Integrand size = 36, antiderivative size = 407 \[ \int \frac {x^2}{\sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\frac {f x \sqrt {a-b x^2} \sqrt {c+d x^2}}{2 (b e+a f) (d e-c f) \left (e+f x^2\right )}+\frac {\sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{2 (b e+a f) (d e-c f) \sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}}}+\frac {\sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{2 f (b e+a f) \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {\sqrt {a} \left (b d e^2+a c f^2\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{2 \sqrt {b} e f (b e+a f) (d e-c f) \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Output:
1/2*f*x*(-b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/(a*f+b*e)/(-c*f+d*e)/(f*x^2+e)+1/ 2*a^(1/2)*b^(1/2)*(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*f+b*e)/(-c*f+d*e)/(-b*x^2+a)^(1/2)/(1+d*x^2/c)^ (1/2)+1/2*a^(1/2)*b^(1/2)*(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))/f/(a*f+b*e)/(-b*x^2+a)^(1/2)/(d*x^2+c)^( 1/2)-1/2*a^(1/2)*(a*c*f^2+b*d*e^2)*(1-b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)*Ell ipticPi(b^(1/2)*x/a^(1/2),-a*f/b/e,(-a*d/b/c)^(1/2))/b^(1/2)/e/f/(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 = 3.93 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.08 \[ \int \frac {x^2}{\sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\frac {\frac {a c f x}{e+f x^2}-\frac {b c f x^3}{e+f x^2}+\frac {a d f x^3}{e+f x^2}-\frac {b d f x^5}{e+f x^2}+i a \sqrt {-\frac {b}{a}} c \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )-\frac {i a \sqrt {-\frac {b}{a}} (-d e+c f) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )}{f}-\frac {i a \sqrt {-\frac {b}{a}} d e \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )}{f}+\frac {i a c f \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )}{\sqrt {-\frac {b}{a}} e}}{2 (b e+a f) (d e-c f) \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[x^2/(Sqrt[a - b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)^2),x]
Output:
((a*c*f*x)/(e + f*x^2) - (b*c*f*x^3)/(e + f*x^2) + (a*d*f*x^3)/(e + f*x^2) - (b*d*f*x^5)/(e + f*x^2) + I*a*Sqrt[-(b/a)]*c*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[-(b/a)]*x], -((a*d)/(b*c))] - (I*a* Sqrt[-(b/a)]*(-(d*e) + c*f)*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*Ellipt icF[I*ArcSinh[Sqrt[-(b/a)]*x], -((a*d)/(b*c))])/f - (I*a*Sqrt[-(b/a)]*d*e* Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticPi[-((a*f)/(b*e)), I*ArcSi nh[Sqrt[-(b/a)]*x], -((a*d)/(b*c))])/f + (I*a*c*f*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))])/(Sqrt[-(b/a)]*e))/(2*(b*e + a*f)*(d*e - c*f)*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.
| \(\displaystyle \int \frac {x^2}{\sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 450 |
| \(\displaystyle \int \frac {x^2}{\sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2}dx\) |
Input:
Int[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. \(724\) vs. \(2(350)=700\).
