3.1125 \(\int x^3 (d+e x^2)^2 (a+b \tan ^{-1}(c x)) \, dx\)

Optimal. Leaf size=185 \[ \frac {1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \tan ^{-1}(c x)\right )-\frac {b e x^5 \left (8 c^2 d-3 e\right )}{120 c^3}-\frac {b \left (6 c^4 d^2-8 c^2 d e+3 e^2\right ) \tan ^{-1}(c x)}{24 c^8}+\frac {b x \left (6 c^4 d^2-8 c^2 d e+3 e^2\right )}{24 c^7}-\frac {b x^3 \left (6 c^4 d^2-8 c^2 d e+3 e^2\right )}{72 c^5}-\frac {b e^2 x^7}{56 c} \]

[Out]

1/24*b*(6*c^4*d^2-8*c^2*d*e+3*e^2)*x/c^7-1/72*b*(6*c^4*d^2-8*c^2*d*e+3*e^2)*x^3/c^5-1/120*b*(8*c^2*d-3*e)*e*x^
5/c^3-1/56*b*e^2*x^7/c-1/24*b*(6*c^4*d^2-8*c^2*d*e+3*e^2)*arctan(c*x)/c^8+1/4*d^2*x^4*(a+b*arctan(c*x))+1/3*d*
e*x^6*(a+b*arctan(c*x))+1/8*e^2*x^8*(a+b*arctan(c*x))

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Rubi [A]  time = 0.19, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {266, 43, 4976, 1261, 203} \[ \frac {1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \tan ^{-1}(c x)\right )-\frac {b x^3 \left (6 c^4 d^2-8 c^2 d e+3 e^2\right )}{72 c^5}+\frac {b x \left (6 c^4 d^2-8 c^2 d e+3 e^2\right )}{24 c^7}-\frac {b \left (6 c^4 d^2-8 c^2 d e+3 e^2\right ) \tan ^{-1}(c x)}{24 c^8}-\frac {b e x^5 \left (8 c^2 d-3 e\right )}{120 c^3}-\frac {b e^2 x^7}{56 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(6*c^4*d^2 - 8*c^2*d*e + 3*e^2)*x)/(24*c^7) - (b*(6*c^4*d^2 - 8*c^2*d*e + 3*e^2)*x^3)/(72*c^5) - (b*(8*c^2*
d - 3*e)*e*x^5)/(120*c^3) - (b*e^2*x^7)/(56*c) - (b*(6*c^4*d^2 - 8*c^2*d*e + 3*e^2)*ArcTan[c*x])/(24*c^8) + (d
^2*x^4*(a + b*ArcTan[c*x]))/4 + (d*e*x^6*(a + b*ArcTan[c*x]))/3 + (e^2*x^8*(a + b*ArcTan[c*x]))/8

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 1261

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] &&
NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rule 4976

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]}, Dist[a + b*ArcTan[c*x], u, x] - Dist[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] &&
!ILtQ[(m - 1)/2, 0]))

Rubi steps

\begin {align*} \int x^3 \left (d+e x^2\right )^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx &=\frac {1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \tan ^{-1}(c x)\right )-(b c) \int \frac {x^4 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )}{24+24 c^2 x^2} \, dx\\ &=\frac {1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \tan ^{-1}(c x)\right )-(b c) \int \left (-\frac {6 c^4 d^2-8 c^2 d e+3 e^2}{24 c^8}+\frac {\left (6 c^4 d^2-8 c^2 d e+3 e^2\right ) x^2}{24 c^6}+\frac {\left (8 c^2 d-3 e\right ) e x^4}{24 c^4}+\frac {e^2 x^6}{8 c^2}+\frac {6 c^4 d^2-8 c^2 d e+3 e^2}{c^8 \left (24+24 c^2 x^2\right )}\right ) \, dx\\ &=\frac {b \left (6 c^4 d^2-8 c^2 d e+3 e^2\right ) x}{24 c^7}-\frac {b \left (6 c^4 d^2-8 c^2 d e+3 e^2\right ) x^3}{72 c^5}-\frac {b \left (8 c^2 d-3 e\right ) e x^5}{120 c^3}-\frac {b e^2 x^7}{56 c}+\frac {1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \tan ^{-1}(c x)\right )-\frac {\left (b \left (6 c^4 d^2-8 c^2 d e+3 e^2\right )\right ) \int \frac {1}{24+24 c^2 x^2} \, dx}{c^7}\\ &=\frac {b \left (6 c^4 d^2-8 c^2 d e+3 e^2\right ) x}{24 c^7}-\frac {b \left (6 c^4 d^2-8 c^2 d e+3 e^2\right ) x^3}{72 c^5}-\frac {b \left (8 c^2 d-3 e\right ) e x^5}{120 c^3}-\frac {b e^2 x^7}{56 c}-\frac {b \left (6 c^4 d^2-8 c^2 d e+3 e^2\right ) \tan ^{-1}(c x)}{24 c^8}+\frac {1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \tan ^{-1}(c x)\right )\\ \end {align*}

