3.453 \(\int \frac{x^2 \tan ^{-1}(a x)^3}{(c+a^2 c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=199 \[ -\frac{14}{9 a^3 c^2 \sqrt{a^2 c x^2+c}}-\frac{4 x \tan ^{-1}(a x)}{3 a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{2 \tan ^{-1}(a x)^2}{3 a^3 c^2 \sqrt{a^2 c x^2+c}}+\frac{2}{27 a^3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac{x^3 \tan ^{-1}(a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}-\frac{2 x^3 \tan ^{-1}(a x)}{9 c \left (a^2 c x^2+c\right )^{3/2}}+\frac{x^2 \tan ^{-1}(a x)^2}{3 a c \left (a^2 c x^2+c\right )^{3/2}} \]

[Out]

2/(27*a^3*c*(c + a^2*c*x^2)^(3/2)) - 14/(9*a^3*c^2*Sqrt[c + a^2*c*x^2]) - (2*x^3*ArcTan[a*x])/(9*c*(c + a^2*c*
x^2)^(3/2)) - (4*x*ArcTan[a*x])/(3*a^2*c^2*Sqrt[c + a^2*c*x^2]) + (x^2*ArcTan[a*x]^2)/(3*a*c*(c + a^2*c*x^2)^(
3/2)) + (2*ArcTan[a*x]^2)/(3*a^3*c^2*Sqrt[c + a^2*c*x^2]) + (x^3*ArcTan[a*x]^3)/(3*c*(c + a^2*c*x^2)^(3/2))

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Rubi [A]  time = 0.406532, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4944, 4940, 4930, 4894, 266, 43} \[ -\frac{14}{9 a^3 c^2 \sqrt{a^2 c x^2+c}}-\frac{4 x \tan ^{-1}(a x)}{3 a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{2 \tan ^{-1}(a x)^2}{3 a^3 c^2 \sqrt{a^2 c x^2+c}}+\frac{2}{27 a^3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac{x^3 \tan ^{-1}(a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}-\frac{2 x^3 \tan ^{-1}(a x)}{9 c \left (a^2 c x^2+c\right )^{3/2}}+\frac{x^2 \tan ^{-1}(a x)^2}{3 a c \left (a^2 c x^2+c\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(x^2*ArcTan[a*x]^3)/(c + a^2*c*x^2)^(5/2),x]

[Out]

2/(27*a^3*c*(c + a^2*c*x^2)^(3/2)) - 14/(9*a^3*c^2*Sqrt[c + a^2*c*x^2]) - (2*x^3*ArcTan[a*x])/(9*c*(c + a^2*c*
x^2)^(3/2)) - (4*x*ArcTan[a*x])/(3*a^2*c^2*Sqrt[c + a^2*c*x^2]) + (x^2*ArcTan[a*x]^2)/(3*a*c*(c + a^2*c*x^2)^(
3/2)) + (2*ArcTan[a*x]^2)/(3*a^3*c^2*Sqrt[c + a^2*c*x^2]) + (x^3*ArcTan[a*x]^3)/(3*c*(c + a^2*c*x^2)^(3/2))

Rule 4944

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp
[((f*x)^(m + 1)*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p)/(d*f*(m + 1)), x] - Dist[(b*c*p)/(f*(m + 1)), Int[(
f*x)^(m + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[e,
c^2*d] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -1]

Rule 4940

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

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 4894

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[b/(c*d*Sqrt[d + e*x^2]),
x] + Simp[(x*(a + b*ArcTan[c*x]))/(d*Sqrt[d + e*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d]

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 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])

