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

Optimal. Leaf size=101 \[ \frac{2}{3 a c^2 \sqrt{a^2 c x^2+c}}+\frac{2 x \tan ^{-1}(a x)}{3 c^2 \sqrt{a^2 c x^2+c}}+\frac{1}{9 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac{x \tan ^{-1}(a x)}{3 c \left (a^2 c x^2+c\right )^{3/2}} \]

[Out]

1/(9*a*c*(c + a^2*c*x^2)^(3/2)) + 2/(3*a*c^2*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x])/(3*c*(c + a^2*c*x^2)^(3/2)
) + (2*x*ArcTan[a*x])/(3*c^2*Sqrt[c + a^2*c*x^2])

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Rubi [A]  time = 0.0539432, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {4896, 4894} \[ \frac{2}{3 a c^2 \sqrt{a^2 c x^2+c}}+\frac{2 x \tan ^{-1}(a x)}{3 c^2 \sqrt{a^2 c x^2+c}}+\frac{1}{9 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac{x \tan ^{-1}(a x)}{3 c \left (a^2 c x^2+c\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

1/(9*a*c*(c + a^2*c*x^2)^(3/2)) + 2/(3*a*c^2*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x])/(3*c*(c + a^2*c*x^2)^(3/2)
) + (2*x*ArcTan[a*x])/(3*c^2*Sqrt[c + a^2*c*x^2])

Rule 4896

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Simp[(b*(d + e*x^2)^(q + 1))/(
4*c*d*(q + 1)^2), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x]), x], x] - Si
mp[(x*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x]))/(2*d*(q + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d
] && LtQ[q, -1] && NeQ[q, -3/2]

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]

Rubi steps

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

Mathematica [A]  time = 0.0520958, size = 63, normalized size = 0.62 \[ \frac{\sqrt{a^2 c x^2+c} \left (6 a^2 x^2+\left (6 a^3 x^3+9 a x\right ) \tan ^{-1}(a x)+7\right )}{9 a c^3 \left (a^2 x^2+1\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.267, size = 240, normalized size = 2.4 \begin{align*} -{\frac{ \left ( i+3\,\arctan \left ( ax \right ) \right ) \left ({a}^{3}{x}^{3}-3\,i{a}^{2}{x}^{2}-3\,ax+i \right ) }{72\, \left ({a}^{2}{x}^{2}+1 \right ) ^{2}a{c}^{3}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{ \left ( 3\,\arctan \left ( ax \right ) +3\,i \right ) \left ( ax-i \right ) }{8\,a{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) }\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{ \left ( 3\,ax+3\,i \right ) \left ( \arctan \left ( ax \right ) -i \right ) }{8\,a{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) }\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{ \left ( -i+3\,\arctan \left ( ax \right ) \right ) \left ({a}^{3}{x}^{3}+3\,i{a}^{2}{x}^{2}-3\,ax-i \right ) }{ \left ( 72\,{a}^{4}{x}^{4}+144\,{a}^{2}{x}^{2}+72 \right ) a{c}^{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(arctan(a*x)/(a^2*c*x^2+c)^(5/2),x)

[Out]

-1/72*(I+3*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/a/c^3+3/8*(arcta
n(a*x)+I)*(a*x-I)*(c*(a*x-I)*(a*x+I))^(1/2)/c^3/a/(a^2*x^2+1)+3/8*(c*(a*x-I)*(a*x+I))^(1/2)*(a*x+I)*(arctan(a*
x)-I)/c^3/a/(a^2*x^2+1)-1/72*(-I+3*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)/a/c^3

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Maxima [A]  time = 1.06625, size = 116, normalized size = 1.15 \begin{align*} \frac{1}{9} \, a{\left (\frac{6}{\sqrt{a^{2} c x^{2} + c} a^{2} c^{2}} + \frac{1}{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} a^{2} c}\right )} + \frac{1}{3} \,{\left (\frac{2 \, x}{\sqrt{a^{2} c x^{2} + c} c^{2}} + \frac{x}{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} c}\right )} \arctan \left (a x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*a*(6/(sqrt(a^2*c*x^2 + c)*a^2*c^2) + 1/((a^2*c*x^2 + c)^(3/2)*a^2*c)) + 1/3*(2*x/(sqrt(a^2*c*x^2 + c)*c^2)
+ x/((a^2*c*x^2 + c)^(3/2)*c))*arctan(a*x)

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Fricas [A]  time = 2.25932, size = 155, normalized size = 1.53 \begin{align*} \frac{\sqrt{a^{2} c x^{2} + c}{\left (6 \, a^{2} x^{2} + 3 \,{\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \arctan \left (a x\right ) + 7\right )}}{9 \,{\left (a^{5} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{atan}{\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(atan(a*x)/(a**2*c*x**2+c)**(5/2),x)

[Out]

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

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Giac [A]  time = 1.21544, size = 95, normalized size = 0.94 \begin{align*} \frac{{\left (\frac{2 \, a^{2} x^{2}}{c} + \frac{3}{c}\right )} x \arctan \left (a x\right )}{3 \,{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}}} + \frac{6 \, a^{2} c x^{2} + 7 \, c}{9 \,{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} a c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(2*a^2*x^2/c + 3/c)*x*arctan(a*x)/(a^2*c*x^2 + c)^(3/2) + 1/9*(6*a^2*c*x^2 + 7*c)/((a^2*c*x^2 + c)^(3/2)*a
*c^2)