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Mathematica [A]  time = 0.15, size = 174, normalized size = 0.94 \[ \frac {105 a c^8 x^4 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )+105 b \tan ^{-1}(c x) \left (c^8 \left (6 d^2 x^4+8 d e x^6+3 e^2 x^8\right )-6 c^4 d^2+8 c^2 d e-3 e^2\right )+b c x \left (-3 c^6 \left (70 d^2 x^2+56 d e x^4+15 e^2 x^6\right )+7 c^4 \left (90 d^2+40 d e x^2+9 e^2 x^4\right )-105 c^2 e \left (8 d+e x^2\right )+315 e^2\right )}{2520 c^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(105*a*c^8*x^4*(6*d^2 + 8*d*e*x^2 + 3*e^2*x^4) + b*c*x*(315*e^2 - 105*c^2*e*(8*d + e*x^2) + 7*c^4*(90*d^2 + 40
*d*e*x^2 + 9*e^2*x^4) - 3*c^6*(70*d^2*x^2 + 56*d*e*x^4 + 15*e^2*x^6)) + 105*b*(-6*c^4*d^2 + 8*c^2*d*e - 3*e^2
+ c^8*(6*d^2*x^4 + 8*d*e*x^6 + 3*e^2*x^8))*ArcTan[c*x])/(2520*c^8)

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fricas [A]  time = 0.42, size = 201, normalized size = 1.09 \[ \frac {315 \, a c^{8} e^{2} x^{8} + 840 \, a c^{8} d e x^{6} - 45 \, b c^{7} e^{2} x^{7} + 630 \, a c^{8} d^{2} x^{4} - 21 \, {\left (8 \, b c^{7} d e - 3 \, b c^{5} e^{2}\right )} x^{5} - 35 \, {\left (6 \, b c^{7} d^{2} - 8 \, b c^{5} d e + 3 \, b c^{3} e^{2}\right )} x^{3} + 105 \, {\left (6 \, b c^{5} d^{2} - 8 \, b c^{3} d e + 3 \, b c e^{2}\right )} x + 105 \, {\left (3 \, b c^{8} e^{2} x^{8} + 8 \, b c^{8} d e x^{6} + 6 \, b c^{8} d^{2} x^{4} - 6 \, b c^{4} d^{2} + 8 \, b c^{2} d e - 3 \, b e^{2}\right )} \arctan \left (c x\right )}{2520 \, c^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2520*(315*a*c^8*e^2*x^8 + 840*a*c^8*d*e*x^6 - 45*b*c^7*e^2*x^7 + 630*a*c^8*d^2*x^4 - 21*(8*b*c^7*d*e - 3*b*c
^5*e^2)*x^5 - 35*(6*b*c^7*d^2 - 8*b*c^5*d*e + 3*b*c^3*e^2)*x^3 + 105*(6*b*c^5*d^2 - 8*b*c^3*d*e + 3*b*c*e^2)*x
+ 105*(3*b*c^8*e^2*x^8 + 8*b*c^8*d*e*x^6 + 6*b*c^8*d^2*x^4 - 6*b*c^4*d^2 + 8*b*c^2*d*e - 3*b*e^2)*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^3*(e*x^2+d)^2*(a+b*arctan(c*x)),x, algorithm="giac")

