\(\int \frac {x^5}{a+\frac {b}{(c+d x^2)^2}} \, dx\) [234]

Optimal result

Integrand size = 19, antiderivative size = 131 \[ \int \frac {x^5}{a+\frac {b}{\left (c+d x^2\right )^2}} \, dx=-\frac {\left (b-a c^2\right ) x^2}{2 a^2 d^2}-\frac {c \left (c+d x^2\right )^2}{2 a d^3}+\frac {\left (c+d x^2\right )^3}{6 a d^3}+\frac {\sqrt {b} \left (b-a c^2\right ) \arctan \left (\frac {\sqrt {a} \left (c+d x^2\right )}{\sqrt {b}}\right )}{2 a^{5/2} d^3}+\frac {b c \log \left (b+a \left (c+d x^2\right )^2\right )}{2 a^2 d^3} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.12 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.05 \[ \int \frac {x^5}{a+\frac {b}{\left (c+d x^2\right )^2}} \, dx=\frac {2 \sqrt {a} d x^2 \left (-3 b+a d^2 x^4\right )-3 i \sqrt {b} \left (\sqrt {b}+i \sqrt {a} c\right )^2 \log \left (-i \sqrt {b}+\sqrt {a} \left (c+d x^2\right )\right )+3 i \sqrt {b} \left (\sqrt {b}-i \sqrt {a} c\right )^2 \log \left (i \sqrt {b}+\sqrt {a} \left (c+d x^2\right )\right )}{12 a^{5/2} d^3} \] Input:

Integrate[x^5/(a + b/(c + d*x^2)^2),x]
 

Output:

(2*Sqrt[a]*d*x^2*(-3*b + a*d^2*x^4) - (3*I)*Sqrt[b]*(Sqrt[b] + I*Sqrt[a]*c 
)^2*Log[(-I)*Sqrt[b] + Sqrt[a]*(c + d*x^2)] + (3*I)*Sqrt[b]*(Sqrt[b] - I*S 
qrt[a]*c)^2*Log[I*Sqrt[b] + Sqrt[a]*(c + d*x^2)])/(12*a^(5/2)*d^3)
 

Rubi [A] (warning: unable to verify)

Time = 0.70 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.78, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {7283, 896, 1776, 525, 25, 523, 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.

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

\(\Big \downarrow \) 7283

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

\(\Big \downarrow \) 896

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

\(\Big \downarrow \) 1776

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

\(\Big \downarrow \) 525

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 523

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

\(\Big \downarrow \) 2009

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

Input:

Int[x^5/(a + b/(c + d*x^2)^2),x]
 

Output:

(x^6/(3*a) - (c*x^4 + (b/a - c^2)*(c + d*x^2) - (Sqrt[b]*(b - a*c^2)*ArcTa 
n[(Sqrt[a]*(c + d*x^2))/Sqrt[b]])/a^(3/2) - (b*c*Log[b + a*x^4])/a)/a)/(2* 
d^3)
 

Defintions of rubi rules used

rule 25

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

rule 523

Int[((x_)^(m_.)*((c_) + (d_.)*(x_)))/((a_) + (b_.)*(x_)^2), x_Symbol] :> In 
t[ExpandIntegrand[x^m*((c + d*x)/(a + b*x^2)), x], x] /; FreeQ[{a, b, c, d} 
, x] && IntegerQ[m]
 

rule 525

Int[((x_)^(m_.)*((c_) + (d_.)*(x_))^(n_))/((a_) + (b_.)*(x_)^2), x_Symbol] 
:> Simp[d^n*(x^(m + n - 1)/(b*(m + n - 1))), x] + Simp[1/b   Int[x^m*(Expan 
dToSum[b*(c + d*x)^n - b*d^n*x^n - a*d^n*x^(n - 2), x]/(a + b*x^2)), x], x] 
 /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 1] && IGtQ[m, -2] && NeQ[m + n - 1, 0 
]
 

rule 896

Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coeff 
icient[v, x, 0], d = Coefficient[v, x, 1]}, Simp[1/d^(m + 1)   Subst[Int[Si 
mplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]] /; 
 FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]
 

rule 1776

Int[((a_.) + (c_.)*(x_)^(mn2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_.))^(q_.), x_S 
ymbol] :> Int[((d + e*x^n)^q*(c + a*x^(2*n))^p)/x^(2*n*p), x] /; FreeQ[{a, 
c, d, e, n, q}, x] && EqQ[mn2, -2*n] && IntegerQ[p]
 

