\(\int \frac {A+B x^2}{\sqrt {a-b x^2} (c+d x^2)^{3/2} (e+f x^2)} \, dx\) [24]

Optimal result

Integrand size = 40, antiderivative size = 294 \[ \int \frac {A+B x^2}{\sqrt {a-b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=-\frac {d (B c-A d) x \sqrt {a-b x^2}}{c (b c+a d) (d e-c f) \sqrt {c+d x^2}}-\frac {\sqrt {a} \sqrt {b} (B c-A d) \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{c (b c+a d) (d e-c f) \sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}}}+\frac {\sqrt {a} (B e-A f) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} e (d e-c f) \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Output:

-d*(-A*d+B*c)*x*(-b*x^2+a)^(1/2)/c/(a*d+b*c)/(-c*f+d*e)/(d*x^2+c)^(1/2)-a^ 
(1/2)*b^(1/2)*(-A*d+B*c)*(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))/c/(a*d+b*c)/(-c*f+d*e)/(-b*x^2+a)^(1/2)/(1+ 
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)*Elli 
pticPi(b^(1/2)*x/a^(1/2),-a*f/b/e,(-a*d/b/c)^(1/2))/b^(1/2)/e/(-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 = 11.31 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.85 \[ \int \frac {A+B x^2}{\sqrt {a-b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\frac {-\sqrt {-\frac {b}{a}} d (-B c+A d) e x \left (a-b x^2\right )-i b c (B c-A d) e \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )+i c (b c+a d) (B e-A f) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )}{\sqrt {-\frac {b}{a}} c (b c+a d) e (-d e+c f) \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Input:

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

Output:

(-(Sqrt[-(b/a)]*d*(-(B*c) + A*d)*e*x*(a - b*x^2)) - I*b*c*(B*c - A*d)*e*Sq 
rt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[-(b/a)]*x], 
 -((a*d)/(b*c))] + I*c*(b*c + a*d)*(B*e - A*f)*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))])/(Sqrt[-(b/a)]*c*(b*c + a*d)*e*(-(d*e) + c*f)*Sqrt[a - b*x^2]*Sqr 
t[c + d*x^2])
 

Rubi [A] (verified)

Time = 1.52 (sec) , antiderivative size = 559, normalized size of antiderivative = 1.90, 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.

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {a} d \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (B e-A f) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x^2} (d e-c f)^2}-\frac {\sqrt {a} \sqrt {b} d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} (B e-A f) E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{c f \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1} (a d+b c) (d e-c f)}-\frac {c^{3/2} f \sqrt {a-b x^2} (B e-A f) \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b c}{a d}+1\right )}{a \sqrt {d} e \sqrt {c+d x^2} (d e-c f)^2 \sqrt {\frac {c \left (a-b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {d^2 x \sqrt {a-b x^2} (B e-A f)}{c f \sqrt {c+d x^2} (a d+b c) (d e-c f)}+\frac {\sqrt {a} \sqrt {b} B \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{c f \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1} (a d+b c)}+\frac {B d x \sqrt {a-b x^2}}{c f \sqrt {c+d x^2} (a d+b c)}\)

Input:

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

Output:

(B*d*x*Sqrt[a - b*x^2])/(c*(b*c + a*d)*f*Sqrt[c + d*x^2]) - (d^2*(B*e - A* 
f)*x*Sqrt[a - b*x^2])/(c*(b*c + a*d)*f*(d*e - c*f)*Sqrt[c + d*x^2]) + (Sqr 
t[a]*Sqrt[b]*B*Sqrt[1 - (b*x^2)/a]*Sqrt[c + d*x^2]*EllipticE[ArcSin[(Sqrt[ 
b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(c*(b*c + a*d)*f*Sqrt[a - b*x^2]*Sqrt[1 + 
 (d*x^2)/c]) - (Sqrt[a]*Sqrt[b]*d*(B*e - A*f)*Sqrt[1 - (b*x^2)/a]*Sqrt[c + 
 d*x^2]*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(c*(b*c + 
a*d)*f*(d*e - c*f)*Sqrt[a - b*x^2]*Sqrt[1 + (d*x^2)/c]) + (Sqrt[a]*d*(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]*(d*e - c*f)^2*Sqrt[a - b*x^2]*Sqrt[c 
 + d*x^2]) - (c^(3/2)*f*(B*e - A*f)*Sqrt[a - b*x^2]*EllipticPi[1 - (c*f)/( 
d*e), ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 + (b*c)/(a*d)])/(a*Sqrt[d]*e*(d*e - c 
*f)^2*Sqrt[(c*(a - b*x^2))/(a*(c + d*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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(535\) vs. \(2(260)=520\).