[Out]

sage0*x

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maple [A]  time = 0.04, size = 203, normalized size = 1.10 \[ \frac {a \,e^{2} x^{8}}{8}+\frac {a e d \,x^{6}}{3}+\frac {a \,x^{4} d^{2}}{4}+\frac {b \arctan \left (c x \right ) e^{2} x^{8}}{8}+\frac {b \arctan \left (c x \right ) e d \,x^{6}}{3}+\frac {b \arctan \left (c x \right ) d^{2} x^{4}}{4}-\frac {b \,e^{2} x^{7}}{56 c}-\frac {b e d \,x^{5}}{15 c}-\frac {b \,d^{2} x^{3}}{12 c}+\frac {b \,x^{5} e^{2}}{40 c^{3}}+\frac {b \,x^{3} d e}{9 c^{3}}+\frac {b \,d^{2} x}{4 c^{3}}-\frac {b \,x^{3} e^{2}}{24 c^{5}}-\frac {b e d x}{3 c^{5}}+\frac {b x \,e^{2}}{8 c^{7}}-\frac {b \,d^{2} \arctan \left (c x \right )}{4 c^{4}}+\frac {b \arctan \left (c x \right ) e d}{3 c^{6}}-\frac {b \arctan \left (c x \right ) e^{2}}{8 c^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*a*e^2*x^8+1/3*a*e*d*x^6+1/4*a*x^4*d^2+1/8*b*arctan(c*x)*e^2*x^8+1/3*b*arctan(c*x)*e*d*x^6+1/4*b*arctan(c*x
)*d^2*x^4-1/56*b*e^2*x^7/c-1/15/c*b*e*d*x^5-1/12*b*d^2*x^3/c+1/40/c^3*b*x^5*e^2+1/9/c^3*b*x^3*d*e+1/4*b*d^2*x/
c^3-1/24/c^5*b*x^3*e^2-1/3/c^5*b*e*d*x+1/8/c^7*b*x*e^2-1/4*b*d^2*arctan(c*x)/c^4+1/3/c^6*b*arctan(c*x)*e*d-1/8
/c^8*b*arctan(c*x)*e^2

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maxima [A]  time = 0.41, size = 184, normalized size = 0.99 \[ \frac {1}{8} \, a e^{2} x^{8} + \frac {1}{3} \, a d e x^{6} + \frac {1}{4} \, a d^{2} x^{4} + \frac {1}{12} \, {\left (3 \, x^{4} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{3} - 3 \, x}{c^{4}} + \frac {3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} b d^{2} + \frac {1}{45} \, {\left (15 \, x^{6} \arctan \left (c x\right ) - c {\left (\frac {3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac {15 \, \arctan \left (c x\right )}{c^{7}}\right )}\right )} b d e + \frac {1}{840} \, {\left (105 \, x^{8} \arctan \left (c x\right ) - c {\left (\frac {15 \, c^{6} x^{7} - 21 \, c^{4} x^{5} + 35 \, c^{2} x^{3} - 105 \, x}{c^{8}} + \frac {105 \, \arctan \left (c x\right )}{c^{9}}\right )}\right )} b e^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*a*e^2*x^8 + 1/3*a*d*e*x^6 + 1/4*a*d^2*x^4 + 1/12*(3*x^4*arctan(c*x) - c*((c^2*x^3 - 3*x)/c^4 + 3*arctan(c*
x)/c^5))*b*d^2 + 1/45*(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
+ 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^2

