3.1139 \(\int x (d+e x^2)^3 (a+b \tan ^{-1}(c x)) \, dx\)

Optimal. Leaf size=158 \[ \frac{\left (d+e x^2\right )^4 \left (a+b \tan ^{-1}(c x)\right )}{8 e}-\frac{b e x^3 \left (6 c^4 d^2-4 c^2 d e+e^2\right )}{24 c^5}-\frac{b x \left (2 c^2 d-e\right ) \left (2 c^4 d^2-2 c^2 d e+e^2\right )}{8 c^7}-\frac{b e^2 x^5 \left (4 c^2 d-e\right )}{40 c^3}-\frac{b \left (c^2 d-e\right )^4 \tan ^{-1}(c x)}{8 c^8 e}-\frac{b e^3 x^7}{56 c} \]

[Out]

-(b*(2*c^2*d - e)*(2*c^4*d^2 - 2*c^2*d*e + e^2)*x)/(8*c^7) - (b*e*(6*c^4*d^2 - 4*c^2*d*e + e^2)*x^3)/(24*c^5)
- (b*(4*c^2*d - e)*e^2*x^5)/(40*c^3) - (b*e^3*x^7)/(56*c) - (b*(c^2*d - e)^4*ArcTan[c*x])/(8*c^8*e) + ((d + e*
x^2)^4*(a + b*ArcTan[c*x]))/(8*e)

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Rubi [A]  time = 0.146663, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {4974, 390, 203} \[ \frac{\left (d+e x^2\right )^4 \left (a+b \tan ^{-1}(c x)\right )}{8 e}-\frac{b e x^3 \left (6 c^4 d^2-4 c^2 d e+e^2\right )}{24 c^5}-\frac{b x \left (2 c^2 d-e\right ) \left (2 c^4 d^2-2 c^2 d e+e^2\right )}{8 c^7}-\frac{b e^2 x^5 \left (4 c^2 d-e\right )}{40 c^3}-\frac{b \left (c^2 d-e\right )^4 \tan ^{-1}(c x)}{8 c^8 e}-\frac{b e^3 x^7}{56 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*(2*c^2*d - e)*(2*c^4*d^2 - 2*c^2*d*e + e^2)*x)/(8*c^7) - (b*e*(6*c^4*d^2 - 4*c^2*d*e + e^2)*x^3)/(24*c^5)
- (b*(4*c^2*d - e)*e^2*x^5)/(40*c^3) - (b*e^3*x^7)/(56*c) - (b*(c^2*d - e)^4*ArcTan[c*x])/(8*c^8*e) + ((d + e*
x^2)^4*(a + b*ArcTan[c*x]))/(8*e)

Rule 4974

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]

Rule 390

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 203

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

Rubi steps

\begin{align*} \int x \left (d+e x^2\right )^3 \left (a+b \tan ^{-1}(c x)\right ) \, dx &=\frac{\left (d+e x^2\right )^4 \left (a+b \tan ^{-1}(c x)\right )}{8 e}-\frac{(b c) \int \frac{\left (d+e x^2\right )^4}{1+c^2 x^2} \, dx}{8 e}\\ &=\frac{\left (d+e x^2\right )^4 \left (a+b \tan ^{-1}(c x)\right )}{8 e}-\frac{(b c) \int \left (\frac{\left (2 c^2 d-e\right ) e \left (2 c^4 d^2-2 c^2 d e+e^2\right )}{c^8}+\frac{e^2 \left (6 c^4 d^2-4 c^2 d e+e^2\right ) x^2}{c^6}+\frac{\left (4 c^2 d-e\right ) e^3 x^4}{c^4}+\frac{e^4 x^6}{c^2}+\frac{c^8 d^4-4 c^6 d^3 e+6 c^4 d^2 e^2-4 c^2 d e^3+e^4}{c^8 \left (1+c^2 x^2\right )}\right ) \, dx}{8 e}\\ &=-\frac{b \left (2 c^2 d-e\right ) \left (2 c^4 d^2-2 c^2 d e+e^2\right ) x}{8 c^7}-\frac{b e \left (6 c^4 d^2-4 c^2 d e+e^2\right ) x^3}{24 c^5}-\frac{b \left (4 c^2 d-e\right ) e^2 x^5}{40 c^3}-\frac{b e^3 x^7}{56 c}+\frac{\left (d+e x^2\right )^4 \left (a+b \tan ^{-1}(c x)\right )}{8 e}-\frac{\left (b \left (c^2 d-e\right )^4\right ) \int \frac{1}{1+c^2 x^2} \, dx}{8 c^7 e}\\ &=-\frac{b \left (2 c^2 d-e\right ) \left (2 c^4 d^2-2 c^2 d e+e^2\right ) x}{8 c^7}-\frac{b e \left (6 c^4 d^2-4 c^2 d e+e^2\right ) x^3}{24 c^5}-\frac{b \left (4 c^2 d-e\right ) e^2 x^5}{40 c^3}-\frac{b e^3 x^7}{56 c}-\frac{b \left (c^2 d-e\right )^4 \tan ^{-1}(c x)}{8 c^8 e}+\frac{\left (d+e x^2\right )^4 \left (a+b \tan ^{-1}(c x)\right )}{8 e}\\ \end{align*}