rule 2009

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

rule 7283

Int[(u_)*(x_)^(m_.), x_Symbol] :> With[{lst = PowerVariableExpn[u, m + 1, x 
]}, Simp[1/lst[[2]]   Subst[Int[NormalizeIntegrand[Simplify[lst[[1]]/x], x] 
, x], x, (lst[[3]]*x)^lst[[2]]], x] /;  !FalseQ[lst] && NeQ[lst[[2]], m + 1 
]] /; IntegerQ[m] && NeQ[m, -1] && NonsumQ[u] && (GtQ[m, 0] ||  !AlgebraicF 
unctionQ[u, x])
 

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.82

method	result	size
default	\(\frac {\frac {1}{3} a \,x^{6} d^{2}-b \,x^{2}}{2 a^{2} d^{2}}+\frac {b \left (\frac {c \ln \left (a \,d^{2} x^{4}+2 a d \,x^{2} c +a \,c^{2}+b \right )}{d}+\frac {\left (-a \,c^{2}+b \right ) \arctan \left (\frac {2 a \,d^{2} x^{2}+2 a c d}{2 d \sqrt {a b}}\right )}{d \sqrt {a b}}\right )}{2 a^{2} d^{2}}\)	\(108\)
risch	\(\frac {x^{6}}{6 a}-\frac {b \,x^{2}}{2 a^{2} d^{2}}+\frac {\ln \left (\left (a^{2} b \,c^{2} d -\sqrt {-a b \left (a \,c^{2}-b \right )^{2}}\, a c d -d a \,b^{2}\right ) x^{2}-\sqrt {-a b \left (a \,c^{2}-b \right )^{2}}\, a \,c^{2}-\sqrt {-a b \left (a \,c^{2}-b \right )^{2}}\, b \right ) b c}{2 a^{2} d^{3}}+\frac {\ln \left (\left (a^{2} b \,c^{2} d -\sqrt {-a b \left (a \,c^{2}-b \right )^{2}}\, a c d -d a \,b^{2}\right ) x^{2}-\sqrt {-a b \left (a \,c^{2}-b \right )^{2}}\, a \,c^{2}-\sqrt {-a b \left (a \,c^{2}-b \right )^{2}}\, b \right ) \sqrt {-a b \left (a \,c^{2}-b \right )^{2}}}{4 a^{3} d^{3}}+\frac {\ln \left (\left (a^{2} b \,c^{2} d +\sqrt {-a b \left (a \,c^{2}-b \right )^{2}}\, a c d -d a \,b^{2}\right ) x^{2}+\sqrt {-a b \left (a \,c^{2}-b \right )^{2}}\, a \,c^{2}+\sqrt {-a b \left (a \,c^{2}-b \right )^{2}}\, b \right ) b c}{2 a^{2} d^{3}}-\frac {\ln \left (\left (a^{2} b \,c^{2} d +\sqrt {-a b \left (a \,c^{2}-b \right )^{2}}\, a c d -d a \,b^{2}\right ) x^{2}+\sqrt {-a b \left (a \,c^{2}-b \right )^{2}}\, a \,c^{2}+\sqrt {-a b \left (a \,c^{2}-b \right )^{2}}\, b \right ) \sqrt {-a b \left (a \,c^{2}-b \right )^{2}}}{4 a^{3} d^{3}}\)	\(438\)

Input:

int(x^5/(a+b/(d*x^2+c)^2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.81 \[ \int \frac {x^5}{a+\frac {b}{\left (c+d x^2\right )^2}} \, dx=\left [\frac {2 \, a d^{3} x^{6} - 6 \, b d x^{2} + 6 \, b c \log \left (a d^{2} x^{4} + 2 \, a c d x^{2} + a c^{2} + b\right ) + 3 \, {\left (a c^{2} - b\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {a d^{2} x^{4} + 2 \, a c d x^{2} + a c^{2} - 2 \, {\left (a d x^{2} + a c\right )} \sqrt {-\frac {b}{a}} - b}{a d^{2} x^{4} + 2 \, a c d x^{2} + a c^{2} + b}\right )}{12 \, a^{2} d^{3}}, \frac {a d^{3} x^{6} - 3 \, b d x^{2} + 3 \, b c \log \left (a d^{2} x^{4} + 2 \, a c d x^{2} + a c^{2} + b\right ) - 3 \, {\left (a c^{2} - b\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {{\left (a d x^{2} + a c\right )} \sqrt {\frac {b}{a}}}{b}\right )}{6 \, a^{2} d^{3}}\right ] \] Input:

integrate(x^5/(a+b/(d*x^2+c)^2),x, algorithm="fricas")
 

