\(\int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} (c+d x^2) (e+f x^2)^2} \, dx\) [6]

Optimal result

Integrand size = 42, antiderivative size = 278 \[ \int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx=\frac {\left (C e^2-B e f+A f^2\right ) x \sqrt {a+b x^2}}{2 e (b e-a f) (d e-c f) \left (e+f x^2\right )}+\frac {\left (c^2 C-B c d+A d^2\right ) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)^2}+\frac {\left (2 b e \left (d e (B e-2 A f)-c \left (C e^2-A f^2\right )\right )-a \left (C e^2 (d e-3 c f)-f (A f (3 d e-c f)-B e (d e+c f))\right )\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{2 e^{3/2} (b e-a f)^{3/2} (d e-c f)^2} \] Output:

1/2*(A*f^2-B*e*f+C*e^2)*x*(b*x^2+a)^(1/2)/e/(-a*f+b*e)/(-c*f+d*e)/(f*x^2+e 
)+(A*d^2-B*c*d+C*c^2)*arctanh((-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x^2+a)^(1/2))/ 
c^(1/2)/(-a*d+b*c)^(1/2)/(-c*f+d*e)^2+1/2*(2*b*e*(d*e*(-2*A*f+B*e)-c*(-A*f 
^2+C*e^2))-a*(C*e^2*(-3*c*f+d*e)-f*(A*f*(-c*f+3*d*e)-B*e*(c*f+d*e))))*arct 
anh((-a*f+b*e)^(1/2)*x/e^(1/2)/(b*x^2+a)^(1/2))/e^(3/2)/(-a*f+b*e)^(3/2)/( 
-c*f+d*e)^2
 

Mathematica [A] (verified)

Time = 2.46 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.06 \[ \int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx=\frac {\frac {(d e-c f) \left (C e^2+f (-B e+A f)\right ) x \sqrt {a+b x^2}}{e (b e-a f) \left (e+f x^2\right )}-\frac {2 \left (c^2 C-B c d+A d^2\right ) \arctan \left (\frac {-d x \sqrt {a+b x^2}+\sqrt {b} \left (c+d x^2\right )}{\sqrt {c} \sqrt {-b c+a d}}\right )}{\sqrt {c} \sqrt {-b c+a d}}-\frac {\left (-2 b e \left (d e (B e-2 A f)+c \left (-C e^2+A f^2\right )\right )+a \left (C e^2 (d e-3 c f)+f (A f (-3 d e+c f)+B e (d e+c f))\right )\right ) \arctan \left (\frac {-f x \sqrt {a+b x^2}+\sqrt {b} \left (e+f x^2\right )}{\sqrt {e} \sqrt {-b e+a f}}\right )}{e^{3/2} (-b e+a f)^{3/2}}}{2 (d e-c f)^2} \] Input:

Integrate[(A + B*x^2 + C*x^4)/(Sqrt[a + b*x^2]*(c + d*x^2)*(e + f*x^2)^2), 
x]
 

Output:

(((d*e - c*f)*(C*e^2 + f*(-(B*e) + A*f))*x*Sqrt[a + b*x^2])/(e*(b*e - a*f) 
*(e + f*x^2)) - (2*(c^2*C - B*c*d + A*d^2)*ArcTan[(-(d*x*Sqrt[a + b*x^2]) 
+ Sqrt[b]*(c + d*x^2))/(Sqrt[c]*Sqrt[-(b*c) + a*d])])/(Sqrt[c]*Sqrt[-(b*c) 
 + a*d]) - ((-2*b*e*(d*e*(B*e - 2*A*f) + c*(-(C*e^2) + A*f^2)) + a*(C*e^2* 
(d*e - 3*c*f) + f*(A*f*(-3*d*e + c*f) + B*e*(d*e + c*f))))*ArcTan[(-(f*x*S 
qrt[a + b*x^2]) + Sqrt[b]*(e + f*x^2))/(Sqrt[e]*Sqrt[-(b*e) + a*f])])/(e^( 
3/2)*(-(b*e) + a*f)^(3/2)))/(2*(d*e - c*f)^2)
 

Rubi [A] (verified)

Time = 1.12 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.14, 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.

