\(\int \frac {x^4}{(a+b x^2) (c+d x^2)^3} \, dx\) [646]

Optimal result

Integrand size = 22, antiderivative size = 157 \[ \int \frac {x^4}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=-\frac {c x}{4 d (b c-a d) \left (c+d x^2\right )^2}+\frac {(b c-5 a d) x}{8 d (b c-a d)^2 \left (c+d x^2\right )}+\frac {a^{3/2} \sqrt {b} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{(b c-a d)^3}+\frac {\left (b^2 c^2-6 a b c d-3 a^2 d^2\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {c} d^{3/2} (b c-a d)^3} \] Output:

-1/4*c*x/d/(-a*d+b*c)/(d*x^2+c)^2+1/8*(-5*a*d+b*c)*x/d/(-a*d+b*c)^2/(d*x^2 
+c)+a^(3/2)*b^(1/2)*arctan(b^(1/2)*x/a^(1/2))/(-a*d+b*c)^3+1/8*(-3*a^2*d^2 
-6*a*b*c*d+b^2*c^2)*arctan(d^(1/2)*x/c^(1/2))/c^(1/2)/d^(3/2)/(-a*d+b*c)^3
 

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.85 \[ \int \frac {x^4}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\frac {\frac {(-b c+a d) x \left (b c \left (c-d x^2\right )+a d \left (3 c+5 d x^2\right )\right )}{d \left (c+d x^2\right )^2}+8 a^{3/2} \sqrt {b} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )+\frac {\left (b^2 c^2-6 a b c d-3 a^2 d^2\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} d^{3/2}}}{8 (b c-a d)^3} \] Input:

Integrate[x^4/((a + b*x^2)*(c + d*x^2)^3),x]
 

Output:

(((-(b*c) + a*d)*x*(b*c*(c - d*x^2) + a*d*(3*c + 5*d*x^2)))/(d*(c + d*x^2) 
^2) + 8*a^(3/2)*Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]] + ((b^2*c^2 - 6*a*b*c* 
d - 3*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*d^(3/2)))/(8*(b*c - a 
*d)^3)
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.18, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {372, 402, 27, 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.

\(\displaystyle \int \frac {x^4}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx\)

\(\Big \downarrow \) 372

\(\displaystyle \frac {\int \frac {(b c-4 a d) x^2+a c}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}dx}{4 d (b c-a d)}-\frac {c x}{4 d \left (c+d x^2\right )^2 (b c-a d)}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\frac {\int \frac {c \left (b (b c-5 a d) x^2+a (b c+3 a d)\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )}dx}{2 c (b c-a d)}+\frac {x (b c-5 a d)}{2 \left (c+d x^2\right ) (b c-a d)}}{4 d (b c-a d)}-\frac {c x}{4 d \left (c+d x^2\right )^2 (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {b (b c-5 a d) x^2+a (b c+3 a d)}{\left (b x^2+a\right ) \left (d x^2+c\right )}dx}{2 (b c-a d)}+\frac {x (b c-5 a d)}{2 \left (c+d x^2\right ) (b c-a d)}}{4 d (b c-a d)}-\frac {c x}{4 d \left (c+d x^2\right )^2 (b c-a d)}\)

\(\Big \downarrow \) 397

\(\displaystyle \frac {\frac {\frac {\left (-3 a^2 d^2-6 a b c d+b^2 c^2\right ) \int \frac {1}{d x^2+c}dx}{b c-a d}+\frac {8 a^2 b d \int \frac {1}{b x^2+a}dx}{b c-a d}}{2 (b c-a d)}+\frac {x (b c-5 a d)}{2 \left (c+d x^2\right ) (b c-a d)}}{4 d (b c-a d)}-\frac {c x}{4 d \left (c+d x^2\right )^2 (b c-a d)}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {\frac {8 a^{3/2} \sqrt {b} d \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b c-a d}+\frac {\left (-3 a^2 d^2-6 a b c d+b^2 c^2\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} \sqrt {d} (b c-a d)}}{2 (b c-a d)}+\frac {x (b c-5 a d)}{2 \left (c+d x^2\right ) (b c-a d)}}{4 d (b c-a d)}-\frac {c x}{4 d \left (c+d x^2\right )^2 (b c-a d)}\)

