\(\int \frac {x^2 (A+B x^2)}{\sqrt {a-b x^2} \sqrt {c+d x^2} (e+f x^2)} \, dx\) [10]

Optimal result

Integrand size = 43, antiderivative size = 311 \[ \int \frac {x^2 \left (A+B x^2\right )}{\sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\frac {\sqrt {a} B \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d f \sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}}}-\frac {\sqrt {a} (B d e+B c f-A d f) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d f^2 \sqrt {a-b x^2} \sqrt {c+d x^2}}+\frac {\sqrt {a} (B e-A f) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} f^2 \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 9.53 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.63 \[ \int \frac {x^2 \left (A+B x^2\right )}{\sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=-\frac {i \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (B c f E\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )-(B d e+B c f-A d f) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )+d (B e-A f) \operatorname {EllipticPi}\left (-\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )\right )}{\sqrt {-\frac {b}{a}} d f^2 \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Input:

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

Output:

((-I)*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*(B*c*f*EllipticE[I*ArcSinh[S 
qrt[-(b/a)]*x], -((a*d)/(b*c))] - (B*d*e + B*c*f - A*d*f)*EllipticF[I*ArcS 
inh[Sqrt[-(b/a)]*x], -((a*d)/(b*c))] + d*(B*e - A*f)*EllipticPi[-((a*f)/(b 
*e)), I*ArcSinh[Sqrt[-(b/a)]*x], -((a*d)/(b*c))]))/(Sqrt[-(b/a)]*d*f^2*Sqr 
t[a - b*x^2]*Sqrt[c + d*x^2])
 

Rubi [A] (verified)

Time = 2.22 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.28, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, Rules used = {7276, 2009}

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.

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (B e-A f) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} f^2 \sqrt {a-b x^2} \sqrt {c+d x^2}}+\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (B e-A f) \operatorname {EllipticPi}\left (-\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} f^2 \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {\sqrt {a} B c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d f \sqrt {a-b x^2} \sqrt {c+d x^2}}+\frac {\sqrt {a} B \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d f \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}\)

Input:

Int[(x^2*(A + B*x^2))/(Sqrt[a - b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)),x]
 

Output:

(Sqrt[a]*B*Sqrt[1 - (b*x^2)/a]*Sqrt[c + d*x^2]*EllipticE[ArcSin[(Sqrt[b]*x 
)/Sqrt[a]], -((a*d)/(b*c))])/(Sqrt[b]*d*f*Sqrt[a - b*x^2]*Sqrt[1 + (d*x^2) 
/c]) - (Sqrt[a]*B*c*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[ArcS 
in[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(Sqrt[b]*d*f*Sqrt[a - b*x^2]*Sqr 
t[c + d*x^2]) - (Sqrt[a]*(B*e - A*f)*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/ 
c]*EllipticF[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(Sqrt[b]*f^2*Sq 
rt[a - b*x^2]*Sqrt[c + d*x^2]) + (Sqrt[a]*(B*e - A*f)*Sqrt[1 - (b*x^2)/a]* 
Sqrt[1 + (d*x^2)/c]*EllipticPi[-((a*f)/(b*e)), ArcSin[(Sqrt[b]*x)/Sqrt[a]] 
, -((a*d)/(b*c))])/(Sqrt[b]*f^2*Sqrt[a - b*x^2]*Sqrt[c + d*x^2])
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

Maple [A] (verified)

Time = 5.91 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.88

method	result	size
default	\(\frac {\left (A \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) d f -A \operatorname {EllipticPi}\left (x \sqrt {\frac {b}{a}}, -\frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {\frac {b}{a}}}\right ) d f -B \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) c f -B \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) d e +B \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) c f +B \operatorname {EllipticPi}\left (x \sqrt {\frac {b}{a}}, -\frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {\frac {b}{a}}}\right ) d e \right ) \sqrt {\frac {x^{2} d +c}{c}}\, \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{f^{2} d \sqrt {\frac {b}{a}}\, \left (-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c \right )}\)	\(275\)
elliptic	\(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {\sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right ) A}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}\, f}-\frac {\sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right ) B e}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}\, f^{2}}-\frac {B c \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{f \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}\, d}+\frac {B c \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{f \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}\, d}-\frac {\sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {b}{a}}, -\frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {\frac {b}{a}}}\right ) A}{f \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}+\frac {e \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {b}{a}}, -\frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {\frac {b}{a}}}\right ) B}{f^{2} \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(623\)

Input:

int(x^2*(B*x^2+A)/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x,method=_RET 
URNVERBOSE)
 

Output:

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

Fricas [F(-1)]

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

integrate(x^2*(B*x^2+A)/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algo 
rithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(x^2*(B*x^2+A)/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algo 
rithm="maxima")
 

Output:

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

Giac [F]

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

integrate(x^2*(B*x^2+A)/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algo 
rithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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