Integrand size = 35, antiderivative size = 430 \[ \int \frac {1}{x^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=-\frac {\sqrt {c+d x^2}}{a c e x \sqrt {a+b x^2}}-\frac {\sqrt {b} \left (2 b^2 c e+a^2 d f-a b (d e+c f)\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{a^{3/2} c (b c-a d) e (b e-a f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {\sqrt {b} \left (b^2 d e^2-a^2 d f^2-a b f (d e-c f)\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {a} c (b c-a d) e (b e-a f)^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} f^3 \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e^2 (b e-a f)^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
-(d*x^2+c)^(1/2)/a/c/e/x/(b*x^2+a)^(1/2)-b^(1/2)*(2*b^2*c*e+a^2*d*f-a*b*(c *f+d*e))*(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/(-a*d+b*c)/e/(-a*f+b*e)/(b*x^2+a)^(1/2)/(a*(d*x^ 2+c)/c/(b*x^2+a))^(1/2)+b^(1/2)*(b^2*d*e^2-a^2*d*f^2-a*b*f*(-c*f+d*e))*(d* x^2+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/ a^(1/2)/c/(-a*d+b*c)/e/(-a*f+b*e)^2/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+ a))^(1/2)-a^(3/2)*f^3*(d*x^2+c)^(1/2)*EllipticPi(b^(1/2)*x/a^(1/2)/(1+b*x^ 2/a)^(1/2),1-a*f/b/e,(1-a*d/b/c)^(1/2))/b^(1/2)/c/e^2/(-a*f+b*e)^2/(b*x^2+ a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)
Result contains complex when optimal does not.
Time = 5.04 (sec) , antiderivative size = 636, normalized size of antiderivative = 1.48 \[ \int \frac {1}{x^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\frac {-a b^2 \sqrt {\frac {b}{a}} c^2 e^2+a^3 \left (\frac {b}{a}\right )^{3/2} c d e^2+a^3 \left (\frac {b}{a}\right )^{3/2} c^2 e f-a^3 \sqrt {\frac {b}{a}} c d e f-2 b^3 \sqrt {\frac {b}{a}} c^2 e^2 x^2+a^3 \left (\frac {b}{a}\right )^{3/2} d^2 e^2 x^2+a b^2 \sqrt {\frac {b}{a}} c^2 e f x^2-a^3 \sqrt {\frac {b}{a}} d^2 e f x^2-2 b^3 \sqrt {\frac {b}{a}} c d e^2 x^4+a b^2 \sqrt {\frac {b}{a}} d^2 e^2 x^4+a b^2 \sqrt {\frac {b}{a}} c d e f x^4-a^3 \left (\frac {b}{a}\right )^{3/2} d^2 e f x^4-i b c e \left (2 b^2 c e+a^2 d f-a b (d e+c f)\right ) x \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i b c (-b c+a d) e (-2 b e+a f) x \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )-i a^2 b c^2 f^2 x \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+i a^3 c d f^2 x \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{a^2 \sqrt {\frac {b}{a}} c (b c-a d) e^2 (b e-a f) x \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[1/(x^2*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2]*(e + f*x^2)),x]
Output:
(-(a*b^2*Sqrt[b/a]*c^2*e^2) + a^3*(b/a)^(3/2)*c*d*e^2 + a^3*(b/a)^(3/2)*c^ 2*e*f - a^3*Sqrt[b/a]*c*d*e*f - 2*b^3*Sqrt[b/a]*c^2*e^2*x^2 + a^3*(b/a)^(3 /2)*d^2*e^2*x^2 + a*b^2*Sqrt[b/a]*c^2*e*f*x^2 - a^3*Sqrt[b/a]*d^2*e*f*x^2 - 2*b^3*Sqrt[b/a]*c*d*e^2*x^4 + a*b^2*Sqrt[b/a]*d^2*e^2*x^4 + a*b^2*Sqrt[b /a]*c*d*e*f*x^4 - a^3*(b/a)^(3/2)*d^2*e*f*x^4 - I*b*c*e*(2*b^2*c*e + a^2*d *f - a*b*(d*e + c*f))*x*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[ I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + I*b*c*(-(b*c) + a*d)*e*(-2*b*e + a* f)*x*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a] *x], (a*d)/(b*c)] - I*a^2*b*c^2*f^2*x*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2) /c]*EllipticPi[(a*f)/(b*e), I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + I*a^3*c *d*f^2*x*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticPi[(a*f)/(b*e), I *ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(a^2*Sqrt[b/a]*c*(b*c - a*d)*e^2*(b*e - a*f)*x*Sqrt[a + b*x^2]*Sqrt[c + d*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.