Input:

Int[x^4/((a + b*x^2)*(c + d*x^2)^3),x]
 

Output:

-1/4*(c*x)/(d*(b*c - a*d)*(c + d*x^2)^2) + (((b*c - 5*a*d)*x)/(2*(b*c - a* 
d)*(c + d*x^2)) + ((8*a^(3/2)*Sqrt[b]*d*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(b*c 
- a*d) + ((b^2*c^2 - 6*a*b*c*d - 3*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/( 
Sqrt[c]*Sqrt[d]*(b*c - a*d)))/(2*(b*c - a*d)))/(4*d*(b*c - a*d))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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 372

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[(-a)*e^3*(e*x)^(m - 3)*(a + b*x^2)^(p + 1)*((c + d*x^2 
)^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] + Simp[e^4/(2*b*(b*c - a*d)*(p + 1 
))   Int[(e*x)^(m - 4)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[a*c*(m - 3) + 
 (a*d*(m + 2*q - 1) + 2*b*c*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, 
e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 3] && IntBinomialQ[a 
, b, c, d, e, m, 2, p, q, x]
 

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 = 0.70 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.98

method	result	size
default	\(-\frac {a^{2} b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\left (a d -b c \right )^{3} \sqrt {a b}}+\frac {\frac {\left (-\frac {5}{8} a^{2} d^{2}+\frac {3}{4} a b c d -\frac {1}{8} b^{2} c^{2}\right ) x^{3}-\frac {c \left (3 a^{2} d^{2}-2 a b c d -b^{2} c^{2}\right ) x}{8 d}}{\left (x^{2} d +c \right )^{2}}+\frac {\left (3 a^{2} d^{2}+6 a b c d -b^{2} c^{2}\right ) \arctan \left (\frac {x d}{\sqrt {c d}}\right )}{8 d \sqrt {c d}}}{\left (a d -b c \right )^{3}}\)	\(154\)
risch	\(\frac {-\frac {\left (5 a d -b c \right ) x^{3}}{8 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {c \left (3 a d +b c \right ) x}{8 d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (x^{2} d +c \right )^{2}}+\frac {\sqrt {-a b}\, a \ln \left (\left (-64 \left (-a b \right )^{\frac {3}{2}} a^{3} d^{4}-64 \left (-a b \right )^{\frac {3}{2}} a^{2} b c \,d^{3}-73 a^{4} \sqrt {-a b}\, d^{4} b -36 \sqrt {-a b}\, a^{3} b^{2} c \,d^{3}-30 \sqrt {-a b}\, a^{2} b^{3} c^{2} d^{2}+12 \sqrt {-a b}\, a \,b^{4} c^{3} d -b^{5} c^{4} \sqrt {-a b}\right ) x -9 a^{5} b \,d^{4}+28 a^{4} b^{2} c \,d^{3}-30 a^{3} b^{3} c^{2} d^{2}+12 a^{2} b^{4} c^{3} d -a \,b^{5} c^{4}\right )}{2 \left (a d -b c \right )^{3}}-\frac {\sqrt {-a b}\, a \ln \left (\left (64 \left (-a b \right )^{\frac {3}{2}} a^{3} d^{4}+64 \left (-a b \right )^{\frac {3}{2}} a^{2} b c \,d^{3}+73 a^{4} \sqrt {-a b}\, d^{4} b +36 \sqrt {-a b}\, a^{3} b^{2} c \,d^{3}+30 \sqrt {-a b}\, a^{2} b^{3} c^{2} d^{2}-12 \sqrt {-a b}\, a \,b^{4} c^{3} d +b^{5} c^{4} \sqrt {-a b}\right ) x -9 a^{5} b \,d^{4}+28 a^{4} b^{2} c \,d^{3}-30 a^{3} b^{3} c^{2} d^{2}+12 a^{2} b^{4} c^{3} d -a \,b^{5} c^{4}\right )}{2 \left (a d -b c \right )^{3}}-\frac {3 d \ln \left (c \,d^{2} x -\left (-c d \right )^{\frac {3}{2}}\right ) a^{2}}{16 \sqrt {-c d}\, \left (a d -b c \right )^{3}}-\frac {3 \ln \left (c \,d^{2} x -\left (-c d \right )^{\frac {3}{2}}\right ) a b c}{8 \sqrt {-c d}\, \left (a d -b c \right )^{3}}+\frac {\ln \left (c \,d^{2} x -\left (-c d \right )^{\frac {3}{2}}\right ) b^{2} c^{2}}{16 \sqrt {-c d}\, \left (a d -b c \right )^{3} d}+\frac {3 d \ln \left (-c \,d^{2} x -\left (-c d \right )^{\frac {3}{2}}\right ) a^{2}}{16 \sqrt {-c d}\, \left (a d -b c \right )^{3}}+\frac {3 \ln \left (-c \,d^{2} x -\left (-c d \right )^{\frac {3}{2}}\right ) a b c}{8 \sqrt {-c d}\, \left (a d -b c \right )^{3}}-\frac {\ln \left (-c \,d^{2} x -\left (-c d \right )^{\frac {3}{2}}\right ) b^{2} c^{2}}{16 \sqrt {-c d}\, \left (a d -b c \right )^{3} d}\)	\(712\)

