\(\int \frac {(a+b x^2)^{3/2} \sqrt {c+d x^2}}{(e+f x^2)^{5/2}} \, dx\) [542]

Optimal result

Integrand size = 34, antiderivative size = 627 \[ \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{\left (e+f x^2\right )^{5/2}} \, dx=-\frac {(b e-a f) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 e f \left (e+f x^2\right )^{3/2}}+\frac {c \sqrt {-b e+a f} (a f (d e-2 c f)+b e (3 d e-2 c f)) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )|\frac {a (d e-c f)}{c (b e-a f)}\right )}{3 \sqrt {a} e^2 f^2 (d e-c f) \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}+\frac {\left (a^2 c d f^3-a b c f^2 (d e+c f)-b^2 e \left (3 d^2 e^2-6 c d e f+2 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{3 \sqrt {a} e f^3 \sqrt {-b e+a f} (d e-c f) \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}+\frac {b^2 d e \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticPi}\left (-\frac {a f}{b e-a f},\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{\sqrt {a} f^3 \sqrt {-b e+a f} \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}} \] Output:

-1/3*(-a*f+b*e)*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/e/f/(f*x^2+e)^(3/2)+1/3* 
c*(a*f-b*e)^(1/2)*(a*f*(-2*c*f+d*e)+b*e*(-2*c*f+3*d*e))*(b*x^2+a)^(1/2)*(e 
*(d*x^2+c)/c/(f*x^2+e))^(1/2)*EllipticE((a*f-b*e)^(1/2)*x/a^(1/2)/(f*x^2+e 
)^(1/2),(a*(-c*f+d*e)/c/(-a*f+b*e))^(1/2))/a^(1/2)/e^2/f^2/(-c*f+d*e)/(d*x 
^2+c)^(1/2)/(e*(b*x^2+a)/a/(f*x^2+e))^(1/2)+1/3*(a^2*c*d*f^3-a*b*c*f^2*(c* 
f+d*e)-b^2*e*(2*c^2*f^2-6*c*d*e*f+3*d^2*e^2))*(b*x^2+a)^(1/2)*(e*(d*x^2+c) 
/c/(f*x^2+e))^(1/2)*EllipticF((a*f-b*e)^(1/2)*x/a^(1/2)/(f*x^2+e)^(1/2),(a 
*(-c*f+d*e)/c/(-a*f+b*e))^(1/2))/a^(1/2)/e/f^3/(a*f-b*e)^(1/2)/(-c*f+d*e)/ 
(d*x^2+c)^(1/2)/(e*(b*x^2+a)/a/(f*x^2+e))^(1/2)+b^2*d*e*(b*x^2+a)^(1/2)*(e 
*(d*x^2+c)/c/(f*x^2+e))^(1/2)*EllipticPi((a*f-b*e)^(1/2)*x/a^(1/2)/(f*x^2+ 
e)^(1/2),-a*f/(-a*f+b*e),(a*(-c*f+d*e)/c/(-a*f+b*e))^(1/2))/a^(1/2)/f^3/(a 
*f-b*e)^(1/2)/(d*x^2+c)^(1/2)/(e*(b*x^2+a)/a/(f*x^2+e))^(1/2)
 

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 434

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

Input:

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

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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