Output:

[1/12*(2*a*d^3*x^6 - 6*b*d*x^2 + 6*b*c*log(a*d^2*x^4 + 2*a*c*d*x^2 + a*c^2 
 + b) + 3*(a*c^2 - b)*sqrt(-b/a)*log((a*d^2*x^4 + 2*a*c*d*x^2 + a*c^2 - 2* 
(a*d*x^2 + a*c)*sqrt(-b/a) - b)/(a*d^2*x^4 + 2*a*c*d*x^2 + a*c^2 + b)))/(a 
^2*d^3), 1/6*(a*d^3*x^6 - 3*b*d*x^2 + 3*b*c*log(a*d^2*x^4 + 2*a*c*d*x^2 + 
a*c^2 + b) - 3*(a*c^2 - b)*sqrt(b/a)*arctan((a*d*x^2 + a*c)*sqrt(b/a)/b))/ 
(a^2*d^3)]
 

Sympy [A] (verification not implemented)

Time = 0.93 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.78 \[ \int \frac {x^5}{a+\frac {b}{\left (c+d x^2\right )^2}} \, dx=\left (\frac {b c}{2 a^{2} d^{3}} - \frac {\sqrt {- a^{5} b} \left (a c^{2} - b\right )}{4 a^{5} d^{3}}\right ) \log {\left (x^{2} + \frac {- 4 a^{2} d^{3} \left (\frac {b c}{2 a^{2} d^{3}} - \frac {\sqrt {- a^{5} b} \left (a c^{2} - b\right )}{4 a^{5} d^{3}}\right ) + a c^{3} + b c}{a c^{2} d - b d} \right )} + \left (\frac {b c}{2 a^{2} d^{3}} + \frac {\sqrt {- a^{5} b} \left (a c^{2} - b\right )}{4 a^{5} d^{3}}\right ) \log {\left (x^{2} + \frac {- 4 a^{2} d^{3} \left (\frac {b c}{2 a^{2} d^{3}} + \frac {\sqrt {- a^{5} b} \left (a c^{2} - b\right )}{4 a^{5} d^{3}}\right ) + a c^{3} + b c}{a c^{2} d - b d} \right )} + \frac {x^{6}}{6 a} - \frac {b x^{2}}{2 a^{2} d^{2}} \] Input:

integrate(x**5/(a+b/(d*x**2+c)**2),x)
 

Output:

(b*c/(2*a**2*d**3) - sqrt(-a**5*b)*(a*c**2 - b)/(4*a**5*d**3))*log(x**2 + 
(-4*a**2*d**3*(b*c/(2*a**2*d**3) - sqrt(-a**5*b)*(a*c**2 - b)/(4*a**5*d**3 
)) + a*c**3 + b*c)/(a*c**2*d - b*d)) + (b*c/(2*a**2*d**3) + sqrt(-a**5*b)* 
(a*c**2 - b)/(4*a**5*d**3))*log(x**2 + (-4*a**2*d**3*(b*c/(2*a**2*d**3) + 
sqrt(-a**5*b)*(a*c**2 - b)/(4*a**5*d**3)) + a*c**3 + b*c)/(a*c**2*d - b*d) 
) + x**6/(6*a) - b*x**2/(2*a**2*d**2)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.81 \[ \int \frac {x^5}{a+\frac {b}{\left (c+d x^2\right )^2}} \, dx=\frac {b c \log \left (a d^{2} x^{4} + 2 \, a c d x^{2} + a c^{2} + b\right )}{2 \, a^{2} d^{3}} + \frac {a d^{2} x^{6} - 3 \, b x^{2}}{6 \, a^{2} d^{2}} - \frac {{\left (a b c^{2} - b^{2}\right )} \arctan \left (\frac {a d^{2} x^{2} + a c d}{\sqrt {a b} d}\right )}{2 \, \sqrt {a b} a^{2} d^{3}} \] Input:

integrate(x^5/(a+b/(d*x^2+c)^2),x, algorithm="maxima")
 

