\(\int \frac {A+B x^2}{\sqrt {a-b x^2} \sqrt {c+d x^2} (e+f x^2)} \, dx\) [11]

Optimal result

Integrand size = 40, antiderivative size = 204 \[ \int \frac {A+B x^2}{\sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\frac {\sqrt {a} B \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} f \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {\sqrt {a} (B e-A f) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} e f \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 8.31 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.74 \[ \int \frac {A+B x^2}{\sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=-\frac {i \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (B e \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )+(-B e+A f) \operatorname {EllipticPi}\left (-\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )\right )}{\sqrt {-\frac {b}{a}} e f \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Input:

Integrate[(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*e*EllipticF[I*ArcSinh[Sqr 
t[-(b/a)]*x], -((a*d)/(b*c))] + (-(B*e) + A*f)*EllipticPi[-((a*f)/(b*e)), 
I*ArcSinh[Sqrt[-(b/a)]*x], -((a*d)/(b*c))]))/(Sqrt[-(b/a)]*e*f*Sqrt[a - b* 
x^2]*Sqrt[c + d*x^2])
 

Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, 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.

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

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

Input:

Int[(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[1 + (d*x^2)/c]*EllipticF[ArcSin[(Sqrt[ 
b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(Sqrt[b]*f*Sqrt[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]*Elliptic 
Pi[-((a*f)/(b*e)), ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(Sqrt[b]* 
e*f*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 = 5.86 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.94

method	result	size
default	\(\frac {\left (A \operatorname {EllipticPi}\left (x \sqrt {\frac {b}{a}}, -\frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {\frac {b}{a}}}\right ) f +B \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) e -B \operatorname {EllipticPi}\left (x \sqrt {\frac {b}{a}}, -\frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {\frac {b}{a}}}\right ) e \right ) \sqrt {\frac {x^{2} d +c}{c}}\, \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{f e \sqrt {\frac {b}{a}}\, \left (-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c \right )}\)	\(192\)
elliptic	\(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {B \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{f \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}+\frac {\sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {b}{a}}, -\frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {\frac {b}{a}}}\right ) A}{e \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}-\frac {\sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {b}{a}}, -\frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {\frac {b}{a}}}\right ) B}{f \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(332\)

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

int((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)/(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))/(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