Integrand size = 36, antiderivative size = 537 \[ \int \frac {1}{x^2 \sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=-\frac {\sqrt {a-b x^2} \sqrt {c+d x^2}}{a c e x \left (e+f x^2\right )}-\frac {f (a f (2 d e-3 c f)+2 b e (d e-c f)) x \sqrt {a-b x^2} \sqrt {c+d x^2}}{2 a c e^2 (b e+a f) (d e-c f) \left (e+f x^2\right )}-\frac {\sqrt {b} (a f (2 d e-3 c f)+2 b e (d e-c f)) \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{2 \sqrt {a} c e^2 (b e+a f) (d e-c f) \sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}}}+\frac {\sqrt {b} (2 b e+3 a f) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{2 \sqrt {a} e^2 (b e+a f) \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {\sqrt {a} f (b e (5 d e-4 c f)+a f (4 d e-3 c f)) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{2 \sqrt {b} e^3 (b e+a f) (d e-c f) \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Output:
-(-b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a/c/e/x/(f*x^2+e)-1/2*f*(a*f*(-3*c*f+2*d *e)+2*b*e*(-c*f+d*e))*x*(-b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a/c/e^2/(a*f+b*e) /(-c*f+d*e)/(f*x^2+e)-1/2*b^(1/2)*(a*f*(-3*c*f+2*d*e)+2*b*e*(-c*f+d*e))*(1 -b*x^2/a)^(1/2)*(d*x^2+c)^(1/2)*EllipticE(b^(1/2)*x/a^(1/2),(-a*d/b/c)^(1/ 2))/a^(1/2)/c/e^2/(a*f+b*e)/(-c*f+d*e)/(-b*x^2+a)^(1/2)/(1+d*x^2/c)^(1/2)+ 1/2*b^(1/2)*(3*a*f+2*b*e)*(1-b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)*EllipticF(b^ (1/2)*x/a^(1/2),(-a*d/b/c)^(1/2))/a^(1/2)/e^2/(a*f+b*e)/(-b*x^2+a)^(1/2)/( d*x^2+c)^(1/2)-1/2*a^(1/2)*f*(b*e*(-4*c*f+5*d*e)+a*f*(-3*c*f+4*d*e))*(1-b* x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)*EllipticPi(b^(1/2)*x/a^(1/2),-a*f/b/e,(-a*d /b/c)^(1/2))/b^(1/2)/e^3/(a*f+b*e)/(-c*f+d*e)/(-b*x^2+a)^(1/2)/(d*x^2+c)^( 1/2)
Result contains complex when optimal does not.
Time = 5.36 (sec) , antiderivative size = 447, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x^2 \sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\frac {-\sqrt {-\frac {b}{a}} e \left (a-b x^2\right ) \left (c+d x^2\right ) \left (-2 b e (d e-c f) \left (e+f x^2\right )+a f \left (-2 d e \left (e+f x^2\right )+c f \left (2 e+3 f x^2\right )\right )\right )+i b c e (2 b e (-d e+c f)+a f (-2 d e+3 c f)) x \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right ) E\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )-i b c e (2 b e+3 a f) (-d e+c f) x \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )+i a c f (a f (-4 d e+3 c f)+b e (-5 d e+4 c f)) x \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right ) \operatorname {EllipticPi}\left (-\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )}{2 a \sqrt {-\frac {b}{a}} c e^3 (b e+a f) (-d e+c f) x \sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \] Input:
Integrate[1/(x^2*Sqrt[a - b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)^2),x]
Output:
(-(Sqrt[-(b/a)]*e*(a - b*x^2)*(c + d*x^2)*(-2*b*e*(d*e - c*f)*(e + f*x^2) + a*f*(-2*d*e*(e + f*x^2) + c*f*(2*e + 3*f*x^2)))) + I*b*c*e*(2*b*e*(-(d*e ) + c*f) + a*f*(-2*d*e + 3*c*f))*x*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c] *(e + f*x^2)*EllipticE[I*ArcSinh[Sqrt[-(b/a)]*x], -((a*d)/(b*c))] - I*b*c* e*(2*b*e + 3*a*f)*(-(d*e) + c*f)*x*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c] *(e + f*x^2)*EllipticF[I*ArcSinh[Sqrt[-(b/a)]*x], -((a*d)/(b*c))] + I*a*c* f*(a*f*(-4*d*e + 3*c*f) + b*e*(-5*d*e + 4*c*f))*x*Sqrt[1 - (b*x^2)/a]*Sqrt [1 + (d*x^2)/c]*(e + f*x^2)*EllipticPi[-((a*f)/(b*e)), I*ArcSinh[Sqrt[-(b/ a)]*x], -((a*d)/(b*c))])/(2*a*Sqrt[-(b/a)]*c*e^3*(b*e + a*f)*(-(d*e) + c*f )*x*Sqrt[a - b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2))
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.
