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

Optimal result

Integrand size = 42, antiderivative size = 353 \[ \int \frac {x^2 \left (A+B x^2\right )}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\frac {B x \sqrt {c+d x^2}}{d f \sqrt {a+b x^2}}-\frac {\sqrt {a} B \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {b} d f \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {a^{3/2} (B e-A f) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c f (b e-a f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} (B e-A f) \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c f (b e-a f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

B*x*(d*x^2+c)^(1/2)/d/f/(b*x^2+a)^(1/2)-a^(1/2)*B*(d*x^2+c)^(1/2)*Elliptic 
E(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/2),(1-a*d/b/c)^(1/2))/b^(1/2)/d/f/(b*x^ 
2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)+a^(3/2)*(-A*f+B*e)*(d*x^2+c)^(1 
/2)*InverseJacobiAM(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/b^(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)-a^(3/2)*(-A* 
f+B*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/f/(-a*f+b*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 = 9.49 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.53 \[ \int \frac {x^2 \left (A+B x^2\right )}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=-\frac {i \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (B c f E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-(B d e+B c f-A d f) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+d (B e-A f) \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{\sqrt {\frac {b}{a}} d f^2 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] 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*ArcSinh[Sq 
rt[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*Sqrt[a + b*x^2]*Sqrt[c + d* 
x^2])
 

Rubi [A] (verified)

Time = 1.59 (sec) , antiderivative size = 331, normalized size of antiderivative = 0.94, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, 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

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

Input:

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

Output:

(B*x*Sqrt[a + b*x^2])/(b*f*Sqrt[c + d*x^2]) - (B*Sqrt[c]*Sqrt[a + b*x^2]*E 
llipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b*Sqrt[d]*f*Sqrt[ 
(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (Sqrt[c]*(B*e - A*f)*S 
qrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a 
*Sqrt[d]*f^2*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (Sqr 
t[-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 = 5.78 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.78

method	result	size
default	\(\frac {\left (A \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) d f -A \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right ) d f -B \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) c f -B \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) d e +B \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) c f +B \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right ) d e \right ) \sqrt {\frac {x^{2} d +c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{f^{2} d \sqrt {-\frac {b}{a}}\, \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right )}\)	\(274\)
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {\sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right ) A}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, f}-\frac {\sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right ) B e}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, f^{2}}-\frac {B c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{f \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}+\frac {B c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{f \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}-\frac {\sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right ) A}{f \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {e \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right ) B}{f^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(607\)

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=_RETU 
RNVERBOSE)
 

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*Ellipt 
icE(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, algor 
ithm="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, algor 
ithm="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, algor 
ithm="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. \[ \int \frac {x^2 \left (A+B x^2\right )}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int \frac {x^2\,\left (B\,x^2+A\right )}{\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}\,\left (f\,x^2+e\right )} \,d x \] 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]

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