\(\int \frac {\sqrt {a+b x^2}}{x^6 (c+d x^2)^{3/2} (e+f x^2)^2} \, dx\) [160]

Optimal result

Integrand size = 35, antiderivative size = 1300 \[ \int \frac {\sqrt {a+b x^2}}{x^6 \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Output:

1/30*(2*b^3*c^2*d*e^3*(-c*f+d*e)^2+2*a*b^2*c*e*(-c*f+d*e)^2*(45*c^2*f^2+49 
*c*d*e*f+32*d^2*e^2)+a^3*d*f*(105*c^4*f^4-80*c^3*d*e*f^3-44*c^2*d^2*e^2*f^ 
2-32*c*d^3*e^3*f+96*d^4*e^4)-a^2*b*(105*c^5*f^5+10*c^4*d*e*f^4-126*c^3*d^2 
*e^2*f^3-72*c^2*d^3*e^3*f^2+32*c*d^4*e^4*f+96*d^5*e^5))*x*(b*x^2+a)^(1/2)/ 
a^2/c^4/e^4/(-a*f+b*e)/(-c*f+d*e)^2/(d*x^2+c)^(1/2)-1/30*b*(4*b^2*c*e+4*a* 
b*(5*c*f+4*d*e)-a^2*(105*c^4*f^4-80*c^3*d*e*f^3-44*c^2*d^2*e^2*f^2-32*c*d^ 
3*e^3*f+96*d^4*e^4)/c/e/(-c*f+d*e)^2)*x*(d*x^2+c)^(1/2)/a^2/c^3/e^3/(b*x^2 
+a)^(1/2)-1/5*(b*x^2+a)^(3/2)/a/c/e/x^5/(d*x^2+c)^(1/2)/(f*x^2+e)+1/15*(7* 
a*c*f+6*a*d*e+2*b*c*e)*(b*x^2+a)^(3/2)/a^2/c^2/e^2/x^3/(d*x^2+c)^(1/2)/(f* 
x^2+e)+1/15*(b*c*e*(2*c*f+d*e)-a*(35*c^2*f^2+34*c*d*e*f+24*d^2*e^2))*(b*x^ 
2+a)^(3/2)/a^2/c^3/e^3/x/(d*x^2+c)^(1/2)/(f*x^2+e)+1/30*f*(2*b^2*c*d*e^3*( 
-c*f+d*e)+a^2*f*(-105*c^3*f^3+10*c^2*d*e*f^2+32*c*d^2*e^2*f+48*d^3*e^3)-2* 
a*b*e*(-45*c^3*f^3+4*c^2*d*e*f^2+17*c*d^2*e^2*f+24*d^3*e^3))*x*(b*x^2+a)^( 
3/2)/a^2/c^3/e^4/(-a*f+b*e)/(-c*f+d*e)/(d*x^2+c)^(1/2)/(f*x^2+e)+1/30*b^(1 
/2)*(4*b^2*c*e+4*a*b*(5*c*f+4*d*e)-a^2*(105*c^4*f^4-80*c^3*d*e*f^3-44*c^2* 
d^2*e^2*f^2-32*c*d^3*e^3*f+96*d^4*e^4)/c/e/(-c*f+d*e)^2)*(d*x^2+c)^(1/2)*E 
llipticE(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^ 
3/e^3/(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*(-c*f+d*e)+a^2*f*(-105*c^3*f^3+10*c^2*d*e*f^2+32*c*d^2*e^2*f+48*d^ 
3*e^3)-2*a*b*e*(-45*c^3*f^3+4*c^2*d*e*f^2+17*c*d^2*e^2*f+24*d^3*e^3))*(...
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.69 (sec) , antiderivative size = 669, normalized size of antiderivative = 0.51 \[ \int \frac {\sqrt {a+b x^2}}{x^6 \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2} \, dx=\frac {-\sqrt {\frac {b}{a}} e \left (a+b x^2\right ) \left (15 a^2 c^4 f^5 x^6 \left (c+d x^2\right )+30 a^2 d^5 e^4 x^6 \left (e+f x^2\right )+6 a^2 c^2 e^2 (d e-c f)^2 \left (c+d x^2\right ) \left (e+f x^2\right )+2 a c e (d e-c f)^2 (b c e-9 a d e-10 a c f) x^2 \left (c+d x^2\right ) \left (e+f x^2\right )+2 (d e-c f)^2 \left (-2 b^2 c^2 e^2-2 a b c e (4 d e+5 c f)+a^2 \left (33 d^2 e^2+50 c d e f+45 c^2 f^2\right )\right ) x^4 \left (c+d x^2\right ) \left (e+f x^2\right )\right )+i c x^5 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right ) \left (b e \left (4 b^2 c^2 e^2 (d e-c f)^2+4 a b c e (d e-c f)^2 (4 d e+5 c f)+a^2 \left (-96 d^4 e^4+32 c d^3 e^3 f+44 c^2 d^2 e^2 f^2+80 c^3 d e f^3-105 c^4 f^4\right )\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-b e (-d e+c f) \left (4 b^2 c^2 e^2 (-d e+c f)+2 a b c e \left (-7 d^2 e^2-3 c d e f+10 c^2 f^2\right )+a^2 \left (48 d^3 e^3+32 c d^2 e^2 f+10 c^2 d e f^2-105 c^3 f^3\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+15 a^2 c^3 f^3 (3 b e (3 d e-2 c f)+a f (-10 d e+7 c f)) \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{30 a^2 \sqrt {\frac {b}{a}} c^4 e^5 (d e-c f)^2 x^5 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \] Input:

Integrate[Sqrt[a + b*x^2]/(x^6*(c + d*x^2)^(3/2)*(e + f*x^2)^2),x]
 

Output:

(-(Sqrt[b/a]*e*(a + b*x^2)*(15*a^2*c^4*f^5*x^6*(c + d*x^2) + 30*a^2*d^5*e^ 
4*x^6*(e + f*x^2) + 6*a^2*c^2*e^2*(d*e - c*f)^2*(c + d*x^2)*(e + f*x^2) + 
2*a*c*e*(d*e - c*f)^2*(b*c*e - 9*a*d*e - 10*a*c*f)*x^2*(c + d*x^2)*(e + f* 
x^2) + 2*(d*e - c*f)^2*(-2*b^2*c^2*e^2 - 2*a*b*c*e*(4*d*e + 5*c*f) + a^2*( 
33*d^2*e^2 + 50*c*d*e*f + 45*c^2*f^2))*x^4*(c + d*x^2)*(e + f*x^2))) + I*c 
*x^5*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*(e + f*x^2)*(b*e*(4*b^2*c^2*e 
^2*(d*e - c*f)^2 + 4*a*b*c*e*(d*e - c*f)^2*(4*d*e + 5*c*f) + a^2*(-96*d^4* 
e^4 + 32*c*d^3*e^3*f + 44*c^2*d^2*e^2*f^2 + 80*c^3*d*e*f^3 - 105*c^4*f^4)) 
*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - b*e*(-(d*e) + c*f)*(4*b^ 
2*c^2*e^2*(-(d*e) + c*f) + 2*a*b*c*e*(-7*d^2*e^2 - 3*c*d*e*f + 10*c^2*f^2) 
 + a^2*(48*d^3*e^3 + 32*c*d^2*e^2*f + 10*c^2*d*e*f^2 - 105*c^3*f^3))*Ellip 
ticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + 15*a^2*c^3*f^3*(3*b*e*(3*d*e - 
 2*c*f) + a*f*(-10*d*e + 7*c*f))*EllipticPi[(a*f)/(b*e), I*ArcSinh[Sqrt[b/ 
a]*x], (a*d)/(b*c)]))/(30*a^2*Sqrt[b/a]*c^4*e^5*(d*e - c*f)^2*x^5*Sqrt[a + 
 b*x^2]*Sqrt[c + d*x^2]*(e + f*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.

