Integrand size = 21, antiderivative size = 158 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^4} \, dx=-\frac {b c d^3}{6 x^2}-\frac {b e^3 x^2}{6 c}-\frac {d^3 (a+b \arctan (c x))}{3 x^3}-\frac {3 d^2 e (a+b \arctan (c x))}{x}+3 d e^2 x (a+b \arctan (c x))+\frac {1}{3} e^3 x^3 (a+b \arctan (c x))-\frac {1}{3} b c d^2 \left (c^2 d-9 e\right ) \log (x)+\frac {b \left (c^2 d+e\right ) \left (c^4 d^2-10 c^2 d e+e^2\right ) \log \left (1+c^2 x^2\right )}{6 c^3} \] Output:
-1/6*b*c*d^3/x^2-1/6*b*e^3*x^2/c-1/3*d^3*(a+b*arctan(c*x))/x^3-3*d^2*e*(a+ b*arctan(c*x))/x+3*d*e^2*x*(a+b*arctan(c*x))+1/3*e^3*x^3*(a+b*arctan(c*x)) -1/3*b*c*d^2*(c^2*d-9*e)*ln(x)+1/6*b*(c^2*d+e)*(c^4*d^2-10*c^2*d*e+e^2)*ln (c^2*x^2+1)/c^3
Time = 0.10 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.05 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^4} \, dx=\frac {1}{6} \left (-\frac {2 a d^3}{x^3}-\frac {b c d^3}{x^2}-\frac {18 a d^2 e}{x}+18 a d e^2 x-\frac {b e^3 x^2}{c}+2 a e^3 x^3+\frac {2 b \left (-d^3-9 d^2 e x^2+9 d e^2 x^4+e^3 x^6\right ) \arctan (c x)}{x^3}-2 b c d^2 \left (c^2 d-9 e\right ) \log (x)+\frac {b \left (c^6 d^3-9 c^4 d^2 e-9 c^2 d e^2+e^3\right ) \log \left (1+c^2 x^2\right )}{c^3}\right ) \] Input:
Integrate[((d + e*x^2)^3*(a + b*ArcTan[c*x]))/x^4,x]
Output:
((-2*a*d^3)/x^3 - (b*c*d^3)/x^2 - (18*a*d^2*e)/x + 18*a*d*e^2*x - (b*e^3*x ^2)/c + 2*a*e^3*x^3 + (2*b*(-d^3 - 9*d^2*e*x^2 + 9*d*e^2*x^4 + e^3*x^6)*Ar cTan[c*x])/x^3 - 2*b*c*d^2*(c^2*d - 9*e)*Log[x] + (b*(c^6*d^3 - 9*c^4*d^2* e - 9*c^2*d*e^2 + e^3)*Log[1 + c^2*x^2])/c^3)/6
Time = 0.60 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.96, 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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^4} \, dx\) |
| \(\Big \downarrow \) 5511 |
| \(\displaystyle -b c \int -\frac {-e^3 x^6-9 d e^2 x^4+9 d^2 e x^2+d^3}{3 x^3 \left (c^2 x^2+1\right )}dx-\frac {d^3 (a+b \arctan (c x))}{3 x^3}-\frac {3 d^2 e (a+b \arctan (c x))}{x}+3 d e^2 x (a+b \arctan (c x))+\frac {1}{3} e^3 x^3 (a+b \arctan (c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} b c \int \frac {-e^3 x^6-9 d e^2 x^4+9 d^2 e x^2+d^3}{x^3 \left (c^2 x^2+1\right )}dx-\frac {d^3 (a+b \arctan (c x))}{3 x^3}-\frac {3 d^2 e (a+b \arctan (c x))}{x}+3 d e^2 x (a+b \arctan (c x))+\frac {1}{3} e^3 x^3 (a+b \arctan (c x))\) |
| \(\Big \downarrow \) 2331 |
