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

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

[Out]

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

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Rubi [A]  time = 0.15, antiderivative size = 158, normalized size of antiderivative = 1.00, 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 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])

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 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]

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*}

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Mathematica [A]  time = 0.17, size = 217, normalized size = 1.37 \[ \frac {c x \left (105 a c^7 x \left (4 d^3+6 d^2 e x^2+4 d e^2 x^4+e^3 x^6\right )-3 b c^6 \left (140 d^3+70 d^2 e x^2+28 d e^2 x^4+5 e^3 x^6\right )+7 b c^4 e \left (90 d^2+20 d e x^2+3 e^2 x^4\right )-35 b c^2 e^2 \left (12 d+e x^2\right )+105 b e^3\right )+105 b \tan ^{-1}(c x) \left (c^8 \left (4 d^3 x^2+6 d^2 e x^4+4 d e^2 x^6+e^3 x^8\right )+4 c^6 d^3-6 c^4 d^2 e+4 c^2 d e^2-e^3\right )}{840 c^8} \]

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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fricas [A]  time = 0.48, size = 258, normalized size = 1.63 \[ \frac {105 \, a c^{8} e^{3} x^{8} + 420 \, a c^{8} d e^{2} x^{6} - 15 \, b c^{7} e^{3} x^{7} + 630 \, a c^{8} d^{2} e x^{4} + 420 \, a c^{8} d^{3} x^{2} - 21 \, {\left (4 \, b c^{7} d e^{2} - b c^{5} e^{3}\right )} x^{5} - 35 \, {\left (6 \, b c^{7} d^{2} e - 4 \, b c^{5} d e^{2} + b c^{3} e^{3}\right )} x^{3} - 105 \, {\left (4 \, b c^{7} d^{3} - 6 \, b c^{5} d^{2} e + 4 \, b c^{3} d e^{2} - b c e^{3}\right )} x + 105 \, {\left (b c^{8} e^{3} x^{8} + 4 \, b c^{8} d e^{2} x^{6} + 6 \, b c^{8} d^{2} e x^{4} + 4 \, b c^{8} d^{3} x^{2} + 4 \, b c^{6} d^{3} - 6 \, b c^{4} d^{2} e + 4 \, b c^{2} d e^{2} - b e^{3}\right )} \arctan \left (c x\right )}{840 \, c^{8}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

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]

sage0*x

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maple [A]  time = 0.04, size = 265, normalized size = 1.68 \[ \frac {a \,e^{3} x^{8}}{8}+\frac {a d \,e^{2} x^{6}}{2}+\frac {3 a \,d^{2} e \,x^{4}}{4}+\frac {a \,x^{2} d^{3}}{2}+\frac {b \arctan \left (c x \right ) e^{3} x^{8}}{8}+\frac {b \arctan \left (c x \right ) d \,e^{2} x^{6}}{2}+\frac {3 b \arctan \left (c x \right ) d^{2} e \,x^{4}}{4}+\frac {b \arctan \left (c x \right ) x^{2} d^{3}}{2}-\frac {b \,e^{3} x^{7}}{56 c}-\frac {b \,x^{5} d \,e^{2}}{10 c}-\frac {b \,x^{3} d^{2} e}{4 c}-\frac {b \,d^{3} x}{2 c}+\frac {b \,x^{5} e^{3}}{40 c^{3}}+\frac {b \,x^{3} d \,e^{2}}{6 c^{3}}+\frac {3 b \,d^{2} e x}{4 c^{3}}-\frac {b \,x^{3} e^{3}}{24 c^{5}}-\frac {b d \,e^{2} x}{2 c^{5}}+\frac {b \,e^{3} x}{8 c^{7}}+\frac {b \arctan \left (c x \right ) d^{3}}{2 c^{2}}-\frac {3 b \arctan \left (c x \right ) d^{2} e}{4 c^{4}}+\frac {b \arctan \left (c x \right ) d \,e^{2}}{2 c^{6}}-\frac {b \arctan \left (c x \right ) e^{3}}{8 c^{8}} \]

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)*x^2*d^3-1/56*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*x^3*e^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 = 0.43, size = 232, normalized size = 1.47 \[ \frac {1}{8} \, a e^{3} x^{8} + \frac {1}{2} \, a d e^{2} x^{6} + \frac {3}{4} \, a d^{2} e x^{4} + \frac {1}{2} \, a d^{3} x^{2} + \frac {1}{2} \, {\left (x^{2} \arctan \left (c x\right ) - c {\left (\frac {x}{c^{2}} - \frac {\arctan \left (c x\right )}{c^{3}}\right )}\right )} b d^{3} + \frac {1}{4} \, {\left (3 \, x^{4} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{3} - 3 \, x}{c^{4}} + \frac {3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} b d^{2} e + \frac {1}{30} \, {\left (15 \, x^{6} \arctan \left (c x\right ) - c {\left (\frac {3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac {15 \, \arctan \left (c x\right )}{c^{7}}\right )}\right )} b d e^{2} + \frac {1}{840} \, {\left (105 \, x^{8} \arctan \left (c x\right ) - c {\left (\frac {15 \, c^{6} x^{7} - 21 \, c^{4} x^{5} + 35 \, c^{2} x^{3} - 105 \, x}{c^{8}} + \frac {105 \, \arctan \left (c x\right )}{c^{9}}\right )}\right )} b e^{3} \]

