Integrand size = 21, antiderivative size = 160 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^2} \, dx=-\frac {b \left (5 c^2 d-e\right ) e^2 x^2}{10 c^3}-\frac {b e^3 x^4}{20 c}-\frac {d^3 (a+b \arctan (c x))}{x}+3 d^2 e x (a+b \arctan (c x))+d e^2 x^3 (a+b \arctan (c x))+\frac {1}{5} e^3 x^5 (a+b \arctan (c x))+b c d^3 \log (x)-\frac {b \left (5 c^6 d^3+15 c^4 d^2 e-5 c^2 d e^2+e^3\right ) \log \left (1+c^2 x^2\right )}{10 c^5} \] Output:
-1/10*b*(5*c^2*d-e)*e^2*x^2/c^3-1/20*b*e^3*x^4/c-d^3*(a+b*arctan(c*x))/x+3 *d^2*e*x*(a+b*arctan(c*x))+d*e^2*x^3*(a+b*arctan(c*x))+1/5*e^3*x^5*(a+b*ar ctan(c*x))+b*c*d^3*ln(x)-1/10*b*(5*c^6*d^3+15*c^4*d^2*e-5*c^2*d*e^2+e^3)*l n(c^2*x^2+1)/c^5
Time = 0.09 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.06 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^2} \, dx=\frac {1}{20} \left (-\frac {20 a d^3}{x}+60 a d^2 e x+\frac {2 b e^2 \left (-5 c^2 d+e\right ) x^2}{c^3}+20 a d e^2 x^3-\frac {b e^3 x^4}{c}+4 a e^3 x^5+\frac {4 b \left (-5 d^3+15 d^2 e x^2+5 d e^2 x^4+e^3 x^6\right ) \arctan (c x)}{x}+20 b c d^3 \log (x)-\frac {2 b \left (5 c^6 d^3+15 c^4 d^2 e-5 c^2 d e^2+e^3\right ) \log \left (1+c^2 x^2\right )}{c^5}\right ) \] Input:
Integrate[((d + e*x^2)^3*(a + b*ArcTan[c*x]))/x^2,x]
Output:
((-20*a*d^3)/x + 60*a*d^2*e*x + (2*b*e^2*(-5*c^2*d + e)*x^2)/c^3 + 20*a*d* e^2*x^3 - (b*e^3*x^4)/c + 4*a*e^3*x^5 + (4*b*(-5*d^3 + 15*d^2*e*x^2 + 5*d* e^2*x^4 + e^3*x^6)*ArcTan[c*x])/x + 20*b*c*d^3*Log[x] - (2*b*(5*c^6*d^3 + 15*c^4*d^2*e - 5*c^2*d*e^2 + e^3)*Log[1 + c^2*x^2])/c^5)/20
Time = 0.59 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {5511, 27, 2331, 2123, 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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^2} \, dx\) |
| \(\Big \downarrow \) 5511 |
| \(\displaystyle -b c \int -\frac {-e^3 x^6-5 d e^2 x^4-15 d^2 e x^2+5 d^3}{5 x \left (c^2 x^2+1\right )}dx-\frac {d^3 (a+b \arctan (c x))}{x}+3 d^2 e x (a+b \arctan (c x))+d e^2 x^3 (a+b \arctan (c x))+\frac {1}{5} e^3 x^5 (a+b \arctan (c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} b c \int \frac {-e^3 x^6-5 d e^2 x^4-15 d^2 e x^2+5 d^3}{x \left (c^2 x^2+1\right )}dx-\frac {d^3 (a+b \arctan (c x))}{x}+3 d^2 e x (a+b \arctan (c x))+d e^2 x^3 (a+b \arctan (c x))+\frac {1}{5} e^3 x^5 (a+b \arctan (c x))\) |
| \(\Big \downarrow \) 2331 |