Mathematica [A]  time = 0.159754, size = 217, normalized size = 1.37 \[ \frac{c x \left (105 a c^7 x \left (6 d^2 e x^2+4 d^3+4 d e^2 x^4+e^3 x^6\right )-3 b c^6 \left (70 d^2 e x^2+140 d^3+28 d e^2 x^4+5 e^3 x^6\right )+7 b c^4 e \left (90 d^2+20 d e x^2+3 e^2 x^4\right )-35 b c^2 e^2 \left (12 d+e x^2\right )+105 b e^3\right )+105 b \tan ^{-1}(c x) \left (c^8 \left (6 d^2 e x^4+4 d^3 x^2+4 d e^2 x^6+e^3 x^8\right )-6 c^4 d^2 e+4 c^6 d^3+4 c^2 d e^2-e^3\right )}{840 c^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*x*(105*b*e^3 - 35*b*c^2*e^2*(12*d + e*x^2) + 7*b*c^4*e*(90*d^2 + 20*d*e*x^2 + 3*e^2*x^4) + 105*a*c^7*x*(4*d
^3 + 6*d^2*e*x^2 + 4*d*e^2*x^4 + e^3*x^6) - 3*b*c^6*(140*d^3 + 70*d^2*e*x^2 + 28*d*e^2*x^4 + 5*e^3*x^6)) + 105
*b*(4*c^6*d^3 - 6*c^4*d^2*e + 4*c^2*d*e^2 - e^3 + c^8*(4*d^3*x^2 + 6*d^2*e*x^4 + 4*d*e^2*x^6 + e^3*x^8))*ArcTa
n[c*x])/(840*c^8)

