\(\int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^6 (e+f x^2)^2} \, dx\) [136]

Optimal result

Integrand size = 35, antiderivative size = 1095 \[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^6 \left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Output:

-1/30*b*(4*b^2*c^2*e^2-4*a*b*c*e*(-5*c*f+d*e)+a^2*(-105*c^2*f^2+20*c*d*e*f 
+4*d^2*e^2))*x*(d*x^2+c)^(1/2)/a^2/c^2/e^4/(b*x^2+a)^(1/2)+1/30*d*(15*a^2* 
c*f^2*(-7*c*f+6*d*e)-4*b^2*c*e^2*(-c*f+d*e)+2*a*b*e*(10*c^2*f^2-11*c*d*e*f 
+d^2*e^2))*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a^2/c^2/e^4/(-c*f+d*e)-1/10*b 
*(5*a^2*c*f^3*(-7*c*f+6*d*e)-2*b^2*d*e^3*(-c*f+d*e)+2*a*b*e*f*(15*c^2*f^2- 
16*c*d*e*f+d^2*e^2))*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/a^2/c^2/e^4/(-a*f+b 
*e)/(-c*f+d*e)-1/5*(b*x^2+a)^(3/2)*(d*x^2+c)^(3/2)/a/c/e/x^5/(f*x^2+e)+1/1 
5*(7*a*c*f+2*a*d*e+2*b*c*e)*(b*x^2+a)^(3/2)*(d*x^2+c)^(3/2)/a^2/c^2/e^2/x^ 
3/(f*x^2+e)+1/15*(a*f*(-35*c*f+2*d*e)-b*e*(-2*c*f+3*d*e))*(b*x^2+a)^(3/2)* 
(d*x^2+c)^(3/2)/a^2/c^2/e^3/x/(f*x^2+e)+1/10*f*(5*a^2*c*f^3*(-7*c*f+6*d*e) 
-2*b^2*d*e^3*(-c*f+d*e)+2*a*b*e*f*(15*c^2*f^2-16*c*d*e*f+d^2*e^2))*x*(b*x^ 
2+a)^(3/2)*(d*x^2+c)^(3/2)/a^2/c^2/e^4/(-a*f+b*e)/(-c*f+d*e)/(f*x^2+e)+1/3 
0*b^(1/2)*(4*b^2*c^2*e^2-4*a*b*c*e*(-5*c*f+d*e)+a^2*(-105*c^2*f^2+20*c*d*e 
*f+4*d^2*e^2))*(d*x^2+c)^(1/2)*EllipticE(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/ 
2),(1-a*d/b/c)^(1/2))/a^(3/2)/c^2/e^4/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^ 
2+a))^(1/2)-1/30*b^(1/2)*(2*b^2*c*d*e^3-a^2*f*(-105*c^2*f^2+55*c*d*e*f+2*d 
^2*e^2)+2*a*b*e*(-45*c^2*f^2+19*c*d*e*f+d^2*e^2))*(d*x^2+c)^(1/2)*InverseJ 
acobiAM(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/a^(1/2)/c^2/e^4/(-a*f 
+b*e)/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)-1/2*a^(3/2)*f^2*(a*f 
*(-7*c*f+6*d*e)-b*e*(-6*c*f+5*d*e))*(d*x^2+c)^(1/2)*EllipticPi(b^(1/2)*...
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.93 (sec) , antiderivative size = 470, normalized size of antiderivative = 0.43 \[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^6 \left (e+f x^2\right )^2} \, dx=\frac {-\frac {e \left (a+b x^2\right ) \left (c+d x^2\right ) \left (15 a^2 c^2 f^3 x^6+6 a^2 c^2 e^2 \left (e+f x^2\right )+2 a c e (b c e+a d e-10 a c f) x^2 \left (e+f x^2\right )-2 \left (2 b^2 c^2 e^2+2 a b c e (-d e+5 c f)+a^2 \left (2 d^2 e^2+10 c d e f-45 c^2 f^2\right )\right ) x^4 \left (e+f x^2\right )\right )}{x^5 \left (e+f x^2\right )}+\frac {i c \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (b e \left (4 b^2 c^2 e^2+4 a b c e (-d e+5 c f)+a^2 \left (4 d^2 e^2+20 c d e f-105 c^2 f^2\right )\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-b e \left (4 b^2 c^2 e^2+2 a b c e (-3 d e+10 c f)+a^2 \left (2 d^2 e^2+55 c d e f-105 c^2 f^2\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+15 a^2 c f (b e (5 d e-6 c f)+a f (-6 d e+7 c f)) \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{\sqrt {\frac {b}{a}}}}{30 a^2 c^2 e^5 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

Integrate[(Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(x^6*(e + f*x^2)^2),x]
 

Output:

(-((e*(a + b*x^2)*(c + d*x^2)*(15*a^2*c^2*f^3*x^6 + 6*a^2*c^2*e^2*(e + f*x 
^2) + 2*a*c*e*(b*c*e + a*d*e - 10*a*c*f)*x^2*(e + f*x^2) - 2*(2*b^2*c^2*e^ 
2 + 2*a*b*c*e*(-(d*e) + 5*c*f) + a^2*(2*d^2*e^2 + 10*c*d*e*f - 45*c^2*f^2) 
)*x^4*(e + f*x^2)))/(x^5*(e + f*x^2))) + (I*c*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + 
 (d*x^2)/c]*(b*e*(4*b^2*c^2*e^2 + 4*a*b*c*e*(-(d*e) + 5*c*f) + a^2*(4*d^2* 
e^2 + 20*c*d*e*f - 105*c^2*f^2))*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/( 
b*c)] - b*e*(4*b^2*c^2*e^2 + 2*a*b*c*e*(-3*d*e + 10*c*f) + a^2*(2*d^2*e^2 
+ 55*c*d*e*f - 105*c^2*f^2))*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c) 
] + 15*a^2*c*f*(b*e*(5*d*e - 6*c*f) + a*f*(-6*d*e + 7*c*f))*EllipticPi[(a* 
f)/(b*e), I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/Sqrt[b/a])/(30*a^2*c^2*e^ 
5*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
 

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.