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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mupad [B]  time = 0.56, size = 442, normalized size = 2.80 \[ x\,\left (\frac {\frac {\frac {b\,e^3}{8\,c^3}-\frac {b\,d\,e^2}{2\,c}}{c^2}+\frac {3\,b\,d^2\,e}{4\,c}}{c^2}-\frac {b\,d^3}{2\,c}\right )-x^6\,\left (\frac {a\,e^3}{6\,c^2}-\frac {a\,e^2\,\left (3\,d\,c^2+e\right )}{6\,c^2}\right )+x^4\,\left (\frac {\frac {a\,e^3}{c^2}-\frac {a\,e^2\,\left (3\,d\,c^2+e\right )}{c^2}}{4\,c^2}+\frac {3\,a\,d\,e\,\left (d\,c^2+e\right )}{4\,c^2}\right )+x^5\,\left (\frac {b\,e^3}{40\,c^3}-\frac {b\,d\,e^2}{10\,c}\right )+\mathrm {atan}\left (c\,x\right )\,\left (\frac {b\,d^3\,x^2}{2}+\frac {3\,b\,d^2\,e\,x^4}{4}+\frac {b\,d\,e^2\,x^6}{2}+\frac {b\,e^3\,x^8}{8}\right )-x^3\,\left (\frac {\frac {b\,e^3}{8\,c^3}-\frac {b\,d\,e^2}{2\,c}}{3\,c^2}+\frac {b\,d^2\,e}{4\,c}\right )-x^2\,\left (\frac {\frac {\frac {a\,e^3}{c^2}-\frac {a\,e^2\,\left (3\,d\,c^2+e\right )}{c^2}}{c^2}+\frac {3\,a\,d\,e\,\left (d\,c^2+e\right )}{c^2}}{2\,c^2}-\frac {a\,d^2\,\left (d\,c^2+3\,e\right )}{2\,c^2}\right )+\frac {a\,e^3\,x^8}{8}-\frac {b\,e^3\,x^7}{56\,c}-\frac {b\,\mathrm {atan}\left (\frac {b\,c\,x\,\left (e-2\,c^2\,d\right )\,\left (2\,c^4\,d^2-2\,c^2\,d\,e+e^2\right )}{-4\,b\,c^6\,d^3+6\,b\,c^4\,d^2\,e-4\,b\,c^2\,d\,e^2+b\,e^3}\right )\,\left (e-2\,c^2\,d\right )\,\left (2\,c^4\,d^2-2\,c^2\,d\,e+e^2\right )}{8\,c^8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*((((b*e^3)/(8*c^3) - (b*d*e^2)/(2*c))/c^2 + (3*b*d^2*e)/(4*c))/c^2 - (b*d^3)/(2*c)) - x^6*((a*e^3)/(6*c^2) -
(a*e^2*(e + 3*c^2*d))/(6*c^2)) + x^4*(((a*e^3)/c^2 - (a*e^2*(e + 3*c^2*d))/c^2)/(4*c^2) + (3*a*d*e*(e + c^2*d
))/(4*c^2)) + x^5*((b*e^3)/(40*c^3) - (b*d*e^2)/(10*c)) + atan(c*x)*((b*d^3*x^2)/2 + (b*e^3*x^8)/8 + (3*b*d^2*
e*x^4)/4 + (b*d*e^2*x^6)/2) - x^3*(((b*e^3)/(8*c^3) - (b*d*e^2)/(2*c))/(3*c^2) + (b*d^2*e)/(4*c)) - x^2*((((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)/(2*c^2) - (a*d^2*(3*e + c^2*d))/(2*c^2)
) + (a*e^3*x^8)/8 - (b*e^3*x^7)/(56*c) - (b*atan((b*c*x*(e - 2*c^2*d)*(e^2 + 2*c^4*d^2 - 2*c^2*d*e))/(b*e^3 -
4*b*c^6*d^3 - 4*b*c^2*d*e^2 + 6*b*c^4*d^2*e))*(e - 2*c^2*d)*(e^2 + 2*c^4*d^2 - 2*c^2*d*e))/(8*c^8)

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sympy [A]  time = 4.57, size = 350, normalized size = 2.22 \[ \begin {cases} \frac {a d^{3} x^{2}}{2} + \frac {3 a d^{2} e x^{4}}{4} + \frac {a d e^{2} x^{6}}{2} + \frac {a e^{3} x^{8}}{8} + \frac {b d^{3} x^{2} \operatorname {atan}{\left (c x \right )}}{2} + \frac {3 b d^{2} e x^{4} \operatorname {atan}{\left (c x \right )}}{4} + \frac {b d e^{2} x^{6} \operatorname {atan}{\left (c x \right )}}{2} + \frac {b e^{3} x^{8} \operatorname {atan}{\left (c x \right )}}{8} - \frac {b d^{3} x}{2 c} - \frac {b d^{2} e x^{3}}{4 c} - \frac {b d e^{2} x^{5}}{10 c} - \frac {b e^{3} x^{7}}{56 c} + \frac {b d^{3} \operatorname {atan}{\left (c x \right )}}{2 c^{2}} + \frac {3 b d^{2} e x}{4 c^{3}} + \frac {b d e^{2} x^{3}}{6 c^{3}} + \frac {b e^{3} x^{5}}{40 c^{3}} - \frac {3 b d^{2} e \operatorname {atan}{\left (c x \right )}}{4 c^{4}} - \frac {b d e^{2} x}{2 c^{5}} - \frac {b e^{3} x^{3}}{24 c^{5}} + \frac {b d e^{2} \operatorname {atan}{\left (c x \right )}}{2 c^{6}} + \frac {b e^{3} x}{8 c^{7}} - \frac {b e^{3} \operatorname {atan}{\left (c x \right )}}{8 c^{8}} & \text {for}\: c \neq 0 \\a \left (\frac {d^{3} x^{2}}{2} + \frac {3 d^{2} e x^{4}}{4} + \frac {d e^{2} x^{6}}{2} + \frac {e^{3} x^{8}}{8}\right ) & \text {otherwise} \end {cases} \]

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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