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Maple [A]  time = 0.037, size = 265, normalized size = 1.7 \begin{align*}{\frac{a{e}^{3}{x}^{8}}{8}}+{\frac{ad{e}^{2}{x}^{6}}{2}}+{\frac{3\,a{d}^{2}e{x}^{4}}{4}}+{\frac{a{x}^{2}{d}^{3}}{2}}+{\frac{b\arctan \left ( cx \right ){e}^{3}{x}^{8}}{8}}+{\frac{b\arctan \left ( cx \right ) d{e}^{2}{x}^{6}}{2}}+{\frac{3\,b\arctan \left ( cx \right ){d}^{2}e{x}^{4}}{4}}+{\frac{b\arctan \left ( cx \right ){d}^{3}{x}^{2}}{2}}-{\frac{b{e}^{3}{x}^{7}}{56\,c}}-{\frac{b{x}^{5}d{e}^{2}}{10\,c}}-{\frac{b{x}^{3}{d}^{2}e}{4\,c}}-{\frac{b{d}^{3}x}{2\,c}}+{\frac{b{x}^{5}{e}^{3}}{40\,{c}^{3}}}+{\frac{b{x}^{3}d{e}^{2}}{6\,{c}^{3}}}+{\frac{3\,b{d}^{2}ex}{4\,{c}^{3}}}-{\frac{b{e}^{3}{x}^{3}}{24\,{c}^{5}}}-{\frac{bd{e}^{2}x}{2\,{c}^{5}}}+{\frac{b{e}^{3}x}{8\,{c}^{7}}}+{\frac{b{d}^{3}\arctan \left ( cx \right ) }{2\,{c}^{2}}}-{\frac{3\,b{d}^{2}\arctan \left ( cx \right ) e}{4\,{c}^{4}}}+{\frac{\arctan \left ( cx \right ) bd{e}^{2}}{2\,{c}^{6}}}-{\frac{b\arctan \left ( cx \right ){e}^{3}}{8\,{c}^{8}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*a*e^3*x^8+1/2*a*d*e^2*x^6+3/4*a*d^2*e*x^4+1/2*a*x^2*d^3+1/8*b*arctan(c*x)*e^3*x^8+1/2*b*arctan(c*x)*d*e^2*
x^6+3/4*b*arctan(c*x)*d^2*e*x^4+1/2*b*arctan(c*x)*d^3*x^2-1/56*b*e^3*x^7/c-1/10/c*b*x^5*d*e^2-1/4/c*b*x^3*d^2*
e-1/2*b*d^3*x/c+1/40/c^3*b*x^5*e^3+1/6/c^3*b*x^3*d*e^2+3/4/c^3*b*d^2*e*x-1/24/c^5*b*e^3*x^3-1/2/c^5*b*d*e^2*x+
1/8/c^7*b*e^3*x+1/2/c^2*b*arctan(c*x)*d^3-3/4/c^4*b*arctan(c*x)*d^2*e+1/2/c^6*b*arctan(c*x)*d*e^2-1/8/c^8*b*ar
ctan(c*x)*e^3

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Maxima [A]  time = 1.46397, size = 313, normalized size = 1.98 \begin{align*} \frac{1}{8} \, a e^{3} x^{8} + \frac{1}{2} \, a d e^{2} x^{6} + \frac{3}{4} \, a d^{2} e x^{4} + \frac{1}{2} \, a d^{3} 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^{3} + \frac{1}{4} \,{\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^{2} e + \frac{1}{30} \,{\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 d e^{2} + \frac{1}{840} \,{\left (105 \, x^{8} \arctan \left (c x\right ) - c{\left (\frac{15 \, c^{6} x^{7} - 21 \, c^{4} x^{5} + 35 \, c^{2} x^{3} - 105 \, x}{c^{8}} + \frac{105 \, \arctan \left (c x\right )}{c^{9}}\right )}\right )} b e^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*a*e^3*x^8 + 1/2*a*d*e^2*x^6 + 3/4*a*d^2*e*x^4 + 1/2*a*d^3*x^2 + 1/2*(x^2*arctan(c*x) - c*(x/c^2 - arctan(c
*x)/c^3))*b*d^3 + 1/4*(3*x^4*arctan(c*x) - c*((c^2*x^3 - 3*x)/c^4 + 3*arctan(c*x)/c^5))*b*d^2*e + 1/30*(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*d*e^2 + 1/840*(105*x^8*arctan(c*
x) - c*((15*c^6*x^7 - 21*c^4*x^5 + 35*c^2*x^3 - 105*x)/c^8 + 105*arctan(c*x)/c^9))*b*e^3