| \(\displaystyle \frac {1}{10} b c \int \frac {-e^3 x^6-5 d e^2 x^4-15 d^2 e x^2+5 d^3}{x^2 \left (c^2 x^2+1\right )}dx^2-\frac {d^3 (a+b \arctan (c x))}{x}+3 d^2 e x (a+b \arctan (c x))+d e^2 x^3 (a+b \arctan (c x))+\frac {1}{5} e^3 x^5 (a+b \arctan (c x))\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle \frac {1}{10} b c \int \left (\frac {5 d^3}{x^2}-\frac {\left (5 c^2 d-e\right ) e^2}{c^4}-\frac {e^3 x^2}{c^2}+\frac {-5 d^3 c^6-15 d^2 e c^4+5 d e^2 c^2-e^3}{c^4 \left (c^2 x^2+1\right )}\right )dx^2-\frac {d^3 (a+b \arctan (c x))}{x}+3 d^2 e x (a+b \arctan (c x))+d e^2 x^3 (a+b \arctan (c x))+\frac {1}{5} e^3 x^5 (a+b \arctan (c x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {d^3 (a+b \arctan (c x))}{x}+3 d^2 e x (a+b \arctan (c x))+d e^2 x^3 (a+b \arctan (c x))+\frac {1}{5} e^3 x^5 (a+b \arctan (c x))+\frac {1}{10} b c \left (-\frac {e^3 x^4}{2 c^2}-\frac {e^2 x^2 \left (5 c^2 d-e\right )}{c^4}-\frac {\left (5 c^6 d^3+15 c^4 d^2 e-5 c^2 d e^2+e^3\right ) \log \left (c^2 x^2+1\right )}{c^6}+5 d^3 \log \left (x^2\right )\right )\) |
Input:
Int[((d + e*x^2)^3*(a + b*ArcTan[c*x]))/x^2,x]
Output:
-((d^3*(a + b*ArcTan[c*x]))/x) + 3*d^2*e*x*(a + b*ArcTan[c*x]) + d*e^2*x^3 *(a + b*ArcTan[c*x]) + (e^3*x^5*(a + b*ArcTan[c*x]))/5 + (b*c*(-(((5*c^2*d - e)*e^2*x^2)/c^4) - (e^3*x^4)/(2*c^2) + 5*d^3*Log[x^2] - ((5*c^6*d^3 + 1 5*c^4*d^2*e - 5*c^2*d*e^2 + e^3)*Log[1 + c^2*x^2])/c^6))/10
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c , d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/2 S ubst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]
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]))
Time = 0.24 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.21
| method | result | size |
| parts | \(a \left (\frac {e^{3} x^{5}}{5}+d \,e^{2} x^{3}+3 d^{2} e x -\frac {d^{3}}{x}\right )+b c \left (\frac {\arctan \left (c x \right ) e^{3} x^{5}}{5 c}+\frac {\arctan \left (c x \right ) x^{3} d \,e^{2}}{c}+\frac {3 \arctan \left (c x \right ) x \,d^{2} e}{c}-\frac {\arctan \left (c x \right ) d^{3}}{c x}-\frac {\frac {c^{4} e^{3} x^{4}}{4}+\frac {5 c^{4} d \,e^{2} x^{2}}{2}-\frac {c^{2} e^{3} x^{2}}{2}+\frac {\left (5 c^{6} d^{3}+15 c^{4} d^{2} e -5 e^{2} d \,c^{2}+e^{3}\right ) \ln \left (c^{2} x^{2}+1\right )}{2}-5 c^{6} d^{3} \ln \left (c x \right )}{5 c^{6}}\right )\) | \(193\) |
| derivativedivides | \(c \left (\frac {a \left (3 c^{5} d^{2} e x +c^{5} d \,e^{2} x^{3}+\frac {e^{3} c^{5} x^{5}}{5}-\frac {c^{5} d^{3}}{x}\right )}{c^{6}}+\frac {b \left (3 \arctan \left (c x \right ) c^{5} x \,d^{2} e +\arctan \left (c x \right ) c^{5} d \,e^{2} x^{3}+\frac {\arctan \left (c x \right ) e^{3} c^{5} x^{5}}{5}-\frac {\arctan \left (c x \right ) c^{5} d^{3}}{x}-\frac {c^{4} d \,e^{2} x^{2}}{2}-\frac {c^{4} e^{3} x^{4}}{20}+\frac {c^{2} e^{3} x^{2}}{10}+c^{6} d^{3} \ln \left (c x \right )-\frac {\left (5 c^{6} d^{3}+15 c^{4} d^{2} e -5 e^{2} d \,c^{2}+e^{3}\right ) \ln \left (c^{2} x^{2}+1\right )}{10}\right )}{c^{6}}\right )\) | \(205\) |