\(\displaystyle \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^6 \left (e+f x^2\right )^2} \, dx\)

\(\Big \downarrow \) 450

\(\displaystyle \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^6 \left (e+f x^2\right )^2}dx\)

Input:

Int[(Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(x^6*(e + f*x^2)^2),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 450

Int[((g_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_)^2)^(p_.)*((c_.) + (d_.)*(x_)^2)^ 
(q_.)*((e_.) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Unintegrable[(g*x)^m*(a + 
b*x^2)^p*(c + d*x^2)^q*(e + f*x^2)^r, x] /; FreeQ[{a, b, c, d, e, f, g, m, 
p, q, r}, x]
 

Maple [A] (verified)

Time = 22.35 (sec) , antiderivative size = 1785, normalized size of antiderivative = 1.63

method	result	size
risch	\(\text {Expression too large to display}\)	\(1785\)
elliptic	\(\text {Expression too large to display}\)	\(1829\)
default	\(\text {Expression too large to display}\)	\(3178\)

Input:

int((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/x^6/(f*x^2+e)^2,x,method=_RETURNVERBOS 
E)
 

Output:

-1/15*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(45*a^2*c^2*f^2*x^4-10*a^2*c*d*e*f*x 
^4-2*a^2*d^2*e^2*x^4-10*a*b*c^2*e*f*x^4+2*a*b*c*d*e^2*x^4-2*b^2*c^2*e^2*x^ 
4-10*a^2*c^2*e*f*x^2+a^2*c*d*e^2*x^2+a*b*c^2*e^2*x^2+3*a^2*c^2*e^2)/a^2/c^ 
2/e^4/x^5+1/15/a^2/c^2/e^4*(-b*(45*a^2*c^2*f^2-10*a^2*c*d*e*f-2*a^2*d^2*e^ 
2-10*a*b*c^2*e*f+2*a*b*c*d*e^2-2*b^2*c^2*e^2)*c/(-b/a)^(1/2)*(1+b*x^2/a)^( 
1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(EllipticF(x*(- 
b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-EllipticE(x*(-b/a)^(1/2),(-1+(a*d+b*c 
)/c/b)^(1/2)))-a*b^2*c^2*d*e^2/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^ 
(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1+(a* 
d+b*c)/c/b)^(1/2))-a^2*b*c*d^2*e^2/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2 
/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1 
+(a*d+b*c)/c/b)^(1/2))-15*a^2*c^2*f*(3*a*c*f^2-2*a*d*e*f-2*b*c*e*f+b*d*e^2 
)/e/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c* 
x^2+a*c)^(1/2)*EllipticPi(x*(-b/a)^(1/2),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/ 
2))+10*a^2*b*c^2*d*e*f/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b 
*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c 
/b)^(1/2))-15*a^2*c^2*e*f*(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)*(1/2*f^2/(a*c* 
f^2-a*d*e*f-b*c*e*f+b*d*e^2)/e*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/(f*x^ 
2+e)-1/2*d*b/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)/(-b/a)^(1/2)*(1+b*x^2/a)^(1 
/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x*(...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^6 \left (e+f x^2\right )^2} \, dx=\text {Timed out} \] Input:

integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/x^6/(f*x^2+e)^2,x, algorithm="fr 
icas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^6 \left (e+f x^2\right )^2} \, dx=\int \frac {\sqrt {a + b x^{2}} \sqrt {c + d x^{2}}}{x^{6} \left (e + f x^{2}\right )^{2}}\, dx \] Input:

integrate((b*x**2+a)**(1/2)*(d*x**2+c)**(1/2)/x**6/(f*x**2+e)**2,x)
 

Output:

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

Maxima [F]

\[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^6 \left (e+f x^2\right )^2} \, dx=\int { \frac {\sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{{\left (f x^{2} + e\right )}^{2} x^{6}} \,d x } \] Input:

integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/x^6/(f*x^2+e)^2,x, algorithm="ma 
xima")
 

Output:

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

Giac [F]

\[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^6 \left (e+f x^2\right )^2} \, dx=\int { \frac {\sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{{\left (f x^{2} + e\right )}^{2} x^{6}} \,d x } \] Input:

integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/x^6/(f*x^2+e)^2,x, algorithm="gi 
ac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^6 \left (e+f x^2\right )^2} \, dx=\int \frac {\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}}{x^6\,{\left (f\,x^2+e\right )}^2} \,d x \] Input:

int(((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2))/(x^6*(e + f*x^2)^2),x)
 

Output:

int(((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2))/(x^6*(e + f*x^2)^2), x)
 

Reduce [F]

\[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^6 \left (e+f x^2\right )^2} \, dx=\int \frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}{x^{6} \left (f \,x^{2}+e \right )^{2}}d x \] Input:

int((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/x^6/(f*x^2+e)^2,x)
 

Output:

int((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/x^6/(f*x^2+e)^2,x)