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Fricas [A]  time = 1.54233, size = 559, normalized size = 3.54 \begin{align*} \frac{105 \, a c^{8} e^{3} x^{8} + 420 \, a c^{8} d e^{2} x^{6} - 15 \, b c^{7} e^{3} x^{7} + 630 \, a c^{8} d^{2} e x^{4} + 420 \, a c^{8} d^{3} x^{2} - 21 \,{\left (4 \, b c^{7} d e^{2} - b c^{5} e^{3}\right )} x^{5} - 35 \,{\left (6 \, b c^{7} d^{2} e - 4 \, b c^{5} d e^{2} + b c^{3} e^{3}\right )} x^{3} - 105 \,{\left (4 \, b c^{7} d^{3} - 6 \, b c^{5} d^{2} e + 4 \, b c^{3} d e^{2} - b c e^{3}\right )} x + 105 \,{\left (b c^{8} e^{3} x^{8} + 4 \, b c^{8} d e^{2} x^{6} + 6 \, b c^{8} d^{2} e x^{4} + 4 \, b c^{8} d^{3} x^{2} + 4 \, b c^{6} d^{3} - 6 \, b c^{4} d^{2} e + 4 \, b c^{2} d e^{2} - b e^{3}\right )} \arctan \left (c x\right )}{840 \, c^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/840*(105*a*c^8*e^3*x^8 + 420*a*c^8*d*e^2*x^6 - 15*b*c^7*e^3*x^7 + 630*a*c^8*d^2*e*x^4 + 420*a*c^8*d^3*x^2 -
21*(4*b*c^7*d*e^2 - b*c^5*e^3)*x^5 - 35*(6*b*c^7*d^2*e - 4*b*c^5*d*e^2 + b*c^3*e^3)*x^3 - 105*(4*b*c^7*d^3 - 6
*b*c^5*d^2*e + 4*b*c^3*d*e^2 - b*c*e^3)*x + 105*(b*c^8*e^3*x^8 + 4*b*c^8*d*e^2*x^6 + 6*b*c^8*d^2*e*x^4 + 4*b*c
^8*d^3*x^2 + 4*b*c^6*d^3 - 6*b*c^4*d^2*e + 4*b*c^2*d*e^2 - b*e^3)*arctan(c*x))/c^8

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Sympy [A]  time = 6.38401, size = 350, normalized size = 2.22 \begin{align*} \begin{cases} \frac{a d^{3} x^{2}}{2} + \frac{3 a d^{2} e x^{4}}{4} + \frac{a d e^{2} x^{6}}{2} + \frac{a e^{3} x^{8}}{8} + \frac{b d^{3} x^{2} \operatorname{atan}{\left (c x \right )}}{2} + \frac{3 b d^{2} e x^{4} \operatorname{atan}{\left (c x \right )}}{4} + \frac{b d e^{2} x^{6} \operatorname{atan}{\left (c x \right )}}{2} + \frac{b e^{3} x^{8} \operatorname{atan}{\left (c x \right )}}{8} - \frac{b d^{3} x}{2 c} - \frac{b d^{2} e x^{3}}{4 c} - \frac{b d e^{2} x^{5}}{10 c} - \frac{b e^{3} x^{7}}{56 c} + \frac{b d^{3} \operatorname{atan}{\left (c x \right )}}{2 c^{2}} + \frac{3 b d^{2} e x}{4 c^{3}} + \frac{b d e^{2} x^{3}}{6 c^{3}} + \frac{b e^{3} x^{5}}{40 c^{3}} - \frac{3 b d^{2} e \operatorname{atan}{\left (c x \right )}}{4 c^{4}} - \frac{b d e^{2} x}{2 c^{5}} - \frac{b e^{3} x^{3}}{24 c^{5}} + \frac{b d e^{2} \operatorname{atan}{\left (c x \right )}}{2 c^{6}} + \frac{b e^{3} x}{8 c^{7}} - \frac{b e^{3} \operatorname{atan}{\left (c x \right )}}{8 c^{8}} & \text{for}\: c \neq 0 \\a \left (\frac{d^{3} x^{2}}{2} + \frac{3 d^{2} e x^{4}}{4} + \frac{d e^{2} x^{6}}{2} + \frac{e^{3} x^{8}}{8}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*d**3*x**2/2 + 3*a*d**2*e*x**4/4 + a*d*e**2*x**6/2 + a*e**3*x**8/8 + b*d**3*x**2*atan(c*x)/2 + 3*b
*d**2*e*x**4*atan(c*x)/4 + b*d*e**2*x**6*atan(c*x)/2 + b*e**3*x**8*atan(c*x)/8 - b*d**3*x/(2*c) - b*d**2*e*x**
3/(4*c) - b*d*e**2*x**5/(10*c) - b*e**3*x**7/(56*c) + b*d**3*atan(c*x)/(2*c**2) + 3*b*d**2*e*x/(4*c**3) + b*d*
e**2*x**3/(6*c**3) + b*e**3*x**5/(40*c**3) - 3*b*d**2*e*atan(c*x)/(4*c**4) - b*d*e**2*x/(2*c**5) - b*e**3*x**3
/(24*c**5) + b*d*e**2*atan(c*x)/(2*c**6) + b*e**3*x/(8*c**7) - b*e**3*atan(c*x)/(8*c**8), Ne(c, 0)), (a*(d**3*
x**2/2 + 3*d**2*e*x**4/4 + d*e**2*x**6/2 + e**3*x**8/8), True))