| \(\displaystyle \frac {1}{6} b c \int \frac {-e^3 x^6-9 d e^2 x^4+9 d^2 e x^2+d^3}{x^4 \left (c^2 x^2+1\right )}dx^2-\frac {d^3 (a+b \arctan (c x))}{3 x^3}-\frac {3 d^2 e (a+b \arctan (c x))}{x}+3 d e^2 x (a+b \arctan (c x))+\frac {1}{3} e^3 x^3 (a+b \arctan (c x))\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle \frac {1}{6} b c \int \left (\frac {d^3}{x^4}-\frac {\left (c^2 d-9 e\right ) d^2}{x^2}-\frac {e^3}{c^2}+\frac {\left (d c^2+e\right ) \left (d^2 c^4-10 d e c^2+e^2\right )}{c^2 \left (c^2 x^2+1\right )}\right )dx^2-\frac {d^3 (a+b \arctan (c x))}{3 x^3}-\frac {3 d^2 e (a+b \arctan (c x))}{x}+3 d e^2 x (a+b \arctan (c x))+\frac {1}{3} e^3 x^3 (a+b \arctan (c x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {d^3 (a+b \arctan (c x))}{3 x^3}-\frac {3 d^2 e (a+b \arctan (c x))}{x}+3 d e^2 x (a+b \arctan (c x))+\frac {1}{3} e^3 x^3 (a+b \arctan (c x))+\frac {1}{6} b c \left (-d^2 \log \left (x^2\right ) \left (c^2 d-9 e\right )-\frac {e^3 x^2}{c^2}+\frac {\left (c^2 d+e\right ) \left (c^4 d^2-10 c^2 d e+e^2\right ) \log \left (c^2 x^2+1\right )}{c^4}-\frac {d^3}{x^2}\right )\) |
Input:
Int[((d + e*x^2)^3*(a + b*ArcTan[c*x]))/x^4,x]
Output:
-1/3*(d^3*(a + b*ArcTan[c*x]))/x^3 - (3*d^2*e*(a + b*ArcTan[c*x]))/x + 3*d *e^2*x*(a + b*ArcTan[c*x]) + (e^3*x^3*(a + b*ArcTan[c*x]))/3 + (b*c*(-(d^3 /x^2) - (e^3*x^2)/c^2 - d^2*(c^2*d - 9*e)*Log[x^2] + ((c^2*d + e)*(c^4*d^2 - 10*c^2*d*e + e^2)*Log[1 + c^2*x^2])/c^4))/6
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 = 0.25 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.23
| method | result | size |
| parts | \(a \left (\frac {e^{3} x^{3}}{3}+3 d \,e^{2} x -\frac {d^{3}}{3 x^{3}}-\frac {3 e \,d^{2}}{x}\right )+b \,c^{3} \left (\frac {\arctan \left (c x \right ) x^{3} e^{3}}{3 c^{3}}+\frac {3 \arctan \left (c x \right ) x \,e^{2} d}{c^{3}}-\frac {\arctan \left (c x \right ) d^{3}}{3 c^{3} x^{3}}-\frac {3 \arctan \left (c x \right ) d^{2} e}{c^{3} x}-\frac {\frac {c^{2} e^{3} x^{2}}{2}+\frac {\left (-c^{6} d^{3}+9 c^{4} d^{2} e +9 e^{2} d \,c^{2}-e^{3}\right ) \ln \left (c^{2} x^{2}+1\right )}{2}+c^{4} d^{2} \left (c^{2} d -9 e \right ) \ln \left (c x \right )+\frac {c^{4} d^{3}}{2 x^{2}}}{3 c^{6}}\right )\) | \(195\) |
| derivativedivides | \(c^{3} \left (\frac {a \left (3 c^{3} x d \,e^{2}+\frac {e^{3} c^{3} x^{3}}{3}-\frac {3 c^{3} d^{2} e}{x}-\frac {c^{3} d^{3}}{3 x^{3}}\right )}{c^{6}}+\frac {b \left (3 \arctan \left (c x \right ) c^{3} x d \,e^{2}+\frac {\arctan \left (c x \right ) e^{3} c^{3} x^{3}}{3}-\frac {3 \arctan \left (c x \right ) c^{3} d^{2} e}{x}-\frac {\arctan \left (c x \right ) c^{3} d^{3}}{3 x^{3}}-\frac {c^{2} e^{3} x^{2}}{6}-\frac {c^{4} d^{2} \left (c^{2} d -9 e \right ) \ln \left (c x \right )}{3}-\frac {c^{4} d^{3}}{6 x^{2}}-\frac {\left (-c^{6} d^{3}+9 c^{4} d^{2} e +9 e^{2} d \,c^{2}-e^{3}\right ) \ln \left (c^{2} x^{2}+1\right )}{6}\right )}{c^{6}}\right )\) | \(209\) |
