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

Optimal. Leaf size=23 \[ \text{Unintegrable}\left (\frac{\left (a^2 c x^2+c\right )^{5/2}}{\tan ^{-1}(a x)},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0680472, 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 = {} \[ \int \frac{\left (c+a^2 c x^2\right )^{5/2}}{\tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.425774, size = 0, normalized size = 0. \[ \int \frac{\left (c+a^2 c x^2\right )^{5/2}}{\tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.766, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\arctan \left ( ax \right ) } \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((a^2*c*x^2+c)^(5/2)/arctan(a*x),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2)*sqrt(a^2*c*x^2 + c)/arctan(a*x), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{5}{2}}}{\operatorname{atan}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((c*(a**2*x**2 + 1))**(5/2)/atan(a*x), x)

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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}}}{\arctan \left (a x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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