\(\int x (d+e x^2)^2 (a+b \arctan (c x)) \, dx\) [1127]

Optimal result

Integrand size = 19, antiderivative size = 115 \[ \int x \left (d+e x^2\right )^2 (a+b \arctan (c x)) \, dx=-\frac {b \left (3 c^4 d^2-3 c^2 d e+e^2\right ) x}{6 c^5}-\frac {b \left (3 c^2 d-e\right ) e x^3}{18 c^3}-\frac {b e^2 x^5}{30 c}-\frac {b \left (c^2 d-e\right )^3 \arctan (c x)}{6 c^6 e}+\frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{6 e} \]

[Out]

-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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {5094, 398, 209} \[ \int x \left (d+e x^2\right )^2 (a+b \arctan (c x)) \, dx=\frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{6 e}-\frac {b \arctan (c x) \left (c^2 d-e\right )^3}{6 c^6 e}-\frac {b e x^3 \left (3 c^2 d-e\right )}{18 c^3}-\frac {b x \left (3 c^4 d^2-3 c^2 d e+e^2\right )}{6 c^5}-\frac {b e^2 x^5}{30 c} \]

[In]

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

[Out]

-1/6*(b*(3*c^4*d^2 - 3*c^2*d*e + e^2)*x)/c^5 - (b*(3*c^2*d - e)*e*x^3)/(18*c^3) - (b*e^2*x^5)/(30*c) - (b*(c^2
*d - e)^3*ArcTan[c*x])/(6*c^6*e) + ((d + e*x^2)^3*(a + b*ArcTan[c*x]))/(6*e)

Rule 209

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

Rule 398

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

Rule 5094

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] - Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{6 e}-\frac {(b c) \int \frac {\left (d+e x^2\right )^3}{1+c^2 x^2} \, dx}{6 e} \\ & = \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{6 e}-\frac {(b c) \int \left (\frac {e \left (3 c^4 d^2-3 c^2 d e+e^2\right )}{c^6}+\frac {\left (3 c^2 d-e\right ) e^2 x^2}{c^4}+\frac {e^3 x^4}{c^2}+\frac {c^6 d^3-3 c^4 d^2 e+3 c^2 d e^2-e^3}{c^6 \left (1+c^2 x^2\right )}\right ) \, dx}{6 e} \\ & = -\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 {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{6 e}-\frac {\left (b \left (c^2 d-e\right )^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{6 c^5 e} \\ & = -\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.22 \[ \int x \left (d+e x^2\right )^2 (a+b \arctan (c x)) \, dx=\frac {c x \left (-15 b e^2+5 b c^2 e \left (9 d+e x^2\right )+15 a c^5 x \left (3 d^2+3 d e x^2+e^2 x^4\right )-3 b c^4 \left (15 d^2+5 d e x^2+e^2 x^4\right )\right )+15 b \left (3 c^4 d^2-3 c^2 d e+e^2+c^6 \left (3 d^2 x^2+3 d e x^4+e^2 x^6\right )\right ) \arctan (c x)}{90 c^6} \]

[In]

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

[Out]

(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)

Maple [A] (verified)

