\(\int \frac {A+B x^2+C x^4}{(a+b x^2)^{3/2} (c+d x^2) (e+f x^2)} \, dx\) [7]

Optimal result

Integrand size = 42, antiderivative size = 205 \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\frac {\left (A b^2-a (b B-a C)\right ) x}{a (b c-a d) (b e-a f) \sqrt {a+b x^2}}-\frac {\left (c^2 C-B c d+A d^2\right ) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} (b c-a d)^{3/2} (d e-c f)}+\frac {\left (C e^2-B e f+A f^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} (b e-a f)^{3/2} (d e-c f)} \] Output:

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

Mathematica [A] (verified)

Time = 1.55 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.16 \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\frac {\left (A b^2+a (-b B+a C)\right ) x}{a (-b c+a d) (-b e+a f) \sqrt {a+b x^2}}-\frac {\left (c^2 C-B c d+A d^2\right ) \arctan \left (\frac {-d x \sqrt {a+b x^2}+\sqrt {b} \left (c+d x^2\right )}{\sqrt {c} \sqrt {-b c+a d}}\right )}{\sqrt {c} (-b c+a d)^{3/2} (d e-c f)}+\frac {\left (C e^2+f (-B e+A f)\right ) \arctan \left (\frac {-f x \sqrt {a+b x^2}+\sqrt {b} \left (e+f x^2\right )}{\sqrt {e} \sqrt {-b e+a f}}\right )}{\sqrt {e} (-b e+a f)^{3/2} (d e-c f)} \] Input:

Integrate[(A + B*x^2 + C*x^4)/((a + b*x^2)^(3/2)*(c + d*x^2)*(e + f*x^2)), 
x]
 

Output:

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

Rubi [A] (verified)

Time = 1.05 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.40, 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.

\(\displaystyle \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right )} \, dx\)

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

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

Input:

Int[(A + B*x^2 + C*x^4)/((a + b*x^2)^(3/2)*(c + d*x^2)*(e + f*x^2)),x]
 

Output:

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

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 = 1.36 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.01

method	result	size
pseudoelliptic	\(\frac {\frac {\left (d^{2} A -c d B +C \,c^{2}\right ) a \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )}{\left (c f -d e \right ) \left (a d -b c \right ) \sqrt {\left (a d -b c \right ) c}}+\frac {\left (b^{2} A -a b B +a^{2} C \right ) x}{\left (a f -b e \right ) \left (a d -b c \right ) \sqrt {b \,x^{2}+a}}-\frac {\left (A \,f^{2}-B e f +C \,e^{2}\right ) a \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )}{\left (c f -d e \right ) \left (a f -b e \right ) \sqrt {\left (a f -b e \right ) e}}}{a}\)	\(208\)
default	\(\text {Expression too large to display}\)	\(1689\)

Input:

int((C*x^4+B*x^2+A)/(b*x^2+a)^(3/2)/(d*x^2+c)/(f*x^2+e),x,method=_RETURNVE 
RBOSE)
 

Output:

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

Fricas [F(-1)]

Timed out. \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:

integrate((C*x^4+B*x^2+A)/(b*x^2+a)^(3/2)/(d*x^2+c)/(f*x^2+e),x, algorithm 
="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:

integrate((C*x**4+B*x**2+A)/(b*x**2+a)**(3/2)/(d*x**2+c)/(f*x**2+e),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((C*x^4+B*x^2+A)/(b*x^2+a)^(3/2)/(d*x^2+c)/(f*x^2+e),x, algorithm 
="maxima")
 

Output:

integrate((C*x^4 + B*x^2 + A)/((b*x^2 + a)^(3/2)*(d*x^2 + c)*(f*x^2 + e)), 
 x)
 

Giac [A] (verification not implemented)

Time = 0.75 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.45 \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\frac {{\left (C a^{2} - B a b + A b^{2}\right )} x}{{\left (a b^{2} c e - a^{2} b d e - a^{2} b c f + a^{3} d f\right )} \sqrt {b x^{2} + a}} + \frac {{\left (C \sqrt {b} c^{2} - B \sqrt {b} c d + A \sqrt {b} d^{2}\right )} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt {-b^{2} c^{2} + a b c d}}\right )}{\sqrt {-b^{2} c^{2} + a b c d} {\left (b c d e - a d^{2} e - b c^{2} f + a c d f\right )}} - \frac {{\left (C \sqrt {b} e^{2} - B \sqrt {b} e f + A \sqrt {b} f^{2}\right )} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} f + 2 \, b e - a f}{2 \, \sqrt {-b^{2} e^{2} + a b e f}}\right )}{\sqrt {-b^{2} e^{2} + a b e f} {\left (b d e^{2} - b c e f - a d e f + a c f^{2}\right )}} \] Input:

integrate((C*x^4+B*x^2+A)/(b*x^2+a)^(3/2)/(d*x^2+c)/(f*x^2+e),x, algorithm 
="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\int \frac {C\,x^4+B\,x^2+A}{{\left (b\,x^2+a\right )}^{3/2}\,\left (d\,x^2+c\right )\,\left (f\,x^2+e\right )} \,d x \] Input:

int((A + B*x^2 + C*x^4)/((a + b*x^2)^(3/2)*(c + d*x^2)*(e + f*x^2)),x)
 

Output:

int((A + B*x^2 + C*x^4)/((a + b*x^2)^(3/2)*(c + d*x^2)*(e + f*x^2)), x)
 

Reduce [B] (verification not implemented)

Time = 1.38 (sec) , antiderivative size = 5297, normalized size of antiderivative = 25.84 \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right )} \, dx =\text {Too large to display} \] Input:

int((C*x^4+B*x^2+A)/(b*x^2+a)^(3/2)/(d*x^2+c)/(f*x^2+e),x)
 

Output:

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