\(\int \frac {x^2 \sqrt {a+b x^2}}{\sqrt {c+d x^2} (e+f x^2)^2} \, dx\) [132]

Optimal result

Integrand size = 35, antiderivative size = 496 \[ \int \frac {x^2 \sqrt {a+b x^2}}{\sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=-\frac {b x \sqrt {c+d x^2}}{2 f (d e-c f) \sqrt {a+b x^2}}+\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 (b e-a f) (d e-c f)}-\frac {f x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{2 (b e-a f) (d e-c f) \left (e+f x^2\right )}+\frac {\sqrt {a} \sqrt {b} \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{2 f (d e-c f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} \sqrt {b} \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{2 c f (b e-a f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {a^{3/2} \left (a c f^2+b e (d e-2 c f)\right ) \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{2 \sqrt {b} c e f (b e-a f) (d e-c f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

-1/2*b*x*(d*x^2+c)^(1/2)/f/(-c*f+d*e)/(b*x^2+a)^(1/2)+1/2*b*x*(b*x^2+a)^(1 
/2)*(d*x^2+c)^(1/2)/(-a*f+b*e)/(-c*f+d*e)-1/2*f*x*(b*x^2+a)^(3/2)*(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)*(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))/f/(-c*f 
+d*e)/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)-1/2*a^(3/2)*b^(1/2)* 
(d*x^2+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2 
))/c/f/(-a*f+b*e)/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)+1/2*a^(3 
/2)*(a*c*f^2+b*e*(-2*c*f+d*e))*(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/f/(-a*f+b*e)/ 
(-c*f+d*e)/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.72 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.51 \[ \int \frac {x^2 \sqrt {a+b x^2}}{\sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\frac {\sqrt {\frac {b}{a}} e f^2 x \left (a+b x^2\right ) \left (c+d x^2\right )+i \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right ) \left (b c e f E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-b e (d e-c f) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+\left (a c f^2+b e (d e-2 c f)\right ) \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{2 \sqrt {\frac {b}{a}} e f^2 (d e-c f) \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \] Input:

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

Output:

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

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

\(\Big \downarrow \) 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
 

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 = 6.75 (sec) , antiderivative size = 812, normalized size of antiderivative = 1.64

method	result	size
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{2 \left (c f -d e \right ) \left (f \,x^{2}+e \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}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, f^{2}}+\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 d e}{2 \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, f^{2} \left (c f -d e \right )}-\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 (c f -d e \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 {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{2 \left (c f -d e \right ) f \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {\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 (c f -d e \right ) e \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\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 c}{\left (c f -d e \right ) f \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 f^{2} \left (c f -d e \right ) \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}}\)	\(812\)
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}-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 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 ) 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}-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 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 ) 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 e \left (f \,x^{2}+e \right ) \sqrt {-\frac {b}{a}}\, f^{2} \left (c f -d e \right ) \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right )}\)	\(922\)

Input:

int(x^2*(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^2,x,method=_RETURNVERBOS 
E)
 

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

Fricas [F(-1)]

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="fr 
icas")
 

Output:

Timed out
 

Sympy [F]

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

Reduce [F]

\[ \int \frac {x^2 \sqrt {a+b x^2}}{\sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{d \,f^{2} x^{6}+c \,f^{2} x^{4}+2 d e f \,x^{4}+2 c e f \,x^{2}+d \,e^{2} x^{2}+c \,e^{2}}d x \] 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)/(c*e**2 + 2*c*e*f*x**2 + c*f* 
*2*x**4 + d*e**2*x**2 + 2*d*e*f*x**4 + d*f**2*x**6),x)