3.159 \(\int x (c+a^2 c x^2)^2 \tan ^{-1}(a x) \, dx\)

Optimal. Leaf size=61 \[ -\frac{1}{30} a^3 c^2 x^5+\frac{c^2 \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)}{6 a^2}-\frac{1}{9} a c^2 x^3-\frac{c^2 x}{6 a} \]

[Out]

-(c^2*x)/(6*a) - (a*c^2*x^3)/9 - (a^3*c^2*x^5)/30 + (c^2*(1 + a^2*x^2)^3*ArcTan[a*x])/(6*a^2)

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Rubi [A]  time = 0.0425797, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {4930, 194} \[ -\frac{1}{30} a^3 c^2 x^5+\frac{c^2 \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)}{6 a^2}-\frac{1}{9} a c^2 x^3-\frac{c^2 x}{6 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c^2*x)/(6*a) - (a*c^2*x^3)/9 - (a^3*c^2*x^5)/30 + (c^2*(1 + a^2*x^2)^3*ArcTan[a*x])/(6*a^2)

Rule 4930

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[((d + e*x^2)^
(q + 1)*(a + b*ArcTan[c*x])^p)/(2*e*(q + 1)), x] - Dist[(b*p)/(2*c*(q + 1)), Int[(d + e*x^2)^q*(a + b*ArcTan[c
*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && NeQ[q, -1]

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int x \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x) \, dx &=\frac{c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{6 a^2}-\frac{\int \left (c+a^2 c x^2\right )^2 \, dx}{6 a}\\ &=\frac{c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{6 a^2}-\frac{\int \left (c^2+2 a^2 c^2 x^2+a^4 c^2 x^4\right ) \, dx}{6 a}\\ &=-\frac{c^2 x}{6 a}-\frac{1}{9} a c^2 x^3-\frac{1}{30} a^3 c^2 x^5+\frac{c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{6 a^2}\\ \end{align*}

Mathematica [A]  time = 0.0483689, size = 98, normalized size = 1.61 \[ -\frac{1}{30} a^3 c^2 x^5+\frac{1}{6} a^4 c^2 x^6 \tan ^{-1}(a x)+\frac{1}{2} a^2 c^2 x^4 \tan ^{-1}(a x)+\frac{c^2 \tan ^{-1}(a x)}{6 a^2}-\frac{1}{9} a c^2 x^3+\frac{1}{2} c^2 x^2 \tan ^{-1}(a x)-\frac{c^2 x}{6 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c^2*x)/(6*a) - (a*c^2*x^3)/9 - (a^3*c^2*x^5)/30 + (c^2*ArcTan[a*x])/(6*a^2) + (c^2*x^2*ArcTan[a*x])/2 + (a^2
*c^2*x^4*ArcTan[a*x])/2 + (a^4*c^2*x^6*ArcTan[a*x])/6

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Maple [A]  time = 0.023, size = 85, normalized size = 1.4 \begin{align*}{\frac{{a}^{4}{c}^{2}\arctan \left ( ax \right ){x}^{6}}{6}}+{\frac{{a}^{2}{c}^{2}\arctan \left ( ax \right ){x}^{4}}{2}}+{\frac{{c}^{2}\arctan \left ( ax \right ){x}^{2}}{2}}-{\frac{{a}^{3}{c}^{2}{x}^{5}}{30}}-{\frac{a{c}^{2}{x}^{3}}{9}}-{\frac{{c}^{2}x}{6\,a}}+{\frac{{c}^{2}\arctan \left ( ax \right ) }{6\,{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a^2*c*x^2+c)^2*arctan(a*x),x)

[Out]

1/6*a^4*c^2*arctan(a*x)*x^6+1/2*a^2*c^2*arctan(a*x)*x^4+1/2*c^2*arctan(a*x)*x^2-1/30*a^3*c^2*x^5-1/9*a*c^2*x^3
-1/6*c^2*x/a+1/6/a^2*c^2*arctan(a*x)

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Maxima [A]  time = 0.968101, size = 84, normalized size = 1.38 \begin{align*} \frac{{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )}{6 \, a^{2} c} - \frac{3 \, a^{4} c^{3} x^{5} + 10 \, a^{2} c^{3} x^{3} + 15 \, c^{3} x}{90 \, a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(a^2*c*x^2 + c)^3*arctan(a*x)/(a^2*c) - 1/90*(3*a^4*c^3*x^5 + 10*a^2*c^3*x^3 + 15*c^3*x)/(a*c)

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Fricas [A]  time = 1.60719, size = 170, normalized size = 2.79 \begin{align*} -\frac{3 \, a^{5} c^{2} x^{5} + 10 \, a^{3} c^{2} x^{3} + 15 \, a c^{2} x - 15 \,{\left (a^{6} c^{2} x^{6} + 3 \, a^{4} c^{2} x^{4} + 3 \, a^{2} c^{2} x^{2} + c^{2}\right )} \arctan \left (a x\right )}{90 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/90*(3*a^5*c^2*x^5 + 10*a^3*c^2*x^3 + 15*a*c^2*x - 15*(a^6*c^2*x^6 + 3*a^4*c^2*x^4 + 3*a^2*c^2*x^2 + c^2)*ar
ctan(a*x))/a^2

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Sympy [A]  time = 2.10347, size = 92, normalized size = 1.51 \begin{align*} \begin{cases} \frac{a^{4} c^{2} x^{6} \operatorname{atan}{\left (a x \right )}}{6} - \frac{a^{3} c^{2} x^{5}}{30} + \frac{a^{2} c^{2} x^{4} \operatorname{atan}{\left (a x \right )}}{2} - \frac{a c^{2} x^{3}}{9} + \frac{c^{2} x^{2} \operatorname{atan}{\left (a x \right )}}{2} - \frac{c^{2} x}{6 a} + \frac{c^{2} \operatorname{atan}{\left (a x \right )}}{6 a^{2}} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a**2*c*x**2+c)**2*atan(a*x),x)

[Out]

Piecewise((a**4*c**2*x**6*atan(a*x)/6 - a**3*c**2*x**5/30 + a**2*c**2*x**4*atan(a*x)/2 - a*c**2*x**3/9 + c**2*
x**2*atan(a*x)/2 - c**2*x/(6*a) + c**2*atan(a*x)/(6*a**2), Ne(a, 0)), (0, True))

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Giac [A]  time = 1.10864, size = 80, normalized size = 1.31 \begin{align*} \frac{{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )}{6 \, a^{2} c} - \frac{3 \, a^{4} c^{2} x^{5} + 10 \, a^{2} c^{2} x^{3} + 15 \, c^{2} x}{90 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(a^2*c*x^2 + c)^3*arctan(a*x)/(a^2*c) - 1/90*(3*a^4*c^2*x^5 + 10*a^2*c^2*x^3 + 15*c^2*x)/a