\(\int \frac {1}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2} (e+f x^2)^3} \, dx\) [203]

Optimal result

Integrand size = 35, antiderivative size = 1129 \[ \int \frac {1}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx =\text {Too large to display} \] Output:

-1/3*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a/c/e/x^3/(f*x^2+e)^2+1/3*(7*a*c*f+2* 
a*d*e+2*b*c*e)*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a^2/c^2/e^2/x/(f*x^2+e)^2+1 
/12*f*(8*b^2*c*e^2*(-c*f+d*e)-a^2*f*(-35*c^2*f^2+24*c*d*e*f+8*d^2*e^2)+8*a 
*b*e*(-3*c^2*f^2+2*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^3/(-a*f+b*e)/(-c*f+d*e)/(f*x^2+e)^2-1/24*(16*b^3*c*e^3*(-c*f+d*e)^2 
+8*a*b^2*e^2*(-c*f+d*e)^2*(5*c*f+2*d*e)+a^3*f^2*(105*c^3*f^3-170*c^2*d*e*f 
^2+40*c*d^2*e^2*f+16*d^3*e^3)-a^2*b*e*f*(170*c^3*f^3-275*c^2*d*e*f^2+64*c* 
d^2*e^2*f+32*d^3*e^3))*x*(b*x^2+a)^(1/2)/a^2/c^2/e^4/(-a*f+b*e)^2/(-c*f+d* 
e)/(d*x^2+c)^(1/2)/(f*x^2+e)+1/24*d^(1/2)*(16*b^3*c*e^3*(-c*f+d*e)^2+8*a*b 
^2*e^2*(-c*f+d*e)^2*(5*c*f+2*d*e)+a^3*f^2*(105*c^3*f^3-170*c^2*d*e*f^2+40* 
c*d^2*e^2*f+16*d^3*e^3)-a^2*b*e*f*(170*c^3*f^3-275*c^2*d*e*f^2+64*c*d^2*e^ 
2*f+32*d^3*e^3))*(b*x^2+a)^(1/2)*EllipticE(d^(1/2)*x/c^(1/2)/(1+d*x^2/c)^( 
1/2),(1-b*c/a/d)^(1/2))/a^2/c^(3/2)/e^4/(-a*f+b*e)^2/(-c*f+d*e)^2/(c*(b*x^ 
2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)-1/24*d^(1/2)*(8*b^2*e^2*(-c*f+d*e) 
^3+3*a^2*c*f^3*(35*c^2*f^2-80*c*d*e*f+48*d^2*e^2)-a*b*e*f*(100*c^3*f^3-219 
*c^2*d*e*f^2+120*c*d^2*e^2*f+8*d^3*e^3))*(b*x^2+a)^(1/2)*InverseJacobiAM(a 
rctan(d^(1/2)*x/c^(1/2)),(1-b*c/a/d)^(1/2))/a^2/c^(1/2)/e^4/(-a*f+b*e)/(-c 
*f+d*e)^3/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)-1/8*c^(3/2)*f^3* 
(3*b^2*e^2*(16*c^2*f^2-36*c*d*e*f+21*d^2*e^2)+a^2*f^2*(35*c^2*f^2-80*c*d*e 
*f+48*d^2*e^2)-2*a*b*e*f*(40*c^2*f^2-91*c*d*e*f+54*d^2*e^2))*(b*x^2+a)^...
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.20 (sec) , antiderivative size = 758, normalized size of antiderivative = 0.67 \[ \int \frac {1}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx=\frac {\sqrt {\frac {b}{a}} e \left (a+b x^2\right ) \left (c+d x^2\right ) \left (6 a^2 c^2 e f^4 (-b e+a f) (-d e+c f) x^4+3 a^2 c^2 f^4 (b e (17 d e-14 c f)+a f (-14 d e+11 c f)) x^4 \left (e+f x^2\right )-8 a c e (b e-a f)^2 (d e-c f)^2 \left (e+f x^2\right )^2+8 (b e-a f)^2 (d e-c f)^2 (2 b c e+2 a d e+9 a c f) x^2 \left (e+f x^2\right )^2\right )-i c x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right )^2 \left (-b e \left (16 b^3 c e^3 (d e-c f)^2+8 a b^2 e^2 (d e-c f)^2 (2 d e+5 c f)+a^3 f^2 \left (16 d^3 e^3+40 c d^2 e^2 f-170 c^2 d e f^2+105 c^3 f^3\right )-a^2 b e f \left (32 d^3 e^3+64 c d^2 e^2 f-275 c^2 d e f^2+170 c^3 f^3\right )\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+b e (d e-c f) \left (-16 b^3 c e^3 (-d e+c f)+a^3 f^2 \left (8 d^2 e^2+100 c d e f-105 c^2 f^2\right )+8 a b^2 e^2 \left (d^2 e^2+4 c d e f-5 c^2 f^2\right )+a^2 b e f \left (-16 d^2 e^2-157 c d e f+170 c^2 f^2\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+3 a^2 c f^2 \left (3 b^2 e^2 \left (21 d^2 e^2-36 c d e f+16 c^2 f^2\right )+a^2 f^2 \left (48 d^2 e^2-80 c d e f+35 c^2 f^2\right )-2 a b e f \left (54 d^2 e^2-91 c d e f+40 c^2 f^2\right )\right ) \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{24 a^2 \sqrt {\frac {b}{a}} c^2 e^5 (b e-a f)^2 (d e-c f)^2 x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \] Input:

Integrate[1/(x^4*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)^3),x]
 

Output:

(Sqrt[b/a]*e*(a + b*x^2)*(c + d*x^2)*(6*a^2*c^2*e*f^4*(-(b*e) + a*f)*(-(d* 
e) + c*f)*x^4 + 3*a^2*c^2*f^4*(b*e*(17*d*e - 14*c*f) + a*f*(-14*d*e + 11*c 
*f))*x^4*(e + f*x^2) - 8*a*c*e*(b*e - a*f)^2*(d*e - c*f)^2*(e + f*x^2)^2 + 
 8*(b*e - a*f)^2*(d*e - c*f)^2*(2*b*c*e + 2*a*d*e + 9*a*c*f)*x^2*(e + f*x^ 
2)^2) - I*c*x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*(e + f*x^2)^2*(-(b 
*e*(16*b^3*c*e^3*(d*e - c*f)^2 + 8*a*b^2*e^2*(d*e - c*f)^2*(2*d*e + 5*c*f) 
 + a^3*f^2*(16*d^3*e^3 + 40*c*d^2*e^2*f - 170*c^2*d*e*f^2 + 105*c^3*f^3) - 
 a^2*b*e*f*(32*d^3*e^3 + 64*c*d^2*e^2*f - 275*c^2*d*e*f^2 + 170*c^3*f^3))* 
EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]) + b*e*(d*e - c*f)*(-16*b^3 
*c*e^3*(-(d*e) + c*f) + a^3*f^2*(8*d^2*e^2 + 100*c*d*e*f - 105*c^2*f^2) + 
8*a*b^2*e^2*(d^2*e^2 + 4*c*d*e*f - 5*c^2*f^2) + a^2*b*e*f*(-16*d^2*e^2 - 1 
57*c*d*e*f + 170*c^2*f^2))*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] 
+ 3*a^2*c*f^2*(3*b^2*e^2*(21*d^2*e^2 - 36*c*d*e*f + 16*c^2*f^2) + a^2*f^2* 
(48*d^2*e^2 - 80*c*d*e*f + 35*c^2*f^2) - 2*a*b*e*f*(54*d^2*e^2 - 91*c*d*e* 
f + 40*c^2*f^2))*EllipticPi[(a*f)/(b*e), I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b* 
c)]))/(24*a^2*Sqrt[b/a]*c^2*e^5*(b*e - a*f)^2*(d*e - c*f)^2*x^3*Sqrt[a + b 
*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)^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.

\(\displaystyle \int \frac {1}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx\)

\(\Big \downarrow \) 450

\(\displaystyle \int \frac {1}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3}dx\)

Input:

Int[1/(x^4*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)^3),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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3323\) vs. \(2(1085)=2170\).

Time = 28.89 (sec) , antiderivative size = 3324, normalized size of antiderivative = 2.94

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(3324\)
risch	\(\text {Expression too large to display}\)	\(3986\)
default	\(\text {Expression too large to display}\)	\(11152\)

Input:

int(1/x^4/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^3,x,method=_RETURNVERB 
OSE)
 

Output:

((b*x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(63/8*f^2/(a*c 
*f^2-a*d*e*f-b*c*e*f+b*d*e^2)^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))*b^2*d^2-1/3/a/c/e^3*(b*d*x^4+a*d*x^2+b*c 
*x^2+a*c)^(1/2)/x^3+1/8*f^4*(11*a*c*f^2-14*a*d*e*f-14*b*c*e*f+17*b*d*e^2)/ 
(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)^2/e^4*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1 
/2)/(f*x^2+e)-17/8*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)*d*f^3*b^2/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2 
)^2/e^2*EllipticE(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-10*f^5/(a*c*f^2 
-a*d*e*f-b*c*e*f+b*d*e^2)^2/e^4/(-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))*a^2*c*d-10*f^5/(a*c*f^2-a*d*e*f-b*c*e*f+b* 
d*e^2)^2/e^4/(-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))*a*b*c^2-27/2*f^3/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)^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)*EllipticPi(x*(-b/a)^(1/2),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2))*a*b* 
d^2-27/2*f^3/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)^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)*EllipticPi 
(x*(-b/a)^(1/2),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2))*b^2*c*d-23/8/(-b/a...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate(1/x^4/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^3,x, algorithm=" 
fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate(1/x**4/(b*x**2+a)**(1/2)/(d*x**2+c)**(1/2)/(f*x**2+e)**3,x)
 

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \frac {1}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx=\int { \frac {1}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{3} x^{4}} \,d x } \] Input:

integrate(1/x^4/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^3,x, algorithm=" 
maxima")
 

Output:

integrate(1/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^3*x^4), x)
 

Giac [F]

\[ \int \frac {1}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx=\int { \frac {1}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{3} x^{4}} \,d x } \] Input:

integrate(1/x^4/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^3,x, algorithm=" 
giac")
 

Output:

integrate(1/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^3*x^4), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx=\int \frac {1}{x^4\,\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}\,{\left (f\,x^2+e\right )}^3} \,d x \] Input:

int(1/(x^4*(a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^3),x)
 

Output:

int(1/(x^4*(a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^3), x)
 

Reduce [F]

\[ \int \frac {1}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx=\int \frac {1}{x^{4} \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (f \,x^{2}+e \right )^{3}}d x \] Input:

int(1/x^4/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^3,x)
 

Output:

int(1/x^4/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^3,x)