| \(\displaystyle \int \frac {1}{x^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx\) |
| \(\Big \downarrow \) 450 |
| \(\displaystyle \int \frac {1}{x^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )}dx\) |
Input:
Int[1/(x^2*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2]*(e + f*x^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]
Time = 22.86 (sec) , antiderivative size = 622, normalized size of antiderivative = 1.45
| method | result | size |
| risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{a^{2} c e x}+\frac {\left (-\frac {b c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {a c e \,b^{2} \left (-\frac {\left (b d \,x^{2}+b c \right ) x}{a \left (a d -b c \right ) \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}+\frac {\left (\frac {1}{a}+\frac {b c}{\left (a d -b c \right ) a}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {b c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\left (a d -b c \right ) a \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )}{a f -b e}-\frac {a^{2} c \,f^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right )}{\left (a f -b e \right ) e \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{a^{2} c e \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(622\) |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {\sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{a^{2} c e x}-\frac {\left (b d \,x^{2}+b c \right ) b^{2} x}{a^{2} \left (a d -b c \right ) \left (a f -b e \right ) \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}+\frac {\sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right ) b^{2}}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, a^{2} \left (a f -b e \right )}-\frac {\sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, b \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, a^{2} e}+\frac {\sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, b \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, a^{2} e}+\frac {c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, b^{3} \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, \left (a d -b c \right ) a^{2} \left (a f -b e \right )}-\frac {f^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right )}{\left (a f -b e \right ) e^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(650\) |
| default | \(\frac {\left (-\sqrt {-\frac {b}{a}}\, a^{2} b \,d^{2} e f \,x^{4}+\sqrt {-\frac {b}{a}}\, a \,b^{2} c d e f \,x^{4}+\sqrt {-\frac {b}{a}}\, a \,b^{2} d^{2} e^{2} x^{4}-2 \sqrt {-\frac {b}{a}}\, b^{3} c d \,e^{2} x^{4}-\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} b c d e f x +\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a \,b^{2} c^{2} e f x +2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a \,b^{2} c d \,e^{2} x -2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{3} c^{2} e^{2} x +\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} b c d e f x -\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a \,b^{2} c^{2} e f x -\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a \,b^{2} c d \,e^{2} x +2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{3} c^{2} e^{2} x -\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right ) a^{3} c d \,f^{2} x +\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right ) a^{2} b \,c^{2} f^{2} x -\sqrt {-\frac {b}{a}}\, a^{3} d^{2} e f \,x^{2}+\sqrt {-\frac {b}{a}}\, a^{2} b \,d^{2} e^{2} x^{2}+\sqrt {-\frac {b}{a}}\, a \,b^{2} c^{2} e f \,x^{2}-2 \sqrt {-\frac {b}{a}}\, b^{3} c^{2} e^{2} x^{2}-\sqrt {-\frac {b}{a}}\, a^{3} c d e f +\sqrt {-\frac {b}{a}}\, a^{2} b \,c^{2} e f +\sqrt {-\frac {b}{a}}\, a^{2} b c d \,e^{2}-\sqrt {-\frac {b}{a}}\, a \,b^{2} c^{2} e^{2}\right ) \sqrt {x^{2} d +c}\, \sqrt {b \,x^{2}+a}}{\left (a f -b e \right ) \left (a d -b c \right ) \sqrt {-\frac {b}{a}}\, x \,e^{2} c \,a^{2} \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right )}\) | \(954\) |