| \(\displaystyle \int \frac {1}{x^2 \sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 450 |
| \(\displaystyle \int \frac {1}{x^2 \sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2}dx\) |
Input:
Int[1/(x^2*Sqrt[a - b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)^2),x]
Output:
$Aborted
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]
Leaf count of result is larger than twice the leaf count of optimal. \(1238\) vs. \(2(476)=952\).
Time = 22.45 (sec) , antiderivative size = 1239, normalized size of antiderivative = 2.31
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1239\) |
| risch | \(\text {Expression too large to display}\) | \(1265\) |
| default | \(\text {Expression too large to display}\) | \(2131\) |
Input:
int(1/x^2/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^2,x,method=_RETURNVER BOSE)
Output:
((-b*x^2+a)*(d*x^2+c))^(1/2)/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(-1/2*f^3/(a *c*f^2-a*d*e*f+b*c*e*f-b*d*e^2)/e^2*x*(-b*d*x^4+a*d*x^2-b*c*x^2+a*c)^(1/2) /(f*x^2+e)-1/a/c/e^2*(-b*d*x^4+a*d*x^2-b*c*x^2+a*c)^(1/2)/x-1/2*b*d*f/(a*c *f^2-a*d*e*f+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)*EllipticF(x*(b/a)^(1/2),(-1-(a* d-b*c)/c/b)^(1/2))+1/2*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)*b*f^2/(a*c*f^2-a*d*e*f+b*c*e*f-b*d*e^2 )/e^2*EllipticF(x*(b/a)^(1/2),(-1-(a*d-b*c)/c/b)^(1/2))-1/2*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)*b *f^2/(a*c*f^2-a*d*e*f+b*c*e*f-b*d*e^2)/e^2*EllipticE(x*(b/a)^(1/2),(-1-(a* d-b*c)/c/b)^(1/2))+1/(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)*b/a/e^2*EllipticF(x*(b/a)^(1/2),(-1-(a*d-b *c)/c/b)^(1/2))-1/(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)*b/a/e^2*EllipticE(x*(b/a)^(1/2),(-1-(a*d-b*c) /c/b)^(1/2))-3/2/(a*c*f^2-a*d*e*f+b*c*e*f-b*d*e^2)/e^3*f^3/(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)*Elli pticPi(x*(b/a)^(1/2),-a*f/b/e,(-1/c*d)^(1/2)/(b/a)^(1/2))*a*c+2/(a*c*f^2-a *d*e*f+b*c*e*f-b*d*e^2)/e^2*f^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*d-2/(a*c*f^2-a*d*e*f+b*c*e*f-b*d*e^2)/...