Time = 8.52 (sec) , antiderivative size = 725, normalized size of antiderivative = 1.78
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {f x \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}{2 \left (a c \,f^{2}-a d e f +b c e f -b d \,e^{2}\right ) \left (f \,x^{2}+e \right )}-\frac {b d e \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{2 \left (a c \,f^{2}-a d e f +b c e f -b d \,e^{2}\right ) f \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}+\frac {b c \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{2 \left (a c \,f^{2}-a d e f +b c e f -b d \,e^{2}\right ) \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}-\frac {b c \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{2 \left (a c \,f^{2}-a d e f +b c e f -b d \,e^{2}\right ) \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}+\frac {f \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {b}{a}}, -\frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {\frac {b}{a}}}\right ) a c}{2 \left (a c \,f^{2}-a d e f +b c e f -b d \,e^{2}\right ) e \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}+\frac {e \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {b}{a}}, -\frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {\frac {b}{a}}}\right ) b d}{2 \left (a c \,f^{2}-a d e f +b c e f -b d \,e^{2}\right ) f \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(725\) |
| default | \(\frac {\left (\sqrt {\frac {b}{a}}\, b d e \,f^{2} x^{5}+\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 c e \,f^{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 d \,e^{2} f \,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 c e \,f^{2} x^{2}+\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 c \,f^{3} x^{2}+\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 ) b d \,e^{2} f \,x^{2}-\sqrt {\frac {b}{a}}\, a d e \,f^{2} x^{3}+\sqrt {\frac {b}{a}}\, b c e \,f^{2} x^{3}+\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 c \,e^{2} f -\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 d \,e^{3}-\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 c \,e^{2} f +\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 c e \,f^{2}+\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 ) b d \,e^{3}-\sqrt {\frac {b}{a}}\, a c e \,f^{2} x \right ) \sqrt {x^{2} d +c}\, \sqrt {-b \,x^{2}+a}}{2 f e \left (f \,x^{2}+e \right ) \sqrt {\frac {b}{a}}\, \left (c f -d e \right ) \left (a f +b e \right ) \left (-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c \right )}\) | \(790\) |
Input:
int(x^2/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^2,x,method=_RETURNVERBO SE)
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/(a*c *f^2-a*d*e*f+b*c*e*f-b*d*e^2)*x*(-b*d*x^4+a*d*x^2-b*c*x^2+a*c)^(1/2)/(f*x^ 2+e)-1/2*b*d*e/(a*c*f^2-a*d*e*f+b*c*e*f-b*d*e^2)/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))+1/2*b/(a*c*f^2-a*d*e*f+b*c*e*f-b*d*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))-1/2*b/( a*c*f^2-a*d*e*f+b*c*e*f-b*d*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)*EllipticE(x*(b/a)^(1/2),(-1- (a*d-b*c)/c/b)^(1/2))+1/2/(a*c*f^2-a*d*e*f+b*c*e*f-b*d*e^2)*f/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))*a*c+1/2/(a* c*f^2-a*d*e*f+b*c*e*f-b*d*e^2)*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)*EllipticPi(x*(b/a)^(1/2),-a* f/b/e,(-1/c*d)^(1/2)/(b/a)^(1/2))*b*d)
Timed out. \[ \int \frac {x^2}{\sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate(x^2/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^2,x, algorithm="f ricas")
Output:
Timed out
\[ \int \frac {x^2}{\sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\int \frac {x^{2}}{\sqrt {a - b x^{2}} \sqrt {c + d x^{2}} \left (e + f x^{2}\right )^{2}}\, dx \] Input:
integrate(x**2/(-b*x**2+a)**(1/2)/(d*x**2+c)**(1/2)/(f*x**2+e)**2,x)
Output:
Integral(x**2/(sqrt(a - b*x**2)*sqrt(c + d*x**2)*(e + f*x**2)**2), x)
\[ \int \frac {x^2}{\sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\int { \frac {x^{2}}{\sqrt {-b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{2}} \,d x } \] Input:
integrate(x^2/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^2,x, algorithm="m axima")
Output:
integrate(x^2/(sqrt(-b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^2), x)
\[ \int \frac {x^2}{\sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\int { \frac {x^{2}}{\sqrt {-b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{2}} \,d x } \] Input:
integrate(x^2/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^2,x, algorithm="g iac")
Output:
integrate(x^2/(sqrt(-b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^2), x)
Timed out. \[ \int \frac {x^2}{\sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\int \frac {x^2}{\sqrt {a-b\,x^2}\,\sqrt {d\,x^2+c}\,{\left (f\,x^2+e\right )}^2} \,d x \] Input:
int(x^2/((a - b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^2),x)
Output:
int(x^2/((a - b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^2), x)
\[ \int \frac {x^2}{\sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,f^{2} x^{8}+a d \,f^{2} x^{6}-b c \,f^{2} x^{6}-2 b d e f \,x^{6}+a c \,f^{2} x^{4}+2 a d e f \,x^{4}-2 b c e f \,x^{4}-b d \,e^{2} x^{4}+2 a c e f \,x^{2}+a d \,e^{2} x^{2}-b c \,e^{2} x^{2}+a c \,e^{2}}d x \] Input:
int(x^2/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^2,x)
Output:
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)