\(\int \frac {\sqrt {a+b x^2} (e+f x^2)^{3/2}}{(c+d x^2)^{9/2}} \, dx\) [425]

Optimal result

Integrand size = 34, antiderivative size = 835 \[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{9/2}} \, dx=\frac {(d e-c f) x \sqrt {a+b x^2} \sqrt {e+f x^2}}{7 c d \left (c+d x^2\right )^{7/2}}-\frac {(2 a d (3 d e+c f)-b c (5 d e+3 c f)) x \sqrt {a+b x^2} \sqrt {e+f x^2}}{35 c^2 d (b c-a d) \left (c+d x^2\right )^{5/2}}-\frac {\left (a b c d \left (43 d^2 e^2-29 c d e f-8 c^2 f^2\right )-3 a^2 d^2 \left (8 d^2 e^2-5 c d e f-2 c^2 f^2\right )-3 b^2 c^2 \left (5 d^2 e^2-2 c d e f-2 c^2 f^2\right )\right ) x \sqrt {a+b x^2} \sqrt {e+f x^2}}{105 c^3 d (b c-a d)^2 (d e-c f) \left (c+d x^2\right )^{3/2}}+\frac {\sqrt {e} \left (3 b^3 c^3 e^2 (5 d e-7 c f)-a b^2 c^2 e \left (103 d^2 e^2-170 c d e f+49 c^2 f^2\right )-6 a^3 d \left (8 d^3 e^3-12 c d^2 e^2 f+2 c^2 d e f^2+c^3 f^3\right )+a^2 b c \left (128 d^3 e^3-197 c d^2 e^2 f+37 c^2 d e f^2+14 c^3 f^3\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{105 c^4 (b c-a d)^3 (d e-c f)^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}+\frac {\sqrt {e} (b e-a f) \left (3 b^2 c^2 e (15 d e-14 c f)-a b c \left (61 d^2 e^2-41 c d e f-14 c^2 f^2\right )+3 a^2 d \left (8 d^2 e^2-5 c d e f-2 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{105 c^3 (b c-a d)^3 (d e-c f)^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}} \] Output:

1/7*(-c*f+d*e)*x*(b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/c/d/(d*x^2+c)^(7/2)-1/35* 
(2*a*d*(c*f+3*d*e)-b*c*(3*c*f+5*d*e))*x*(b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/c^ 
2/d/(-a*d+b*c)/(d*x^2+c)^(5/2)-1/105*(a*b*c*d*(-8*c^2*f^2-29*c*d*e*f+43*d^ 
2*e^2)-3*a^2*d^2*(-2*c^2*f^2-5*c*d*e*f+8*d^2*e^2)-3*b^2*c^2*(-2*c^2*f^2-2* 
c*d*e*f+5*d^2*e^2))*x*(b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/c^3/d/(-a*d+b*c)^2/( 
-c*f+d*e)/(d*x^2+c)^(3/2)+1/105*e^(1/2)*(3*b^3*c^3*e^2*(-7*c*f+5*d*e)-a*b^ 
2*c^2*e*(49*c^2*f^2-170*c*d*e*f+103*d^2*e^2)-6*a^3*d*(c^3*f^3+2*c^2*d*e*f^ 
2-12*c*d^2*e^2*f+8*d^3*e^3)+a^2*b*c*(14*c^3*f^3+37*c^2*d*e*f^2-197*c*d^2*e 
^2*f+128*d^3*e^3))*(b*x^2+a)^(1/2)*(c*(f*x^2+e)/e/(d*x^2+c))^(1/2)*Ellipti 
cE((-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)^3/(-c*f+d*e)^(3/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/ 
(f*x^2+e)^(1/2)+1/105*e^(1/2)*(-a*f+b*e)*(3*b^2*c^2*e*(-14*c*f+15*d*e)-a*b 
*c*(-14*c^2*f^2-41*c*d*e*f+61*d^2*e^2)+3*a^2*d*(-2*c^2*f^2-5*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)*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)^3/(-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]

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

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

Output:

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

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

\(\Big \downarrow \) 434

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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