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mupad [B]  time = 0.54, size = 374, normalized size = 2.02 \[ x^4\,\left (\frac {\frac {a\,e^2}{c^2}-\frac {a\,e\,\left (2\,d\,c^2+e\right )}{c^2}}{4\,c^2}+\frac {a\,d\,\left (d\,c^2+2\,e\right )}{4\,c^2}\right )-x^6\,\left (\frac {a\,e^2}{6\,c^2}-\frac {a\,e\,\left (2\,d\,c^2+e\right )}{6\,c^2}\right )+x^5\,\left (\frac {b\,e^2}{40\,c^3}-\frac {b\,d\,e}{15\,c}\right )+\mathrm {atan}\left (c\,x\right )\,\left (\frac {b\,d^2\,x^4}{4}+\frac {b\,d\,e\,x^6}{3}+\frac {b\,e^2\,x^8}{8}\right )-x^2\,\left (\frac {\frac {\frac {a\,e^2}{c^2}-\frac {a\,e\,\left (2\,d\,c^2+e\right )}{c^2}}{c^2}+\frac {a\,d\,\left (d\,c^2+2\,e\right )}{c^2}}{2\,c^2}-\frac {a\,d^2}{2\,c^2}\right )-x^3\,\left (\frac {\frac {b\,e^2}{8\,c^3}-\frac {b\,d\,e}{3\,c}}{3\,c^2}+\frac {b\,d^2}{12\,c}\right )+\frac {x\,\left (\frac {\frac {b\,e^2}{8\,c^3}-\frac {b\,d\,e}{3\,c}}{c^2}+\frac {b\,d^2}{4\,c}\right )}{c^2}+\frac {a\,e^2\,x^8}{8}-\frac {b\,\mathrm {atan}\left (\frac {b\,c\,x\,\left (6\,c^4\,d^2-8\,c^2\,d\,e+3\,e^2\right )}{6\,b\,c^4\,d^2-8\,b\,c^2\,d\,e+3\,b\,e^2}\right )\,\left (6\,c^4\,d^2-8\,c^2\,d\,e+3\,e^2\right )}{24\,c^8}-\frac {b\,e^2\,x^7}{56\,c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^4*(((a*e^2)/c^2 - (a*e*(e + 2*c^2*d))/c^2)/(4*c^2) + (a*d*(2*e + c^2*d))/(4*c^2)) - x^6*((a*e^2)/(6*c^2) - (
a*e*(e + 2*c^2*d))/(6*c^2)) + x^5*((b*e^2)/(40*c^3) - (b*d*e)/(15*c)) + atan(c*x)*((b*d^2*x^4)/4 + (b*e^2*x^8)
/8 + (b*d*e*x^6)/3) - x^2*((((a*e^2)/c^2 - (a*e*(e + 2*c^2*d))/c^2)/c^2 + (a*d*(2*e + c^2*d))/c^2)/(2*c^2) - (
a*d^2)/(2*c^2)) - x^3*(((b*e^2)/(8*c^3) - (b*d*e)/(3*c))/(3*c^2) + (b*d^2)/(12*c)) + (x*(((b*e^2)/(8*c^3) - (b
*d*e)/(3*c))/c^2 + (b*d^2)/(4*c)))/c^2 + (a*e^2*x^8)/8 - (b*atan((b*c*x*(3*e^2 + 6*c^4*d^2 - 8*c^2*d*e))/(3*b*
e^2 + 6*b*c^4*d^2 - 8*b*c^2*d*e))*(3*e^2 + 6*c^4*d^2 - 8*c^2*d*e))/(24*c^8) - (b*e^2*x^7)/(56*c)

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sympy [A]  time = 3.80, size = 260, normalized size = 1.41 \[ \begin {cases} \frac {a d^{2} x^{4}}{4} + \frac {a d e x^{6}}{3} + \frac {a e^{2} x^{8}}{8} + \frac {b d^{2} x^{4} \operatorname {atan}{\left (c x \right )}}{4} + \frac {b d e x^{6} \operatorname {atan}{\left (c x \right )}}{3} + \frac {b e^{2} x^{8} \operatorname {atan}{\left (c x \right )}}{8} - \frac {b d^{2} x^{3}}{12 c} - \frac {b d e x^{5}}{15 c} - \frac {b e^{2} x^{7}}{56 c} + \frac {b d^{2} x}{4 c^{3}} + \frac {b d e x^{3}}{9 c^{3}} + \frac {b e^{2} x^{5}}{40 c^{3}} - \frac {b d^{2} \operatorname {atan}{\left (c x \right )}}{4 c^{4}} - \frac {b d e x}{3 c^{5}} - \frac {b e^{2} x^{3}}{24 c^{5}} + \frac {b d e \operatorname {atan}{\left (c x \right )}}{3 c^{6}} + \frac {b e^{2} x}{8 c^{7}} - \frac {b e^{2} \operatorname {atan}{\left (c x \right )}}{8 c^{8}} & \text {for}\: c \neq 0 \\a \left (\frac {d^{2} x^{4}}{4} + \frac {d e x^{6}}{3} + \frac {e^{2} x^{8}}{8}\right ) & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*d**2*x**4/4 + a*d*e*x**6/3 + a*e**2*x**8/8 + b*d**2*x**4*atan(c*x)/4 + b*d*e*x**6*atan(c*x)/3 + b
*e**2*x**8*atan(c*x)/8 - b*d**2*x**3/(12*c) - b*d*e*x**5/(15*c) - b*e**2*x**7/(56*c) + b*d**2*x/(4*c**3) + b*d
*e*x**3/(9*c**3) + b*e**2*x**5/(40*c**3) - b*d**2*atan(c*x)/(4*c**4) - b*d*e*x/(3*c**5) - b*e**2*x**3/(24*c**5
) + b*d*e*atan(c*x)/(3*c**6) + b*e**2*x/(8*c**7) - b*e**2*atan(c*x)/(8*c**8), Ne(c, 0)), (a*(d**2*x**4/4 + d*e
*x**6/3 + e**2*x**8/8), True))

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