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

Optimal. Leaf size=18 \[ \frac{2 \tan ^{-1}(a x)^{7/2}}{7 a c} \]

[Out]

(2*ArcTan[a*x]^(7/2))/(7*a*c)

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Rubi [A]  time = 0.0252277, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {4884} \[ \frac{2 \tan ^{-1}(a x)^{7/2}}{7 a c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*ArcTan[a*x]^(7/2))/(7*a*c)

Rule 4884

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

Rubi steps

\begin{align*} \int \frac{\tan ^{-1}(a x)^{5/2}}{c+a^2 c x^2} \, dx &=\frac{2 \tan ^{-1}(a x)^{7/2}}{7 a c}\\ \end{align*}

Mathematica [A]  time = 0.0029866, size = 18, normalized size = 1. \[ \frac{2 \tan ^{-1}(a x)^{7/2}}{7 a c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*ArcTan[a*x]^(7/2))/(7*a*c)

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Maple [A]  time = 0.079, size = 15, normalized size = 0.8 \begin{align*}{\frac{2}{7\,ac} \left ( \arctan \left ( ax \right ) \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*arctan(a*x)^(7/2)/a/c

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.61919, size = 39, normalized size = 2.17 \begin{align*} \frac{2 \, \arctan \left (a x\right )^{\frac{7}{2}}}{7 \, a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*arctan(a*x)^(7/2)/(a*c)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.08444, size = 19, normalized size = 1.06 \begin{align*} \frac{2 \, \arctan \left (a x\right )^{\frac{7}{2}}}{7 \, a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*arctan(a*x)^(7/2)/(a*c)