\(\int \frac {(a+b x^2)^{5/2}}{(c+d x^2)^{7/2} (e+f x^2)^{3/2}} \, dx\) [525]

Optimal result

Integrand size = 34, antiderivative size = 784 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )^{3/2}} \, dx=\frac {(b c-a d)^2 x \sqrt {a+b x^2}}{5 c d (d e-c f) \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}}-\frac {(b c-a d) (2 a d (2 d e-5 c f)+b c (7 d e-c f)) x \sqrt {a+b x^2}}{15 c^2 d (d e-c f)^2 \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}+\frac {\left (a b c d \left (7 d^2 e^2-35 c d e f-20 c^2 f^2\right )+2 b^2 c^2 \left (4 d^2 e^2+9 c d e f-c^2 f^2\right )+a^2 d^2 \left (8 d^2 e^2-29 c d e f+45 c^2 f^2\right )\right ) x \sqrt {a+b x^2}}{15 c^3 d (d e-c f)^3 \sqrt {c+d x^2} \sqrt {e+f x^2}}+\frac {\sqrt {-b e+a f} \left (8 b^2 c^2 e^2 (d e+5 c f)+a b c e \left (7 d^2 e^2-38 c d e f-65 c^2 f^2\right )+a^2 \left (8 d^3 e^3-33 c d^2 e^2 f+58 c^2 d e f^2+15 c^3 f^3\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )|\frac {a (d e-c f)}{c (b e-a f)}\right )}{15 \sqrt {a} c^2 e (d e-c f)^4 \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}-\frac {\sqrt {-b e+a f} \left (8 b^2 c^2 e (d e+5 c f)+3 a b c \left (d^2 e^2-18 c d e f-15 c^2 f^2\right )+4 a^2 d \left (d^2 e^2-4 c d e f+15 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{15 \sqrt {a} c^2 (d e-c f)^4 \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}} \] Output:

1/5*(-a*d+b*c)^2*x*(b*x^2+a)^(1/2)/c/d/(-c*f+d*e)/(d*x^2+c)^(5/2)/(f*x^2+e 
)^(1/2)-1/15*(-a*d+b*c)*(2*a*d*(-5*c*f+2*d*e)+b*c*(-c*f+7*d*e))*x*(b*x^2+a 
)^(1/2)/c^2/d/(-c*f+d*e)^2/(d*x^2+c)^(3/2)/(f*x^2+e)^(1/2)+1/15*(a*b*c*d*( 
-20*c^2*f^2-35*c*d*e*f+7*d^2*e^2)+2*b^2*c^2*(-c^2*f^2+9*c*d*e*f+4*d^2*e^2) 
+a^2*d^2*(45*c^2*f^2-29*c*d*e*f+8*d^2*e^2))*x*(b*x^2+a)^(1/2)/c^3/d/(-c*f+ 
d*e)^3/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2)+1/15*(a*f-b*e)^(1/2)*(8*b^2*c^2*e^2 
*(5*c*f+d*e)+a*b*c*e*(-65*c^2*f^2-38*c*d*e*f+7*d^2*e^2)+a^2*(15*c^3*f^3+58 
*c^2*d*e*f^2-33*c*d^2*e^2*f+8*d^3*e^3))*(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)/c^2/e/(-c*f+d*e)^4/(d*x^2+c)^(1/2)/(e*( 
b*x^2+a)/a/(f*x^2+e))^(1/2)-1/15*(a*f-b*e)^(1/2)*(8*b^2*c^2*e*(5*c*f+d*e)+ 
3*a*b*c*(-15*c^2*f^2-18*c*d*e*f+d^2*e^2)+4*a^2*d*(15*c^2*f^2-4*c*d*e*f+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 
)/c^2/(-c*f+d*e)^4/(d*x^2+c)^(1/2)/(e*(b*x^2+a)/a/(f*x^2+e))^(1/2)
 

Mathematica [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )^{3/2}} \, dx=\int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )^{3/2}} \, dx \] Input:

Integrate[(a + b*x^2)^(5/2)/((c + d*x^2)^(7/2)*(e + f*x^2)^(3/2)),x]
 

Output:

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

\(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 434

\(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )^{3/2}}dx\)

Input:

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

\[\int \frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{\left (x^{2} d +c \right )^{\frac {7}{2}} \left (f \,x^{2}+e \right )^{\frac {3}{2}}}d x\]

Input:

int((b*x^2+a)^(5/2)/(d*x^2+c)^(7/2)/(f*x^2+e)^(3/2),x)
 

Output:

int((b*x^2+a)^(5/2)/(d*x^2+c)^(7/2)/(f*x^2+e)^(3/2),x)
 

Fricas [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{{\left (d x^{2} + c\right )}^{\frac {7}{2}} {\left (f x^{2} + e\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(5/2)/(d*x^2+c)^(7/2)/(f*x^2+e)^(3/2),x, algorithm="fr 
icas")
 

Output:

integral((b^2*x^4 + 2*a*b*x^2 + a^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt( 
f*x^2 + e)/(d^4*f^2*x^12 + 2*(d^4*e*f + 2*c*d^3*f^2)*x^10 + (d^4*e^2 + 8*c 
*d^3*e*f + 6*c^2*d^2*f^2)*x^8 + 4*(c*d^3*e^2 + 3*c^2*d^2*e*f + c^3*d*f^2)* 
x^6 + c^4*e^2 + (6*c^2*d^2*e^2 + 8*c^3*d*e*f + c^4*f^2)*x^4 + 2*(2*c^3*d*e 
^2 + c^4*e*f)*x^2), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate((b*x**2+a)**(5/2)/(d*x**2+c)**(7/2)/(f*x**2+e)**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{{\left (d x^{2} + c\right )}^{\frac {7}{2}} {\left (f x^{2} + e\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(5/2)/(d*x^2+c)^(7/2)/(f*x^2+e)^(3/2),x, algorithm="ma 
xima")
 

Output:

integrate((b*x^2 + a)^(5/2)/((d*x^2 + c)^(7/2)*(f*x^2 + e)^(3/2)), x)
 

Giac [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{{\left (d x^{2} + c\right )}^{\frac {7}{2}} {\left (f x^{2} + e\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(5/2)/(d*x^2+c)^(7/2)/(f*x^2+e)^(3/2),x, algorithm="gi 
ac")
 

Output:

integrate((b*x^2 + a)^(5/2)/((d*x^2 + c)^(7/2)*(f*x^2 + e)^(3/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )^{3/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{5/2}}{{\left (d\,x^2+c\right )}^{7/2}\,{\left (f\,x^2+e\right )}^{3/2}} \,d x \] Input:

int((a + b*x^2)^(5/2)/((c + d*x^2)^(7/2)*(e + f*x^2)^(3/2)),x)
 

Output:

int((a + b*x^2)^(5/2)/((c + d*x^2)^(7/2)*(e + f*x^2)^(3/2)), x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )^{3/2}} \, dx=\int \frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{\left (d \,x^{2}+c \right )^{\frac {7}{2}} \left (f \,x^{2}+e \right )^{\frac {3}{2}}}d x \] Input:

int((b*x^2+a)^(5/2)/(d*x^2+c)^(7/2)/(f*x^2+e)^(3/2),x)
 

Output:

int((b*x^2+a)^(5/2)/(d*x^2+c)^(7/2)/(f*x^2+e)^(3/2),x)