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

Optimal. Leaf size=24 \[ \text{Unintegrable}\left (\frac{x \left (a^2 c x^2+c\right )^{5/2}}{\tan ^{-1}(a x)^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.077516, 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 = {} \[ \int \frac{x \left (c+a^2 c x^2\right )^{5/2}}{\tan ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.94567, size = 0, normalized size = 0. \[ \int \frac{x \left (c+a^2 c x^2\right )^{5/2}}{\tan ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.776, size = 0, normalized size = 0. \begin{align*} \int{\frac{x}{ \left ( \arctan \left ( ax \right ) \right ) ^{2}} \left ({a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x*(a^2*c*x^2+c)^(5/2)/arctan(a*x)^2,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{5}{2}} x}{\arctan \left (a x\right )^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (a^{4} c^{2} x^{5} + 2 \, a^{2} c^{2} x^{3} + c^{2} x\right )} \sqrt{a^{2} c x^{2} + c}}{\arctan \left (a x\right )^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a^4*c^2*x^5 + 2*a^2*c^2*x^3 + c^2*x)*sqrt(a^2*c*x^2 + c)/arctan(a*x)^2, 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*(a**2*c*x**2+c)**(5/2)/atan(a*x)**2,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{5}{2}} x}{\arctan \left (a x\right )^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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