\(\int \frac {(a+b x^2)^{5/2} \sqrt {e+f x^2}}{(c+d x^2)^{7/2}} \, dx\) [399]

Optimal result

Integrand size = 34, antiderivative size = 812 \[ \int \frac {\left (a+b x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{7/2}} \, dx=\frac {(b c-a d)^2 x \sqrt {a+b x^2} \sqrt {e+f x^2}}{5 c d^2 \left (c+d x^2\right )^{5/2}}-\frac {(b c-a d) (b c (7 d e-8 c f)+a d (4 d e-3 c f)) x \sqrt {a+b x^2} \sqrt {e+f x^2}}{15 c^2 d^2 (d e-c f) \left (c+d x^2\right )^{3/2}}+\frac {\sqrt {e} \left (a^2 d^2 \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )+a b c d \left (7 d^2 e^2-8 c d e f+5 c^2 f^2\right )+b^2 c^2 \left (8 d^2 e^2-25 c d e f+15 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{15 c^3 d^3 (d e-c f)^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}+\frac {\sqrt {e} \left (a^3 d^3 f (4 d e-3 c f)-15 b^3 c^3 f (d e-c f)-a^2 b d^2 \left (4 d^2 e^2-7 c d e f+5 c^2 f^2\right )-a b^2 c d \left (4 d^2 e^2-20 c d e f+15 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{15 a c^2 d^4 (d e-c f)^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}+\frac {b^3 c \sqrt {e} f \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticPi}\left (\frac {d e}{d e-c f},\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{a d^4 \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}} \] Output:

1/5*(-a*d+b*c)^2*x*(b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/c/d^2/(d*x^2+c)^(5/2)-1 
/15*(-a*d+b*c)*(b*c*(-8*c*f+7*d*e)+a*d*(-3*c*f+4*d*e))*x*(b*x^2+a)^(1/2)*( 
f*x^2+e)^(1/2)/c^2/d^2/(-c*f+d*e)/(d*x^2+c)^(3/2)+1/15*e^(1/2)*(a^2*d^2*(3 
*c^2*f^2-13*c*d*e*f+8*d^2*e^2)+a*b*c*d*(5*c^2*f^2-8*c*d*e*f+7*d^2*e^2)+b^2 
*c^2*(15*c^2*f^2-25*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)*EllipticE((-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/d^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)+1/15*e^(1/2)*(a^3*d^3*f*(-3*c*f+4*d*e)-15*b^3* 
c^3*f*(-c*f+d*e)-a^2*b*d^2*(5*c^2*f^2-7*c*d*e*f+4*d^2*e^2)-a*b^2*c*d*(15*c 
^2*f^2-20*c*d*e*f+4*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))/a/c^2/d^4/(-c*f+d*e)^(3/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1 
/2)/(f*x^2+e)^(1/2)+b^3*c*e^(1/2)*f*(b*x^2+a)^(1/2)*(c*(f*x^2+e)/e/(d*x^2+ 
c))^(1/2)*EllipticPi((-c*f+d*e)^(1/2)*x/e^(1/2)/(d*x^2+c)^(1/2),d*e/(-c*f+ 
d*e),(-(-a*d+b*c)*e/a/(-c*f+d*e))^(1/2))/a/d^4/(-c*f+d*e)^(1/2)/(c*(b*x^2+ 
a)/a/(d*x^2+c))^(1/2)/(f*x^2+e)^(1/2)
 

Mathematica [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{7/2}} \, dx=\int \frac {\left (a+b x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{7/2}} \, dx \] Input:

Integrate[((a + b*x^2)^(5/2)*Sqrt[e + f*x^2])/(c + d*x^2)^(7/2),x]
 

Output:

Integrate[((a + b*x^2)^(5/2)*Sqrt[e + f*x^2])/(c + d*x^2)^(7/2), 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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 434

\(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{7/2}}dx\)

Input:

Int[((a + b*x^2)^(5/2)*Sqrt[e + f*x^2])/(c + d*x^2)^(7/2),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]

\[\int \frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} \sqrt {f \,x^{2}+e}}{\left (x^{2} d +c \right )^{\frac {7}{2}}}d x\]

Input:

int((b*x^2+a)^(5/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(7/2),x)
 

Output:

int((b*x^2+a)^(5/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(7/2),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{7/2}} \, dx=\text {Timed out} \] Input:

integrate((b*x^2+a)^(5/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(7/2),x, algorithm="fr 
icas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{7/2}} \, dx=\text {Timed out} \] Input:

integrate((b*x**2+a)**(5/2)*(f*x**2+e)**(1/2)/(d*x**2+c)**(7/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{7/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} \sqrt {f x^{2} + e}}{{\left (d x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(5/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(7/2),x, algorithm="ma 
xima")
 

Output:

integrate((b*x^2 + a)^(5/2)*sqrt(f*x^2 + e)/(d*x^2 + c)^(7/2), x)
 

Giac [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{7/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} \sqrt {f x^{2} + e}}{{\left (d x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(5/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(7/2),x, algorithm="gi 
ac")
 

Output:

integrate((b*x^2 + a)^(5/2)*sqrt(f*x^2 + e)/(d*x^2 + c)^(7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{7/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{5/2}\,\sqrt {f\,x^2+e}}{{\left (d\,x^2+c\right )}^{7/2}} \,d x \] Input:

int(((a + b*x^2)^(5/2)*(e + f*x^2)^(1/2))/(c + d*x^2)^(7/2),x)
 

Output:

int(((a + b*x^2)^(5/2)*(e + f*x^2)^(1/2))/(c + d*x^2)^(7/2), x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{7/2}} \, dx=\int \frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} \sqrt {f \,x^{2}+e}}{\left (d \,x^{2}+c \right )^{\frac {7}{2}}}d x \] Input:

int((b*x^2+a)^(5/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(7/2),x)
 

Output:

int((b*x^2+a)^(5/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(7/2),x)