\(\int \frac {(a+b x^2)^{5/2} (e+f x^2)^{3/2}}{(c+d x^2)^{3/2}} \, dx\) [435]

Optimal result

Integrand size = 34, antiderivative size = 934 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {\left (33 a^2 d^2 f^2+68 a b d f (d e-c f)+b^2 \left (3 d^2 e^2-38 c d e f+35 c^2 f^2\right )\right ) x \sqrt {a+b x^2} \sqrt {e+f x^2}}{48 d^3 f \sqrt {c+d x^2}}+\frac {b (7 b d e-7 b c f+13 a d f) x^3 \sqrt {a+b x^2} \sqrt {e+f x^2}}{24 d^2 \sqrt {c+d x^2}}+\frac {b^2 f x^5 \sqrt {a+b x^2} \sqrt {e+f x^2}}{6 d \sqrt {c+d x^2}}-\frac {\sqrt {e} \sqrt {d e-c f} \left (2 a b c d f (82 d e-95 c f)-3 a^2 d^2 f (16 d e-27 c f)+b^2 c \left (3 d^2 e^2-100 c d e f+105 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{48 c d^4 f \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}+\frac {\sqrt {e} \sqrt {d e-c f} \left (96 a^3 d^3 f^2+a^2 b d^2 f (96 d e-325 c f)-110 a b^2 c d f (d e-3 c f)+3 b^3 c \left (d^2 e^2+10 c d e f-35 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{48 a d^5 f \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}+\frac {c \sqrt {e} \left (5 a^3 d^3 f^3+45 a^2 b d^2 f^2 (d e-c f)+15 a b^2 d f \left (d^2 e^2-6 c d e f+5 c^2 f^2\right )-b^3 \left (d^3 e^3+9 c d^2 e^2 f-45 c^2 d e f^2+35 c^3 f^3\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticPi}\left (\frac {d e}{d e-c f},\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{16 a d^5 f \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}} \] Output:

1/48*(33*a^2*d^2*f^2+68*a*b*d*f*(-c*f+d*e)+b^2*(35*c^2*f^2-38*c*d*e*f+3*d^ 
2*e^2))*x*(b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/d^3/f/(d*x^2+c)^(1/2)+1/24*b*(13 
*a*d*f-7*b*c*f+7*b*d*e)*x^3*(b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/d^2/(d*x^2+c)^ 
(1/2)+1/6*b^2*f*x^5*(b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/d/(d*x^2+c)^(1/2)-1/48 
*e^(1/2)*(-c*f+d*e)^(1/2)*(2*a*b*c*d*f*(-95*c*f+82*d*e)-3*a^2*d^2*f*(-27*c 
*f+16*d*e)+b^2*c*(105*c^2*f^2-100*c*d*e*f+3*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/d^4/f/(c*(b*x^2+a)/a/(d*x^2+c 
))^(1/2)/(f*x^2+e)^(1/2)+1/48*e^(1/2)*(-c*f+d*e)^(1/2)*(96*a^3*d^3*f^2+a^2 
*b*d^2*f*(-325*c*f+96*d*e)-110*a*b^2*c*d*f*(-3*c*f+d*e)+3*b^3*c*(-35*c^2*f 
^2+10*c*d*e*f+d^2*e^2))*(b*x^2+a)^(1/2)*(c*(f*x^2+e)/e/(d*x^2+c))^(1/2)*El 
lipticF((-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/d^5/f/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(f*x^2+e)^(1/2)+1/16* 
c*e^(1/2)*(5*a^3*d^3*f^3+45*a^2*b*d^2*f^2*(-c*f+d*e)+15*a*b^2*d*f*(5*c^2*f 
^2-6*c*d*e*f+d^2*e^2)-b^3*(35*c^3*f^3-45*c^2*d*e*f^2+9*c*d^2*e^2*f+d^3*e^3 
))*(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^5/f/(-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]

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

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

Output:

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

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

\(\Big \downarrow \) 434

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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