\(\int \frac {(a+b x^2)^3}{(c+d x^2) (e+f x^2)} \, dx\) [247]

Optimal result

Integrand size = 28, antiderivative size = 132 \[ \int \frac {\left (a+b x^2\right )^3}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=-\frac {b^2 (b d e+b c f-3 a d f) x}{d^2 f^2}+\frac {b^3 x^3}{3 d f}-\frac {(b c-a d)^3 \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} d^{5/2} (d e-c f)}+\frac {(b e-a f)^3 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} f^{5/2} (d e-c f)} \] Output:

-b^2*(-3*a*d*f+b*c*f+b*d*e)*x/d^2/f^2+1/3*b^3*x^3/d/f-(-a*d+b*c)^3*arctan( 
d^(1/2)*x/c^(1/2))/c^(1/2)/d^(5/2)/(-c*f+d*e)+(-a*f+b*e)^3*arctan(f^(1/2)* 
x/e^(1/2))/e^(1/2)/f^(5/2)/(-c*f+d*e)
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b x^2\right )^3}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=-\frac {b^2 (b d e+b c f-3 a d f) x}{d^2 f^2}+\frac {b^3 x^3}{3 d f}+\frac {(b c-a d)^3 \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} d^{5/2} (-d e+c f)}+\frac {(b e-a f)^3 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} f^{5/2} (d e-c f)} \] Input:

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

Output:

-((b^2*(b*d*e + b*c*f - 3*a*d*f)*x)/(d^2*f^2)) + (b^3*x^3)/(3*d*f) + ((b*c 
 - a*d)^3*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*d^(5/2)*(-(d*e) + c*f)) + 
((b*e - a*f)^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*f^(5/2)*(d*e - c*f))
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.72, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {420, 300, 420, 299, 218, 397, 218, 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.

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

\(\Big \downarrow \) 420

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

\(\Big \downarrow \) 300

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

\(\Big \downarrow \) 420

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

\(\Big \downarrow \) 299

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 397

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b \left (\frac {(b e-a f)^2 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} f^{5/2}}-\frac {b x (b e-2 a f)}{f^2}+\frac {b^2 x^3}{3 f}\right )}{d}-\frac {(b c-a d) \left (\frac {b \left (\frac {b x}{f}-\frac {(b e-a f) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} f^{3/2}}\right )}{d}-\frac {(b c-a d) \left (\frac {(b e-a f) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} \sqrt {f} (d e-c f)}-\frac {(b c-a d) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} \sqrt {d} (d e-c f)}\right )}{d}\right )}{d}\)

Input:

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

Output:

(b*(-((b*(b*e - 2*a*f)*x)/f^2) + (b^2*x^3)/(3*f) + ((b*e - a*f)^2*ArcTan[( 
Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*f^(5/2))))/d - ((b*c - a*d)*((b*((b*x)/f - ( 
(b*e - a*f)*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*f^(3/2))))/d - ((b*c - a 
*d)*(-(((b*c - a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*Sqrt[d]*(d*e - c 
*f))) + ((b*e - a*f)*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*Sqrt[f]*(d*e - 
c*f))))/d))/d
 

Defintions of rubi rules used

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 299

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x 
*((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 
*p + 3))   Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && NeQ[2*p + 3, 0]
 

rule 300

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Int 
[PolynomialDivide[(a + b*x^2)^p, (c + d*x^2)^(-q), x], x] /; FreeQ[{a, b, c 
, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && ILtQ[q, 0] && GeQ[p, -q]
 

