*(f*x^2+e 
)^(1/2)/c^3/d/(-c*f+d*e)^3/(d*x^2+c)^(3/2)+1/105*e^(1/2)*(8*b^3*c^3*e^2*(- 
7*c*f+d*e)+a*b^2*c^2*e*(161*c^2*f^2-26*c*d*e*f+9*d^2*e^2)+a^2*b*c*(-161*c^ 
3*f^3+58*c^2*d*e*f^2-57*c*d^2*e^2*f+16*d^3*e^3)-8*a^3*d*(-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)/(-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)*(4*b^2*c^2*e*(-7* 
c*f+d*e)+a*b*c*(56*c^2*f^2-13*c*d*e*f+5*d^2*e^2)-a^2*d*(71*c^2*f^2-71*c*d* 
e*f+24*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)/(-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]

\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{9/2} \sqrt {e+f x^2}} \, dx=\int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{9/2} \sqrt {e+f x^2}} \, dx \] Input:

Integrate[(a + b*x^2)^(5/2)/((c + d*x^2)^(9/2)*Sqrt[e + f*x^2]),x]
 

Output:

Integrate[(a + b*x^2)^(5/2)/((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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{9/2} \sqrt {e+f x^2}} \, dx\)

\(\Big \downarrow \) 434

\(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{9/2} \sqrt {e+f x^2}}dx\)

Input:

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

\[\int \frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{\left (x^{2} d +c \right )^{\frac {9}{2}} \sqrt {f \,x^{2}+e}}d x\]

Input:

int((b*x^2+a)^(5/2)/(d*x^2+c)^(9/2)/(f*x^2+e)^(1/2),x)
 

Output:

int((b*x^2+a)^(5/2)/(d*x^2+c)^(9/2)/(f*x^2+e)^(1/2),x)
 

Fricas [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{9/2} \sqrt {e+f x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{{\left (d x^{2} + c\right )}^{\frac {9}{2}} \sqrt {f x^{2} + e}} \,d x } \] Input:

integrate((b*x^2+a)^(5/2)/(d*x^2+c)^(9/2)/(f*x^2+e)^(1/2),x, algorithm="fr 
icas")
 

Output:

integral((b^2*x^4 + 2*a*b*x^2 + a^2)*sqrt(b*x^2 + a)*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. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{9/2} \sqrt {e+f x^2}} \, dx=\text {Timed out} \] Input:

integrate((b*x**2+a)**(5/2)/(d*x**2+c)**(9/2)/(f*x**2+e)**(1/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{9/2} \sqrt {e+f x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{{\left (d x^{2} + c\right )}^{\frac {9}{2}} \sqrt {f x^{2} + e}} \,d x } \] Input:

integrate((b*x^2+a)^(5/2)/(d*x^2+c)^(9/2)/(f*x^2+e)^(1/2),x, algorithm="ma 
xima")
 

Output:

integrate((b*x^2 + a)^(5/2)/((d*x^2 + c)^(9/2)*sqrt(f*x^2 + e)), x)
 

Giac [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{9/2} \sqrt {e+f x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{{\left (d x^{2} + c\right )}^{\frac {9}{2}} \sqrt {f x^{2} + e}} \,d x } \] Input:

integrate((b*x^2+a)^(5/2)/(d*x^2+c)^(9/2)/(f*x^2+e)^(1/2),x, algorithm="gi 
ac")
 

Output:

integrate((b*x^2 + a)^(5/2)/((d*x^2 + c)^(9/2)*sqrt(f*x^2 + e)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{9/2} \sqrt {e+f x^2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{5/2}}{{\left (d\,x^2+c\right )}^{9/2}\,\sqrt {f\,x^2+e}} \,d x \] Input:

int((a + b*x^2)^(5/2)/((c + d*x^2)^(9/2)*(e + f*x^2)^(1/2)),x)
 

Output:

int((a + b*x^2)^(5/2)/((c + d*x^2)^(9/2)*(e + f*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{9/2} \sqrt {e+f x^2}} \, dx=\int \frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{\left (d \,x^{2}+c \right )^{\frac {9}{2}} \sqrt {f \,x^{2}+e}}d x \] Input:

int((b*x^2+a)^(5/2)/(d*x^2+c)^(9/2)/(f*x^2+e)^(1/2),x)
 

Output:

int((b*x^2+a)^(5/2)/(d*x^2+c)^(9/2)/(f*x^2+e)^(1/2),x)