\(\int \frac {a+b x^2}{(c+d x^2) (e+f x^2)^2} \, dx\) [220]

Optimal result

Integrand size = 26, antiderivative size = 140 \[ \int \frac {a+b x^2}{\left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx=\frac {(b e-a f) x}{2 e (d e-c f) \left (e+f x^2\right )}-\frac {\sqrt {d} (b c-a d) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (d e-c f)^2}-\frac {(a f (3 d e-c f)-b e (d e+c f)) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{2 e^{3/2} \sqrt {f} (d e-c f)^2} \] Output:

1/2*(-a*f+b*e)*x/e/(-c*f+d*e)/(f*x^2+e)-d^(1/2)*(-a*d+b*c)*arctan(d^(1/2)* 
x/c^(1/2))/c^(1/2)/(-c*f+d*e)^2-1/2*(a*f*(-c*f+3*d*e)-b*e*(c*f+d*e))*arcta 
n(f^(1/2)*x/e^(1/2))/e^(3/2)/f^(1/2)/(-c*f+d*e)^2
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.89 \[ \int \frac {a+b x^2}{\left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx=\frac {\frac {(b e-a f) (d e-c f) x}{e \left (e+f x^2\right )}+\frac {2 \sqrt {d} (-b c+a d) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c}}+\frac {(a f (-3 d e+c f)+b e (d e+c f)) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{e^{3/2} \sqrt {f}}}{2 (d e-c f)^2} \] Input:

Integrate[(a + b*x^2)/((c + d*x^2)*(e + f*x^2)^2),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.11, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {402, 25, 397, 218}

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.

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 397

\(\displaystyle \frac {x (b e-a f)}{2 e \left (e+f x^2\right ) (d e-c f)}-\frac {\frac {2 d e (b c-a d) \int \frac {1}{d x^2+c}dx}{d e-c f}+\frac {(a f (3 d e-c f)-b e (c f+d e)) \int \frac {1}{f x^2+e}dx}{d e-c f}}{2 e (d e-c f)}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {x (b e-a f)}{2 e \left (e+f x^2\right ) (d e-c f)}-\frac {\frac {2 \sqrt {d} e (b c-a d) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (d e-c f)}+\frac {\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) (a f (3 d e-c f)-b e (c f+d e))}{\sqrt {e} \sqrt {f} (d e-c f)}}{2 e (d e-c f)}\)

Input:

Int[(a + b*x^2)/((c + d*x^2)*(e + f*x^2)^2),x]
 

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 397

Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ 
Symbol] :> Simp[(b*e - a*f)/(b*c - a*d)   Int[1/(a + b*x^2), x], x] - Simp[ 
(d*e - c*f)/(b*c - a*d)   Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e 
, f}, x]
 

rule 402

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x 
_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ 
(q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) 
 Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) 
*(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b 
, c, d, e, f, q}, x] && LtQ[p, -1]
 

Maple [A] (verified)

Time = 0.68 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.94

method	result	size
default	\(\frac {\frac {\left (a c \,f^{2}-a d e f -b c e f +b d \,e^{2}\right ) x}{2 e \left (f \,x^{2}+e \right )}+\frac {\left (a c \,f^{2}-3 a d e f +b c e f +b d \,e^{2}\right ) \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{2 e \sqrt {e f}}}{\left (c f -d e \right )^{2}}+\frac {\left (a d -b c \right ) d \arctan \left (\frac {x d}{\sqrt {c d}}\right )}{\left (c f -d e \right )^{2} \sqrt {c d}}\)	\(131\)
risch	\(\text {Expression too large to display}\)	\(2213\)

Input:

int((b*x^2+a)/(d*x^2+c)/(f*x^2+e)^2,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.60 (sec) , antiderivative size = 997, normalized size of antiderivative = 7.12 \[ \int \frac {a+b x^2}{\left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/4*(2*((b*c - a*d)*e^2*f^2*x^2 + (b*c - a*d)*e^3*f)*sqrt(-d/c)*log((d*x 
^2 + 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)) + (b*d*e^3 + a*c*e*f^2 + (b*c - 3* 
a*d)*e^2*f + (b*d*e^2*f + a*c*f^3 + (b*c - 3*a*d)*e*f^2)*x^2)*sqrt(-e*f)*l 
og((f*x^2 - 2*sqrt(-e*f)*x - e)/(f*x^2 + e)) - 2*(b*d*e^3*f + a*c*e*f^3 - 
(b*c + a*d)*e^2*f^2)*x)/(d^2*e^5*f - 2*c*d*e^4*f^2 + c^2*e^3*f^3 + (d^2*e^ 
4*f^2 - 2*c*d*e^3*f^3 + c^2*e^2*f^4)*x^2), 1/2*((b*d*e^3 + a*c*e*f^2 + (b* 
c - 3*a*d)*e^2*f + (b*d*e^2*f + a*c*f^3 + (b*c - 3*a*d)*e*f^2)*x^2)*sqrt(e 
*f)*arctan(sqrt(e*f)*x/e) - ((b*c - a*d)*e^2*f^2*x^2 + (b*c - a*d)*e^3*f)* 
sqrt(-d/c)*log((d*x^2 + 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)) + (b*d*e^3*f + 
a*c*e*f^3 - (b*c + a*d)*e^2*f^2)*x)/(d^2*e^5*f - 2*c*d*e^4*f^2 + c^2*e^3*f 
^3 + (d^2*e^4*f^2 - 2*c*d*e^3*f^3 + c^2*e^2*f^4)*x^2), -1/4*(4*((b*c - a*d 
)*e^2*f^2*x^2 + (b*c - a*d)*e^3*f)*sqrt(d/c)*arctan(x*sqrt(d/c)) + (b*d*e^ 
3 + a*c*e*f^2 + (b*c - 3*a*d)*e^2*f + (b*d*e^2*f + a*c*f^3 + (b*c - 3*a*d) 
*e*f^2)*x^2)*sqrt(-e*f)*log((f*x^2 - 2*sqrt(-e*f)*x - e)/(f*x^2 + e)) - 2* 
(b*d*e^3*f + a*c*e*f^3 - (b*c + a*d)*e^2*f^2)*x)/(d^2*e^5*f - 2*c*d*e^4*f^ 
2 + c^2*e^3*f^3 + (d^2*e^4*f^2 - 2*c*d*e^3*f^3 + c^2*e^2*f^4)*x^2), -1/2*( 
2*((b*c - a*d)*e^2*f^2*x^2 + (b*c - a*d)*e^3*f)*sqrt(d/c)*arctan(x*sqrt(d/ 
c)) - (b*d*e^3 + a*c*e*f^2 + (b*c - 3*a*d)*e^2*f + (b*d*e^2*f + a*c*f^3 + 
(b*c - 3*a*d)*e*f^2)*x^2)*sqrt(e*f)*arctan(sqrt(e*f)*x/e) - (b*d*e^3*f + a 
*c*e*f^3 - (b*c + a*d)*e^2*f^2)*x)/(d^2*e^5*f - 2*c*d*e^4*f^2 + c^2*e^3...
 

Sympy [F(-1)]

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

integrate((b*x**2+a)/(d*x**2+c)/(f*x**2+e)**2,x)
 

Output:

Timed out
 

Maxima [F(-2)]

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

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.08 \[ \int \frac {a+b x^2}{\left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx=-\frac {{\left (b c d - a d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (d^{2} e^{2} - 2 \, c d e f + c^{2} f^{2}\right )} \sqrt {c d}} + \frac {{\left (b d e^{2} + b c e f - 3 \, a d e f + a c f^{2}\right )} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{2 \, {\left (d^{2} e^{3} - 2 \, c d e^{2} f + c^{2} e f^{2}\right )} \sqrt {e f}} + \frac {b e x - a f x}{2 \, {\left (d e^{2} - c e f\right )} {\left (f x^{2} + e\right )}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 4.00 (sec) , antiderivative size = 5567, normalized size of antiderivative = 39.76 \[ \int \frac {a+b x^2}{\left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