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 420

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

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.68 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.27

method	result	size
default	\(\frac {b^{2} \left (\frac {1}{3} b d f \,x^{3}+3 a d f x -b c f x -b d e x \right )}{d^{2} f^{2}}+\frac {\left (a^{3} f^{3}-3 a^{2} b e \,f^{2}+3 a \,b^{2} e^{2} f -b^{3} e^{3}\right ) \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{f^{2} \left (c f -d e \right ) \sqrt {e f}}+\frac {\left (-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \arctan \left (\frac {x d}{\sqrt {c d}}\right )}{d^{2} \left (c f -d e \right ) \sqrt {c d}}\)	\(167\)
risch	\(\frac {b^{3} x^{3}}{3 d f}+\frac {3 b^{2} a x}{d f}-\frac {b^{3} c x}{d^{2} f}-\frac {b^{3} e x}{d \,f^{2}}-\frac {f \ln \left (e \,f^{2} x -\left (-e f \right )^{\frac {3}{2}}\right ) a^{3}}{2 \sqrt {-e f}\, \left (c f -d e \right )}+\frac {3 \ln \left (e \,f^{2} x -\left (-e f \right )^{\frac {3}{2}}\right ) a^{2} b e}{2 \sqrt {-e f}\, \left (c f -d e \right )}-\frac {3 \ln \left (e \,f^{2} x -\left (-e f \right )^{\frac {3}{2}}\right ) a \,b^{2} e^{2}}{2 f \sqrt {-e f}\, \left (c f -d e \right )}+\frac {\ln \left (e \,f^{2} x -\left (-e f \right )^{\frac {3}{2}}\right ) b^{3} e^{3}}{2 f^{2} \sqrt {-e f}\, \left (c f -d e \right )}+\frac {f \ln \left (-e \,f^{2} x -\left (-e f \right )^{\frac {3}{2}}\right ) a^{3}}{2 \sqrt {-e f}\, \left (c f -d e \right )}-\frac {3 \ln \left (-e \,f^{2} x -\left (-e f \right )^{\frac {3}{2}}\right ) a^{2} b e}{2 \sqrt {-e f}\, \left (c f -d e \right )}+\frac {3 \ln \left (-e \,f^{2} x -\left (-e f \right )^{\frac {3}{2}}\right ) a \,b^{2} e^{2}}{2 f \sqrt {-e f}\, \left (c f -d e \right )}-\frac {\ln \left (-e \,f^{2} x -\left (-e f \right )^{\frac {3}{2}}\right ) b^{3} e^{3}}{2 f^{2} \sqrt {-e f}\, \left (c f -d e \right )}-\frac {d \ln \left (-c \,d^{2} x -\left (-c d \right )^{\frac {3}{2}}\right ) a^{3}}{2 \sqrt {-c d}\, \left (c f -d e \right )}+\frac {3 \ln \left (-c \,d^{2} x -\left (-c d \right )^{\frac {3}{2}}\right ) a^{2} b c}{2 \sqrt {-c d}\, \left (c f -d e \right )}-\frac {3 \ln \left (-c \,d^{2} x -\left (-c d \right )^{\frac {3}{2}}\right ) a \,b^{2} c^{2}}{2 d \sqrt {-c d}\, \left (c f -d e \right )}+\frac {\ln \left (-c \,d^{2} x -\left (-c d \right )^{\frac {3}{2}}\right ) b^{3} c^{3}}{2 d^{2} \sqrt {-c d}\, \left (c f -d e \right )}+\frac {d \ln \left (c \,d^{2} x -\left (-c d \right )^{\frac {3}{2}}\right ) a^{3}}{2 \sqrt {-c d}\, \left (c f -d e \right )}-\frac {3 \ln \left (c \,d^{2} x -\left (-c d \right )^{\frac {3}{2}}\right ) a^{2} b c}{2 \sqrt {-c d}\, \left (c f -d e \right )}+\frac {3 \ln \left (c \,d^{2} x -\left (-c d \right )^{\frac {3}{2}}\right ) a \,b^{2} c^{2}}{2 d \sqrt {-c d}\, \left (c f -d e \right )}-\frac {\ln \left (c \,d^{2} x -\left (-c d \right )^{\frac {3}{2}}\right ) b^{3} c^{3}}{2 d^{2} \sqrt {-c d}\, \left (c f -d e \right )}\)	\(719\)

Input:

int((b*x^2+a)^3/(d*x^2+c)/(f*x^2+e),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 251 vs. \(2 (114) = 228\).