Output:

1/2*b*c*log(a*d^2*x^4 + 2*a*c*d*x^2 + a*c^2 + b)/(a^2*d^3) + 1/6*(a*d^2*x^ 
6 - 3*b*x^2)/(a^2*d^2) - 1/2*(a*b*c^2 - b^2)*arctan((a*d^2*x^2 + a*c*d)/(s 
qrt(a*b)*d))/(sqrt(a*b)*a^2*d^3)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x^5}{a+\frac {b}{\left (c+d x^2\right )^2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^5/(a+b/(d*x^2+c)^2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{%%{[1,0]:[1,0,%%%{1,[1,1]%%%}]%%},[0,1]%%%}+%%%{%%%{1,[0,1 
]%%%},[0,
 

Mupad [B] (verification not implemented)

Time = 9.05 (sec) , antiderivative size = 633, normalized size of antiderivative = 4.83 \[ \int \frac {x^5}{a+\frac {b}{\left (c+d x^2\right )^2}} \, dx=\frac {x^6}{6\,a}-x^2\,\left (\frac {a\,c^2+b}{2\,a^2\,d^2}-\frac {c^2}{2\,a\,d^2}\right )+\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {a^3\,d^2\,\left (x^2\,\left (\frac {\frac {2\,b^{3/2}\,c^2\,\left (b-a\,c^2\right )}{a^{3/2}\,d}+\frac {\sqrt {b}\,\left (b-a\,c^2\right )\,\left (\frac {2\,\left (2\,a^2\,b^2\,d^3-10\,a^3\,b\,c^2\,d^3\right )}{a^2\,d}+8\,a\,b\,c^2\,d^2\right )}{4\,a^{5/2}\,d^3}}{a\,c^2+b}+\frac {\sqrt {a}\,c\,\left (\frac {2\,\left (b^3\,c-3\,a\,b^2\,c^3\right )}{a^2\,d}+\frac {b\,c\,{\left (b-a\,c^2\right )}^2}{a^2\,d}-\frac {b\,c\,\left (\frac {2\,\left (2\,a^2\,b^2\,d^3-10\,a^3\,b\,c^2\,d^3\right )}{a^2\,d}+8\,a\,b\,c^2\,d^2\right )}{2\,a^2\,d^3}\right )}{\sqrt {b}\,\left (a\,c^2+b\right )}\right )-\frac {\frac {\sqrt {b}\,\left (b-a\,c^2\right )\,\left (\frac {4\,\left (4\,a^3\,b\,c^3\,d^3+4\,a^2\,b^2\,c\,d^3\right )}{a^2\,d^2}-\frac {2\,b\,c\,\left (4\,a^5\,c^2\,d^6+4\,b\,a^4\,d^6\right )}{a^4\,d^5}\right )}{4\,a^{5/2}\,d^3}-\frac {b^{3/2}\,c\,\left (b-a\,c^2\right )\,\left (4\,a^5\,c^2\,d^6+4\,b\,a^4\,d^6\right )}{2\,a^{13/2}\,d^8}}{a\,c^2+b}+\frac {\sqrt {a}\,c\,\left (\frac {b\,{\left (b-a\,c^2\right )}^2\,\left (4\,a^5\,c^2\,d^6+4\,b\,a^4\,d^6\right )}{4\,a^7\,d^8}-\frac {4\,\left (b^3\,c^2+a\,b^2\,c^4\right )}{a^2\,d^2}+\frac {b\,c\,\left (\frac {4\,\left (4\,a^3\,b\,c^3\,d^3+4\,a^2\,b^2\,c\,d^3\right )}{a^2\,d^2}-\frac {2\,b\,c\,\left (4\,a^5\,c^2\,d^6+4\,b\,a^4\,d^6\right )}{a^4\,d^5}\right )}{2\,a^2\,d^3}\right )}{\sqrt {b}\,\left (a\,c^2+b\right )}\right )}{a^2\,b\,c^4-2\,a\,b^2\,c^2+b^3}\right )\,\left (b-a\,c^2\right )}{2\,a^{5/2}\,d^3}+\frac {b\,c\,\ln \left (a\,c^2+2\,a\,c\,d\,x^2+a\,d^2\,x^4+b\right )}{2\,a^2\,d^3} \] Input:

int(x^5/(a + b/(c + d*x^2)^2),x)
 

