\(\int \frac {1}{x^2 \sqrt {a-b x^2} \sqrt {c+d x^2} (e+f x^2)} \, dx\) [223]

Optimal result

Integrand size = 36, antiderivative size = 324 \[ \int \frac {1}{x^2 \sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=-\frac {\sqrt {a-b x^2} \sqrt {c+d x^2}}{a c e x}-\frac {\sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {a} c e \sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}}}+\frac {\sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {a} e \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {\sqrt {a} f \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} e^2 \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Output:

-(-b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a/c/e/x-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^(1/2)/c/e/(-b*x^ 
2+a)^(1/2)/(1+d*x^2/c)^(1/2)+b^(1/2)*(1-b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)*E 
llipticF(b^(1/2)*x/a^(1/2),(-a*d/b/c)^(1/2))/a^(1/2)/e/(-b*x^2+a)^(1/2)/(d 
*x^2+c)^(1/2)-a^(1/2)*f*(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^2/(-b*x^2+a)^(1/2)/(d* 
x^2+c)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.14 (sec) , antiderivative size = 280, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x^2 \sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\frac {-\sqrt {-\frac {b}{a}} e \left (a-b x^2\right ) \left (c+d x^2\right )+i b c e x \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )-i b c e x \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )+i a c f x \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )}{a \sqrt {-\frac {b}{a}} c e^2 x \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Input:

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

Output:

(-(Sqrt[-(b/a)]*e*(a - b*x^2)*(c + d*x^2)) + I*b*c*e*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*e*x*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sq 
rt[-(b/a)]*x], -((a*d)/(b*c))] + I*a*c*f*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*Sqrt[-(b/a)]*c*e^2*x*Sqrt[a - b*x^2]*Sqrt[c + d*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.

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

\(\Big \downarrow \) 450

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

Input:

Int[1/(x^2*Sqrt[a - b*x^2]*Sqrt[c + d*x^2]*(e + f*x^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 = 8.66 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.93

method	result	size
default	\(\frac {\left (\sqrt {\frac {b}{a}}\, b d e \,x^{4}+\sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) b c e x -\sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) b c e x -\sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {b}{a}}, -\frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {\frac {b}{a}}}\right ) a c f x -\sqrt {\frac {b}{a}}\, a d e \,x^{2}+\sqrt {\frac {b}{a}}\, b c e \,x^{2}-\sqrt {\frac {b}{a}}\, a c e \right ) \sqrt {x^{2} d +c}\, \sqrt {-b \,x^{2}+a}}{\sqrt {\frac {b}{a}}\, x \,e^{2} c a \left (-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c \right )}\)	\(300\)
risch	\(-\frac {\sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{a c e x}-\frac {\left (-\frac {b c \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}+\frac {a c f \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {b}{a}}, -\frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {\frac {b}{a}}}\right )}{e \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}\right ) \sqrt {\left (-b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{a c e \sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(310\)
elliptic	\(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {\sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}{a c e x}+\frac {b \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{a e \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}-\frac {b \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{a e \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}-\frac {f \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {b}{a}}, -\frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {\frac {b}{a}}}\right )}{e^{2} \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(373\)

Input:

int(1/x^2/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x,method=_RETURNVERBO 
SE)
 

Output:

((b/a)^(1/2)*b*d*e*x^4+((-b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF( 
x*(b/a)^(1/2),(-a*d/b/c)^(1/2))*b*c*e*x-((-b*x^2+a)/a)^(1/2)*((d*x^2+c)/c) 
^(1/2)*EllipticE(x*(b/a)^(1/2),(-a*d/b/c)^(1/2))*b*c*e*x-((-b*x^2+a)/a)^(1 
/2)*((d*x^2+c)/c)^(1/2)*EllipticPi(x*(b/a)^(1/2),-a*f/b/e,(-1/c*d)^(1/2)/( 
b/a)^(1/2))*a*c*f*x-(b/a)^(1/2)*a*d*e*x^2+(b/a)^(1/2)*b*c*e*x^2-(b/a)^(1/2 
)*a*c*e)*(d*x^2+c)^(1/2)*(-b*x^2+a)^(1/2)/(b/a)^(1/2)/x/e^2/c/a/(-b*d*x^4+ 
a*d*x^2-b*c*x^2+a*c)
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

Integral(1/(x**2*sqrt(a - b*x**2)*sqrt(c + d*x**2)*(e + f*x**2)), x)
 

Maxima [F]

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

integrate(1/x^2/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algorithm="m 
axima")
 

Output:

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

Giac [F]

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

integrate(1/x^2/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algorithm="g 
iac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{x^2 \sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}}{-b d f \,x^{8}+a d f \,x^{6}-b c f \,x^{6}-b d e \,x^{6}+a c f \,x^{4}+a d e \,x^{4}-b c e \,x^{4}+a c e \,x^{2}}d x \] Input:

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

Output:

int((sqrt(c + d*x**2)*sqrt(a - b*x**2))/(a*c*e*x**2 + a*c*f*x**4 + a*d*e*x 
**4 + a*d*f*x**6 - b*c*e*x**4 - b*c*f*x**6 - b*d*e*x**6 - b*d*f*x**8),x)