Time = 10.63 (sec) , antiderivative size = 1081, normalized size of antiderivative = 8.19 \[ \int \frac {\left (a+b x^2\right )^3}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/6*(3*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(-c*d)*e* 
f^3*log((d*x^2 + 2*sqrt(-c*d)*x - c)/(d*x^2 + c)) - 2*(b^3*c*d^3*e^2*f^2 - 
 b^3*c^2*d^2*e*f^3)*x^3 + 3*(b^3*c*d^3*e^3 - 3*a*b^2*c*d^3*e^2*f + 3*a^2*b 
*c*d^3*e*f^2 - a^3*c*d^3*f^3)*sqrt(-e*f)*log((f*x^2 - 2*sqrt(-e*f)*x - e)/ 
(f*x^2 + e)) + 6*(b^3*c*d^3*e^3*f - 3*a*b^2*c*d^3*e^2*f^2 - (b^3*c^3*d - 3 
*a*b^2*c^2*d^2)*e*f^3)*x)/(c*d^4*e^2*f^3 - c^2*d^3*e*f^4), -1/6*(3*(b^3*c^ 
3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(-c*d)*e*f^3*log((d*x^2 + 
 2*sqrt(-c*d)*x - c)/(d*x^2 + c)) - 2*(b^3*c*d^3*e^2*f^2 - b^3*c^2*d^2*e*f 
^3)*x^3 - 6*(b^3*c*d^3*e^3 - 3*a*b^2*c*d^3*e^2*f + 3*a^2*b*c*d^3*e*f^2 - a 
^3*c*d^3*f^3)*sqrt(e*f)*arctan(sqrt(e*f)*x/e) + 6*(b^3*c*d^3*e^3*f - 3*a*b 
^2*c*d^3*e^2*f^2 - (b^3*c^3*d - 3*a*b^2*c^2*d^2)*e*f^3)*x)/(c*d^4*e^2*f^3 
- c^2*d^3*e*f^4), -1/6*(6*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d 
^3)*sqrt(c*d)*e*f^3*arctan(sqrt(c*d)*x/c) - 2*(b^3*c*d^3*e^2*f^2 - b^3*c^2 
*d^2*e*f^3)*x^3 + 3*(b^3*c*d^3*e^3 - 3*a*b^2*c*d^3*e^2*f + 3*a^2*b*c*d^3*e 
*f^2 - a^3*c*d^3*f^3)*sqrt(-e*f)*log((f*x^2 - 2*sqrt(-e*f)*x - e)/(f*x^2 + 
 e)) + 6*(b^3*c*d^3*e^3*f - 3*a*b^2*c*d^3*e^2*f^2 - (b^3*c^3*d - 3*a*b^2*c 
^2*d^2)*e*f^3)*x)/(c*d^4*e^2*f^3 - c^2*d^3*e*f^4), -1/3*(3*(b^3*c^3 - 3*a* 
b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(c*d)*e*f^3*arctan(sqrt(c*d)*x/c) 
 - (b^3*c*d^3*e^2*f^2 - b^3*c^2*d^2*e*f^3)*x^3 - 3*(b^3*c*d^3*e^3 - 3*a*b^ 
2*c*d^3*e^2*f + 3*a^2*b*c*d^3*e*f^2 - a^3*c*d^3*f^3)*sqrt(e*f)*arctan(s...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x^2\right )^3}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((b*x^2+a)^3/(d*x^2+c)/(f*x^2+e),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 = 191, normalized size of antiderivative = 1.45 \[ \int \frac {\left (a+b x^2\right )^3}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=-\frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (d^{3} e - c d^{2} f\right )} \sqrt {c d}} + \frac {{\left (b^{3} e^{3} - 3 \, a b^{2} e^{2} f + 3 \, a^{2} b e f^{2} - a^{3} f^{3}\right )} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{{\left (d e f^{2} - c f^{3}\right )} \sqrt {e f}} + \frac {b^{3} d^{2} f^{2} x^{3} - 3 \, b^{3} d^{2} e f x - 3 \, b^{3} c d f^{2} x + 9 \, a b^{2} d^{2} f^{2} x}{3 \, d^{3} f^{3}} \] Input:

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

Output:

-(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*arctan(d*x/sqrt(c*d)) 
/((d^3*e - c*d^2*f)*sqrt(c*d)) + (b^3*e^3 - 3*a*b^2*e^2*f + 3*a^2*b*e*f^2 
- a^3*f^3)*arctan(f*x/sqrt(e*f))/((d*e*f^2 - c*f^3)*sqrt(e*f)) + 1/3*(b^3* 
d^2*f^2*x^3 - 3*b^3*d^2*e*f*x - 3*b^3*c*d*f^2*x + 9*a*b^2*d^2*f^2*x)/(d^3* 
f^3)
 

Mupad [B] (verification not implemented)

Time = 3.04 (sec) , antiderivative size = 5412, normalized size of antiderivative = 41.00 \[ \int \frac {\left (a+b x^2\right )^3}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

x*((3*a*b^2)/(d*f) - (b^3*(c*f + d*e))/(d^2*f^2)) + (b^3*x^3)/(3*d*f) + (a 
tan(((((((4*a^3*c^2*d^5*f^7 + 4*a^3*d^7*e^2*f^5 - 4*b^3*c^2*d^5*e^3*f^4 - 
4*b^3*c^3*d^4*e^2*f^5 - 8*a^3*c*d^6*e*f^6 + 4*b^3*c*d^6*e^4*f^3 + 4*b^3*c^ 
4*d^3*e*f^6 - 12*a*b^2*c*d^6*e^3*f^4 - 12*a*b^2*c^3*d^4*e*f^6 + 24*a*b^2*c 
^2*d^5*e^2*f^5)/(d^3*f^3) + (x*(-c*d^5)^(1/2)*(a*d - b*c)^3*(4*c^3*d^5*f^8 
 + 4*d^8*e^3*f^5 - 4*c*d^7*e^2*f^6 - 4*c^2*d^6*e*f^7))/(d^3*f^3*(c^2*d^5*f 
 - c*d^6*e)))*(-c*d^5)^(1/2)*(a*d - b*c)^3)/(2*(c^2*d^5*f - c*d^6*e)) + (2 
*x*(2*a^6*d^6*f^6 + b^6*c^6*f^6 + b^6*d^6*e^6 + 15*a^2*b^4*c^4*d^2*f^6 - 2 
0*a^3*b^3*c^3*d^3*f^6 + 15*a^4*b^2*c^2*d^4*f^6 + 15*a^2*b^4*d^6*e^4*f^2 - 
20*a^3*b^3*d^6*e^3*f^3 + 15*a^4*b^2*d^6*e^2*f^4 - 6*a*b^5*c^5*d*f^6 - 6*a^ 
5*b*c*d^5*f^6 - 6*a*b^5*d^6*e^5*f - 6*a^5*b*d^6*e*f^5))/(d^3*f^3))*(-c*d^5 
)^(1/2)*(a*d - b*c)^3*1i)/(2*(c^2*d^5*f - c*d^6*e)) - (((((4*a^3*c^2*d^5*f 
^7 + 4*a^3*d^7*e^2*f^5 - 4*b^3*c^2*d^5*e^3*f^4 - 4*b^3*c^3*d^4*e^2*f^5 - 8 
*a^3*c*d^6*e*f^6 + 4*b^3*c*d^6*e^4*f^3 + 4*b^3*c^4*d^3*e*f^6 - 12*a*b^2*c* 
d^6*e^3*f^4 - 12*a*b^2*c^3*d^4*e*f^6 + 24*a*b^2*c^2*d^5*e^2*f^5)/(d^3*f^3) 
 - (x*(-c*d^5)^(1/2)*(a*d - b*c)^3*(4*c^3*d^5*f^8 + 4*d^8*e^3*f^5 - 4*c*d^ 
7*e^2*f^6 - 4*c^2*d^6*e*f^7))/(d^3*f^3*(c^2*d^5*f - c*d^6*e)))*(-c*d^5)^(1 
/2)*(a*d - b*c)^3)/(2*(c^2*d^5*f - c*d^6*e)) - (2*x*(2*a^6*d^6*f^6 + b^6*c 
^6*f^6 + b^6*d^6*e^6 + 15*a^2*b^4*c^4*d^2*f^6 - 20*a^3*b^3*c^3*d^3*f^6 + 1 