| default | \(c \left (\frac {a \left (3 c^{5} d^{2} e x +c^{5} d \,e^{2} x^{3}+\frac {e^{3} c^{5} x^{5}}{5}-\frac {c^{5} d^{3}}{x}\right )}{c^{6}}+\frac {b \left (3 \arctan \left (c x \right ) c^{5} x \,d^{2} e +\arctan \left (c x \right ) c^{5} d \,e^{2} x^{3}+\frac {\arctan \left (c x \right ) e^{3} c^{5} x^{5}}{5}-\frac {\arctan \left (c x \right ) c^{5} d^{3}}{x}-\frac {c^{4} d \,e^{2} x^{2}}{2}-\frac {c^{4} e^{3} x^{4}}{20}+\frac {c^{2} e^{3} x^{2}}{10}+c^{6} d^{3} \ln \left (c x \right )-\frac {\left (5 c^{6} d^{3}+15 c^{4} d^{2} e -5 e^{2} d \,c^{2}+e^{3}\right ) \ln \left (c^{2} x^{2}+1\right )}{10}\right )}{c^{6}}\right )\) | \(205\) |
| parallelrisch | \(\frac {4 x^{6} \arctan \left (c x \right ) b \,c^{5} e^{3}+4 a \,c^{5} e^{3} x^{6}+20 x^{4} \arctan \left (c x \right ) b \,c^{5} d \,e^{2}-b \,c^{4} e^{3} x^{5}+20 a \,c^{5} d \,e^{2} x^{4}+20 b \,c^{6} d^{3} \ln \left (x \right ) x -10 \ln \left (c^{2} x^{2}+1\right ) b \,c^{6} d^{3} x +60 x^{2} \arctan \left (c x \right ) b \,c^{5} d^{2} e -10 b \,c^{4} d \,e^{2} x^{3}+60 a \,c^{5} d^{2} e \,x^{2}-30 \ln \left (c^{2} x^{2}+1\right ) b \,c^{4} d^{2} e x -20 \arctan \left (c x \right ) b \,c^{5} d^{3}+2 b \,c^{2} e^{3} x^{3}-20 a \,c^{5} d^{3}+10 \ln \left (c^{2} x^{2}+1\right ) b \,c^{2} d \,e^{2} x -2 \ln \left (c^{2} x^{2}+1\right ) b \,e^{3} x}{20 x \,c^{5}}\) | \(248\) |
| risch | \(\frac {i b \left (-x^{6} e^{3}-5 x^{4} e^{2} d -15 x^{2} e \,d^{2}+5 d^{3}\right ) \ln \left (i c x +1\right )}{10 x}+\frac {10 i b \,c^{5} d \,e^{2} x^{4} \ln \left (-i c x +1\right )+2 i b \,c^{5} e^{3} x^{6} \ln \left (-i c x +1\right )+4 a \,c^{5} e^{3} x^{6}-10 i b \,c^{5} d^{3} \ln \left (-i c x +1\right )+20 a \,c^{5} d \,e^{2} x^{4}-b \,c^{4} e^{3} x^{5}+20 b \,c^{6} d^{3} \ln \left (x \right ) x -10 \ln \left (c^{2} x^{2}+1\right ) b \,c^{6} d^{3} x +30 i b \,c^{5} d^{2} e \,x^{2} \ln \left (-i c x +1\right )+60 a \,c^{5} d^{2} e \,x^{2}-10 b \,c^{4} d \,e^{2} x^{3}-30 \ln \left (c^{2} x^{2}+1\right ) b \,c^{4} d^{2} e x -20 a \,c^{5} d^{3}+2 b \,c^{2} e^{3} x^{3}+10 \ln \left (c^{2} x^{2}+1\right ) b \,c^{2} d \,e^{2} x -2 \ln \left (c^{2} x^{2}+1\right ) b \,e^{3} x}{20 c^{5} x}\) | \(316\) |
Input:
int((e*x^2+d)^3*(a+b*arctan(c*x))/x^2,x,method=_RETURNVERBOSE)
Output:
a*(1/5*e^3*x^5+d*e^2*x^3+3*d^2*e*x-d^3/x)+b*c*(1/5*arctan(c*x)/c*e^3*x^5+a rctan(c*x)/c*x^3*d*e^2+3*arctan(c*x)/c*x*d^2*e-arctan(c*x)*d^3/c/x-1/5/c^6 *(1/4*c^4*e^3*x^4+5/2*c^4*d*e^2*x^2-1/2*c^2*e^3*x^2+1/2*(5*c^6*d^3+15*c^4* d^2*e-5*c^2*d*e^2+e^3)*ln(c^2*x^2+1)-5*c^6*d^3*ln(c*x)))
