Integrand size = 19, antiderivative size = 115 \[ \int x \left (d+e x^2\right )^2 (a+b \arctan (c x)) \, dx=-\frac {b \left (3 c^4 d^2-3 c^2 d e+e^2\right ) x}{6 c^5}-\frac {b \left (3 c^2 d-e\right ) e x^3}{18 c^3}-\frac {b e^2 x^5}{30 c}-\frac {b \left (c^2 d-e\right )^3 \arctan (c x)}{6 c^6 e}+\frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{6 e} \] Output:
-1/6*b*(3*c^4*d^2-3*c^2*d*e+e^2)*x/c^5-1/18*b*(3*c^2*d-e)*e*x^3/c^3-1/30*b *e^2*x^5/c-1/6*b*(c^2*d-e)^3*arctan(c*x)/c^6/e+1/6*(e*x^2+d)^3*(a+b*arctan (c*x))/e
Time = 0.05 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.22 \[ \int x \left (d+e x^2\right )^2 (a+b \arctan (c x)) \, dx=\frac {c x \left (-15 b e^2+5 b c^2 e \left (9 d+e x^2\right )+15 a c^5 x \left (3 d^2+3 d e x^2+e^2 x^4\right )-3 b c^4 \left (15 d^2+5 d e x^2+e^2 x^4\right )\right )+15 b \left (3 c^4 d^2-3 c^2 d e+e^2+c^6 \left (3 d^2 x^2+3 d e x^4+e^2 x^6\right )\right ) \arctan (c x)}{90 c^6} \] Input:
Integrate[x*(d + e*x^2)^2*(a + b*ArcTan[c*x]),x]
Output:
(c*x*(-15*b*e^2 + 5*b*c^2*e*(9*d + e*x^2) + 15*a*c^5*x*(3*d^2 + 3*d*e*x^2 + e^2*x^4) - 3*b*c^4*(15*d^2 + 5*d*e*x^2 + e^2*x^4)) + 15*b*(3*c^4*d^2 - 3 *c^2*d*e + e^2 + c^6*(3*d^2*x^2 + 3*d*e*x^4 + e^2*x^6))*ArcTan[c*x])/(90*c ^6)
Time = 0.33 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {5509, 300, 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.
| \(\displaystyle \int x \left (d+e x^2\right )^2 (a+b \arctan (c x)) \, dx\) |
| \(\Big \downarrow \) 5509 |
| \(\displaystyle \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{6 e}-\frac {b c \int \frac {\left (e x^2+d\right )^3}{c^2 x^2+1}dx}{6 e}\) |
| \(\Big \downarrow \) 300 |
| \(\displaystyle \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{6 e}-\frac {b c \int \left (\frac {e^3 x^4}{c^2}+\frac {\left (3 c^2 d-e\right ) e^2 x^2}{c^4}+\frac {e \left (3 d^2 c^4-3 d e c^2+e^2\right )}{c^6}+\frac {d^3 c^6-3 d^2 e c^4+3 d e^2 c^2-e^3}{c^6 \left (c^2 x^2+1\right )}\right )dx}{6 e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{6 e}-\frac {b c \left (\frac {\arctan (c x) \left (c^2 d-e\right )^3}{c^7}+\frac {e^3 x^5}{5 c^2}+\frac {e^2 x^3 \left (3 c^2 d-e\right )}{3 c^4}+\frac {e x \left (3 c^4 d^2-3 c^2 d e+e^2\right )}{c^6}\right )}{6 e}\) |
Input:
Int[x*(d + e*x^2)^2*(a + b*ArcTan[c*x]),x]
Output:
((d + e*x^2)^3*(a + b*ArcTan[c*x]))/(6*e) - (b*c*((e*(3*c^4*d^2 - 3*c^2*d* e + e^2)*x)/c^6 + ((3*c^2*d - e)*e^2*x^3)/(3*c^4) + (e^3*x^5)/(5*c^2) + (( c^2*d - e)^3*ArcTan[c*x])/c^7))/(6*e)
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Int [PolynomialDivide[(a + b*x^2)^p, (c + d*x^2)^(-q), x], x] /; FreeQ[{a, b, c , d}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && ILtQ[q, 0] && GeQ[p, -q]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*(x_)*((d_.) + (e_.)*(x_)^2)^(q_.), x _Symbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])/(2*e*(q + 1))), x ] - Simp[b*(c/(2*e*(q + 1))) Int[(d + e*x^2)^(q + 1)/(1 + c^2*x^2), x], x ] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[q, -1]
Time = 0.45 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.62
| method | result | size |
| parallelrisch | \(\frac {15 x^{6} \arctan \left (c x \right ) b \,c^{6} e^{2}+15 x^{6} a \,c^{6} e^{2}+45 x^{4} \arctan \left (c x \right ) b \,c^{6} d e -3 b \,c^{5} e^{2} x^{5}+45 x^{4} a \,c^{6} d e +45 x^{2} \arctan \left (c x \right ) b \,c^{6} d^{2}-15 b \,c^{5} d e \,x^{3}+45 x^{2} a \,c^{6} d^{2}+5 b \,c^{3} e^{2} x^{3}-45 b \,c^{5} d^{2} x +45 b \,c^{4} d^{2} \arctan \left (c x \right )+45 b \,c^{3} d e x -45 b \,c^{2} d e \arctan \left (c x \right )-15 b c \,e^{2} x +15 b \,e^{2} \arctan \left (c x \right )}{90 c^{6}}\) | \(186\) |
| parts | \(\frac {a \left (e \,x^{2}+d \right )^{3}}{6 e}+\frac {b \left (\frac {\arctan \left (c x \right ) c^{2} e^{2} x^{6}}{6}+\frac {\arctan \left (c x \right ) c^{2} e \,x^{4} d}{2}+\frac {\arctan \left (c x \right ) c^{2} x^{2} d^{2}}{2}+\frac {\arctan \left (c x \right ) c^{2} d^{3}}{6 e}-\frac {3 c^{5} d^{2} e x +c^{5} d \,e^{2} x^{3}+\frac {e^{3} c^{5} x^{5}}{5}-3 c^{3} x d \,e^{2}-\frac {e^{3} c^{3} x^{3}}{3}+c x \,e^{3}+\left (c^{6} d^{3}-3 d^{2} e \,c^{4}+3 e^{2} d \,c^{2}-e^{3}\right ) \arctan \left (c x \right )}{6 c^{4} e}\right )}{c^{2}}\) | \(186\) |
| derivativedivides | \(\frac {\frac {a \left (c^{2} e \,x^{2}+c^{2} d \right )^{3}}{6 c^{4} e}+\frac {b \left (\frac {\arctan \left (c x \right ) c^{6} d^{3}}{6 e}+\frac {\arctan \left (c x \right ) c^{6} d^{2} x^{2}}{2}+\frac {\arctan \left (c x \right ) e \,c^{6} d \,x^{4}}{2}+\frac {\arctan \left (c x \right ) e^{2} c^{6} x^{6}}{6}-\frac {3 c^{5} d^{2} e x +c^{5} d \,e^{2} x^{3}+\frac {e^{3} c^{5} x^{5}}{5}-3 c^{3} x d \,e^{2}-\frac {e^{3} c^{3} x^{3}}{3}+c x \,e^{3}+\left (c^{6} d^{3}-3 d^{2} e \,c^{4}+3 e^{2} d \,c^{2}-e^{3}\right ) \arctan \left (c x \right )}{6 e}\right )}{c^{4}}}{c^{2}}\) | \(197\) |
