\(\int \frac {\sqrt {a+b x^2} (A+B x^2+C x^4)}{(c+d x^2) (e+f x^2)^2} \, dx\) [5]

Optimal result

Integrand size = 42, antiderivative size = 310 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4\right )}{\left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx=-\frac {\left (C e^2-B e f+A f^2\right ) x \sqrt {a+b x^2}}{2 e f (d e-c f) \left (e+f x^2\right )}+\frac {\sqrt {b} C \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d f^2}-\frac {\sqrt {b c-a d} \left (c^2 C-B c d+A d^2\right ) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d (d e-c f)^2}-\frac {\left (2 b e^2 \left ((B c-A d) f^2+C e (d e-2 c f)\right )-a f \left (C e^2 (d e-3 c f)-f (A f (3 d e-c f)-B e (d e+c f))\right )\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{2 e^{3/2} f^2 \sqrt {b e-a f} (d e-c f)^2} \] Output:

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

Mathematica [A] (verified)

Time = 2.29 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.11 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4\right )}{\left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx=\frac {1}{2} \left (-\frac {\left (C e^2+f (-B e+A f)\right ) x \sqrt {a+b x^2}}{e f (d e-c f) \left (e+f x^2\right )}-\frac {2 \sqrt {-b c+a d} \left (c^2 C-B c d+A d^2\right ) \arctan \left (\frac {-d x \sqrt {a+b x^2}+\sqrt {b} \left (c+d x^2\right )}{\sqrt {c} \sqrt {-b c+a d}}\right )}{\sqrt {c} d (d e-c f)^2}+\frac {\left (2 b e^2 \left ((B c-A d) f^2+C e (d e-2 c f)\right )-a f \left (C e^2 (d e-3 c f)+f (A f (-3 d e+c f)+B e (d e+c f))\right )\right ) \arctan \left (\frac {-f x \sqrt {a+b x^2}+\sqrt {b} \left (e+f x^2\right )}{\sqrt {e} \sqrt {-b e+a f}}\right )}{e^{3/2} f^2 \sqrt {-b e+a f} (d e-c f)^2}-\frac {2 \sqrt {b} C \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{d f^2}\right ) \] Input:

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

Output:

(-(((C*e^2 + f*(-(B*e) + A*f))*x*Sqrt[a + b*x^2])/(e*f*(d*e - c*f)*(e + f* 
x^2))) - (2*Sqrt[-(b*c) + a*d]*(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] 
*d*(d*e - c*f)^2) + ((2*b*e^2*((B*c - A*d)*f^2 + C*e*(d*e - 2*c*f)) - a*f* 
(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*Sqrt[a + b*x^2]) + Sqrt[b]*(e + f*x^2))/(Sqrt[e]*Sqrt[-(b*e) + a*f]) 
])/(e^(3/2)*f^2*Sqrt[-(b*e) + a*f]*(d*e - c*f)^2) - (2*Sqrt[b]*C*Log[-(Sqr 
t[b]*x) + Sqrt[a + b*x^2]])/(d*f^2))/2
 

Rubi [A] (verified)

Time = 1.17 (sec) , antiderivative size = 422, normalized size of antiderivative = 1.36, 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.

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-1/2*((C*e^2 - B*e*f + A*f^2)*x*Sqrt[a + b*x^2])/(e*f*(d*e - c*f)*(e + f*x 
^2)) + (Sqrt[b]*(c^2*C - B*c*d + A*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2 
]])/(d*(d*e - c*f)^2) + (Sqrt[b]*((B*c - A*d)*f^2 + C*e*(d*e - 2*c*f))*Arc 
Tanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(f^2*(d*e - c*f)^2) - (Sqrt[b*c - a*d]* 
(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]*d*(d*e - c*f)^2) - (a*(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*Sqrt[b*e - a*f]*(d 
*e - c*f)) - (Sqrt[b*e - a*f]*((B*c - A*d)*f^2 + C*e*(d*e - 2*c*f))*ArcTan 
h[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(Sqrt[e]*f^2*(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 = 1.56 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.12

method	result	size
pseudoelliptic	\(\frac {-\sqrt {\left (a d -b c \right ) c}\, \left (f \,x^{2}+e \right ) \left (-2 C b d \,e^{4}+C f \left (a d +4 b c \right ) e^{3}+2 f^{2} \left (\left (-B b -\frac {3 C a}{2}\right ) c +d \left (A b +\frac {B a}{2}\right )\right ) e^{2}-3 \left (A d -\frac {B c}{3}\right ) f^{3} a e +A a c \,f^{4}\right ) d \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )+\left (-2 e \,f^{2} \left (f \,x^{2}+e \right ) \left (a d -b c \right ) \left (d^{2} A -c d B +C \,c^{2}\right ) \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )+\sqrt {\left (a d -b c \right ) c}\, \left (2 \sqrt {b}\, C e \left (f \,x^{2}+e \right ) \left (c f -d e \right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )+x \sqrt {b \,x^{2}+a}\, d f \left (A \,f^{2}-B e f +C \,e^{2}\right )\right ) \left (c f -d e \right )\right ) \sqrt {\left (a f -b e \right ) e}}{2 \sqrt {\left (a d -b c \right ) c}\, \sqrt {\left (a f -b e \right ) e}\, d \left (c f -d e \right )^{2} e \left (f \,x^{2}+e \right ) f^{2}}\)	\(348\)
default	\(\text {Expression too large to display}\)	\(2994\)

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate((b*x^2+a)^(1/2)*(C*x^4+B*x^2+A)/(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 [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 1.52 (sec) , antiderivative size = 4615, normalized size of antiderivative = 14.89 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4\right )}{\left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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