Time = 0.13 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.29 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^2} \, dx=\frac {4 \, a c^{5} e^{3} x^{6} + 20 \, a c^{5} d e^{2} x^{4} - b c^{4} e^{3} x^{5} + 20 \, b c^{6} d^{3} x \log \left (x\right ) + 60 \, a c^{5} d^{2} e x^{2} - 20 \, a c^{5} d^{3} - 2 \, {\left (5 \, b c^{4} d e^{2} - b c^{2} e^{3}\right )} x^{3} - 2 \, {\left (5 \, b c^{6} d^{3} + 15 \, b c^{4} d^{2} e - 5 \, b c^{2} d e^{2} + b e^{3}\right )} x \log \left (c^{2} x^{2} + 1\right ) + 4 \, {\left (b c^{5} e^{3} x^{6} + 5 \, b c^{5} d e^{2} x^{4} + 15 \, b c^{5} d^{2} e x^{2} - 5 \, b c^{5} d^{3}\right )} \arctan \left (c x\right )}{20 \, c^{5} x} \] Input:
integrate((e*x^2+d)^3*(a+b*arctan(c*x))/x^2,x, algorithm="fricas")
Output:
1/20*(4*a*c^5*e^3*x^6 + 20*a*c^5*d*e^2*x^4 - b*c^4*e^3*x^5 + 20*b*c^6*d^3* x*log(x) + 60*a*c^5*d^2*e*x^2 - 20*a*c^5*d^3 - 2*(5*b*c^4*d*e^2 - b*c^2*e^ 3)*x^3 - 2*(5*b*c^6*d^3 + 15*b*c^4*d^2*e - 5*b*c^2*d*e^2 + b*e^3)*x*log(c^ 2*x^2 + 1) + 4*(b*c^5*e^3*x^6 + 5*b*c^5*d*e^2*x^4 + 15*b*c^5*d^2*e*x^2 - 5 *b*c^5*d^3)*arctan(c*x))/(c^5*x)
Time = 0.58 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.61 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^2} \, dx=\begin {cases} - \frac {a d^{3}}{x} + 3 a d^{2} e x + a d e^{2} x^{3} + \frac {a e^{3} x^{5}}{5} + b c d^{3} \log {\left (x \right )} - \frac {b c d^{3} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{2} - \frac {b d^{3} \operatorname {atan}{\left (c x \right )}}{x} + 3 b d^{2} e x \operatorname {atan}{\left (c x \right )} + b d e^{2} x^{3} \operatorname {atan}{\left (c x \right )} + \frac {b e^{3} x^{5} \operatorname {atan}{\left (c x \right )}}{5} - \frac {3 b d^{2} e \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{2 c} - \frac {b d e^{2} x^{2}}{2 c} - \frac {b e^{3} x^{4}}{20 c} + \frac {b d e^{2} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{2 c^{3}} + \frac {b e^{3} x^{2}}{10 c^{3}} - \frac {b e^{3} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{10 c^{5}} & \text {for}\: c \neq 0 \\a \left (- \frac {d^{3}}{x} + 3 d^{2} e x + d e^{2} x^{3} + \frac {e^{3} x^{5}}{5}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((e*x**2+d)**3*(a+b*atan(c*x))/x**2,x)
Output:
Piecewise((-a*d**3/x + 3*a*d**2*e*x + a*d*e**2*x**3 + a*e**3*x**5/5 + b*c* d**3*log(x) - b*c*d**3*log(x**2 + c**(-2))/2 - b*d**3*atan(c*x)/x + 3*b*d* *2*e*x*atan(c*x) + b*d*e**2*x**3*atan(c*x) + b*e**3*x**5*atan(c*x)/5 - 3*b *d**2*e*log(x**2 + c**(-2))/(2*c) - b*d*e**2*x**2/(2*c) - b*e**3*x**4/(20* c) + b*d*e**2*log(x**2 + c**(-2))/(2*c**3) + b*e**3*x**2/(10*c**3) - b*e** 3*log(x**2 + c**(-2))/(10*c**5), Ne(c, 0)), (a*(-d**3/x + 3*d**2*e*x + d*e **2*x**3 + e**3*x**5/5), True))
