\(\int \frac {x^2 \sqrt {a+b x^2}}{(c-d x^2)^{3/2} (e+f x^2)} \, dx\) [162]

Optimal result

Integrand size = 36, antiderivative size = 357 \[ \int \frac {x^2 \sqrt {a+b x^2}}{\left (c-d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\frac {x \sqrt {a+b x^2}}{(d e+c f) \sqrt {c-d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} \sqrt {1-\frac {d x^2}{c}} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{\sqrt {d} (d e+c f) \sqrt {1+\frac {b x^2}{a}} \sqrt {c-d x^2}}-\frac {\sqrt {c} (b e-a f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{\sqrt {d} f (d e+c f) \sqrt {a+b x^2} \sqrt {c-d x^2}}+\frac {\sqrt {c} (b e-a f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {c f}{d e},\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{\sqrt {d} f (d e+c f) \sqrt {a+b x^2} \sqrt {c-d x^2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.57 (sec) , antiderivative size = 354, normalized size of antiderivative = 0.99 \[ \int \frac {x^2 \sqrt {a+b x^2}}{\left (c-d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\frac {a \sqrt {\frac {b}{a}} d f x+b \sqrt {\frac {b}{a}} d f x^3-i b c f \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )+i b (d e+c f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )-i b d e \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )+i a d f \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )}{\sqrt {\frac {b}{a}} d f (d e+c f) \sqrt {a+b x^2} \sqrt {c-d x^2}} \] Input:

Integrate[(x^2*Sqrt[a + b*x^2])/((c - d*x^2)^(3/2)*(e + f*x^2)),x]
 

Output:

(a*Sqrt[b/a]*d*f*x + b*Sqrt[b/a]*d*f*x^3 - I*b*c*f*Sqrt[1 + (b*x^2)/a]*Sqr 
t[1 - (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], -((a*d)/(b*c))] + I*b*( 
d*e + c*f)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*EllipticF[I*ArcSinh[Sqr 
t[b/a]*x], -((a*d)/(b*c))] - I*b*d*e*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*d 
*f*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))])/(Sqrt[b/a]*d*f*(d*e + c*f)*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.

\(\displaystyle \int \frac {x^2 \sqrt {a+b x^2}}{\left (c-d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx\)

\(\Big \downarrow \) 450

\(\displaystyle \int \frac {x^2 \sqrt {a+b x^2}}{\left (c-d x^2\right )^{3/2} \left (e+f x^2\right )}dx\)

Input:

Int[(x^2*Sqrt[a + b*x^2])/((c - d*x^2)^(3/2)*(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 = 6.38 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.07

method	result	size
default	\(\frac {\left (\sqrt {\frac {d}{c}}\, b f \,x^{3}+\sqrt {\frac {-x^{2} d +c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) a f -\sqrt {\frac {-x^{2} d +c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) b e -\sqrt {\frac {-x^{2} d +c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) a f -\sqrt {\frac {-x^{2} d +c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {d}{c}}, -\frac {c f}{d e}, \frac {\sqrt {-\frac {b}{a}}}{\sqrt {\frac {d}{c}}}\right ) a f +\sqrt {\frac {-x^{2} d +c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {d}{c}}, -\frac {c f}{d e}, \frac {\sqrt {-\frac {b}{a}}}{\sqrt {\frac {d}{c}}}\right ) b e +\sqrt {\frac {d}{c}}\, a f x \right ) \sqrt {-x^{2} d +c}\, \sqrt {b \,x^{2}+a}}{f \sqrt {\frac {d}{c}}\, \left (c f +d e \right ) \left (-b d \,x^{4}-a d \,x^{2}+x^{2} b c +a c \right )}\)	\(382\)
elliptic	\(\frac {\sqrt {\left (-x^{2} d +c \right ) \left (b \,x^{2}+a \right )}\, \left (-\frac {\left (-b d \,x^{2}-a d \right ) x}{d \left (c f +d e \right ) \sqrt {\left (x^{2}-\frac {c}{d}\right ) \left (-b d \,x^{2}-a d \right )}}-\frac {\sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right ) b}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+x^{2} b c +a c}\, d f}+\frac {\sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right ) b c}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+x^{2} b c +a c}\, \left (c f +d e \right ) d}+\frac {a \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )}{\left (c f +d e \right ) \sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+x^{2} b c +a c}}-\frac {a \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )}{\left (c f +d e \right ) \sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+x^{2} b c +a c}}-\frac {\sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {d}{c}}, -\frac {c f}{d e}, \frac {\sqrt {-\frac {b}{a}}}{\sqrt {\frac {d}{c}}}\right ) a}{\left (c f +d e \right ) \sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+x^{2} b c +a c}}+\frac {e \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {d}{c}}, -\frac {c f}{d e}, \frac {\sqrt {-\frac {b}{a}}}{\sqrt {\frac {d}{c}}}\right ) b}{\left (c f +d e \right ) f \sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+x^{2} b c +a c}}\right )}{\sqrt {-x^{2} d +c}\, \sqrt {b \,x^{2}+a}}\)	\(706\)