Input:

int(x^4/(b*x^2+a)/(d*x^2+c)^3,x,method=_RETURNVERBOSE)
 

Output:

-a^2*b/(a*d-b*c)^3/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))+1/(a*d-b*c)^3*(((-5 
/8*a^2*d^2+3/4*a*b*c*d-1/8*b^2*c^2)*x^3-1/8*c*(3*a^2*d^2-2*a*b*c*d-b^2*c^2 
)/d*x)/(d*x^2+c)^2+1/8*(3*a^2*d^2+6*a*b*c*d-b^2*c^2)/d/(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. 373 vs. \(2 (135) = 270\).

Time = 0.32 (sec) , antiderivative size = 1573, normalized size of antiderivative = 10.02 \[ \int \frac {x^4}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/16*(2*(b^2*c^3*d^2 - 6*a*b*c^2*d^3 + 5*a^2*c*d^4)*x^3 - 8*(a*c*d^4*x^4 
+ 2*a*c^2*d^3*x^2 + a*c^3*d^2)*sqrt(-a*b)*log((b*x^2 - 2*sqrt(-a*b)*x - a) 
/(b*x^2 + a)) - (b^2*c^4 - 6*a*b*c^3*d - 3*a^2*c^2*d^2 + (b^2*c^2*d^2 - 6* 
a*b*c*d^3 - 3*a^2*d^4)*x^4 + 2*(b^2*c^3*d - 6*a*b*c^2*d^2 - 3*a^2*c*d^3)*x 
^2)*sqrt(-c*d)*log((d*x^2 - 2*sqrt(-c*d)*x - c)/(d*x^2 + c)) - 2*(b^2*c^4* 
d + 2*a*b*c^3*d^2 - 3*a^2*c^2*d^3)*x)/(b^3*c^6*d^2 - 3*a*b^2*c^5*d^3 + 3*a 
^2*b*c^4*d^4 - a^3*c^3*d^5 + (b^3*c^4*d^4 - 3*a*b^2*c^3*d^5 + 3*a^2*b*c^2* 
d^6 - a^3*c*d^7)*x^4 + 2*(b^3*c^5*d^3 - 3*a*b^2*c^4*d^4 + 3*a^2*b*c^3*d^5 
- a^3*c^2*d^6)*x^2), 1/8*((b^2*c^3*d^2 - 6*a*b*c^2*d^3 + 5*a^2*c*d^4)*x^3 
+ (b^2*c^4 - 6*a*b*c^3*d - 3*a^2*c^2*d^2 + (b^2*c^2*d^2 - 6*a*b*c*d^3 - 3* 
a^2*d^4)*x^4 + 2*(b^2*c^3*d - 6*a*b*c^2*d^2 - 3*a^2*c*d^3)*x^2)*sqrt(c*d)* 
arctan(sqrt(c*d)*x/c) - 4*(a*c*d^4*x^4 + 2*a*c^2*d^3*x^2 + a*c^3*d^2)*sqrt 
(-a*b)*log((b*x^2 - 2*sqrt(-a*b)*x - a)/(b*x^2 + a)) - (b^2*c^4*d + 2*a*b* 
c^3*d^2 - 3*a^2*c^2*d^3)*x)/(b^3*c^6*d^2 - 3*a*b^2*c^5*d^3 + 3*a^2*b*c^4*d 
^4 - a^3*c^3*d^5 + (b^3*c^4*d^4 - 3*a*b^2*c^3*d^5 + 3*a^2*b*c^2*d^6 - a^3* 
c*d^7)*x^4 + 2*(b^3*c^5*d^3 - 3*a*b^2*c^4*d^4 + 3*a^2*b*c^3*d^5 - a^3*c^2* 
d^6)*x^2), 1/16*(2*(b^2*c^3*d^2 - 6*a*b*c^2*d^3 + 5*a^2*c*d^4)*x^3 + 16*(a 
*c*d^4*x^4 + 2*a*c^2*d^3*x^2 + a*c^3*d^2)*sqrt(a*b)*arctan(sqrt(a*b)*x/a) 