| default | \(\frac {\frac {a \left (c^{2} e \,x^{2}+c^{2} d \right )^{3}}{6 c^{4} e}+\frac {b \left (\frac {\arctan \left (c x \right ) c^{6} d^{3}}{6 e}+\frac {\arctan \left (c x \right ) c^{6} d^{2} x^{2}}{2}+\frac {\arctan \left (c x \right ) e \,c^{6} d \,x^{4}}{2}+\frac {\arctan \left (c x \right ) e^{2} c^{6} x^{6}}{6}-\frac {3 c^{5} d^{2} e x +c^{5} d \,e^{2} x^{3}+\frac {e^{3} c^{5} x^{5}}{5}-3 c^{3} x d \,e^{2}-\frac {e^{3} c^{3} x^{3}}{3}+c x \,e^{3}+\left (c^{6} d^{3}-3 d^{2} e \,c^{4}+3 e^{2} d \,c^{2}-e^{3}\right ) \arctan \left (c x \right )}{6 e}\right )}{c^{4}}}{c^{2}}\) | \(197\) |
| orering | \(\frac {\left (15 c^{6} e^{3} x^{8}+69 c^{6} d \,e^{2} x^{6}+165 c^{6} d^{2} e \,x^{4}-5 c^{4} e^{3} x^{6}+45 c^{6} d^{3} x^{2}-76 c^{4} d \,e^{2} x^{4}+120 c^{4} d^{2} e \,x^{2}+25 c^{2} e^{3} x^{4}+45 c^{4} d^{3}-130 c^{2} d \,e^{2} x^{2}-45 c^{2} d^{2} e +45 e^{3} x^{2}+15 e^{2} d \right ) \left (a +b \arctan \left (c x \right )\right )}{45 c^{6} \left (e \,x^{2}+d \right )}-\frac {\left (3 c^{4} e^{2} x^{4}+15 c^{4} d e \,x^{2}+45 c^{4} d^{2}-5 c^{2} e^{2} x^{2}-45 c^{2} d e +15 e^{2}\right ) \left (c^{2} x^{2}+1\right ) \left (\left (e \,x^{2}+d \right )^{2} \left (a +b \arctan \left (c x \right )\right )+4 x^{2} \left (e \,x^{2}+d \right ) \left (a +b \arctan \left (c x \right )\right ) e +\frac {x \left (e \,x^{2}+d \right )^{2} b c}{c^{2} x^{2}+1}\right )}{90 c^{6} \left (e \,x^{2}+d \right )^{2}}\) | \(300\) |
| risch | \(\frac {i e^{2} b \,x^{6} \ln \left (-i c x +1\right )}{12}-\frac {i \left (e \,x^{2}+d \right )^{3} b \ln \left (i c x +1\right )}{12 e}-\frac {e b d \arctan \left (\frac {\left (c^{7} d^{3}-6 c^{5} d^{2} e +6 c^{3} d \,e^{2}-2 c \,e^{3}\right ) x}{c^{6} d^{3}-6 d^{2} e \,c^{4}+6 e^{2} d \,c^{2}-2 e^{3}}\right )}{4 c^{4}}+\frac {b \,d^{2} \arctan \left (\frac {\left (c^{7} d^{3}-6 c^{5} d^{2} e +6 c^{3} d \,e^{2}-2 c \,e^{3}\right ) x}{c^{6} d^{3}-6 d^{2} e \,c^{4}+6 e^{2} d \,c^{2}-2 e^{3}}\right )}{4 c^{2}}+\frac {e b d \arctan \left (\frac {\left (-c^{7} d^{3}+6 c^{5} d^{2} e -6 c^{3} d \,e^{2}+2 c \,e^{3}\right ) x}{c^{6} d^{3}-6 d^{2} e \,c^{4}+6 e^{2} d \,c^{2}-2 e^{3}}\right )}{4 c^{4}}-\frac {b \,d^{2} \arctan \left (\frac {\left (-c^{7} d^{3}+6 c^{5} d^{2} e -6 c^{3} d \,e^{2}+2 c \,e^{3}\right ) x}{c^{6} d^{3}-6 d^{2} e \,c^{4}+6 e^{2} d \,c^{2}-2 e^{3}}\right )}{4 c^{2}}+\frac {i b \,d^{2} x^{2} \ln \left (-i c x +1\right )}{4}+\frac {i b \,d^{3} \ln \left (c^{14} d^{6} x^{2}-12 c^{12} d^{5} e \,x^{2}+c^{12} d^{6}+48 c^{10} d^{4} e^{2} x^{2}-12 c^{10} d^{5} e -76 c^{8} d^{3} e^{3} x^{2}+48 c^{8} d^{4} e^{2}+60 c^{6} d^{2} e^{4} x^{2}-76 c^{6} d^{3} e^{3}-24 c^{4} d \,e^{5} x^{2}+60 c^{4} d^{2} e^{4}+4 c^{2} e^{6} x^{2}-24 c^{2} d \,e^{5}+4 e^{6}\right )}{24 e}+\frac {e^{2} b \,x^{3}}{18 c^{3}}-\frac {e^{2} b x}{6 c^{5}}+\frac {x^{6} e^{2} a}{6}-\frac {b \,e^{2} x^{5}}{30 c}+\frac {e^{2} b \arctan \left (\frac {\left (c^{7} d^{3}-6 c^{5} d^{2} e +6 c^{3} d \,e^{2}-2 c \,e^{3}\right ) x}{c^{6} d^{3}-6 d^{2} e \,c^{4}+6 e^{2} d \,c^{2}-2 e^{3}}\right )}{12 c^{6}}+\frac {x^{4} e d a}{2}+\frac {x^{2} d^{2} a}{2}-\frac {e b d \,x^{3}}{6 c}-\frac {b \,d^{2} x}{2 c}+\frac {e b d x}{2 c^{3}}-\frac {e^{2} b \arctan \left (\frac {\left (-c^{7} d^{3}+6 c^{5} d^{2} e -6 c^{3} d \,e^{2}+2 c \,e^{3}\right ) x}{c^{6} d^{3}-6 d^{2} e \,c^{4}+6 e^{2} d \,c^{2}-2 e^{3}}\right )}{12 c^{6}}+\frac {i e b d \,x^{4} \ln \left (-i c x +1\right )}{4}+\frac {b \,d^{3} \arctan \left (\frac {\left (-c^{7} d^{3}+6 c^{5} d^{2} e -6 c^{3} d \,e^{2}+2 c \,e^{3}\right ) x}{c^{6} d^{3}-6 d^{2} e \,c^{4}+6 e^{2} d \,c^{2}-2 e^{3}}\right )}{12 e}\) | \(872\) |
Input:
int(x*(e*x^2+d)^2*(a+b*arctan(c*x)),x,method=_RETURNVERBOSE)
Output:
1/90*(15*x^6*arctan(c*x)*b*c^6*e^2+15*x^6*a*c^6*e^2+45*x^4*arctan(c*x)*b*c ^6*d*e-3*b*c^5*e^2*x^5+45*x^4*a*c^6*d*e+45*x^2*arctan(c*x)*b*c^6*d^2-15*b* c^5*d*e*x^3+45*x^2*a*c^6*d^2+5*b*c^3*e^2*x^3-45*b*c^5*d^2*x+45*b*c^4*d^2*a rctan(c*x)+45*b*c^3*d*e*x-45*b*c^2*d*e*arctan(c*x)-15*b*c*e^2*x+15*b*e^2*a rctan(c*x))/c^6
Time = 0.10 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.44 \[ \int x \left (d+e x^2\right )^2 (a+b \arctan (c x)) \, dx=\frac {15 \, a c^{6} e^{2} x^{6} + 45 \, a c^{6} d e x^{4} - 3 \, b c^{5} e^{2} x^{5} + 45 \, a c^{6} d^{2} x^{2} - 5 \, {\left (3 \, b c^{5} d e - b c^{3} e^{2}\right )} x^{3} - 15 \, {\left (3 \, b c^{5} d^{2} - 3 \, b c^{3} d e + b c e^{2}\right )} x + 15 \, {\left (b c^{6} e^{2} x^{6} + 3 \, b c^{6} d e x^{4} + 3 \, b c^{6} d^{2} x^{2} + 3 \, b c^{4} d^{2} - 3 \, b c^{2} d e + b e^{2}\right )} \arctan \left (c x\right )}{90 \, c^{6}} \] Input:
integrate(x*(e*x^2+d)^2*(a+b*arctan(c*x)),x, algorithm="fricas")
Output:
1/90*(15*a*c^6*e^2*x^6 + 45*a*c^6*d*e*x^4 - 3*b*c^5*e^2*x^5 + 45*a*c^6*d^2 *x^2 - 5*(3*b*c^5*d*e - b*c^3*e^2)*x^3 - 15*(3*b*c^5*d^2 - 3*b*c^3*d*e + b *c*e^2)*x + 15*(b*c^6*e^2*x^6 + 3*b*c^6*d*e*x^4 + 3*b*c^6*d^2*x^2 + 3*b*c^ 4*d^2 - 3*b*c^2*d*e + b*e^2)*arctan(c*x))/c^6
Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (102) = 204\).
Time = 0.40 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.90 \[ \int x \left (d+e x^2\right )^2 (a+b \arctan (c x)) \, dx=\begin {cases} \frac {a d^{2} x^{2}}{2} + \frac {a d e x^{4}}{2} + \frac {a e^{2} x^{6}}{6} + \frac {b d^{2} x^{2} \operatorname {atan}{\left (c x \right )}}{2} + \frac {b d e x^{4} \operatorname {atan}{\left (c x \right )}}{2} + \frac {b e^{2} x^{6} \operatorname {atan}{\left (c x \right )}}{6} - \frac {b d^{2} x}{2 c} - \frac {b d e x^{3}}{6 c} - \frac {b e^{2} x^{5}}{30 c} + \frac {b d^{2} \operatorname {atan}{\left (c x \right )}}{2 c^{2}} + \frac {b d e x}{2 c^{3}} + \frac {b e^{2} x^{3}}{18 c^{3}} - \frac {b d e \operatorname {atan}{\left (c x \right )}}{2 c^{4}} - \frac {b e^{2} x}{6 c^{5}} + \frac {b e^{2} \operatorname {atan}{\left (c x \right )}}{6 c^{6}} & \text {for}\: c \neq 0 \\a \left (\frac {d^{2} x^{2}}{2} + \frac {d e x^{4}}{2} + \frac {e^{2} x^{6}}{6}\right ) & \text {otherwise} \end {cases} \] Input:
integrate(x*(e*x**2+d)**2*(a+b*atan(c*x)),x)
Output:
Piecewise((a*d**2*x**2/2 + a*d*e*x**4/2 + a*e**2*x**6/6 + b*d**2*x**2*atan (c*x)/2 + b*d*e*x**4*atan(c*x)/2 + b*e**2*x**6*atan(c*x)/6 - b*d**2*x/(2*c ) - b*d*e*x**3/(6*c) - b*e**2*x**5/(30*c) + b*d**2*atan(c*x)/(2*c**2) + b* d*e*x/(2*c**3) + b*e**2*x**3/(18*c**3) - b*d*e*atan(c*x)/(2*c**4) - b*e**2 *x/(6*c**5) + b*e**2*atan(c*x)/(6*c**6), Ne(c, 0)), (a*(d**2*x**2/2 + d*e* x**4/2 + e**2*x**6/6), True))