Rubi steps

\begin{align*} \int \frac{x^2 \tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx &=\frac{x^3 \tan ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}-a \int \frac{x^3 \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx\\ &=-\frac{2 x^3 \tan ^{-1}(a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{x^2 \tan ^{-1}(a x)^2}{3 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac{x^3 \tan ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{1}{9} (2 a) \int \frac{x^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx-\frac{2 \int \frac{x \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 a c}\\ &=-\frac{2 x^3 \tan ^{-1}(a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{x^2 \tan ^{-1}(a x)^2}{3 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac{2 \tan ^{-1}(a x)^2}{3 a^3 c^2 \sqrt{c+a^2 c x^2}}+\frac{x^3 \tan ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{1}{9} a \operatorname{Subst}\left (\int \frac{x}{\left (c+a^2 c x\right )^{5/2}} \, dx,x,x^2\right )-\frac{4 \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 a^2 c}\\ &=-\frac{4}{3 a^3 c^2 \sqrt{c+a^2 c x^2}}-\frac{2 x^3 \tan ^{-1}(a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{4 x \tan ^{-1}(a x)}{3 a^2 c^2 \sqrt{c+a^2 c x^2}}+\frac{x^2 \tan ^{-1}(a x)^2}{3 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac{2 \tan ^{-1}(a x)^2}{3 a^3 c^2 \sqrt{c+a^2 c x^2}}+\frac{x^3 \tan ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{1}{9} a \operatorname{Subst}\left (\int \left (-\frac{1}{a^2 \left (c+a^2 c x\right )^{5/2}}+\frac{1}{a^2 c \left (c+a^2 c x\right )^{3/2}}\right ) \, dx,x,x^2\right )\\ &=\frac{2}{27 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{14}{9 a^3 c^2 \sqrt{c+a^2 c x^2}}-\frac{2 x^3 \tan ^{-1}(a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{4 x \tan ^{-1}(a x)}{3 a^2 c^2 \sqrt{c+a^2 c x^2}}+\frac{x^2 \tan ^{-1}(a x)^2}{3 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac{2 \tan ^{-1}(a x)^2}{3 a^3 c^2 \sqrt{c+a^2 c x^2}}+\frac{x^3 \tan ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.101736, size = 95, normalized size = 0.48 \[ \frac{\sqrt{a^2 c x^2+c} \left (-42 a^2 x^2+9 a^3 x^3 \tan ^{-1}(a x)^3+9 \left (3 a^2 x^2+2\right ) \tan ^{-1}(a x)^2-6 a x \left (7 a^2 x^2+6\right ) \tan ^{-1}(a x)-40\right )}{27 a^3 c^3 \left (a^2 x^2+1\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^2*ArcTan[a*x]^3)/(c + a^2*c*x^2)^(5/2),x]

[Out]

(Sqrt[c + a^2*c*x^2]*(-40 - 42*a^2*x^2 - 6*a*x*(6 + 7*a^2*x^2)*ArcTan[a*x] + 9*(2 + 3*a^2*x^2)*ArcTan[a*x]^2 +
9*a^3*x^3*ArcTan[a*x]^3))/(27*a^3*c^3*(1 + a^2*x^2)^2)

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Maple [C]  time = 0.733, size = 308, normalized size = 1.6 \begin{align*}{\frac{ \left ( 9\,i \left ( \arctan \left ( ax \right ) \right ) ^{2}+9\, \left ( \arctan \left ( ax \right ) \right ) ^{3}-2\,i-6\,\arctan \left ( ax \right ) \right ) \left ({a}^{3}{x}^{3}-3\,i{a}^{2}{x}^{2}-3\,ax+i \right ) }{216\, \left ({a}^{2}{x}^{2}+1 \right ) ^{2}{c}^{3}{a}^{3}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{ \left ( \left ( \arctan \left ( ax \right ) \right ) ^{3}-6\,\arctan \left ( ax \right ) +3\,i \left ( \arctan \left ( ax \right ) \right ) ^{2}-6\,i \right ) \left ( ax-i \right ) }{8\,{c}^{3}{a}^{3} \left ({a}^{2}{x}^{2}+1 \right ) }\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{ \left ( ax+i \right ) \left ( \left ( \arctan \left ( ax \right ) \right ) ^{3}-6\,\arctan \left ( ax \right ) -3\,i \left ( \arctan \left ( ax \right ) \right ) ^{2}+6\,i \right ) }{8\,{c}^{3}{a}^{3} \left ({a}^{2}{x}^{2}+1 \right ) }\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{ \left ( -9\,i \left ( \arctan \left ( ax \right ) \right ) ^{2}+9\, \left ( \arctan \left ( ax \right ) \right ) ^{3}+2\,i-6\,\arctan \left ( ax \right ) \right ) \left ({a}^{3}{x}^{3}+3\,i{a}^{2}{x}^{2}-3\,ax-i \right ) }{ \left ( 216\,{a}^{4}{x}^{4}+432\,{a}^{2}{x}^{2}+216 \right ){c}^{3}{a}^{3}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*arctan(a*x)^3/(a^2*c*x^2+c)^(5/2),x)