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Giac [B]  time = 1.46809, size = 416, normalized size = 2.63 \begin{align*} \frac{105 \, b c^{8} x^{8} \arctan \left (c x\right ) e^{3} + 105 \, a c^{8} x^{8} e^{3} + 420 \, b c^{8} d x^{6} \arctan \left (c x\right ) e^{2} + 420 \, a c^{8} d x^{6} e^{2} + 630 \, b c^{8} d^{2} x^{4} \arctan \left (c x\right ) e - 15 \, b c^{7} x^{7} e^{3} + 630 \, a c^{8} d^{2} x^{4} e + 420 \, b c^{8} d^{3} x^{2} \arctan \left (c x\right ) - 84 \, b c^{7} d x^{5} e^{2} + 420 \, a c^{8} d^{3} x^{2} - 210 \, b c^{7} d^{2} x^{3} e - 420 \, \pi b c^{6} d^{3} \mathrm{sgn}\left (c\right ) \mathrm{sgn}\left (x\right ) - 420 \, b c^{7} d^{3} x + 21 \, b c^{5} x^{5} e^{3} + 420 \, b c^{6} d^{3} \arctan \left (c x\right ) + 140 \, b c^{5} d x^{3} e^{2} + 630 \, b c^{5} d^{2} x e - 630 \, b c^{4} d^{2} \arctan \left (c x\right ) e - 35 \, b c^{3} x^{3} e^{3} - 420 \, \pi b c^{2} d e^{2} \mathrm{sgn}\left (c\right ) \mathrm{sgn}\left (x\right ) - 420 \, b c^{3} d x e^{2} + 420 \, b c^{2} d \arctan \left (c x\right ) e^{2} + 105 \, b c x e^{3} - 105 \, b \arctan \left (c x\right ) e^{3}}{840 \, c^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/840*(105*b*c^8*x^8*arctan(c*x)*e^3 + 105*a*c^8*x^8*e^3 + 420*b*c^8*d*x^6*arctan(c*x)*e^2 + 420*a*c^8*d*x^6*e
^2 + 630*b*c^8*d^2*x^4*arctan(c*x)*e - 15*b*c^7*x^7*e^3 + 630*a*c^8*d^2*x^4*e + 420*b*c^8*d^3*x^2*arctan(c*x)
- 84*b*c^7*d*x^5*e^2 + 420*a*c^8*d^3*x^2 - 210*b*c^7*d^2*x^3*e - 420*pi*b*c^6*d^3*sgn(c)*sgn(x) - 420*b*c^7*d^
3*x + 21*b*c^5*x^5*e^3 + 420*b*c^6*d^3*arctan(c*x) + 140*b*c^5*d*x^3*e^2 + 630*b*c^5*d^2*x*e - 630*b*c^4*d^2*a
rctan(c*x)*e - 35*b*c^3*x^3*e^3 - 420*pi*b*c^2*d*e^2*sgn(c)*sgn(x) - 420*b*c^3*d*x*e^2 + 420*b*c^2*d*arctan(c*
x)*e^2 + 105*b*c*x*e^3 - 105*b*arctan(c*x)*e^3)/c^8