\(\int \frac {x^3 (a+b \arctan (c x))}{(d+e x^2)^3} \, dx\) [1165]

Optimal result

Integrand size = 21, antiderivative size = 130 \[ \int \frac {x^3 (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=\frac {b c x}{8 \left (c^2 d-e\right ) e \left (d+e x^2\right )}-\frac {b \arctan (c x)}{4 d \left (c^2 d-e\right )^2}+\frac {x^4 (a+b \arctan (c x))}{4 d \left (d+e x^2\right )^2}-\frac {b c \left (c^2 d-3 e\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 \sqrt {d} \left (c^2 d-e\right )^2 e^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 3.12 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.22 \[ \int \frac {x^3 (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=\frac {\frac {2 a d}{\left (d+e x^2\right )^2}+\frac {-4 a c^2 d+4 a e+b c e x}{\left (c^2 d-e\right ) \left (d+e x^2\right )}+\frac {2 b c^2 \left (c^2 d-2 e\right ) \arctan (c x)}{\left (-c^2 d+e\right )^2}-\frac {2 b \left (d+2 e x^2\right ) \arctan (c x)}{\left (d+e x^2\right )^2}-\frac {b c \left (c^2 d-3 e\right ) \sqrt {e} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \left (-c^2 d+e\right )^2}}{8 e^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.16, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5511, 27, 372, 397, 216, 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.

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

\(\Big \downarrow \) 5511

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 372

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

\(\Big \downarrow \) 397

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

\(\Big \downarrow \) 216

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

\(\Big \downarrow \) 218

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

Input:

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

Output:

(x^4*(a + b*ArcTan[c*x]))/(4*d*(d + e*x^2)^2) - (b*c*(-1/2*(d*x)/((c^2*d - 
 e)*e*(d + e*x^2)) + ((2*e*ArcTan[c*x])/(c*(c^2*d - e)) + (Sqrt[d]*(c^2*d 
- 3*e)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/((c^2*d - e)*Sqrt[e]))/(2*(c^2*d - e)* 
e)))/(4*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 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

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 5511

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x 
_)^2)^(q_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^q, x]}, Sim 
p[(a + b*ArcTan[c*x])   u, x] - Simp[b*c   Int[SimplifyIntegrand[u/(1 + c^2 
*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && ((IGtQ[q, 0] && 
  !(ILtQ[(m - 1)/2, 0] && GtQ[m + 2*q + 3, 0])) || (IGtQ[(m + 1)/2, 0] && 
!(ILtQ[q, 0] && GtQ[m + 2*q + 3, 0])) || (ILtQ[(m + 2*q + 1)/2, 0] &&  !ILt 
Q[(m - 1)/2, 0]))
 

Maple [A] (verified)

Time = 0.88 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.62

method	result	size
parts	\(a \left (\frac {d}{4 e^{2} \left (e \,x^{2}+d \right )^{2}}-\frac {1}{2 e^{2} \left (e \,x^{2}+d \right )}\right )+\frac {b \left (-\frac {\arctan \left (c x \right ) c^{6}}{2 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {\arctan \left (c x \right ) c^{8} d}{4 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )^{2}}-\frac {c^{6} \left (\frac {\left (-c^{2} d +2 e \right ) \arctan \left (c x \right )}{\left (c^{2} d -e \right )^{2}}+\frac {e^{2} \left (-\frac {\left (c^{2} d -e \right ) c x}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {\left (c^{2} d -3 e \right ) \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 e c \sqrt {d e}}\right )}{\left (c^{2} d -e \right )^{2}}\right )}{4 e^{2}}\right )}{c^{4}}\)	\(211\)
derivativedivides	\(\frac {a \,c^{6} \left (\frac {d \,c^{2}}{4 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )^{2}}-\frac {1}{2 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}\right )+b \,c^{6} \left (\frac {\arctan \left (c x \right ) d \,c^{2}}{4 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )^{2}}-\frac {\arctan \left (c x \right )}{2 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {\frac {\left (-c^{2} d +2 e \right ) \arctan \left (c x \right )}{\left (c^{2} d -e \right )^{2}}+\frac {e^{2} \left (-\frac {\left (c^{2} d -e \right ) c x}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {\left (c^{2} d -3 e \right ) \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 e c \sqrt {d e}}\right )}{\left (c^{2} d -e \right )^{2}}}{4 e^{2}}\right )}{c^{4}}\)	\(229\)
default	\(\frac {a \,c^{6} \left (\frac {d \,c^{2}}{4 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )^{2}}-\frac {1}{2 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}\right )+b \,c^{6} \left (\frac {\arctan \left (c x \right ) d \,c^{2}}{4 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )^{2}}-\frac {\arctan \left (c x \right )}{2 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {\frac {\left (-c^{2} d +2 e \right ) \arctan \left (c x \right )}{\left (c^{2} d -e \right )^{2}}+\frac {e^{2} \left (-\frac {\left (c^{2} d -e \right ) c x}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {\left (c^{2} d -3 e \right ) \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 e c \sqrt {d e}}\right )}{\left (c^{2} d -e \right )^{2}}}{4 e^{2}}\right )}{c^{4}}\)	\(229\)
risch	\(\text {Expression too large to display}\)	\(1153\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 338 vs. \(2 (114) = 228\).