- (b^2*c^4 - 6*a*b*c^3*d - 3*a^2*c^2*d^2 + (b^2*c^2*d^2 - 6*a*b*c*d^3 - 3* 
a^2*d^4)*x^4 + 2*(b^2*c^3*d - 6*a*b*c^2*d^2 - 3*a^2*c*d^3)*x^2)*sqrt(-c...
 

Sympy [F(-1)]

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

integrate(x**4/(b*x**2+a)/(d*x**2+c)**3,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.68 \[ \int \frac {x^4}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\frac {a^{2} b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {a b}} + \frac {{\left (b^{2} c^{2} - 6 \, a b c d - 3 \, a^{2} d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} \sqrt {c d}} + \frac {{\left (b c d - 5 \, a d^{2}\right )} x^{3} - {\left (b c^{2} + 3 \, a c d\right )} x}{8 \, {\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3} + {\left (b^{2} c^{2} d^{3} - 2 \, a b c d^{4} + a^{2} d^{5}\right )} x^{4} + 2 \, {\left (b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x^{2}\right )}} \] Input:

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

Output:

a^2*b*arctan(b*x/sqrt(a*b))/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^ 
3*d^3)*sqrt(a*b)) + 1/8*(b^2*c^2 - 6*a*b*c*d - 3*a^2*d^2)*arctan(d*x/sqrt( 
c*d))/((b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*sqrt(c*d)) 
+ 1/8*((b*c*d - 5*a*d^2)*x^3 - (b*c^2 + 3*a*c*d)*x)/(b^2*c^4*d - 2*a*b*c^3 
*d^2 + a^2*c^2*d^3 + (b^2*c^2*d^3 - 2*a*b*c*d^4 + a^2*d^5)*x^4 + 2*(b^2*c^ 
3*d^2 - 2*a*b*c^2*d^3 + a^2*c*d^4)*x^2)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.30 \[ \int \frac {x^4}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\frac {a^{2} b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {a b}} + \frac {{\left (b^{2} c^{2} - 6 \, a b c d - 3 \, a^{2} d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} \sqrt {c d}} + \frac {b c d x^{3} - 5 \, a d^{2} x^{3} - b c^{2} x - 3 \, a c d x}{8 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} {\left (d x^{2} + c\right )}^{2}} \] Input:

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

Output:

a^2*b*arctan(b*x/sqrt(a*b))/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^ 
3*d^3)*sqrt(a*b)) + 1/8*(b^2*c^2 - 6*a*b*c*d - 3*a^2*d^2)*arctan(d*x/sqrt( 
c*d))/((b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*sqrt(c*d)) 
+ 1/8*(b*c*d*x^3 - 5*a*d^2*x^3 - b*c^2*x - 3*a*c*d*x)/((b^2*c^2*d - 2*a*b* 
c*d^2 + a^2*d^3)*(d*x^2 + c)^2)
 

Mupad [B] (verification not implemented)

Time = 2.33 (sec) , antiderivative size = 5754, normalized size of antiderivative = 36.65 \[ \int \frac {x^4}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \] Input:

int(x^4/((a + b*x^2)*(c + d*x^2)^3),x)
 

Output:

(atan((((-c*d^3)^(1/2)*((x*(b^7*c^4 + 73*a^4*b^3*d^4 + 36*a^3*b^4*c*d^3 + 
30*a^2*b^5*c^2*d^2 - 12*a*b^6*c^3*d))/(32*(a^4*d^5 + b^4*c^4*d - 4*a*b^3*c 
^3*d^2 + 6*a^2*b^2*c^2*d^3 - 4*a^3*b*c*d^4)) - (((96*a^8*b^2*d^9 + 32*a*b^ 
9*c^7*d^2 - 544*a^7*b^3*c*d^8 - 96*a^2*b^8*c^6*d^3 - 96*a^3*b^7*c^5*d^4 + 
800*a^4*b^6*c^4*d^5 - 1440*a^5*b^5*c^3*d^6 + 1248*a^6*b^4*c^2*d^7)/(64*(a^ 
6*d^7 + b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 20*a^3*b^3*c^3* 
d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6)) - (x*(-c*d^3)^(1/2)*(3*a^2*d^2 
- b^2*c^2 + 6*a*b*c*d)*(256*a^7*b^2*d^10 + 256*b^9*c^7*d^3 - 1280*a*b^8*c^ 
6*d^4 - 1280*a^6*b^3*c*d^9 + 2304*a^2*b^7*c^5*d^5 - 1280*a^3*b^6*c^4*d^6 - 
 1280*a^4*b^5*c^3*d^7 + 2304*a^5*b^4*c^2*d^8))/(512*(a^3*c*d^6 - b^3*c^4*d 
^3 + 3*a*b^2*c^3*d^4 - 3*a^2*b*c^2*d^5)*(a^4*d^5 + b^4*c^4*d - 4*a*b^3*c^3 
*d^2 + 6*a^2*b^2*c^2*d^3 - 4*a^3*b*c*d^4)))*(-c*d^3)^(1/2)*(3*a^2*d^2 - b^ 
2*c^2 + 6*a*b*c*d))/(16*(a^3*c*d^6 - b^3*c^4*d^3 + 3*a*b^2*c^3*d^4 - 3*a^2 
*b*c^2*d^5)))*(3*a^2*d^2 - b^2*c^2 + 6*a*b*c*d)*1i)/(16*(a^3*c*d^6 - b^3*c 
^4*d^3 + 3*a*b^2*c^3*d^4 - 3*a^2*b*c^2*d^5)) + ((-c*d^3)^(1/2)*((x*(b^7*c^ 
4 + 73*a^4*b^3*d^4 + 36*a^3*b^4*c*d^3 + 30*a^2*b^5*c^2*d^2 - 12*a*b^6*c^3* 
d))/(32*(a^4*d^5 + b^4*c^4*d - 4*a*b^3*c^3*d^2 + 6*a^2*b^2*c^2*d^3 - 4*a^3 
*b*c*d^4)) + (((96*a^8*b^2*d^9 + 32*a*b^9*c^7*d^2 - 544*a^7*b^3*c*d^8 - 96 
*a^2*b^8*c^6*d^3 - 96*a^3*b^7*c^5*d^4 + 800*a^4*b^6*c^4*d^5 - 1440*a^5*b^5 
*c^3*d^6 + 1248*a^6*b^4*c^2*d^7)/(64*(a^6*d^7 + b^6*c^6*d - 6*a*b^5*c^5...
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 552, normalized size of antiderivative = 3.52 \[ \int \frac {x^4}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\frac {-8 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,c^{3} d^{2}-16 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,c^{2} d^{3} x^{2}-8 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a c \,d^{4} x^{4}+3 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) a^{2} c^{2} d^{2}+6 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) a^{2} c \,d^{3} x^{2}+3 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) a^{2} d^{4} x^{4}+6 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) a b \,c^{3} d +12 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) a b \,c^{2} d^{2} x^{2}+6 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) a b c \,d^{3} x^{4}-\sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) b^{2} c^{4}-2 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) b^{2} c^{3} d \,x^{2}-\sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) b^{2} c^{2} d^{2} x^{4}-3 a^{2} c^{2} d^{3} x -5 a^{2} c \,d^{4} x^{3}+2 a b \,c^{3} d^{2} x +6 a b \,c^{2} d^{3} x^{3}+b^{2} c^{4} d x -b^{2} c^{3} d^{2} x^{3}}{8 c \,d^{2} \left (a^{3} d^{5} x^{4}-3 a^{2} b c \,d^{4} x^{4}+3 a \,b^{2} c^{2} d^{3} x^{4}-b^{3} c^{3} d^{2} x^{4}+2 a^{3} c \,d^{4} x^{2}-6 a^{2} b \,c^{2} d^{3} x^{2}+6 a \,b^{2} c^{3} d^{2} x^{2}-2 b^{3} c^{4} d \,x^{2}+a^{3} c^{2} d^{3}-3 a^{2} b \,c^{3} d^{2}+3 a \,b^{2} c^{4} d -b^{3} c^{5}\right )} \] Input:

int(x^4/(b*x^2+a)/(d*x^2+c)^3,x)
 

Output:

( - 8*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*c**3*d**2 - 16*sqrt( 
b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*c**2*d**3*x**2 - 8*sqrt(b)*sqrt 
(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*c*d**4*x**4 + 3*sqrt(d)*sqrt(c)*atan(( 
d*x)/(sqrt(d)*sqrt(c)))*a**2*c**2*d**2 + 6*sqrt(d)*sqrt(c)*atan((d*x)/(sqr 
t(d)*sqrt(c)))*a**2*c*d**3*x**2 + 3*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sq 
rt(c)))*a**2*d**4*x**4 + 6*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a 
*b*c**3*d + 12*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a*b*c**2*d**2 
*x**2 + 6*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a*b*c*d**3*x**4 - 
sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*b**2*c**4 - 2*sqrt(d)*sqrt(c 
)*atan((d*x)/(sqrt(d)*sqrt(c)))*b**2*c**3*d*x**2 - sqrt(d)*sqrt(c)*atan((d 
*x)/(sqrt(d)*sqrt(c)))*b**2*c**2*d**2*x**4 - 3*a**2*c**2*d**3*x - 5*a**2*c 
*d**4*x**3 + 2*a*b*c**3*d**2*x + 6*a*b*c**2*d**3*x**3 + b**2*c**4*d*x - b* 
*2*c**3*d**2*x**3)/(8*c*d**2*(a**3*c**2*d**3 + 2*a**3*c*d**4*x**2 + a**3*d 
**5*x**4 - 3*a**2*b*c**3*d**2 - 6*a**2*b*c**2*d**3*x**2 - 3*a**2*b*c*d**4* 
x**4 + 3*a*b**2*c**4*d + 6*a*b**2*c**3*d**2*x**2 + 3*a*b**2*c**2*d**3*x**4 
 - b**3*c**5 - 2*b**3*c**4*d*x**2 - b**3*c**3*d**2*x**4))