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

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

[Out]

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

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Rubi [A]  time = 0.0643157, 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 = {} \[ \int \frac{x^3 \tan ^{-1}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 3.87674, size = 0, normalized size = 0. \[ \int \frac{x^3 \tan ^{-1}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.523, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3}}{ \left ({a}^{2}c{x}^{2}+c \right ) ^{2}} \left ( \arctan \left ( ax \right ) \right ) ^{{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^3*arctan(a*x)^(3/2)/(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)^(3/2)/(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)^(3/2)/(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} \operatorname{atan}^{\frac{3}{2}}{\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)**(3/2)/(a**2*c*x**2+c)**2,x)

[Out]

Integral(x**3*atan(a*x)**(3/2)/(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} \arctan \left (a x\right )^{\frac{3}{2}}}{{\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)^(3/2)/(a^2*c*x^2+c)^2,x, algorithm="giac")

[Out]

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