\(\displaystyle \int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx\)

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

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

Input:

Int[(A + B*x^2 + C*x^4)/(Sqrt[a + b*x^2]*(c + d*x^2)*(e + f*x^2)^2),x]
 

Output:

((C*e^2 - B*e*f + A*f^2)*x*Sqrt[a + b*x^2])/(2*e*(b*e - a*f)*(d*e - c*f)*( 
e + f*x^2)) + ((c^2*C - B*c*d + A*d^2)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c 
]*Sqrt[a + b*x^2])])/(Sqrt[c]*Sqrt[b*c - a*d]*(d*e - c*f)^2) - ((2*b*e - a 
*f)*(C*e^2 - B*e*f + A*f^2)*ArcTanh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + 
b*x^2])])/(2*e^(3/2)*f*(b*e - a*f)^(3/2)*(d*e - c*f)) + (((B*c - A*d)*f^2 
+ C*e*(d*e - 2*c*f))*ArcTanh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2]) 
])/(Sqrt[e]*f*Sqrt[b*e - a*f]*(d*e - c*f)^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 = 1.52 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.09

method	result	size
pseudoelliptic	\(\frac {-\left (f \,x^{2}+e \right ) \sqrt {\left (a d -b c \right ) c}\, \left (\left (-2 B b d +C a d +2 C b c \right ) e^{3}+4 f \left (-\frac {3 C a c}{4}+d \left (A b +\frac {B a}{4}\right )\right ) e^{2}-3 f^{2} \left (\frac {\left (2 A b -B a \right ) c}{3}+A a d \right ) e +A a c \,f^{3}\right ) \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )+\left (-2 e \left (f \,x^{2}+e \right ) \left (a f -b e \right ) \left (d^{2} A -c d B +C \,c^{2}\right ) \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )+x \sqrt {\left (a d -b c \right ) c}\, \left (A \,f^{2}-B e f +C \,e^{2}\right ) \left (c f -d e \right ) \sqrt {b \,x^{2}+a}\right ) \sqrt {\left (a f -b e \right ) e}}{2 \sqrt {\left (a d -b c \right ) c}\, \sqrt {\left (a f -b e \right ) e}\, \left (c f -d e \right )^{2} \left (a f -b e \right ) e \left (f \,x^{2}+e \right )}\)	\(302\)
default	\(\text {Expression too large to display}\)	\(1514\)

Input:

int((C*x^4+B*x^2+A)/(b*x^2+a)^(1/2)/(d*x^2+c)/(f*x^2+e)^2,x,method=_RETURN 
VERBOSE)
 

Output:

1/2/((a*d-b*c)*c)^(1/2)*(-(f*x^2+e)*((a*d-b*c)*c)^(1/2)*((-2*B*b*d+C*a*d+2 
*C*b*c)*e^3+4*f*(-3/4*C*a*c+d*(A*b+1/4*B*a))*e^2-3*f^2*(1/3*(2*A*b-B*a)*c+ 
A*a*d)*e+A*a*c*f^3)*arctan(e*(b*x^2+a)^(1/2)/x/((a*f-b*e)*e)^(1/2))+(-2*e* 
(f*x^2+e)*(a*f-b*e)*(A*d^2-B*c*d+C*c^2)*arctan(c*(b*x^2+a)^(1/2)/x/((a*d-b 
*c)*c)^(1/2))+x*((a*d-b*c)*c)^(1/2)*(A*f^2-B*e*f+C*e^2)*(c*f-d*e)*(b*x^2+a 
)^(1/2))*((a*f-b*e)*e)^(1/2))/((a*f-b*e)*e)^(1/2)/(c*f-d*e)^2/(a*f-b*e)/e/ 
(f*x^2+e)
 