Timed out. \[ \int \frac {1}{x^2 \sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate(1/x^2/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^2,x, algorithm= "fricas")
Output:
Timed out
\[ \int \frac {1}{x^2 \sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\int \frac {1}{x^{2} \sqrt {a - b x^{2}} \sqrt {c + d x^{2}} \left (e + f x^{2}\right )^{2}}\, dx \] Input:
integrate(1/x**2/(-b*x**2+a)**(1/2)/(d*x**2+c)**(1/2)/(f*x**2+e)**2,x)
Output:
Integral(1/(x**2*sqrt(a - b*x**2)*sqrt(c + d*x**2)*(e + f*x**2)**2), x)
\[ \int \frac {1}{x^2 \sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\int { \frac {1}{\sqrt {-b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{2} x^{2}} \,d x } \] Input:
integrate(1/x^2/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^2,x, algorithm= "maxima")
Output:
integrate(1/(sqrt(-b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^2*x^2), x)
\[ \int \frac {1}{x^2 \sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\int { \frac {1}{\sqrt {-b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{2} x^{2}} \,d x } \] Input:
integrate(1/x^2/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^2,x, algorithm= "giac")
Output:
integrate(1/(sqrt(-b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^2*x^2), x)
Timed out. \[ \int \frac {1}{x^2 \sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\int \frac {1}{x^2\,\sqrt {a-b\,x^2}\,\sqrt {d\,x^2+c}\,{\left (f\,x^2+e\right )}^2} \,d x \] Input:
int(1/(x^2*(a - b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^2),x)
Output:
int(1/(x^2*(a - b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^2), x)
\[ \int \frac {1}{x^2 \sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Input:
int(1/x^2/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^2,x)
Output:
( - sqrt(c + d*x**2)*sqrt(a - b*x**2) + int((sqrt(c + d*x**2)*sqrt(a - b*x **2)*x**4)/(a*c*e**2 + 2*a*c*e*f*x**2 + a*c*f**2*x**4 + a*d*e**2*x**2 + 2* a*d*e*f*x**4 + a*d*f**2*x**6 - b*c*e**2*x**2 - 2*b*c*e*f*x**4 - b*c*f**2*x **6 - b*d*e**2*x**4 - 2*b*d*e*f*x**6 - b*d*f**2*x**8),x)*b*d*e*f*x + int(( sqrt(c + d*x**2)*sqrt(a - b*x**2)*x**4)/(a*c*e**2 + 2*a*c*e*f*x**2 + a*c*f **2*x**4 + a*d*e**2*x**2 + 2*a*d*e*f*x**4 + a*d*f**2*x**6 - b*c*e**2*x**2 - 2*b*c*e*f*x**4 - b*c*f**2*x**6 - b*d*e**2*x**4 - 2*b*d*e*f*x**6 - b*d*f* *2*x**8),x)*b*d*f**2*x**3 - 2*int((sqrt(c + d*x**2)*sqrt(a - b*x**2)*x**2) /(a*c*e**2 + 2*a*c*e*f*x**2 + a*c*f**2*x**4 + a*d*e**2*x**2 + 2*a*d*e*f*x* *4 + a*d*f**2*x**6 - b*c*e**2*x**2 - 2*b*c*e*f*x**4 - b*c*f**2*x**6 - b*d* e**2*x**4 - 2*b*d*e*f*x**6 - b*d*f**2*x**8),x)*a*d*e*f*x - 2*int((sqrt(c + d*x**2)*sqrt(a - b*x**2)*x**2)/(a*c*e**2 + 2*a*c*e*f*x**2 + a*c*f**2*x**4 + a*d*e**2*x**2 + 2*a*d*e*f*x**4 + a*d*f**2*x**6 - b*c*e**2*x**2 - 2*b*c* e*f*x**4 - b*c*f**2*x**6 - b*d*e**2*x**4 - 2*b*d*e*f*x**6 - b*d*f**2*x**8) ,x)*a*d*f**2*x**3 + 2*int((sqrt(c + d*x**2)*sqrt(a - b*x**2)*x**2)/(a*c*e* *2 + 2*a*c*e*f*x**2 + a*c*f**2*x**4 + a*d*e**2*x**2 + 2*a*d*e*f*x**4 + a*d *f**2*x**6 - b*c*e**2*x**2 - 2*b*c*e*f*x**4 - b*c*f**2*x**6 - b*d*e**2*x** 4 - 2*b*d*e*f*x**6 - b*d*f**2*x**8),x)*b*c*e*f*x + 2*int((sqrt(c + d*x**2) *sqrt(a - b*x**2)*x**2)/(a*c*e**2 + 2*a*c*e*f*x**2 + a*c*f**2*x**4 + a*d*e **2*x**2 + 2*a*d*e*f*x**4 + a*d*f**2*x**6 - b*c*e**2*x**2 - 2*b*c*e*f*x...