Time = 0.12 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.36 \[ \int x \left (d+e x^2\right )^2 (a+b \arctan (c x)) \, dx=\frac {1}{6} \, a e^{2} x^{6} + \frac {1}{2} \, a d e x^{4} + \frac {1}{2} \, a d^{2} x^{2} + \frac {1}{2} \, {\left (x^{2} \arctan \left (c x\right ) - c {\left (\frac {x}{c^{2}} - \frac {\arctan \left (c x\right )}{c^{3}}\right )}\right )} b d^{2} + \frac {1}{6} \, {\left (3 \, x^{4} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{3} - 3 \, x}{c^{4}} + \frac {3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} b d e + \frac {1}{90} \, {\left (15 \, x^{6} \arctan \left (c x\right ) - c {\left (\frac {3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac {15 \, \arctan \left (c x\right )}{c^{7}}\right )}\right )} b e^{2} \] Input:
integrate(x*(e*x^2+d)^2*(a+b*arctan(c*x)),x, algorithm="maxima")
Output:
1/6*a*e^2*x^6 + 1/2*a*d*e*x^4 + 1/2*a*d^2*x^2 + 1/2*(x^2*arctan(c*x) - c*( x/c^2 - arctan(c*x)/c^3))*b*d^2 + 1/6*(3*x^4*arctan(c*x) - c*((c^2*x^3 - 3 *x)/c^4 + 3*arctan(c*x)/c^5))*b*d*e + 1/90*(15*x^6*arctan(c*x) - c*((3*c^4 *x^5 - 5*c^2*x^3 + 15*x)/c^6 - 15*arctan(c*x)/c^7))*b*e^2
Leaf count of result is larger than twice the leaf count of optimal. 223 vs. \(2 (105) = 210\).
Time = 0.24 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.94 \[ \int x \left (d+e x^2\right )^2 (a+b \arctan (c x)) \, dx=\frac {15 \, b c^{6} e^{2} x^{6} \arctan \left (c x\right ) + 15 \, a c^{6} e^{2} x^{6} + 45 \, b c^{6} d e x^{4} \arctan \left (c x\right ) + 45 \, a c^{6} d e x^{4} - 3 \, b c^{5} e^{2} x^{5} + 45 \, b c^{6} d^{2} x^{2} \arctan \left (c x\right ) + 45 \, a c^{6} d^{2} x^{2} - 15 \, b c^{5} d e x^{3} - 45 \, \pi b c^{4} d^{2} \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (x\right ) - 45 \, b c^{5} d^{2} x + 5 \, b c^{3} e^{2} x^{3} + 45 \, b c^{4} d^{2} \arctan \left (c x\right ) + 45 \, \pi b c^{2} d e \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (x\right ) + 45 \, b c^{3} d e x - 45 \, b c^{2} d e \arctan \left (c x\right ) - 15 \, \pi b e^{2} \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (x\right ) - 15 \, b c e^{2} x + 15 \, b e^{2} \arctan \left (c x\right )}{90 \, c^{6}} \] Input:
integrate(x*(e*x^2+d)^2*(a+b*arctan(c*x)),x, algorithm="giac")
Output:
1/90*(15*b*c^6*e^2*x^6*arctan(c*x) + 15*a*c^6*e^2*x^6 + 45*b*c^6*d*e*x^4*a rctan(c*x) + 45*a*c^6*d*e*x^4 - 3*b*c^5*e^2*x^5 + 45*b*c^6*d^2*x^2*arctan( c*x) + 45*a*c^6*d^2*x^2 - 15*b*c^5*d*e*x^3 - 45*pi*b*c^4*d^2*sgn(c)*sgn(x) - 45*b*c^5*d^2*x + 5*b*c^3*e^2*x^3 + 45*b*c^4*d^2*arctan(c*x) + 45*pi*b*c ^2*d*e*sgn(c)*sgn(x) + 45*b*c^3*d*e*x - 45*b*c^2*d*e*arctan(c*x) - 15*pi*b *e^2*sgn(c)*sgn(x) - 15*b*c*e^2*x + 15*b*e^2*arctan(c*x))/c^6
