\(\int \frac {(e+f x^2)^{3/2}}{\sqrt {a+b x^2} (c+d x^2)^{7/2}} \, dx\) [445]

Optimal result

Integrand size = 34, antiderivative size = 583 \[ \int \frac {\left (e+f x^2\right )^{3/2}}{\sqrt {a+b x^2} \left (c+d x^2\right )^{7/2}} \, dx=-\frac {(d e-c f) x \sqrt {a+b x^2} \sqrt {e+f x^2}}{5 c (b c-a d) \left (c+d x^2\right )^{5/2}}-\frac {2 (b c (4 d e-c f)-a d (2 d e+c f)) x \sqrt {a+b x^2} \sqrt {e+f x^2}}{15 c^2 (b c-a d)^2 \left (c+d x^2\right )^{3/2}}-\frac {\sqrt {e} \left (b^2 c^2 e (23 d e-20 c f)-a b c \left (23 d^2 e^2-7 c d e f-10 c^2 f^2\right )+a^2 d \left (8 d^2 e^2-3 c d e f-2 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{15 c^3 (b c-a d)^3 \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}+\frac {\sqrt {e} (b e-a f) \left (15 b^2 c^2 e+2 a^2 d (2 d e+c f)-a b c (11 d e+10 c f)\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{15 a c^2 (b c-a d)^3 \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}} \] Output:

-1/5*(-c*f+d*e)*x*(b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/c/(-a*d+b*c)/(d*x^2+c)^( 
5/2)-2/15*(b*c*(-c*f+4*d*e)-a*d*(c*f+2*d*e))*x*(b*x^2+a)^(1/2)*(f*x^2+e)^( 
1/2)/c^2/(-a*d+b*c)^2/(d*x^2+c)^(3/2)-1/15*e^(1/2)*(b^2*c^2*e*(-20*c*f+23* 
d*e)-a*b*c*(-10*c^2*f^2-7*c*d*e*f+23*d^2*e^2)+a^2*d*(-2*c^2*f^2-3*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/(-a*d+b*c)^3/(-c*f+d*e)^(1/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(f*x^2+ 
e)^(1/2)+1/15*e^(1/2)*(-a*f+b*e)*(15*b^2*c^2*e+2*a^2*d*(c*f+2*d*e)-a*b*c*( 
10*c*f+11*d*e))*(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/(-a*d+b*c)^3/(-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]

\[ \int \frac {\left (e+f x^2\right )^{3/2}}{\sqrt {a+b x^2} \left (c+d x^2\right )^{7/2}} \, dx=\int \frac {\left (e+f x^2\right )^{3/2}}{\sqrt {a+b x^2} \left (c+d x^2\right )^{7/2}} \, dx \] Input:

Integrate[(e + f*x^2)^(3/2)/(Sqrt[a + b*x^2]*(c + d*x^2)^(7/2)),x]
 

Output:

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

\(\displaystyle \int \frac {\left (e+f x^2\right )^{3/2}}{\sqrt {a+b x^2} \left (c+d x^2\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 434

\(\displaystyle \int \frac {\left (e+f x^2\right )^{3/2}}{\sqrt {a+b x^2} \left (c+d x^2\right )^{7/2}}dx\)

Input:

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

\[\int \frac {\left (f \,x^{2}+e \right )^{\frac {3}{2}}}{\sqrt {b \,x^{2}+a}\, \left (x^{2} d +c \right )^{\frac {7}{2}}}d x\]

Input:

int((f*x^2+e)^(3/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(7/2),x)
 

Output:

int((f*x^2+e)^(3/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(7/2),x)
 

Fricas [F]

\[ \int \frac {\left (e+f x^2\right )^{3/2}}{\sqrt {a+b x^2} \left (c+d x^2\right )^{7/2}} \, dx=\int { \frac {{\left (f x^{2} + e\right )}^{\frac {3}{2}}}{\sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((f*x^2+e)^(3/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(7/2),x, algorithm="fr 
icas")
 

Output:

integral(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^(3/2)/(b*d^4*x^10 + ( 
4*b*c*d^3 + a*d^4)*x^8 + 2*(3*b*c^2*d^2 + 2*a*c*d^3)*x^6 + a*c^4 + 2*(2*b* 
c^3*d + 3*a*c^2*d^2)*x^4 + (b*c^4 + 4*a*c^3*d)*x^2), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (e+f x^2\right )^{3/2}}{\sqrt {a+b x^2} \left (c+d x^2\right )^{7/2}} \, dx=\text {Timed out} \] Input:

integrate((f*x**2+e)**(3/2)/(b*x**2+a)**(1/2)/(d*x**2+c)**(7/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (e+f x^2\right )^{3/2}}{\sqrt {a+b x^2} \left (c+d x^2\right )^{7/2}} \, dx=\int { \frac {{\left (f x^{2} + e\right )}^{\frac {3}{2}}}{\sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((f*x^2+e)^(3/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(7/2),x, algorithm="ma 
xima")
 

Output:

integrate((f*x^2 + e)^(3/2)/(sqrt(b*x^2 + a)*(d*x^2 + c)^(7/2)), x)
 

Giac [F]

\[ \int \frac {\left (e+f x^2\right )^{3/2}}{\sqrt {a+b x^2} \left (c+d x^2\right )^{7/2}} \, dx=\int { \frac {{\left (f x^{2} + e\right )}^{\frac {3}{2}}}{\sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((f*x^2+e)^(3/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(7/2),x, algorithm="gi 
ac")
 

Output:

integrate((f*x^2 + e)^(3/2)/(sqrt(b*x^2 + a)*(d*x^2 + c)^(7/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (e+f x^2\right )^{3/2}}{\sqrt {a+b x^2} \left (c+d x^2\right )^{7/2}} \, dx=\int \frac {{\left (f\,x^2+e\right )}^{3/2}}{\sqrt {b\,x^2+a}\,{\left (d\,x^2+c\right )}^{7/2}} \,d x \] Input:

int((e + f*x^2)^(3/2)/((a + b*x^2)^(1/2)*(c + d*x^2)^(7/2)),x)
 

Output:

int((e + f*x^2)^(3/2)/((a + b*x^2)^(1/2)*(c + d*x^2)^(7/2)), x)
 

Reduce [F]

\[ \int \frac {\left (e+f x^2\right )^{3/2}}{\sqrt {a+b x^2} \left (c+d x^2\right )^{7/2}} \, dx=\int \frac {\left (f \,x^{2}+e \right )^{\frac {3}{2}}}{\sqrt {b \,x^{2}+a}\, \left (d \,x^{2}+c \right )^{\frac {7}{2}}}d x \] Input:

int((f*x^2+e)^(3/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(7/2),x)
 

Output:

int((f*x^2+e)^(3/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(7/2),x)