Fricas [F(-1)]

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

integrate((C*x^4+B*x^2+A)/(b*x^2+a)^(1/2)/(d*x^2+c)/(f*x^2+e)^2,x, algorit 
hm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

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

integrate((C*x**4+B*x**2+A)/(b*x**2+a)**(1/2)/(d*x**2+c)/(f*x**2+e)**2,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((C*x^4+B*x^2+A)/(b*x^2+a)^(1/2)/(d*x^2+c)/(f*x^2+e)^2,x, algorit 
hm="maxima")
 

Output:

integrate((C*x^4 + B*x^2 + A)/(sqrt(b*x^2 + a)*(d*x^2 + c)*(f*x^2 + e)^2), 
 x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 678 vs. \(2 (251) = 502\).

Time = 0.86 (sec) , antiderivative size = 678, normalized size of antiderivative = 2.44 \[ \int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx=-\frac {{\left (C \sqrt {b} c^{2} - B \sqrt {b} c d + A \sqrt {b} d^{2}\right )} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt {-b^{2} c^{2} + a b c d}}\right )}{\sqrt {-b^{2} c^{2} + a b c d} {\left (d^{2} e^{2} - 2 \, c d e f + c^{2} f^{2}\right )}} + \frac {{\left (2 \, C b^{\frac {3}{2}} c e^{3} + C a \sqrt {b} d e^{3} - 2 \, B b^{\frac {3}{2}} d e^{3} - 3 \, C a \sqrt {b} c e^{2} f + B a \sqrt {b} d e^{2} f + 4 \, A b^{\frac {3}{2}} d e^{2} f + B a \sqrt {b} c e f^{2} - 2 \, A b^{\frac {3}{2}} c e f^{2} - 3 \, A a \sqrt {b} d e f^{2} + A a \sqrt {b} c f^{3}\right )} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} f + 2 \, b e - a f}{2 \, \sqrt {-b^{2} e^{2} + a b e f}}\right )}{2 \, {\left (b d^{2} e^{4} - 2 \, b c d e^{3} f - a d^{2} e^{3} f + b c^{2} e^{2} f^{2} + 2 \, a c d e^{2} f^{2} - a c^{2} e f^{3}\right )} \sqrt {-b^{2} e^{2} + a b e f}} + \frac {2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} C b^{\frac {3}{2}} e^{3} - {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} C a \sqrt {b} e^{2} f - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B b^{\frac {3}{2}} e^{2} f + {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a \sqrt {b} e f^{2} + 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A b^{\frac {3}{2}} e f^{2} - {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a \sqrt {b} f^{3} + C a^{2} \sqrt {b} e^{2} f - B a^{2} \sqrt {b} e f^{2} + A a^{2} \sqrt {b} f^{3}}{{\left (b d e^{3} f - b c e^{2} f^{2} - a d e^{2} f^{2} + a c e f^{3}\right )} {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} f + 4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b e - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a f + a^{2} f\right )}} \] Input:

integrate((C*x^4+B*x^2+A)/(b*x^2+a)^(1/2)/(d*x^2+c)/(f*x^2+e)^2,x, algorit 
hm="giac")
 

Output:

-(C*sqrt(b)*c^2 - B*sqrt(b)*c*d + A*sqrt(b)*d^2)*arctan(1/2*((sqrt(b)*x - 
sqrt(b*x^2 + a))^2*d + 2*b*c - a*d)/sqrt(-b^2*c^2 + a*b*c*d))/(sqrt(-b^2*c 
^2 + a*b*c*d)*(d^2*e^2 - 2*c*d*e*f + c^2*f^2)) + 1/2*(2*C*b^(3/2)*c*e^3 + 
C*a*sqrt(b)*d*e^3 - 2*B*b^(3/2)*d*e^3 - 3*C*a*sqrt(b)*c*e^2*f + B*a*sqrt(b 
)*d*e^2*f + 4*A*b^(3/2)*d*e^2*f + B*a*sqrt(b)*c*e*f^2 - 2*A*b^(3/2)*c*e*f^ 
2 - 3*A*a*sqrt(b)*d*e*f^2 + A*a*sqrt(b)*c*f^3)*arctan(1/2*((sqrt(b)*x - sq 
rt(b*x^2 + a))^2*f + 2*b*e - a*f)/sqrt(-b^2*e^2 + a*b*e*f))/((b*d^2*e^4 - 
2*b*c*d*e^3*f - a*d^2*e^3*f + b*c^2*e^2*f^2 + 2*a*c*d*e^2*f^2 - a*c^2*e*f^ 
3)*sqrt(-b^2*e^2 + a*b*e*f)) + (2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*C*b^(3/2 
)*e^3 - (sqrt(b)*x - sqrt(b*x^2 + a))^2*C*a*sqrt(b)*e^2*f - 2*(sqrt(b)*x - 
 sqrt(b*x^2 + a))^2*B*b^(3/2)*e^2*f + (sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a* 
sqrt(b)*e*f^2 + 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*b^(3/2)*e*f^2 - (sqrt( 
b)*x - sqrt(b*x^2 + a))^2*A*a*sqrt(b)*f^3 + C*a^2*sqrt(b)*e^2*f - B*a^2*sq 
rt(b)*e*f^2 + A*a^2*sqrt(b)*f^3)/((b*d*e^3*f - b*c*e^2*f^2 - a*d*e^2*f^2 + 
 a*c*e*f^3)*((sqrt(b)*x - sqrt(b*x^2 + a))^4*f + 4*(sqrt(b)*x - sqrt(b*x^2 
 + a))^2*b*e - 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a*f + a^2*f))
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

int((A + B*x^2 + C*x^4)/((a + b*x^2)^(1/2)*(c + d*x^2)*(e + f*x^2)^2),x)
 

Output:

int((A + B*x^2 + C*x^4)/((a + b*x^2)^(1/2)*(c + d*x^2)*(e + f*x^2)^2), x)
 

Reduce [B] (verification not implemented)

Time = 1.39 (sec) , antiderivative size = 6592, normalized size of antiderivative = 23.71 \[ \int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Input:

int((C*x^4+B*x^2+A)/(b*x^2+a)^(1/2)/(d*x^2+c)/(f*x^2+e)^2,x)
 

Output:

( - 2*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x 
**2) - sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))*a**3*d**2*e**3*f**2 - 2*sqrt( 
c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt 
(d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))*a**3*d**2*e**2*f**3*x**2 + 2*sqrt(c)*sqr 
t(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)*sq 
rt(b)*x)/(sqrt(c)*sqrt(b)))*a**2*b*c*d*e**3*f**2 + 2*sqrt(c)*sqrt(a*d - b* 
c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)*sqrt(b)*x)/( 
sqrt(c)*sqrt(b)))*a**2*b*c*d*e**2*f**3*x**2 + 4*sqrt(c)*sqrt(a*d - b*c)*at 
an((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)*sqrt(b)*x)/(sqrt( 
c)*sqrt(b)))*a**2*b*d**2*e**4*f + 4*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d 
 - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqrt(b))) 
*a**2*b*d**2*e**3*f**2*x**2 - 2*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b 
*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))*a** 
2*c**3*e**3*f**2 - 2*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt( 
d)*sqrt(a + b*x**2) - sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))*a**2*c**3*e**2 
*f**3*x**2 - 4*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqr 
t(a + b*x**2) - sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))*a*b**2*c*d*e**4*f - 
4*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) 
 - sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))*a*b**2*c*d*e**3*f**2*x**2 - 2*sqr 
t(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) -...