Time = 0.51 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.45 \[ \int x \left (d+e x^2\right )^2 (a+b \arctan (c x)) \, dx=\frac {a\,d^2\,x^2}{2}+\frac {a\,e^2\,x^6}{6}-\frac {b\,e^2\,x^5}{30\,c}+\frac {b\,e^2\,x^3}{18\,c^3}+\frac {a\,d\,e\,x^4}{2}-\frac {b\,d^2\,x}{2\,c}-\frac {b\,e^2\,x}{6\,c^5}+\frac {b\,d^2\,\mathrm {atan}\left (c\,x\right )}{2\,c^2}+\frac {b\,e^2\,\mathrm {atan}\left (c\,x\right )}{6\,c^6}+\frac {b\,d^2\,x^2\,\mathrm {atan}\left (c\,x\right )}{2}+\frac {b\,e^2\,x^6\,\mathrm {atan}\left (c\,x\right )}{6}-\frac {b\,d\,e\,x^3}{6\,c}+\frac {b\,d\,e\,x}{2\,c^3}-\frac {b\,d\,e\,\mathrm {atan}\left (c\,x\right )}{2\,c^4}+\frac {b\,d\,e\,x^4\,\mathrm {atan}\left (c\,x\right )}{2} \] Input:
int(x*(a + b*atan(c*x))*(d + e*x^2)^2,x)
Output:
(a*d^2*x^2)/2 + (a*e^2*x^6)/6 - (b*e^2*x^5)/(30*c) + (b*e^2*x^3)/(18*c^3) + (a*d*e*x^4)/2 - (b*d^2*x)/(2*c) - (b*e^2*x)/(6*c^5) + (b*d^2*atan(c*x))/ (2*c^2) + (b*e^2*atan(c*x))/(6*c^6) + (b*d^2*x^2*atan(c*x))/2 + (b*e^2*x^6 *atan(c*x))/6 - (b*d*e*x^3)/(6*c) + (b*d*e*x)/(2*c^3) - (b*d*e*atan(c*x))/ (2*c^4) + (b*d*e*x^4*atan(c*x))/2
Time = 0.19 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.61 \[ \int x \left (d+e x^2\right )^2 (a+b \arctan (c x)) \, dx=\frac {45 \mathit {atan} \left (c x \right ) b \,c^{6} d^{2} x^{2}+45 \mathit {atan} \left (c x \right ) b \,c^{6} d e \,x^{4}+15 \mathit {atan} \left (c x \right ) b \,c^{6} e^{2} x^{6}+45 \mathit {atan} \left (c x \right ) b \,c^{4} d^{2}-45 \mathit {atan} \left (c x \right ) b \,c^{2} d e +15 \mathit {atan} \left (c x \right ) b \,e^{2}+45 a \,c^{6} d^{2} x^{2}+45 a \,c^{6} d e \,x^{4}+15 a \,c^{6} e^{2} x^{6}-45 b \,c^{5} d^{2} x -15 b \,c^{5} d e \,x^{3}-3 b \,c^{5} e^{2} x^{5}+45 b \,c^{3} d e x +5 b \,c^{3} e^{2} x^{3}-15 b c \,e^{2} x}{90 c^{6}} \] Input:
int(x*(e*x^2+d)^2*(a+b*atan(c*x)),x)
Output:
(45*atan(c*x)*b*c**6*d**2*x**2 + 45*atan(c*x)*b*c**6*d*e*x**4 + 15*atan(c* x)*b*c**6*e**2*x**6 + 45*atan(c*x)*b*c**4*d**2 - 45*atan(c*x)*b*c**2*d*e + 15*atan(c*x)*b*e**2 + 45*a*c**6*d**2*x**2 + 45*a*c**6*d*e*x**4 + 15*a*c** 6*e**2*x**6 - 45*b*c**5*d**2*x - 15*b*c**5*d*e*x**3 - 3*b*c**5*e**2*x**5 + 45*b*c**3*d*e*x + 5*b*c**3*e**2*x**3 - 15*b*c*e**2*x)/(90*c**6)