Time = 0.05 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.23 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^2} \, dx=\frac {1}{5} \, a e^{3} x^{5} + a d e^{2} x^{3} - \frac {1}{2} \, {\left (c {\left (\log \left (c^{2} x^{2} + 1\right ) - \log \left (x^{2}\right )\right )} + \frac {2 \, \arctan \left (c x\right )}{x}\right )} b d^{3} + \frac {1}{2} \, {\left (2 \, x^{3} \arctan \left (c x\right ) - c {\left (\frac {x^{2}}{c^{2}} - \frac {\log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )}\right )} b d e^{2} + \frac {1}{20} \, {\left (4 \, x^{5} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{4} - 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} + 1\right )}{c^{6}}\right )}\right )} b e^{3} + 3 \, a d^{2} e x + \frac {3 \, {\left (2 \, c x \arctan \left (c x\right ) - \log \left (c^{2} x^{2} + 1\right )\right )} b d^{2} e}{2 \, c} - \frac {a d^{3}}{x} \] Input:
integrate((e*x^2+d)^3*(a+b*arctan(c*x))/x^2,x, algorithm="maxima")
Output:
1/5*a*e^3*x^5 + a*d*e^2*x^3 - 1/2*(c*(log(c^2*x^2 + 1) - log(x^2)) + 2*arc tan(c*x)/x)*b*d^3 + 1/2*(2*x^3*arctan(c*x) - c*(x^2/c^2 - log(c^2*x^2 + 1) /c^4))*b*d*e^2 + 1/20*(4*x^5*arctan(c*x) - c*((c^2*x^4 - 2*x^2)/c^4 + 2*lo g(c^2*x^2 + 1)/c^6))*b*e^3 + 3*a*d^2*e*x + 3/2*(2*c*x*arctan(c*x) - log(c^ 2*x^2 + 1))*b*d^2*e/c - a*d^3/x
Time = 0.13 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.54 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^2} \, dx=\frac {4 \, b c^{5} e^{3} x^{6} \arctan \left (c x\right ) + 4 \, a c^{5} e^{3} x^{6} + 20 \, b c^{5} d e^{2} x^{4} \arctan \left (c x\right ) + 20 \, a c^{5} d e^{2} x^{4} - b c^{4} e^{3} x^{5} + 60 \, b c^{5} d^{2} e x^{2} \arctan \left (c x\right ) - 10 \, b c^{6} d^{3} x \log \left (c^{2} x^{2} + 1\right ) + 20 \, b c^{6} d^{3} x \log \left (x\right ) + 60 \, a c^{5} d^{2} e x^{2} - 10 \, b c^{4} d e^{2} x^{3} - 20 \, b c^{5} d^{3} \arctan \left (c x\right ) - 30 \, b c^{4} d^{2} e x \log \left (c^{2} x^{2} + 1\right ) - 20 \, a c^{5} d^{3} + 2 \, b c^{2} e^{3} x^{3} + 10 \, b c^{2} d e^{2} x \log \left (c^{2} x^{2} + 1\right ) - 2 \, b e^{3} x \log \left (c^{2} x^{2} + 1\right )}{20 \, c^{5} x} \] Input:
integrate((e*x^2+d)^3*(a+b*arctan(c*x))/x^2,x, algorithm="giac")
Output:
1/20*(4*b*c^5*e^3*x^6*arctan(c*x) + 4*a*c^5*e^3*x^6 + 20*b*c^5*d*e^2*x^4*a rctan(c*x) + 20*a*c^5*d*e^2*x^4 - b*c^4*e^3*x^5 + 60*b*c^5*d^2*e*x^2*arcta n(c*x) - 10*b*c^6*d^3*x*log(c^2*x^2 + 1) + 20*b*c^6*d^3*x*log(x) + 60*a*c^ 5*d^2*e*x^2 - 10*b*c^4*d*e^2*x^3 - 20*b*c^5*d^3*arctan(c*x) - 30*b*c^4*d^2 *e*x*log(c^2*x^2 + 1) - 20*a*c^5*d^3 + 2*b*c^2*e^3*x^3 + 10*b*c^2*d*e^2*x* log(c^2*x^2 + 1) - 2*b*e^3*x*log(c^2*x^2 + 1))/(c^5*x)