Time = 0.20 (sec) , antiderivative size = 697, normalized size of antiderivative = 5.36 \[ \int \frac {x^3 (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=\left [-\frac {4 \, a c^{4} d^{4} - 8 \, a c^{2} d^{3} e + 4 \, a d^{2} e^{2} - 2 \, {\left (b c^{3} d^{2} e^{2} - b c d e^{3}\right )} x^{3} + 8 \, {\left (a c^{4} d^{3} e - 2 \, a c^{2} d^{2} e^{2} + a d e^{3}\right )} x^{2} - {\left (b c^{3} d^{3} - 3 \, b c d^{2} e + {\left (b c^{3} d e^{2} - 3 \, b c e^{3}\right )} x^{4} + 2 \, {\left (b c^{3} d^{2} e - 3 \, b c d e^{2}\right )} x^{2}\right )} \sqrt {-d e} \log \left (\frac {e x^{2} - 2 \, \sqrt {-d e} x - d}{e x^{2} + d}\right ) - 2 \, {\left (b c^{3} d^{3} e - b c d^{2} e^{2}\right )} x + 4 \, {\left (2 \, b d e^{3} x^{2} + b d^{2} e^{2} - {\left (b c^{4} d^{2} e^{2} - 2 \, b c^{2} d e^{3}\right )} x^{4}\right )} \arctan \left (c x\right )}{16 \, {\left (c^{4} d^{5} e^{2} - 2 \, c^{2} d^{4} e^{3} + d^{3} e^{4} + {\left (c^{4} d^{3} e^{4} - 2 \, c^{2} d^{2} e^{5} + d e^{6}\right )} x^{4} + 2 \, {\left (c^{4} d^{4} e^{3} - 2 \, c^{2} d^{3} e^{4} + d^{2} e^{5}\right )} x^{2}\right )}}, -\frac {2 \, a c^{4} d^{4} - 4 \, a c^{2} d^{3} e + 2 \, a d^{2} e^{2} - {\left (b c^{3} d^{2} e^{2} - b c d e^{3}\right )} x^{3} + 4 \, {\left (a c^{4} d^{3} e - 2 \, a c^{2} d^{2} e^{2} + a d e^{3}\right )} x^{2} + {\left (b c^{3} d^{3} - 3 \, b c d^{2} e + {\left (b c^{3} d e^{2} - 3 \, b c e^{3}\right )} x^{4} + 2 \, {\left (b c^{3} d^{2} e - 3 \, b c d e^{2}\right )} x^{2}\right )} \sqrt {d e} \arctan \left (\frac {\sqrt {d e} x}{d}\right ) - {\left (b c^{3} d^{3} e - b c d^{2} e^{2}\right )} x + 2 \, {\left (2 \, b d e^{3} x^{2} + b d^{2} e^{2} - {\left (b c^{4} d^{2} e^{2} - 2 \, b c^{2} d e^{3}\right )} x^{4}\right )} \arctan \left (c x\right )}{8 \, {\left (c^{4} d^{5} e^{2} - 2 \, c^{2} d^{4} e^{3} + d^{3} e^{4} + {\left (c^{4} d^{3} e^{4} - 2 \, c^{2} d^{2} e^{5} + d e^{6}\right )} x^{4} + 2 \, {\left (c^{4} d^{4} e^{3} - 2 \, c^{2} d^{3} e^{4} + d^{2} e^{5}\right )} x^{2}\right )}}\right ] \] Input:

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate(x^3*(a+b*arctan(c*x))/(e*x^2+d)^3,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 [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.27 (sec) , antiderivative size = 762, normalized size of antiderivative = 5.86 \[ \int \frac {x^3 (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 3.92 (sec) , antiderivative size = 273, normalized size of antiderivative = 2.10 \[ \int \frac {x^3 (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=\frac {b\,c^4\,d\,\mathrm {atan}\left (c\,x\right )}{4\,e^2\,{\left (e-c^2\,d\right )}^2}-\frac {a\,d}{4\,e^2\,{\left (e\,x^2+d\right )}^2}-\frac {b\,d\,\mathrm {atan}\left (c\,x\right )}{4\,e^2\,{\left (e\,x^2+d\right )}^2}-\frac {b\,c\,x^3}{8\,\left (e-c^2\,d\right )\,{\left (e\,x^2+d\right )}^2}-\frac {b\,c^2\,\mathrm {atan}\left (c\,x\right )}{2\,e\,{\left (e-c^2\,d\right )}^2}-\frac {b\,x^2\,\mathrm {atan}\left (c\,x\right )}{2\,e\,{\left (e\,x^2+d\right )}^2}-\frac {b\,c^3\,\mathrm {atan}\left (\frac {x\,\sqrt {-d\,e^3}\,1{}\mathrm {i}}{d\,e}\right )\,\sqrt {-d\,e^3}\,1{}\mathrm {i}}{8\,e^3\,{\left (e-c^2\,d\right )}^2}-\frac {a\,x^2}{2\,e\,{\left (e\,x^2+d\right )}^2}-\frac {b\,c\,d\,x}{8\,e\,\left (e-c^2\,d\right )\,{\left (e\,x^2+d\right )}^2}+\frac {b\,c\,\mathrm {atan}\left (\frac {x\,\sqrt {-d\,e^3}\,1{}\mathrm {i}}{d\,e}\right )\,\sqrt {-d\,e^3}\,3{}\mathrm {i}}{8\,d\,e^2\,{\left (e-c^2\,d\right )}^2} \] Input:

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

Output:

(b*c^4*d*atan(c*x))/(4*e^2*(e - c^2*d)^2) - (a*d)/(4*e^2*(d + e*x^2)^2) - 
(b*d*atan(c*x))/(4*e^2*(d + e*x^2)^2) - (b*c*x^3)/(8*(e - c^2*d)*(d + e*x^ 
2)^2) - (b*c^2*atan(c*x))/(2*e*(e - c^2*d)^2) - (b*x^2*atan(c*x))/(2*e*(d 
+ e*x^2)^2) - (b*c^3*atan((x*(-d*e^3)^(1/2)*1i)/(d*e))*(-d*e^3)^(1/2)*1i)/ 
(8*e^3*(e - c^2*d)^2) - (a*x^2)/(2*e*(d + e*x^2)^2) - (b*c*d*x)/(8*e*(e - 
c^2*d)*(d + e*x^2)^2) + (b*c*atan((x*(-d*e^3)^(1/2)*1i)/(d*e))*(-d*e^3)^(1 
/2)*3i)/(8*d*e^2*(e - c^2*d)^2)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 408, normalized size of antiderivative = 3.14 \[ \int \frac {x^3 (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=\frac {-\sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) b \,c^{3} d^{3}-2 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) b \,c^{3} d^{2} e \,x^{2}-\sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) b \,c^{3} d \,e^{2} x^{4}+3 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) b c \,d^{2} e +6 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) b c d \,e^{2} x^{2}+3 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) b c \,e^{3} x^{4}+2 \mathit {atan} \left (c x \right ) b \,c^{4} d^{2} e^{2} x^{4}-4 \mathit {atan} \left (c x \right ) b \,c^{2} d \,e^{3} x^{4}-2 \mathit {atan} \left (c x \right ) b \,d^{2} e^{2}-4 \mathit {atan} \left (c x \right ) b d \,e^{3} x^{2}+2 a \,c^{4} d^{2} e^{2} x^{4}-4 a \,c^{2} d \,e^{3} x^{4}+2 a \,e^{4} x^{4}+b \,c^{3} d^{3} e x +b \,c^{3} d^{2} e^{2} x^{3}-b c \,d^{2} e^{2} x -b c d \,e^{3} x^{3}}{8 d \,e^{2} \left (c^{4} d^{2} e^{2} x^{4}+2 c^{4} d^{3} e \,x^{2}-2 c^{2} d \,e^{3} x^{4}+c^{4} d^{4}-4 c^{2} d^{2} e^{2} x^{2}+e^{4} x^{4}-2 c^{2} d^{3} e +2 d \,e^{3} x^{2}+d^{2} e^{2}\right )} \] Input:

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

Output:

( - sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*b*c**3*d**3 - 2*sqrt(e)* 
sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*b*c**3*d**2*e*x**2 - sqrt(e)*sqrt(d) 
*atan((e*x)/(sqrt(e)*sqrt(d)))*b*c**3*d*e**2*x**4 + 3*sqrt(e)*sqrt(d)*atan 
((e*x)/(sqrt(e)*sqrt(d)))*b*c*d**2*e + 6*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt( 
e)*sqrt(d)))*b*c*d*e**2*x**2 + 3*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt( 
d)))*b*c*e**3*x**4 + 2*atan(c*x)*b*c**4*d**2*e**2*x**4 - 4*atan(c*x)*b*c** 
2*d*e**3*x**4 - 2*atan(c*x)*b*d**2*e**2 - 4*atan(c*x)*b*d*e**3*x**2 + 2*a* 
c**4*d**2*e**2*x**4 - 4*a*c**2*d*e**3*x**4 + 2*a*e**4*x**4 + b*c**3*d**3*e 
*x + b*c**3*d**2*e**2*x**3 - b*c*d**2*e**2*x - b*c*d*e**3*x**3)/(8*d*e**2* 
(c**4*d**4 + 2*c**4*d**3*e*x**2 + c**4*d**2*e**2*x**4 - 2*c**2*d**3*e - 4* 
c**2*d**2*e**2*x**2 - 2*c**2*d*e**3*x**4 + d**2*e**2 + 2*d*e**3*x**2 + e** 
4*x**4))