(x*(a*f - b*e))/(2*e*(e + f*x^2)*(c*f - d*e)) - (atan(((((x*(b^2*d^5*e^4*f 
 + a^2*c^2*d^3*f^5 + 13*a^2*d^5*e^2*f^3 + 5*b^2*c^2*d^3*e^2*f^3 - 6*a*b*d^ 
5*e^3*f^2 - 6*a^2*c*d^4*e*f^4 + 2*b^2*c*d^4*e^3*f^2 - 12*a*b*c*d^4*e^2*f^3 
 + 2*a*b*c^2*d^3*e*f^4))/(2*(d^2*e^4 + c^2*e^2*f^2 - 2*c*d*e^3*f)) - ((-e^ 
3*f)^(1/2)*((18*a*c*d^6*e^5*f^3 - 4*a*d^7*e^6*f^2 + 2*a*c^5*d^2*e*f^7 + 2* 
b*c*d^6*e^6*f^2 - 32*a*c^2*d^5*e^4*f^4 + 28*a*c^3*d^4*e^3*f^5 - 12*a*c^4*d 
^3*e^2*f^6 - 8*b*c^2*d^5*e^5*f^3 + 12*b*c^3*d^4*e^4*f^4 - 8*b*c^4*d^3*e^3* 
f^5 + 2*b*c^5*d^2*e^2*f^6)/(d^3*e^5 - c^3*e^2*f^3 + 3*c^2*d*e^3*f^2 - 3*c* 
d^2*e^4*f) - (x*(-e^3*f)^(1/2)*(a*c*f^2 + b*d*e^2 - 3*a*d*e*f + b*c*e*f)*( 
16*d^7*e^7*f^2 - 48*c*d^6*e^6*f^3 + 32*c^2*d^5*e^5*f^4 + 32*c^3*d^4*e^4*f^ 
5 - 48*c^4*d^3*e^3*f^6 + 16*c^5*d^2*e^2*f^7))/(8*(d^2*e^5*f + c^2*e^3*f^3 
- 2*c*d*e^4*f^2)*(d^2*e^4 + c^2*e^2*f^2 - 2*c*d*e^3*f)))*(a*c*f^2 + b*d*e^ 
2 - 3*a*d*e*f + b*c*e*f))/(4*(d^2*e^5*f + c^2*e^3*f^3 - 2*c*d*e^4*f^2)))*( 
-e^3*f)^(1/2)*(a*c*f^2 + b*d*e^2 - 3*a*d*e*f + b*c*e*f)*1i)/(4*(d^2*e^5*f 
+ c^2*e^3*f^3 - 2*c*d*e^4*f^2)) + (((x*(b^2*d^5*e^4*f + a^2*c^2*d^3*f^5 + 
13*a^2*d^5*e^2*f^3 + 5*b^2*c^2*d^3*e^2*f^3 - 6*a*b*d^5*e^3*f^2 - 6*a^2*c*d 
^4*e*f^4 + 2*b^2*c*d^4*e^3*f^2 - 12*a*b*c*d^4*e^2*f^3 + 2*a*b*c^2*d^3*e*f^ 
4))/(2*(d^2*e^4 + c^2*e^2*f^2 - 2*c*d*e^3*f)) + ((-e^3*f)^(1/2)*((18*a*c*d 
^6*e^5*f^3 - 4*a*d^7*e^6*f^2 + 2*a*c^5*d^2*e*f^7 + 2*b*c*d^6*e^6*f^2 - 32* 
a*c^2*d^5*e^4*f^4 + 28*a*c^3*d^4*e^3*f^5 - 12*a*c^4*d^3*e^2*f^6 - 8*b*c...
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 428, normalized size of antiderivative = 3.06 \[ \int \frac {a+b x^2}{\left (c+d x^2\right ) \left (e+f x^2\right )^2} \, dx=\frac {2 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) a d \,e^{3} f +2 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) a d \,e^{2} f^{2} x^{2}-2 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) b c \,e^{3} f -2 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) b c \,e^{2} f^{2} x^{2}+\sqrt {f}\, \sqrt {e}\, \mathit {atan} \left (\frac {f x}{\sqrt {f}\, \sqrt {e}}\right ) a \,c^{2} e \,f^{2}+\sqrt {f}\, \sqrt {e}\, \mathit {atan} \left (\frac {f x}{\sqrt {f}\, \sqrt {e}}\right ) a \,c^{2} f^{3} x^{2}-3 \sqrt {f}\, \sqrt {e}\, \mathit {atan} \left (\frac {f x}{\sqrt {f}\, \sqrt {e}}\right ) a c d \,e^{2} f -3 \sqrt {f}\, \sqrt {e}\, \mathit {atan} \left (\frac {f x}{\sqrt {f}\, \sqrt {e}}\right ) a c d e \,f^{2} x^{2}+\sqrt {f}\, \sqrt {e}\, \mathit {atan} \left (\frac {f x}{\sqrt {f}\, \sqrt {e}}\right ) b \,c^{2} e^{2} f +\sqrt {f}\, \sqrt {e}\, \mathit {atan} \left (\frac {f x}{\sqrt {f}\, \sqrt {e}}\right ) b \,c^{2} e \,f^{2} x^{2}+\sqrt {f}\, \sqrt {e}\, \mathit {atan} \left (\frac {f x}{\sqrt {f}\, \sqrt {e}}\right ) b c d \,e^{3}+\sqrt {f}\, \sqrt {e}\, \mathit {atan} \left (\frac {f x}{\sqrt {f}\, \sqrt {e}}\right ) b c d \,e^{2} f \,x^{2}+a \,c^{2} e \,f^{3} x -a c d \,e^{2} f^{2} x -b \,c^{2} e^{2} f^{2} x +b c d \,e^{3} f x}{2 c \,e^{2} f \left (c^{2} f^{3} x^{2}-2 c d e \,f^{2} x^{2}+d^{2} e^{2} f \,x^{2}+c^{2} e \,f^{2}-2 c d \,e^{2} f +d^{2} e^{3}\right )} \] Input:

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

Output:

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