| default | \(c^{3} \left (\frac {a \left (3 c^{3} x d \,e^{2}+\frac {e^{3} c^{3} x^{3}}{3}-\frac {3 c^{3} d^{2} e}{x}-\frac {c^{3} d^{3}}{3 x^{3}}\right )}{c^{6}}+\frac {b \left (3 \arctan \left (c x \right ) c^{3} x d \,e^{2}+\frac {\arctan \left (c x \right ) e^{3} c^{3} x^{3}}{3}-\frac {3 \arctan \left (c x \right ) c^{3} d^{2} e}{x}-\frac {\arctan \left (c x \right ) c^{3} d^{3}}{3 x^{3}}-\frac {c^{2} e^{3} x^{2}}{6}-\frac {c^{4} d^{2} \left (c^{2} d -9 e \right ) \ln \left (c x \right )}{3}-\frac {c^{4} d^{3}}{6 x^{2}}-\frac {\left (-c^{6} d^{3}+9 c^{4} d^{2} e +9 e^{2} d \,c^{2}-e^{3}\right ) \ln \left (c^{2} x^{2}+1\right )}{6}\right )}{c^{6}}\right )\) | \(209\) |
| parallelrisch | \(-\frac {2 \ln \left (x \right ) b \,c^{6} d^{3} x^{3}-\ln \left (c^{2} x^{2}+1\right ) x^{3} b \,c^{6} d^{3}-2 x^{6} \arctan \left (c x \right ) b \,c^{3} e^{3}-2 a \,c^{3} e^{3} x^{6}-b \,c^{6} d^{3} x^{3}-18 \ln \left (x \right ) b \,c^{4} d^{2} e \,x^{3}+9 \ln \left (c^{2} x^{2}+1\right ) x^{3} b \,c^{4} d^{2} e -18 x^{4} \arctan \left (c x \right ) b \,c^{3} d \,e^{2}+b \,c^{2} e^{3} x^{5}-18 a \,c^{3} d \,e^{2} x^{4}+9 \ln \left (c^{2} x^{2}+1\right ) x^{3} b \,c^{2} d \,e^{2}+18 x^{2} \arctan \left (c x \right ) b \,c^{3} d^{2} e +18 a \,c^{3} d^{2} e \,x^{2}+b \,c^{4} d^{3} x -\ln \left (c^{2} x^{2}+1\right ) x^{3} b \,e^{3}+2 \arctan \left (c x \right ) b \,c^{3} d^{3}+2 a \,c^{3} d^{3}}{6 x^{3} c^{3}}\) | \(268\) |
| risch | \(\frac {i b \left (-x^{6} e^{3}-9 x^{4} e^{2} d +9 x^{2} e \,d^{2}+d^{3}\right ) \ln \left (i c x +1\right )}{6 x^{3}}-\frac {-i b \,c^{3} e^{3} x^{6} \ln \left (-i c x +1\right )+2 \ln \left (x \right ) b \,c^{6} d^{3} x^{3}-\ln \left (-c^{2} x^{2}-1\right ) b \,c^{6} d^{3} x^{3}-9 i b \,c^{3} d \,e^{2} x^{4} \ln \left (-i c x +1\right )-2 a \,c^{3} e^{3} x^{6}-18 \ln \left (x \right ) b \,c^{4} d^{2} e \,x^{3}+9 \ln \left (-c^{2} x^{2}-1\right ) b \,c^{4} d^{2} e \,x^{3}+i b \,c^{3} d^{3} \ln \left (-i c x +1\right )-18 a \,c^{3} d \,e^{2} x^{4}+b \,c^{2} e^{3} x^{5}+9 \ln \left (-c^{2} x^{2}-1\right ) b \,c^{2} d \,e^{2} x^{3}+9 i b \,c^{3} d^{2} e \,x^{2} \ln \left (-i c x +1\right )+18 a \,c^{3} d^{2} e \,x^{2}+b \,c^{4} d^{3} x -\ln \left (-c^{2} x^{2}-1\right ) b \,e^{3} x^{3}+2 a \,c^{3} d^{3}}{6 c^{3} x^{3}}\) | \(326\) |
Input:
int((e*x^2+d)^3*(a+b*arctan(c*x))/x^4,x,method=_RETURNVERBOSE)
Output:
a*(1/3*e^3*x^3+3*d*e^2*x-1/3*d^3/x^3-3*e*d^2/x)+b*c^3*(1/3*arctan(c*x)/c^3 *x^3*e^3+3*arctan(c*x)/c^3*x*e^2*d-1/3*arctan(c*x)*d^3/c^3/x^3-3*arctan(c* x)/c^3*d^2*e/x-1/3/c^6*(1/2*c^2*e^3*x^2+1/2*(-c^6*d^3+9*c^4*d^2*e+9*c^2*d* e^2-e^3)*ln(c^2*x^2+1)+c^4*d^2*(c^2*d-9*e)*ln(c*x)+1/2*c^4*d^3/x^2))