Input:
int(1/x^2/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x,method=_RETURNVERBOS E)
Output:
-1/a^2/c/e*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/x+1/a^2/c/e*(-b*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)*(E llipticF(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*c*e*b^2/(a*f-b*e)*(-(b*d*x^2+b*c)/a/(a*d-b*c )*x/((x^2+a/b)*(b*d*x^2+b*c))^(1/2)+(1/a+b*c/(a*d-b*c)/a)/(-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)*Ellip ticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-b/(a*d-b*c)/a*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^2*c*f^2/(a*f-b*e)/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*x^2+a)*(d*x^2+c)) ^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)
Timed out. \[ \int \frac {1}{x^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:
integrate(1/x^2/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algorithm="fr icas")
Output:
Timed out
\[ \int \frac {1}{x^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int \frac {1}{x^{2} \left (a + b x^{2}\right )^{\frac {3}{2}} \sqrt {c + d x^{2}} \left (e + f x^{2}\right )}\, dx \] Input:
integrate(1/x**2/(b*x**2+a)**(3/2)/(d*x**2+c)**(1/2)/(f*x**2+e),x)
Output:
Integral(1/(x**2*(a + b*x**2)**(3/2)*sqrt(c + d*x**2)*(e + f*x**2)), x)
\[ \int \frac {1}{x^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )} x^{2}} \,d x } \] Input:
integrate(1/x^2/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algorithm="ma xima")
Output:
integrate(1/((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*(f*x^2 + e)*x^2), x)
\[ \int \frac {1}{x^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )} x^{2}} \,d x } \] Input:
integrate(1/x^2/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algorithm="gi ac")
Output:
integrate(1/((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*(f*x^2 + e)*x^2), x)
Timed out. \[ \int \frac {1}{x^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int \frac {1}{x^2\,{\left (b\,x^2+a\right )}^{3/2}\,\sqrt {d\,x^2+c}\,\left (f\,x^2+e\right )} \,d x \] Input:
int(1/(x^2*(a + b*x^2)^(3/2)*(c + d*x^2)^(1/2)*(e + f*x^2)),x)
Output:
int(1/(x^2*(a + b*x^2)^(3/2)*(c + d*x^2)^(1/2)*(e + f*x^2)), x)
\[ \int \frac {1}{x^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx =\text {Too large to display} \] Input:
int(1/x^2/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e),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**2*c*e + a**2*c*f*x**2 + a**2*d*e*x**2 + a**2*d*f*x**4 + 2*a *b*c*e*x**2 + 2*a*b*c*f*x**4 + 2*a*b*d*e*x**4 + 2*a*b*d*f*x**6 + b**2*c*e* x**4 + b**2*c*f*x**6 + b**2*d*e*x**6 + b**2*d*f*x**8),x)*a*b*d*f*x - int(( sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**2*c*e + a**2*c*f*x**2 + a**2*d *e*x**2 + a**2*d*f*x**4 + 2*a*b*c*e*x**2 + 2*a*b*c*f*x**4 + 2*a*b*d*e*x**4 + 2*a*b*d*f*x**6 + b**2*c*e*x**4 + b**2*c*f*x**6 + b**2*d*e*x**6 + b**2*d *f*x**8),x)*b**2*d*f*x**3 - 2*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2) /(a**2*c*e + a**2*c*f*x**2 + a**2*d*e*x**2 + a**2*d*f*x**4 + 2*a*b*c*e*x** 2 + 2*a*b*c*f*x**4 + 2*a*b*d*e*x**4 + 2*a*b*d*f*x**6 + b**2*c*e*x**4 + b** 2*c*f*x**6 + b**2*d*e*x**6 + b**2*d*f*x**8),x)*a*b*c*f*x - int((sqrt(c + d *x**2)*sqrt(a + b*x**2)*x**2)/(a**2*c*e + a**2*c*f*x**2 + a**2*d*e*x**2 + a**2*d*f*x**4 + 2*a*b*c*e*x**2 + 2*a*b*c*f*x**4 + 2*a*b*d*e*x**4 + 2*a*b*d *f*x**6 + b**2*c*e*x**4 + b**2*c*f*x**6 + b**2*d*e*x**6 + b**2*d*f*x**8),x )*a*b*d*e*x - 2*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a**2*c*e + a **2*c*f*x**2 + a**2*d*e*x**2 + a**2*d*f*x**4 + 2*a*b*c*e*x**2 + 2*a*b*c*f* x**4 + 2*a*b*d*e*x**4 + 2*a*b*d*f*x**6 + b**2*c*e*x**4 + b**2*c*f*x**6 + b **2*d*e*x**6 + b**2*d*f*x**8),x)*b**2*c*f*x**3 - int((sqrt(c + d*x**2)*sqr t(a + b*x**2)*x**2)/(a**2*c*e + a**2*c*f*x**2 + a**2*d*e*x**2 + a**2*d*f*x **4 + 2*a*b*c*e*x**2 + 2*a*b*c*f*x**4 + 2*a*b*d*e*x**4 + 2*a*b*d*f*x**6...