\(\int \frac {\sqrt {a+b x^2} (c+d x^2)^{3/2}}{\sqrt {e+f x^2}} \, dx\) [458]

Optimal result

Integrand size = 34, antiderivative size = 648 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{\sqrt {e+f x^2}} \, dx=-\frac {(3 b d e-5 b c f-a d f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{8 f^2 \sqrt {a+b x^2}}+\frac {d x \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}}{4 f}+\frac {\sqrt {b c-a d} e (3 b d e-5 b c f-a d f) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )|\frac {c (b e-a f)}{(b c-a d) e}\right )}{8 b \sqrt {c} f^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}-\frac {a \sqrt {b c-a d} (b d e-5 b c f+a d f) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {c (b e-a f)}{(b c-a d) e}\right )}{8 b^2 \sqrt {c} f \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}-\frac {a \left (a^2 d^2 f^2+2 a b d f (d e-3 c f)-3 b^2 (d e-c f)^2\right ) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticPi}\left (\frac {b c}{b c-a d},\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {c (b e-a f)}{(b c-a d) e}\right )}{8 b^2 \sqrt {c} \sqrt {b c-a d} f^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}} \] Output:

-1/8*(-a*d*f-5*b*c*f+3*b*d*e)*x*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/f^2/(b*x^2 
+a)^(1/2)+1/4*d*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/f+1/8*(- 
a*d+b*c)^(1/2)*e*(-a*d*f-5*b*c*f+3*b*d*e)*(d*x^2+c)^(1/2)*(a*(f*x^2+e)/e/( 
b*x^2+a))^(1/2)*EllipticE((-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x^2+a)^(1/2),(c*(- 
a*f+b*e)/(-a*d+b*c)/e)^(1/2))/b/c^(1/2)/f^2/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2 
)/(f*x^2+e)^(1/2)-1/8*a*(-a*d+b*c)^(1/2)*(a*d*f-5*b*c*f+b*d*e)*(d*x^2+c)^( 
1/2)*(a*(f*x^2+e)/e/(b*x^2+a))^(1/2)*EllipticF((-a*d+b*c)^(1/2)*x/c^(1/2)/ 
(b*x^2+a)^(1/2),(c*(-a*f+b*e)/(-a*d+b*c)/e)^(1/2))/b^2/c^(1/2)/f/(a*(d*x^2 
+c)/c/(b*x^2+a))^(1/2)/(f*x^2+e)^(1/2)-1/8*a*(a^2*d^2*f^2+2*a*b*d*f*(-3*c* 
f+d*e)-3*b^2*(-c*f+d*e)^2)*(d*x^2+c)^(1/2)*(a*(f*x^2+e)/e/(b*x^2+a))^(1/2) 
*EllipticPi((-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x^2+a)^(1/2),b*c/(-a*d+b*c),(c*( 
-a*f+b*e)/(-a*d+b*c)/e)^(1/2))/b^2/c^(1/2)/(-a*d+b*c)^(1/2)/f^2/(a*(d*x^2+ 
c)/c/(b*x^2+a))^(1/2)/(f*x^2+e)^(1/2)
 

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 434

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(e + f*x**2)*sqrt(c + d*x**2)*sqrt(a + b*x**2)*d*x + int((sqrt(e + f* 
x**2)*sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c*e + a*c*f*x**2 + a*d*e* 
x**2 + a*d*f*x**4 + b*c*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)* 
a*d**2*f + 5*int((sqrt(e + f*x**2)*sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4) 
/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b*c*e*x**2 + b*c*f*x**4 + 
 b*d*e*x**4 + b*d*f*x**6),x)*b*c*d*f - 3*int((sqrt(e + f*x**2)*sqrt(c + d* 
x**2)*sqrt(a + b*x**2)*x**4)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 
 + b*c*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*b*d**2*e + 6*int( 
(sqrt(e + f*x**2)*sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c*e + a*c*f*x 
**2 + a*d*e*x**2 + a*d*f*x**4 + b*c*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d 
*f*x**6),x)*a*c*d*f - 2*int((sqrt(e + f*x**2)*sqrt(c + d*x**2)*sqrt(a + b* 
x**2)*x**2)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b*c*e*x**2 + b 
*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*a*d**2*e + 4*int((sqrt(e + f*x**2) 
*sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 
+ a*d*f*x**4 + b*c*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*b*c** 
2*f - 2*int((sqrt(e + f*x**2)*sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c 
*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b*c*e*x**2 + b*c*f*x**4 + b*d* 
e*x**4 + b*d*f*x**6),x)*b*c*d*e + 4*int((sqrt(e + f*x**2)*sqrt(c + d*x**2) 
*sqrt(a + b*x**2))/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b*c*e*x 
**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*a*c**2*f - int((sqrt(e +...