Time = 0.14 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.30 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^4} \, dx=\frac {2 \, a c^{3} e^{3} x^{6} + 18 \, a c^{3} d e^{2} x^{4} - b c^{2} e^{3} x^{5} - b c^{4} d^{3} x - 18 \, a c^{3} d^{2} e x^{2} - 2 \, a c^{3} d^{3} + {\left (b c^{6} d^{3} - 9 \, b c^{4} d^{2} e - 9 \, b c^{2} d e^{2} + b e^{3}\right )} x^{3} \log \left (c^{2} x^{2} + 1\right ) - 2 \, {\left (b c^{6} d^{3} - 9 \, b c^{4} d^{2} e\right )} x^{3} \log \left (x\right ) + 2 \, {\left (b c^{3} e^{3} x^{6} + 9 \, b c^{3} d e^{2} x^{4} - 9 \, b c^{3} d^{2} e x^{2} - b c^{3} d^{3}\right )} \arctan \left (c x\right )}{6 \, c^{3} x^{3}} \] Input:
integrate((e*x^2+d)^3*(a+b*arctan(c*x))/x^4,x, algorithm="fricas")
Output:
1/6*(2*a*c^3*e^3*x^6 + 18*a*c^3*d*e^2*x^4 - b*c^2*e^3*x^5 - b*c^4*d^3*x - 18*a*c^3*d^2*e*x^2 - 2*a*c^3*d^3 + (b*c^6*d^3 - 9*b*c^4*d^2*e - 9*b*c^2*d* e^2 + b*e^3)*x^3*log(c^2*x^2 + 1) - 2*(b*c^6*d^3 - 9*b*c^4*d^2*e)*x^3*log( x) + 2*(b*c^3*e^3*x^6 + 9*b*c^3*d*e^2*x^4 - 9*b*c^3*d^2*e*x^2 - b*c^3*d^3) *arctan(c*x))/(c^3*x^3)
Time = 0.59 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.72 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^4} \, dx=\begin {cases} - \frac {a d^{3}}{3 x^{3}} - \frac {3 a d^{2} e}{x} + 3 a d e^{2} x + \frac {a e^{3} x^{3}}{3} - \frac {b c^{3} d^{3} \log {\left (x \right )}}{3} + \frac {b c^{3} d^{3} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{6} - \frac {b c d^{3}}{6 x^{2}} + 3 b c d^{2} e \log {\left (x \right )} - \frac {3 b c d^{2} e \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{2} - \frac {b d^{3} \operatorname {atan}{\left (c x \right )}}{3 x^{3}} - \frac {3 b d^{2} e \operatorname {atan}{\left (c x \right )}}{x} + 3 b d e^{2} x \operatorname {atan}{\left (c x \right )} + \frac {b e^{3} x^{3} \operatorname {atan}{\left (c x \right )}}{3} - \frac {3 b d e^{2} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{2 c} - \frac {b e^{3} x^{2}}{6 c} + \frac {b e^{3} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{6 c^{3}} & \text {for}\: c \neq 0 \\a \left (- \frac {d^{3}}{3 x^{3}} - \frac {3 d^{2} e}{x} + 3 d e^{2} x + \frac {e^{3} x^{3}}{3}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((e*x**2+d)**3*(a+b*atan(c*x))/x**4,x)
Output:
Piecewise((-a*d**3/(3*x**3) - 3*a*d**2*e/x + 3*a*d*e**2*x + a*e**3*x**3/3 - b*c**3*d**3*log(x)/3 + b*c**3*d**3*log(x**2 + c**(-2))/6 - b*c*d**3/(6*x **2) + 3*b*c*d**2*e*log(x) - 3*b*c*d**2*e*log(x**2 + c**(-2))/2 - b*d**3*a tan(c*x)/(3*x**3) - 3*b*d**2*e*atan(c*x)/x + 3*b*d*e**2*x*atan(c*x) + b*e* *3*x**3*atan(c*x)/3 - 3*b*d*e**2*log(x**2 + c**(-2))/(2*c) - b*e**3*x**2/( 6*c) + b*e**3*log(x**2 + c**(-2))/(6*c**3), Ne(c, 0)), (a*(-d**3/(3*x**3) - 3*d**2*e/x + 3*d*e**2*x + e**3*x**3/3), True))
