\(\int \frac {\sqrt {c+d x^2} (e+f x^2)^{3/2}}{(a+b x^2)^{3/2}} \, dx\) [448]

Optimal result

Integrand size = 34, antiderivative size = 575 \[ \int \frac {\sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {f x \sqrt {c+d x^2} \sqrt {e+f x^2}}{2 b \sqrt {a+b x^2}}+\frac {\sqrt {b c-a d} e (2 b e-3 a f) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )|\frac {c (b e-a f)}{(b c-a d) e}\right )}{2 a b^2 \sqrt {c} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}+\frac {\left (3 a^2 d f^2+2 b^2 e (d e+c f)-a b f (6 d e+c f)\right ) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {c (b e-a f)}{(b c-a d) e}\right )}{2 b^3 \sqrt {c} \sqrt {b c-a d} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}+\frac {a f (3 b d e+b c f-3 a d f) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticPi}\left (\frac {b c}{b c-a d},\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {c (b e-a f)}{(b c-a d) e}\right )}{2 b^3 \sqrt {c} \sqrt {b c-a d} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}} \] Output:

1/2*f*x*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/b/(b*x^2+a)^(1/2)+1/2*(-a*d+b*c)^( 
1/2)*e*(-3*a*f+2*b*e)*(d*x^2+c)^(1/2)*(a*(f*x^2+e)/e/(b*x^2+a))^(1/2)*Elli 
pticE((-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))/a/b^2/c^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)/(f*x^2+e)^(1/2)+1/ 
2*(3*a^2*d*f^2+2*b^2*e*(c*f+d*e)-a*b*f*(c*f+6*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^3/c^(1/2)/(-a*d+b*c)^(1/2)/(a*(d 
*x^2+c)/c/(b*x^2+a))^(1/2)/(f*x^2+e)^(1/2)+1/2*a*f*(-3*a*d*f+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)*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^3/c^(1/2)/(-a*d+b*c)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)/(f*x^2+ 
e)^(1/2)
 

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 434

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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