Time = 0.31 (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	\(\frac {a \left (e \,x^{2}+d \right )^{3}}{6 e}+\frac {b \left (\frac {\arctan \left (c x \right ) c^{2} e^{2} x^{6}}{6}+\frac {\arctan \left (c x \right ) c^{2} e \,x^{4} d}{2}+\frac {\arctan \left (c x \right ) c^{2} x^{2} d^{2}}{2}+\frac {\arctan \left (c x \right ) c^{2} d^{3}}{6 e}-\frac {3 c^{5} d^{2} e x +c^{5} d \,e^{2} x^{3}+\frac {e^{3} c^{5} x^{5}}{5}-3 c^{3} x d \,e^{2}-\frac {e^{3} c^{3} x^{3}}{3}+c x \,e^{3}+\left (c^{6} d^{3}-3 e \,d^{2} c^{4}+3 e^{2} d \,c^{2}-e^{3}\right ) \arctan \left (c x \right )}{6 c^{4} e}\right )}{c^{2}}\)	\(186\)
derivativedivides	\(\frac {\frac {a \left (e \,c^{2} x^{2}+c^{2} d \right )^{3}}{6 c^{4} e}+\frac {b \left (\frac {\arctan \left (c x \right ) c^{6} d^{3}}{6 e}+\frac {\arctan \left (c x \right ) c^{6} d^{2} x^{2}}{2}+\frac {\arctan \left (c x \right ) e \,c^{6} d \,x^{4}}{2}+\frac {\arctan \left (c x \right ) e^{2} c^{6} x^{6}}{6}-\frac {3 c^{5} d^{2} e x +c^{5} d \,e^{2} x^{3}+\frac {e^{3} c^{5} x^{5}}{5}-3 c^{3} x d \,e^{2}-\frac {e^{3} c^{3} x^{3}}{3}+c x \,e^{3}+\left (c^{6} d^{3}-3 e \,d^{2} c^{4}+3 e^{2} d \,c^{2}-e^{3}\right ) \arctan \left (c x \right )}{6 e}\right )}{c^{4}}}{c^{2}}\)	\(197\)
default	\(\frac {\frac {a \left (e \,c^{2} x^{2}+c^{2} d \right )^{3}}{6 c^{4} e}+\frac {b \left (\frac {\arctan \left (c x \right ) c^{6} d^{3}}{6 e}+\frac {\arctan \left (c x \right ) c^{6} d^{2} x^{2}}{2}+\frac {\arctan \left (c x \right ) e \,c^{6} d \,x^{4}}{2}+\frac {\arctan \left (c x \right ) e^{2} c^{6} x^{6}}{6}-\frac {3 c^{5} d^{2} e x +c^{5} d \,e^{2} x^{3}+\frac {e^{3} c^{5} x^{5}}{5}-3 c^{3} x d \,e^{2}-\frac {e^{3} c^{3} x^{3}}{3}+c x \,e^{3}+\left (c^{6} d^{3}-3 e \,d^{2} c^{4}+3 e^{2} d \,c^{2}-e^{3}\right ) \arctan \left (c x \right )}{6 e}\right )}{c^{4}}}{c^{2}}\)	\(197\)
risch	\(\frac {i e^{2} b \,x^{6} \ln \left (-i c x +1\right )}{12}-\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 e \,d^{2} c^{4}+6 e^{2} d \,c^{2}-2 e^{3}}\right )}{12 c^{6}}+\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 e \,d^{2} c^{4}+6 e^{2} d \,c^{2}-2 e^{3}}\right )}{12 c^{6}}+\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 {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 e \,d^{2} c^{4}+6 e^{2} d \,c^{2}-2 e^{3}}\right )}{12 e}+\frac {i b \,d^{2} x^{2} \ln \left (-i c x +1\right )}{4}+\frac {x^{6} e^{2} a}{6}-\frac {b \,e^{2} x^{5}}{30 c}-\frac {i \left (e \,x^{2}+d \right )^{3} b \ln \left (i c x +1\right )}{12 e}+\frac {i e b d \,x^{4} \ln \left (-i c x +1\right )}{4}+\frac {e^{2} b \,x^{3}}{18 c^{3}}-\frac {e^{2} b x}{6 c^{5}}-\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 e \,d^{2} 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 e \,d^{2} c^{4}+6 e^{2} d \,c^{2}-2 e^{3}}\right )}{4 c^{4}}+\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 e \,d^{2} c^{4}+6 e^{2} d \,c^{2}-2 e^{3}}\right )}{4 c^{4}}-\frac {b \,d^{2} x}{2 c}+\frac {e b d x}{2 c^{3}}+\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} \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 e \,d^{2} c^{4}+6 e^{2} d \,c^{2}-2 e^{3}}\right )}{4 c^{2}}\)	\(872\)

[In]

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

[Out]

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
*arctan(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*arctan(c*x))/c^6

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.44 \[ \int x \left (d+e x^2\right )^2 (a+b \arctan (c x)) \, dx=\frac {15 \, a c^{6} e^{2} x^{6} + 45 \, a c^{6} d e x^{4} - 3 \, b c^{5} e^{2} x^{5} + 45 \, a c^{6} d^{2} x^{2} - 5 \, {\left (3 \, b c^{5} d e - b c^{3} e^{2}\right )} x^{3} - 15 \, {\left (3 \, b c^{5} d^{2} - 3 \, b c^{3} d e + b c e^{2}\right )} x + 15 \, {\left (b c^{6} e^{2} x^{6} + 3 \, b c^{6} d e x^{4} + 3 \, b c^{6} d^{2} x^{2} + 3 \, b c^{4} d^{2} - 3 \, b c^{2} d e + b e^{2}\right )} \arctan \left (c x\right )}{90 \, c^{6}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (102) = 204\).