[Out]

1/216*(9*I*arctan(a*x)^2+9*arctan(a*x)^3-2*I-6*arctan(a*x))*(a^3*x^3-3*I*a^2*x^2-3*a*x+I)*(c*(a*x-I)*(a*x+I))^
(1/2)/(a^2*x^2+1)^2/c^3/a^3+1/8*(arctan(a*x)^3-6*arctan(a*x)+3*I*arctan(a*x)^2-6*I)*(a*x-I)*(c*(a*x-I)*(a*x+I)
)^(1/2)/a^3/c^3/(a^2*x^2+1)+1/8*(c*(a*x-I)*(a*x+I))^(1/2)*(a*x+I)*(arctan(a*x)^3-6*arctan(a*x)-3*I*arctan(a*x)
^2+6*I)/a^3/c^3/(a^2*x^2+1)+1/216*(-9*I*arctan(a*x)^2+9*arctan(a*x)^3+2*I-6*arctan(a*x))*(c*(a*x-I)*(a*x+I))^(
1/2)*(a^3*x^3+3*I*a^2*x^2-3*a*x-I)/(a^4*x^4+2*a^2*x^2+1)/c^3/a^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arctan(a*x)^3/(a^2*c*x^2+c)^(5/2),x, algorithm="maxima")

[Out]

integrate(x^2*arctan(a*x)^3/(a^2*c*x^2 + c)^(5/2), x)

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Fricas [A]  time = 2.12679, size = 243, normalized size = 1.22 \begin{align*} \frac{{\left (9 \, a^{3} x^{3} \arctan \left (a x\right )^{3} - 42 \, a^{2} x^{2} + 9 \,{\left (3 \, a^{2} x^{2} + 2\right )} \arctan \left (a x\right )^{2} - 6 \,{\left (7 \, a^{3} x^{3} + 6 \, a x\right )} \arctan \left (a x\right ) - 40\right )} \sqrt{a^{2} c x^{2} + c}}{27 \,{\left (a^{7} c^{3} x^{4} + 2 \, a^{5} c^{3} x^{2} + a^{3} c^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arctan(a*x)^3/(a^2*c*x^2+c)^(5/2),x, algorithm="fricas")

[Out]

1/27*(9*a^3*x^3*arctan(a*x)^3 - 42*a^2*x^2 + 9*(3*a^2*x^2 + 2)*arctan(a*x)^2 - 6*(7*a^3*x^3 + 6*a*x)*arctan(a*
x) - 40)*sqrt(a^2*c*x^2 + c)/(a^7*c^3*x^4 + 2*a^5*c^3*x^2 + a^3*c^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \operatorname{atan}^{3}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*atan(a*x)**3/(a**2*c*x**2+c)**(5/2),x)

[Out]

Integral(x**2*atan(a*x)**3/(c*(a**2*x**2 + 1))**(5/2), x)

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Giac [A]  time = 1.3084, size = 192, normalized size = 0.96 \begin{align*} \frac{x^{3} \arctan \left (a x\right )^{3}}{3 \,{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} c} - \frac{1}{27} \,{\left (\frac{6 \, x{\left (\frac{7 \, x^{2}}{a c} + \frac{6}{a^{3} c}\right )} \arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}}} - \frac{9 \,{\left (3 \, a^{2} c x^{2} + 2 \, c\right )} \arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} a^{4} c^{2}} + \frac{2 \,{\left (21 \, a^{2} c x^{2} + 20 \, c\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} a^{4} c^{2}}\right )} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arctan(a*x)^3/(a^2*c*x^2+c)^(5/2),x, algorithm="giac")

[Out]

1/3*x^3*arctan(a*x)^3/((a^2*c*x^2 + c)^(3/2)*c) - 1/27*(6*x*(7*x^2/(a*c) + 6/(a^3*c))*arctan(a*x)/(a^2*c*x^2 +
c)^(3/2) - 9*(3*a^2*c*x^2 + 2*c)*arctan(a*x)^2/((a^2*c*x^2 + c)^(3/2)*a^4*c^2) + 2*(21*a^2*c*x^2 + 20*c)/((a^
2*c*x^2 + c)^(3/2)*a^4*c^2))*a