Time = 1.00 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.48 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^2} \, dx=x\,\left (\frac {\frac {a\,e^3}{c^2}-\frac {a\,e^2\,\left (3\,d\,c^2+e\right )}{c^2}}{c^2}+\frac {3\,a\,d\,e\,\left (d\,c^2+e\right )}{c^2}\right )-x^3\,\left (\frac {a\,e^3}{3\,c^2}-\frac {a\,e^2\,\left (3\,d\,c^2+e\right )}{3\,c^2}\right )+x^2\,\left (\frac {b\,e^3}{10\,c^3}-\frac {b\,d\,e^2}{2\,c}\right )-\frac {a\,d^3}{x}+\frac {a\,e^3\,x^5}{5}-\frac {\ln \left (c^2\,x^2+1\right )\,\left (5\,b\,c^6\,d^3+15\,b\,c^4\,d^2\,e-5\,b\,c^2\,d\,e^2+b\,e^3\right )}{10\,c^5}+\frac {\mathrm {atan}\left (c\,x\right )\,\left (-b\,d^3+3\,b\,d^2\,e\,x^2+b\,d\,e^2\,x^4+\frac {b\,e^3\,x^6}{5}\right )}{x}-\frac {b\,e^3\,x^4}{20\,c}+b\,c\,d^3\,\ln \left (x\right ) \] Input:
int(((a + b*atan(c*x))*(d + e*x^2)^3)/x^2,x)
Output:
x*(((a*e^3)/c^2 - (a*e^2*(e + 3*c^2*d))/c^2)/c^2 + (3*a*d*e*(e + c^2*d))/c ^2) - x^3*((a*e^3)/(3*c^2) - (a*e^2*(e + 3*c^2*d))/(3*c^2)) + x^2*((b*e^3) /(10*c^3) - (b*d*e^2)/(2*c)) - (a*d^3)/x + (a*e^3*x^5)/5 - (log(c^2*x^2 + 1)*(b*e^3 + 5*b*c^6*d^3 - 5*b*c^2*d*e^2 + 15*b*c^4*d^2*e))/(10*c^5) + (ata n(c*x)*((b*e^3*x^6)/5 - b*d^3 + 3*b*d^2*e*x^2 + b*d*e^2*x^4))/x - (b*e^3*x ^4)/(20*c) + b*c*d^3*log(x)
Time = 0.19 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.54 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^2} \, dx=\frac {-20 \mathit {atan} \left (c x \right ) b \,c^{5} d^{3}+60 \mathit {atan} \left (c x \right ) b \,c^{5} d^{2} e \,x^{2}+20 \mathit {atan} \left (c x \right ) b \,c^{5} d \,e^{2} x^{4}+4 \mathit {atan} \left (c x \right ) b \,c^{5} e^{3} x^{6}-10 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) b \,c^{6} d^{3} x -30 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) b \,c^{4} d^{2} e x +10 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) b \,c^{2} d \,e^{2} x -2 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) b \,e^{3} x +20 \,\mathrm {log}\left (x \right ) b \,c^{6} d^{3} x -20 a \,c^{5} d^{3}+60 a \,c^{5} d^{2} e \,x^{2}+20 a \,c^{5} d \,e^{2} x^{4}+4 a \,c^{5} e^{3} x^{6}-10 b \,c^{4} d \,e^{2} x^{3}-b \,c^{4} e^{3} x^{5}+2 b \,c^{2} e^{3} x^{3}}{20 c^{5} x} \] Input:
int((e*x^2+d)^3*(a+b*atan(c*x))/x^2,x)
Output:
( - 20*atan(c*x)*b*c**5*d**3 + 60*atan(c*x)*b*c**5*d**2*e*x**2 + 20*atan(c *x)*b*c**5*d*e**2*x**4 + 4*atan(c*x)*b*c**5*e**3*x**6 - 10*log(c**2*x**2 + 1)*b*c**6*d**3*x - 30*log(c**2*x**2 + 1)*b*c**4*d**2*e*x + 10*log(c**2*x* *2 + 1)*b*c**2*d*e**2*x - 2*log(c**2*x**2 + 1)*b*e**3*x + 20*log(x)*b*c**6 *d**3*x - 20*a*c**5*d**3 + 60*a*c**5*d**2*e*x**2 + 20*a*c**5*d*e**2*x**4 + 4*a*c**5*e**3*x**6 - 10*b*c**4*d*e**2*x**3 - b*c**4*e**3*x**5 + 2*b*c**2* e**3*x**3)/(20*c**5*x)