Time = 8.68 (sec) , antiderivative size = 536, normalized size of antiderivative = 1.82

method	result	size
default	\(\frac {\left (A \sqrt {\frac {b}{a}}\, b \,d^{2} e \,x^{3}-B \sqrt {\frac {b}{a}}\, b c d e \,x^{3}-A \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) b c d e +A \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {b}{a}}, -\frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {\frac {b}{a}}}\right ) a c d f +A \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {b}{a}}, -\frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {\frac {b}{a}}}\right ) b \,c^{2} f +B \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) b \,c^{2} e -B \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {b}{a}}, -\frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {\frac {b}{a}}}\right ) a c d e -B \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {b}{a}}, -\frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {\frac {b}{a}}}\right ) b \,c^{2} e -A \sqrt {\frac {b}{a}}\, a \,d^{2} e x +B \sqrt {\frac {b}{a}}\, a c d e x \right ) \sqrt {x^{2} d +c}\, \sqrt {-b \,x^{2}+a}}{c \left (a d +b c \right ) e \sqrt {\frac {b}{a}}\, \left (c f -d e \right ) \left (-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c \right )}\)	\(536\)
elliptic	\(\text {Expression too large to display}\)	\(1203\)

Input:

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

Output:

(A*(b/a)^(1/2)*b*d^2*e*x^3-B*(b/a)^(1/2)*b*c*d*e*x^3-A*((-b*x^2+a)/a)^(1/2 
)*((d*x^2+c)/c)^(1/2)*EllipticE(x*(b/a)^(1/2),(-a*d/b/c)^(1/2))*b*c*d*e+A* 
((-b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticPi(x*(b/a)^(1/2),-a*f/b/e 
,(-1/c*d)^(1/2)/(b/a)^(1/2))*a*c*d*f+A*((-b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^ 
(1/2)*EllipticPi(x*(b/a)^(1/2),-a*f/b/e,(-1/c*d)^(1/2)/(b/a)^(1/2))*b*c^2* 
f+B*((-b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticE(x*(b/a)^(1/2),(-a*d 
/b/c)^(1/2))*b*c^2*e-B*((-b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticPi 
(x*(b/a)^(1/2),-a*f/b/e,(-1/c*d)^(1/2)/(b/a)^(1/2))*a*c*d*e-B*((-b*x^2+a)/ 
a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticPi(x*(b/a)^(1/2),-a*f/b/e,(-1/c*d)^(1 
/2)/(b/a)^(1/2))*b*c^2*e-A*(b/a)^(1/2)*a*d^2*e*x+B*(b/a)^(1/2)*a*c*d*e*x)* 
(d*x^2+c)^(1/2)*(-b*x^2+a)^(1/2)/c/(a*d+b*c)/e/(b/a)^(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. \[ \int \frac {A+B x^2}{\sqrt {a-b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:

integrate((B*x^2+A)/(-b*x^2+a)^(1/2)/(d*x^2+c)^(3/2)/(f*x^2+e),x, algorith 
m="fricas")
 

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((B*x^2+A)/(-b*x^2+a)^(1/2)/(d*x^2+c)^(3/2)/(f*x^2+e),x, algorith 
m="maxima")
 

Output:

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

Giac [F]

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

integrate((B*x^2+A)/(-b*x^2+a)^(1/2)/(d*x^2+c)^(3/2)/(f*x^2+e),x, algorith 
m="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {A+B x^2}{\sqrt {a-b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b \,d^{2} f \,x^{8}+a \,d^{2} f \,x^{6}-2 b c d f \,x^{6}-b \,d^{2} e \,x^{6}+2 a c d f \,x^{4}+a \,d^{2} e \,x^{4}-b \,c^{2} f \,x^{4}-2 b c d e \,x^{4}+a \,c^{2} f \,x^{2}+2 a c d e \,x^{2}-b \,c^{2} e \,x^{2}+a \,c^{2} e}d x \right ) b +\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}}{-b \,d^{2} f \,x^{8}+a \,d^{2} f \,x^{6}-2 b c d f \,x^{6}-b \,d^{2} e \,x^{6}+2 a c d f \,x^{4}+a \,d^{2} e \,x^{4}-b \,c^{2} f \,x^{4}-2 b c d e \,x^{4}+a \,c^{2} f \,x^{2}+2 a c d e \,x^{2}-b \,c^{2} e \,x^{2}+a \,c^{2} e}d x \right ) a \] Input:

int((B*x^2+A)/(-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)/(a*c**2*e + a*c**2*f*x**2 + 2 
*a*c*d*e*x**2 + 2*a*c*d*f*x**4 + a*d**2*e*x**4 + a*d**2*f*x**6 - b*c**2*e* 
x**2 - b*c**2*f*x**4 - 2*b*c*d*e*x**4 - 2*b*c*d*f*x**6 - b*d**2*e*x**6 - b 
*d**2*f*x**8),x)*b + int((sqrt(c + d*x**2)*sqrt(a - b*x**2))/(a*c**2*e + a 
*c**2*f*x**2 + 2*a*c*d*e*x**2 + 2*a*c*d*f*x**4 + a*d**2*e*x**4 + a*d**2*f* 
x**6 - b*c**2*e*x**2 - b*c**2*f*x**4 - 2*b*c*d*e*x**4 - 2*b*c*d*f*x**6 - b 
*d**2*e*x**6 - b*d**2*f*x**8),x)*a