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

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

[Out]

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

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Rubi [A]  time = 0.113165, 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 = {} \[ \int \frac{x^m \left (c+a^2 c x^2\right )^{3/2}}{\tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.770978, size = 0, normalized size = 0. \[ \int \frac{x^m \left (c+a^2 c x^2\right )^{3/2}}{\tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.585, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m}}{ \left ( \arctan \left ( ax \right ) \right ) ^{3}} \left ({a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} x^{m}}{\arctan \left (a x\right )^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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