Input:

int(x^2*(b*x^2+a)^(1/2)/(-d*x^2+c)^(3/2)/(f*x^2+e),x,method=_RETURNVERBOSE 
)
 

Output:

((1/c*d)^(1/2)*b*f*x^3+((-d*x^2+c)/c)^(1/2)*((b*x^2+a)/a)^(1/2)*EllipticF( 
x*(1/c*d)^(1/2),(-b*c/a/d)^(1/2))*a*f-((-d*x^2+c)/c)^(1/2)*((b*x^2+a)/a)^( 
1/2)*EllipticF(x*(1/c*d)^(1/2),(-b*c/a/d)^(1/2))*b*e-((-d*x^2+c)/c)^(1/2)* 
((b*x^2+a)/a)^(1/2)*EllipticE(x*(1/c*d)^(1/2),(-b*c/a/d)^(1/2))*a*f-((-d*x 
^2+c)/c)^(1/2)*((b*x^2+a)/a)^(1/2)*EllipticPi(x*(1/c*d)^(1/2),-c*f/d/e,(-b 
/a)^(1/2)/(1/c*d)^(1/2))*a*f+((-d*x^2+c)/c)^(1/2)*((b*x^2+a)/a)^(1/2)*Elli 
pticPi(x*(1/c*d)^(1/2),-c*f/d/e,(-b/a)^(1/2)/(1/c*d)^(1/2))*b*e+(1/c*d)^(1 
/2)*a*f*x)*(-d*x^2+c)^(1/2)*(b*x^2+a)^(1/2)/f/(1/c*d)^(1/2)/(c*f+d*e)/(-b* 
d*x^4-a*d*x^2+b*c*x^2+a*c)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {x^2 \sqrt {a+b x^2}}{\left (c-d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:

integrate(x^2*(b*x^2+a)^(1/2)/(-d*x^2+c)^(3/2)/(f*x^2+e),x, algorithm="fri 
cas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {x^2 \sqrt {a+b x^2}}{\left (c-d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\int \frac {x^{2} \sqrt {a + b x^{2}}}{\left (c - d x^{2}\right )^{\frac {3}{2}} \left (e + f x^{2}\right )}\, dx \] Input:

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

Output:

Integral(x**2*sqrt(a + b*x**2)/((c - d*x**2)**(3/2)*(e + f*x**2)), x)
 

Maxima [F]

\[ \int \frac {x^2 \sqrt {a+b x^2}}{\left (c-d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\int { \frac {\sqrt {b x^{2} + a} x^{2}}{{\left (-d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}} \,d x } \] Input:

integrate(x^2*(b*x^2+a)^(1/2)/(-d*x^2+c)^(3/2)/(f*x^2+e),x, algorithm="max 
ima")
 

Output:

integrate(sqrt(b*x^2 + a)*x^2/((-d*x^2 + c)^(3/2)*(f*x^2 + e)), x)
 

Giac [F]

\[ \int \frac {x^2 \sqrt {a+b x^2}}{\left (c-d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\int { \frac {\sqrt {b x^{2} + a} x^{2}}{{\left (-d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}} \,d x } \] Input:

integrate(x^2*(b*x^2+a)^(1/2)/(-d*x^2+c)^(3/2)/(f*x^2+e),x, algorithm="gia 
c")
 

Output:

integrate(sqrt(b*x^2 + a)*x^2/((-d*x^2 + c)^(3/2)*(f*x^2 + e)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \sqrt {a+b x^2}}{\left (c-d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\int \frac {x^2\,\sqrt {b\,x^2+a}}{{\left (c-d\,x^2\right )}^{3/2}\,\left (f\,x^2+e\right )} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {x^2 \sqrt {a+b x^2}}{\left (c-d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\int \frac {\sqrt {-d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{d^{2} f \,x^{6}-2 c d f \,x^{4}+d^{2} e \,x^{4}+c^{2} f \,x^{2}-2 c d e \,x^{2}+c^{2} e}d x \] Input:

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

Output:

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