3.708 \(\int \frac{x^3 \sqrt{\tan ^{-1}(a x)}}{(c+a^2 c x^2)^2} \, dx\)

Optimal. Leaf size=26 \[ \text{Unintegrable}\left (\frac{x^3 \sqrt{\tan ^{-1}(a x)}}{\left (a^2 c x^2+c\right )^2},x\right ) \]

[Out]

Unintegrable[(x^3*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^2, x]

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Rubi [A]  time = 0.0637315, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{x^3 \sqrt{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 3.68064, size = 0, normalized size = 0. \[ \int \frac{x^3 \sqrt{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.562, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3}}{ \left ({a}^{2}c{x}^{2}+c \right ) ^{2}}\sqrt{\arctan \left ( ax \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*arctan(a*x)^(1/2)/(a^2*c*x^2+c)^2,x)

[Out]

int(x^3*arctan(a*x)^(1/2)/(a^2*c*x^2+c)^2,x)

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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(x^3*arctan(a*x)^(1/2)/(a^2*c*x^2+c)^2,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arctan(a*x)^(1/2)/(a^2*c*x^2+c)^2,x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{x^{3} \sqrt{\operatorname{atan}{\left (a x \right )}}}{a^{4} x^{4} + 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3*sqrt(atan(a*x))/(a**4*x**4 + 2*a**2*x**2 + 1), x)/c**2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arctan(a*x)^(1/2)/(a^2*c*x^2+c)^2,x, algorithm="giac")

[Out]

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