\(\int \frac {(c+d x^2)^{3/2} \sqrt {e+f x^2}}{(a+b x^2)^{5/2}} \, dx\) [414]

Optimal result

Integrand size = 34, antiderivative size = 629 \[ \int \frac {\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {(b c-a d) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {\sqrt {b c-a d} e \left (2 b^2 c e+2 a b d e-a b c f-3 a^2 d f\right ) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )|\frac {c (b e-a f)}{(b c-a d) e}\right )}{3 a^2 b^2 \sqrt {c} (b e-a f) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}-\frac {\left (6 a^2 b d^2 e f-3 a^3 d^2 f^2-b^3 c e (d e-c f)-a b^2 d e (2 d e+c f)\right ) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {c (b e-a f)}{(b c-a d) e}\right )}{3 a b^3 \sqrt {c} \sqrt {b c-a d} (b e-a f) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}+\frac {a d^2 f \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticPi}\left (\frac {b c}{b c-a d},\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {c (b e-a f)}{(b c-a d) e}\right )}{b^3 \sqrt {c} \sqrt {b c-a d} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}} \] Output:

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

\[ \int \frac {\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{\left (a+b x^2\right )^{5/2}} \, dx \] Input:

Integrate[((c + d*x^2)^(3/2)*Sqrt[e + f*x^2])/(a + b*x^2)^(5/2),x]
 

Output:

Integrate[((c + d*x^2)^(3/2)*Sqrt[e + f*x^2])/(a + b*x^2)^(5/2), 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.

\(\displaystyle \int \frac {\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{\left (a+b x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 434

\(\displaystyle \int \frac {\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{\left (a+b x^2\right )^{5/2}}dx\)

Input:

Int[((c + d*x^2)^(3/2)*Sqrt[e + f*x^2])/(a + b*x^2)^(5/2),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]

\[\int \frac {\left (x^{2} d +c \right )^{\frac {3}{2}} \sqrt {f \,x^{2}+e}}{\left (b \,x^{2}+a \right )^{\frac {5}{2}}}d x\]

Input:

int((d*x^2+c)^(3/2)*(f*x^2+e)^(1/2)/(b*x^2+a)^(5/2),x)
 

Output:

int((d*x^2+c)^(3/2)*(f*x^2+e)^(1/2)/(b*x^2+a)^(5/2),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{\left (a+b x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((d*x^2+c)^(3/2)*(f*x^2+e)^(1/2)/(b*x^2+a)^(5/2),x, algorithm="fr 
icas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {\left (c + d x^{2}\right )^{\frac {3}{2}} \sqrt {e + f x^{2}}}{\left (a + b x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:

integrate((d*x**2+c)**(3/2)*(f*x**2+e)**(1/2)/(b*x**2+a)**(5/2),x)
 

Output:

Integral((c + d*x**2)**(3/2)*sqrt(e + f*x**2)/(a + b*x**2)**(5/2), x)
 

Maxima [F]

\[ \int \frac {\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{\left (a+b x^2\right )^{5/2}} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} \sqrt {f x^{2} + e}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((d*x^2+c)^(3/2)*(f*x^2+e)^(1/2)/(b*x^2+a)^(5/2),x, algorithm="ma 
xima")
 

Output:

integrate((d*x^2 + c)^(3/2)*sqrt(f*x^2 + e)/(b*x^2 + a)^(5/2), x)
 

Giac [F]

\[ \int \frac {\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{\left (a+b x^2\right )^{5/2}} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} \sqrt {f x^{2} + e}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((d*x^2+c)^(3/2)*(f*x^2+e)^(1/2)/(b*x^2+a)^(5/2),x, algorithm="gi 
ac")
 

Output:

integrate((d*x^2 + c)^(3/2)*sqrt(f*x^2 + e)/(b*x^2 + a)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {{\left (d\,x^2+c\right )}^{3/2}\,\sqrt {f\,x^2+e}}{{\left (b\,x^2+a\right )}^{5/2}} \,d x \] Input:

int(((c + d*x^2)^(3/2)*(e + f*x^2)^(1/2))/(a + b*x^2)^(5/2),x)
 

Output:

int(((c + d*x^2)^(3/2)*(e + f*x^2)^(1/2))/(a + b*x^2)^(5/2), x)
 

Reduce [F]

\[ \int \frac {\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}} \sqrt {f \,x^{2}+e}}{\left (b \,x^{2}+a \right )^{\frac {5}{2}}}d x \] Input:

int((d*x^2+c)^(3/2)*(f*x^2+e)^(1/2)/(b*x^2+a)^(5/2),x)
 

Output:

int((d*x^2+c)^(3/2)*(f*x^2+e)^(1/2)/(b*x^2+a)^(5/2),x)