Time = 0.05 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.22 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^4} \, dx=\frac {1}{3} \, a e^{3} x^{3} + \frac {1}{6} \, {\left ({\left (c^{2} \log \left (c^{2} x^{2} + 1\right ) - c^{2} \log \left (x^{2}\right ) - \frac {1}{x^{2}}\right )} c - \frac {2 \, \arctan \left (c x\right )}{x^{3}}\right )} b d^{3} - \frac {3}{2} \, {\left (c {\left (\log \left (c^{2} x^{2} + 1\right ) - \log \left (x^{2}\right )\right )} + \frac {2 \, \arctan \left (c x\right )}{x}\right )} b d^{2} e + \frac {1}{6} \, {\left (2 \, x^{3} \arctan \left (c x\right ) - c {\left (\frac {x^{2}}{c^{2}} - \frac {\log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )}\right )} b e^{3} + 3 \, a d e^{2} x + \frac {3 \, {\left (2 \, c x \arctan \left (c x\right ) - \log \left (c^{2} x^{2} + 1\right )\right )} b d e^{2}}{2 \, c} - \frac {3 \, a d^{2} e}{x} - \frac {a d^{3}}{3 \, x^{3}} \] Input:
integrate((e*x^2+d)^3*(a+b*arctan(c*x))/x^4,x, algorithm="maxima")
Output:
1/3*a*e^3*x^3 + 1/6*((c^2*log(c^2*x^2 + 1) - c^2*log(x^2) - 1/x^2)*c - 2*a rctan(c*x)/x^3)*b*d^3 - 3/2*(c*(log(c^2*x^2 + 1) - log(x^2)) + 2*arctan(c* x)/x)*b*d^2*e + 1/6*(2*x^3*arctan(c*x) - c*(x^2/c^2 - log(c^2*x^2 + 1)/c^4 ))*b*e^3 + 3*a*d*e^2*x + 3/2*(2*c*x*arctan(c*x) - log(c^2*x^2 + 1))*b*d*e^ 2/c - 3*a*d^2*e/x - 1/3*a*d^3/x^3
Time = 0.12 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.61 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^4} \, dx=\frac {2 \, b c^{3} e^{3} x^{6} \arctan \left (c x\right ) + b c^{6} d^{3} x^{3} \log \left (c^{2} x^{2} + 1\right ) - 2 \, b c^{6} d^{3} x^{3} \log \left (x\right ) + 2 \, a c^{3} e^{3} x^{6} + 18 \, b c^{3} d e^{2} x^{4} \arctan \left (c x\right ) - 9 \, b c^{4} d^{2} e x^{3} \log \left (c^{2} x^{2} + 1\right ) + 18 \, b c^{4} d^{2} e x^{3} \log \left (x\right ) + 18 \, a c^{3} d e^{2} x^{4} - b c^{2} e^{3} x^{5} - 18 \, b c^{3} d^{2} e x^{2} \arctan \left (c x\right ) - 9 \, b c^{2} d e^{2} x^{3} \log \left (c^{2} x^{2} + 1\right ) - b c^{4} d^{3} x - 18 \, a c^{3} d^{2} e x^{2} - 2 \, b c^{3} d^{3} \arctan \left (c x\right ) + b e^{3} x^{3} \log \left (c^{2} x^{2} + 1\right ) - 2 \, a c^{3} d^{3}}{6 \, c^{3} x^{3}} \] Input:
integrate((e*x^2+d)^3*(a+b*arctan(c*x))/x^4,x, algorithm="giac")
Output:
1/6*(2*b*c^3*e^3*x^6*arctan(c*x) + b*c^6*d^3*x^3*log(c^2*x^2 + 1) - 2*b*c^ 6*d^3*x^3*log(x) + 2*a*c^3*e^3*x^6 + 18*b*c^3*d*e^2*x^4*arctan(c*x) - 9*b* c^4*d^2*e*x^3*log(c^2*x^2 + 1) + 18*b*c^4*d^2*e*x^3*log(x) + 18*a*c^3*d*e^ 2*x^4 - b*c^2*e^3*x^5 - 18*b*c^3*d^2*e*x^2*arctan(c*x) - 9*b*c^2*d*e^2*x^3 *log(c^2*x^2 + 1) - b*c^4*d^3*x - 18*a*c^3*d^2*e*x^2 - 2*b*c^3*d^3*arctan( c*x) + b*e^3*x^3*log(c^2*x^2 + 1) - 2*a*c^3*d^3)/(c^3*x^3)