\(\displaystyle \int \frac {\sqrt {a+b x^2}}{x^6 \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2} \, dx\)

\(\Big \downarrow \) 450

\(\displaystyle \int \frac {\sqrt {a+b x^2}}{x^6 \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2}dx\)

Input:

Int[Sqrt[a + b*x^2]/(x^6*(c + d*x^2)^(3/2)*(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 = 28.91 (sec) , antiderivative size = 2108, normalized size of antiderivative = 1.62

method	result	size
risch	\(\text {Expression too large to display}\)	\(2108\)
elliptic	\(\text {Expression too large to display}\)	\(2377\)
default	\(\text {Expression too large to display}\)	\(5749\)

Input:

int((b*x^2+a)^(1/2)/x^6/(d*x^2+c)^(3/2)/(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+50*a^2*c*d*e*f*x 
^4+33*a^2*d^2*e^2*x^4-10*a*b*c^2*e*f*x^4-8*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^2-9*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^4/e^4/x^5+1/15/e^4/a^2/c^4*(-b*(45*a^2*c^2*f^2+50*a^2*c*d*e*f+33*a^2*d^ 
2*e^2-10*a*b*c^2*e*f-8*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))+9*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))+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^4*f^3*(3*a*c*f^2-4*a*d*e*f-2* 
b*c*e*f+3*b*d*e^2)/(c*f-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))-15*a^2*c*d^4*e^4*(a*d-b*c)/(c*f-d*e)^2*(( 
b*d*x^2+a*d)/c/(a*d-b*c)*x/((x^2+c/d)*(b*d*x^2+a*d))^(1/2)+(1/c-1/(a*d-b*c 
)/c*a*d)/(-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))+...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^2}}{x^6 \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2} \, dx=\text {Timed out} \] Input:

integrate((b*x^2+a)^(1/2)/x^6/(d*x^2+c)^(3/2)/(f*x^2+e)^2,x, algorithm="fr 
icas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\sqrt {a+b x^2}}{x^6 \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2} \, dx=\int \frac {\sqrt {a + b x^{2}}}{x^{6} \left (c + d x^{2}\right )^{\frac {3}{2}} \left (e + f x^{2}\right )^{2}}\, dx \] Input:

integrate((b*x**2+a)**(1/2)/x**6/(d*x**2+c)**(3/2)/(f*x**2+e)**2,x)
 

Output:

Integral(sqrt(a + b*x**2)/(x**6*(c + d*x**2)**(3/2)*(e + f*x**2)**2), x)
 

Maxima [F]

\[ \int \frac {\sqrt {a+b x^2}}{x^6 \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2} \, dx=\int { \frac {\sqrt {b x^{2} + a}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{2} x^{6}} \,d x } \] Input:

integrate((b*x^2+a)^(1/2)/x^6/(d*x^2+c)^(3/2)/(f*x^2+e)^2,x, algorithm="ma 
xima")
 

Output:

integrate(sqrt(b*x^2 + a)/((d*x^2 + c)^(3/2)*(f*x^2 + e)^2*x^6), x)
 

Giac [F]

\[ \int \frac {\sqrt {a+b x^2}}{x^6 \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2} \, dx=\int { \frac {\sqrt {b x^{2} + a}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{2} x^{6}} \,d x } \] Input:

integrate((b*x^2+a)^(1/2)/x^6/(d*x^2+c)^(3/2)/(f*x^2+e)^2,x, algorithm="gi 
ac")
 

Output:

integrate(sqrt(b*x^2 + a)/((d*x^2 + c)^(3/2)*(f*x^2 + e)^2*x^6), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^2}}{x^6 \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2} \, dx=\int \frac {\sqrt {b\,x^2+a}}{x^6\,{\left (d\,x^2+c\right )}^{3/2}\,{\left (f\,x^2+e\right )}^2} \,d x \] Input:

int((a + b*x^2)^(1/2)/(x^6*(c + d*x^2)^(3/2)*(e + f*x^2)^2),x)
 

Output:

int((a + b*x^2)^(1/2)/(x^6*(c + d*x^2)^(3/2)*(e + f*x^2)^2), x)
 

Reduce [F]

\[ \int \frac {\sqrt {a+b x^2}}{x^6 \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2} \, dx=\int \frac {\sqrt {b \,x^{2}+a}}{x^{6} \left (d \,x^{2}+c \right )^{\frac {3}{2}} \left (f \,x^{2}+e \right )^{2}}d x \] Input:

int((b*x^2+a)^(1/2)/x^6/(d*x^2+c)^(3/2)/(f*x^2+e)^2,x)
 

Output:

int((b*x^2+a)^(1/2)/x^6/(d*x^2+c)^(3/2)/(f*x^2+e)^2,x)