5*a^4*b^2*c^2*d^4*f^6 + 15*a^2*b^4*d^6*e^4*f^2 - 20*a^3*b^3*d^6*e^3*f^3...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 354, normalized size of antiderivative = 2.68 \[ \int \frac {\left (a+b x^2\right )^3}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\frac {-3 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) a^{3} d^{3} e \,f^{3}+9 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) a^{2} b c \,d^{2} e \,f^{3}-9 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) a \,b^{2} c^{2} d e \,f^{3}+3 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) b^{3} c^{3} e \,f^{3}+3 \sqrt {f}\, \sqrt {e}\, \mathit {atan} \left (\frac {f x}{\sqrt {f}\, \sqrt {e}}\right ) a^{3} c \,d^{3} f^{3}-9 \sqrt {f}\, \sqrt {e}\, \mathit {atan} \left (\frac {f x}{\sqrt {f}\, \sqrt {e}}\right ) a^{2} b c \,d^{3} e \,f^{2}+9 \sqrt {f}\, \sqrt {e}\, \mathit {atan} \left (\frac {f x}{\sqrt {f}\, \sqrt {e}}\right ) a \,b^{2} c \,d^{3} e^{2} f -3 \sqrt {f}\, \sqrt {e}\, \mathit {atan} \left (\frac {f x}{\sqrt {f}\, \sqrt {e}}\right ) b^{3} c \,d^{3} e^{3}+9 a \,b^{2} c^{2} d^{2} e \,f^{3} x -9 a \,b^{2} c \,d^{3} e^{2} f^{2} x -3 b^{3} c^{3} d e \,f^{3} x +b^{3} c^{2} d^{2} e \,f^{3} x^{3}+3 b^{3} c \,d^{3} e^{3} f x -b^{3} c \,d^{3} e^{2} f^{2} x^{3}}{3 c \,d^{3} e \,f^{3} \left (c f -d e \right )} \] Input:

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

Output:

( - 3*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**3*d**3*e*f**3 + 9*s 
qrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2*b*c*d**2*e*f**3 - 9*sqrt 
(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a*b**2*c**2*d*e*f**3 + 3*sqrt(d) 
*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*b**3*c**3*e*f**3 + 3*sqrt(f)*sqrt(e 
)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**3*c*d**3*f**3 - 9*sqrt(f)*sqrt(e)*atan( 
(f*x)/(sqrt(f)*sqrt(e)))*a**2*b*c*d**3*e*f**2 + 9*sqrt(f)*sqrt(e)*atan((f* 
x)/(sqrt(f)*sqrt(e)))*a*b**2*c*d**3*e**2*f - 3*sqrt(f)*sqrt(e)*atan((f*x)/ 
(sqrt(f)*sqrt(e)))*b**3*c*d**3*e**3 + 9*a*b**2*c**2*d**2*e*f**3*x - 9*a*b* 
*2*c*d**3*e**2*f**2*x - 3*b**3*c**3*d*e*f**3*x + b**3*c**2*d**2*e*f**3*x** 
3 + 3*b**3*c*d**3*e**3*f*x - b**3*c*d**3*e**2*f**2*x**3)/(3*c*d**3*e*f**3* 
(c*f - d*e))