Time = 0.98 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.28 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^4} \, dx=\frac {a\,e^3\,x^3}{3}-\ln \left (x\right )\,\left (\frac {b\,c^3\,d^3}{3}-3\,b\,c\,d^2\,e\right )-\frac {\frac {b\,c^2\,d^3\,x}{2}+a\,c\,d^3+9\,a\,e\,c\,d^2\,x^2}{3\,c\,x^3}-x\,\left (\frac {a\,e^3}{c^2}-\frac {a\,e^2\,\left (3\,d\,c^2+e\right )}{c^2}\right )+\frac {\ln \left (c^2\,x^2+1\right )\,\left (b\,c^6\,d^3-9\,b\,c^4\,d^2\,e-9\,b\,c^2\,d\,e^2+b\,e^3\right )}{6\,c^3}-\frac {\mathrm {atan}\left (c\,x\right )\,\left (\frac {b\,d^3}{3}+3\,b\,d^2\,e\,x^2-3\,b\,d\,e^2\,x^4-\frac {b\,e^3\,x^6}{3}\right )}{x^3}-\frac {b\,e^3\,x^2}{6\,c} \] Input:
int(((a + b*atan(c*x))*(d + e*x^2)^3)/x^4,x)
Output:
(a*e^3*x^3)/3 - log(x)*((b*c^3*d^3)/3 - 3*b*c*d^2*e) - (a*c*d^3 + (b*c^2*d ^3*x)/2 + 9*a*c*d^2*e*x^2)/(3*c*x^3) - x*((a*e^3)/c^2 - (a*e^2*(e + 3*c^2* d))/c^2) + (log(c^2*x^2 + 1)*(b*e^3 + b*c^6*d^3 - 9*b*c^2*d*e^2 - 9*b*c^4* d^2*e))/(6*c^3) - (atan(c*x)*((b*d^3)/3 - (b*e^3*x^6)/3 + 3*b*d^2*e*x^2 - 3*b*d*e^2*x^4))/x^3 - (b*e^3*x^2)/(6*c)
Time = 0.21 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.61 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^4} \, dx=\frac {-2 \mathit {atan} \left (c x \right ) b \,c^{3} d^{3}-18 \mathit {atan} \left (c x \right ) b \,c^{3} d^{2} e \,x^{2}+18 \mathit {atan} \left (c x \right ) b \,c^{3} d \,e^{2} x^{4}+2 \mathit {atan} \left (c x \right ) b \,c^{3} e^{3} x^{6}+\mathrm {log}\left (c^{2} x^{2}+1\right ) b \,c^{6} d^{3} x^{3}-9 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) b \,c^{4} d^{2} e \,x^{3}-9 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) b \,c^{2} d \,e^{2} x^{3}+\mathrm {log}\left (c^{2} x^{2}+1\right ) b \,e^{3} x^{3}-2 \,\mathrm {log}\left (x \right ) b \,c^{6} d^{3} x^{3}+18 \,\mathrm {log}\left (x \right ) b \,c^{4} d^{2} e \,x^{3}-2 a \,c^{3} d^{3}-18 a \,c^{3} d^{2} e \,x^{2}+18 a \,c^{3} d \,e^{2} x^{4}+2 a \,c^{3} e^{3} x^{6}-b \,c^{4} d^{3} x -b \,c^{2} e^{3} x^{5}}{6 c^{3} x^{3}} \] Input:
int((e*x^2+d)^3*(a+b*atan(c*x))/x^4,x)
Output:
( - 2*atan(c*x)*b*c**3*d**3 - 18*atan(c*x)*b*c**3*d**2*e*x**2 + 18*atan(c* x)*b*c**3*d*e**2*x**4 + 2*atan(c*x)*b*c**3*e**3*x**6 + log(c**2*x**2 + 1)* b*c**6*d**3*x**3 - 9*log(c**2*x**2 + 1)*b*c**4*d**2*e*x**3 - 9*log(c**2*x* *2 + 1)*b*c**2*d*e**2*x**3 + log(c**2*x**2 + 1)*b*e**3*x**3 - 2*log(x)*b*c **6*d**3*x**3 + 18*log(x)*b*c**4*d**2*e*x**3 - 2*a*c**3*d**3 - 18*a*c**3*d **2*e*x**2 + 18*a*c**3*d*e**2*x**4 + 2*a*c**3*e**3*x**6 - b*c**4*d**3*x - b*c**2*e**3*x**5)/(6*c**3*x**3)