Time = 0.41 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.90 \[ \int x \left (d+e x^2\right )^2 (a+b \arctan (c x)) \, dx=\begin {cases} \frac {a d^{2} x^{2}}{2} + \frac {a d e x^{4}}{2} + \frac {a e^{2} x^{6}}{6} + \frac {b d^{2} x^{2} \operatorname {atan}{\left (c x \right )}}{2} + \frac {b d e x^{4} \operatorname {atan}{\left (c x \right )}}{2} + \frac {b e^{2} x^{6} \operatorname {atan}{\left (c x \right )}}{6} - \frac {b d^{2} x}{2 c} - \frac {b d e x^{3}}{6 c} - \frac {b e^{2} x^{5}}{30 c} + \frac {b d^{2} \operatorname {atan}{\left (c x \right )}}{2 c^{2}} + \frac {b d e x}{2 c^{3}} + \frac {b e^{2} x^{3}}{18 c^{3}} - \frac {b d e \operatorname {atan}{\left (c x \right )}}{2 c^{4}} - \frac {b e^{2} x}{6 c^{5}} + \frac {b e^{2} \operatorname {atan}{\left (c x \right )}}{6 c^{6}} & \text {for}\: c \neq 0 \\a \left (\frac {d^{2} x^{2}}{2} + \frac {d e x^{4}}{2} + \frac {e^{2} x^{6}}{6}\right ) & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.36 \[ \int x \left (d+e x^2\right )^2 (a+b \arctan (c x)) \, dx=\frac {1}{6} \, a e^{2} x^{6} + \frac {1}{2} \, a d e x^{4} + \frac {1}{2} \, a d^{2} x^{2} + \frac {1}{2} \, {\left (x^{2} \arctan \left (c x\right ) - c {\left (\frac {x}{c^{2}} - \frac {\arctan \left (c x\right )}{c^{3}}\right )}\right )} b d^{2} + \frac {1}{6} \, {\left (3 \, x^{4} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{3} - 3 \, x}{c^{4}} + \frac {3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} b d e + \frac {1}{90} \, {\left (15 \, x^{6} \arctan \left (c x\right ) - c {\left (\frac {3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac {15 \, \arctan \left (c x\right )}{c^{7}}\right )}\right )} b e^{2} \]

[In]

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

[Out]

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

Giac [F]

\[ \int x \left (d+e x^2\right )^2 (a+b \arctan (c x)) \, dx=\int { {\left (e x^{2} + d\right )}^{2} {\left (b \arctan \left (c x\right ) + a\right )} x \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 0.51 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.45 \[ \int x \left (d+e x^2\right )^2 (a+b \arctan (c x)) \, dx=\frac {a\,d^2\,x^2}{2}+\frac {a\,e^2\,x^6}{6}-\frac {b\,e^2\,x^5}{30\,c}+\frac {b\,e^2\,x^3}{18\,c^3}+\frac {a\,d\,e\,x^4}{2}-\frac {b\,d^2\,x}{2\,c}-\frac {b\,e^2\,x}{6\,c^5}+\frac {b\,d^2\,\mathrm {atan}\left (c\,x\right )}{2\,c^2}+\frac {b\,e^2\,\mathrm {atan}\left (c\,x\right )}{6\,c^6}+\frac {b\,d^2\,x^2\,\mathrm {atan}\left (c\,x\right )}{2}+\frac {b\,e^2\,x^6\,\mathrm {atan}\left (c\,x\right )}{6}-\frac {b\,d\,e\,x^3}{6\,c}+\frac {b\,d\,e\,x}{2\,c^3}-\frac {b\,d\,e\,\mathrm {atan}\left (c\,x\right )}{2\,c^4}+\frac {b\,d\,e\,x^4\,\mathrm {atan}\left (c\,x\right )}{2} \]

[In]

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

[Out]

(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