Output:

x^6/(6*a) - x^2*((b + a*c^2)/(2*a^2*d^2) - c^2/(2*a*d^2)) + (b^(1/2)*atan( 
(a^3*d^2*(x^2*(((2*b^(3/2)*c^2*(b - a*c^2))/(a^(3/2)*d) + (b^(1/2)*(b - a* 
c^2)*((2*(2*a^2*b^2*d^3 - 10*a^3*b*c^2*d^3))/(a^2*d) + 8*a*b*c^2*d^2))/(4* 
a^(5/2)*d^3))/(b + a*c^2) + (a^(1/2)*c*((2*(b^3*c - 3*a*b^2*c^3))/(a^2*d) 
+ (b*c*(b - a*c^2)^2)/(a^2*d) - (b*c*((2*(2*a^2*b^2*d^3 - 10*a^3*b*c^2*d^3 
))/(a^2*d) + 8*a*b*c^2*d^2))/(2*a^2*d^3)))/(b^(1/2)*(b + a*c^2))) - ((b^(1 
/2)*(b - a*c^2)*((4*(4*a^2*b^2*c*d^3 + 4*a^3*b*c^3*d^3))/(a^2*d^2) - (2*b* 
c*(4*a^4*b*d^6 + 4*a^5*c^2*d^6))/(a^4*d^5)))/(4*a^(5/2)*d^3) - (b^(3/2)*c* 
(b - a*c^2)*(4*a^4*b*d^6 + 4*a^5*c^2*d^6))/(2*a^(13/2)*d^8))/(b + a*c^2) + 
 (a^(1/2)*c*((b*(b - a*c^2)^2*(4*a^4*b*d^6 + 4*a^5*c^2*d^6))/(4*a^7*d^8) - 
 (4*(b^3*c^2 + a*b^2*c^4))/(a^2*d^2) + (b*c*((4*(4*a^2*b^2*c*d^3 + 4*a^3*b 
*c^3*d^3))/(a^2*d^2) - (2*b*c*(4*a^4*b*d^6 + 4*a^5*c^2*d^6))/(a^4*d^5)))/( 
2*a^2*d^3)))/(b^(1/2)*(b + a*c^2))))/(b^3 - 2*a*b^2*c^2 + a^2*b*c^4))*(b - 
 a*c^2))/(2*a^(5/2)*d^3) + (b*c*log(b + a*c^2 + a*d^2*x^4 + 2*a*c*d*x^2))/ 
(2*a^2*d^3)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 496, normalized size of antiderivative = 3.79 \[ \int \frac {x^5}{a+\frac {b}{\left (c+d x^2\right )^2}} \, dx=\frac {3 \sqrt {\sqrt {a}\, \sqrt {a \,c^{2}+b}+a c}\, \sqrt {\sqrt {a}\, \sqrt {a \,c^{2}+b}-a c}\, \mathit {atan} \left (\frac {\sqrt {d}\, \sqrt {\sqrt {a}\, \sqrt {a \,c^{2}+b}-a c}\, \sqrt {2}-2 \sqrt {a}\, d x}{\sqrt {d}\, \sqrt {\sqrt {a}\, \sqrt {a \,c^{2}+b}+a c}\, \sqrt {2}}\right ) a \,c^{2}-3 \sqrt {\sqrt {a}\, \sqrt {a \,c^{2}+b}+a c}\, \sqrt {\sqrt {a}\, \sqrt {a \,c^{2}+b}-a c}\, \mathit {atan} \left (\frac {\sqrt {d}\, \sqrt {\sqrt {a}\, \sqrt {a \,c^{2}+b}-a c}\, \sqrt {2}-2 \sqrt {a}\, d x}{\sqrt {d}\, \sqrt {\sqrt {a}\, \sqrt {a \,c^{2}+b}+a c}\, \sqrt {2}}\right ) b +3 \sqrt {\sqrt {a}\, \sqrt {a \,c^{2}+b}+a c}\, \sqrt {\sqrt {a}\, \sqrt {a \,c^{2}+b}-a c}\, \mathit {atan} \left (\frac {\sqrt {d}\, \sqrt {\sqrt {a}\, \sqrt {a \,c^{2}+b}-a c}\, \sqrt {2}+2 \sqrt {a}\, d x}{\sqrt {d}\, \sqrt {\sqrt {a}\, \sqrt {a \,c^{2}+b}+a c}\, \sqrt {2}}\right ) a \,c^{2}-3 \sqrt {\sqrt {a}\, \sqrt {a \,c^{2}+b}+a c}\, \sqrt {\sqrt {a}\, \sqrt {a \,c^{2}+b}-a c}\, \mathit {atan} \left (\frac {\sqrt {d}\, \sqrt {\sqrt {a}\, \sqrt {a \,c^{2}+b}-a c}\, \sqrt {2}+2 \sqrt {a}\, d x}{\sqrt {d}\, \sqrt {\sqrt {a}\, \sqrt {a \,c^{2}+b}+a c}\, \sqrt {2}}\right ) b +3 \,\mathrm {log}\left (-\sqrt {d}\, \sqrt {\sqrt {a}\, \sqrt {a \,c^{2}+b}-a c}\, \sqrt {2}\, x +\sqrt {a \,c^{2}+b}+\sqrt {a}\, d \,x^{2}\right ) a b c +3 \,\mathrm {log}\left (\sqrt {d}\, \sqrt {\sqrt {a}\, \sqrt {a \,c^{2}+b}-a c}\, \sqrt {2}\, x +\sqrt {a \,c^{2}+b}+\sqrt {a}\, d \,x^{2}\right ) a b c +a^{2} d^{3} x^{6}-3 a b d \,x^{2}}{6 a^{3} d^{3}} \] Input:

int(x^5/(a+b/(d*x^2+c)^2),x)
 

Output:

(3*sqrt(sqrt(a)*sqrt(a*c**2 + b) + a*c)*sqrt(sqrt(a)*sqrt(a*c**2 + b) - a* 
c)*atan((sqrt(d)*sqrt(sqrt(a)*sqrt(a*c**2 + b) - a*c)*sqrt(2) - 2*sqrt(a)* 
d*x)/(sqrt(d)*sqrt(sqrt(a)*sqrt(a*c**2 + b) + a*c)*sqrt(2)))*a*c**2 - 3*sq 
rt(sqrt(a)*sqrt(a*c**2 + b) + a*c)*sqrt(sqrt(a)*sqrt(a*c**2 + b) - a*c)*at 
an((sqrt(d)*sqrt(sqrt(a)*sqrt(a*c**2 + b) - a*c)*sqrt(2) - 2*sqrt(a)*d*x)/ 
(sqrt(d)*sqrt(sqrt(a)*sqrt(a*c**2 + b) + a*c)*sqrt(2)))*b + 3*sqrt(sqrt(a) 
*sqrt(a*c**2 + b) + a*c)*sqrt(sqrt(a)*sqrt(a*c**2 + b) - a*c)*atan((sqrt(d 
)*sqrt(sqrt(a)*sqrt(a*c**2 + b) - a*c)*sqrt(2) + 2*sqrt(a)*d*x)/(sqrt(d)*s 
qrt(sqrt(a)*sqrt(a*c**2 + b) + a*c)*sqrt(2)))*a*c**2 - 3*sqrt(sqrt(a)*sqrt 
(a*c**2 + b) + a*c)*sqrt(sqrt(a)*sqrt(a*c**2 + b) - a*c)*atan((sqrt(d)*sqr 
t(sqrt(a)*sqrt(a*c**2 + b) - a*c)*sqrt(2) + 2*sqrt(a)*d*x)/(sqrt(d)*sqrt(s 
qrt(a)*sqrt(a*c**2 + b) + a*c)*sqrt(2)))*b + 3*log( - sqrt(d)*sqrt(sqrt(a) 
*sqrt(a*c**2 + b) - a*c)*sqrt(2)*x + sqrt(a*c**2 + b) + sqrt(a)*d*x**2)*a* 
b*c + 3*log(sqrt(d)*sqrt(sqrt(a)*sqrt(a*c**2 + b) - a*c)*sqrt(2)*x + sqrt( 
a*c**2 + b) + sqrt(a)*d*x**2)*a*b*c + a**2*d**3*x**6 - 3*a*b*d*x**2)/(6*a* 
*3*d**3)