3.652 \(\int \frac{(c+a^2 c x^2)^{5/2}}{x \tan ^{-1}(a x)^3} \, dx\)

Optimal. Leaf size=26 \[ \text{Unintegrable}\left (\frac{\left (a^2 c x^2+c\right )^{5/2}}{x \tan ^{-1}(a x)^3},x\right ) \]

[Out]

Unintegrable[(c + a^2*c*x^2)^(5/2)/(x*ArcTan[a*x]^3), x]

________________________________________________________________________________________

Rubi [A]  time = 0.126013, 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 = {} \[ \int \frac{\left (c+a^2 c x^2\right )^{5/2}}{x \tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(c + a^2*c*x^2)^(5/2)/(x*ArcTan[a*x]^3),x]

[Out]

Defer[Int][(c + a^2*c*x^2)^(5/2)/(x*ArcTan[a*x]^3), x]

Rubi steps

\begin{align*} \int \frac{\left (c+a^2 c x^2\right )^{5/2}}{x \tan ^{-1}(a x)^3} \, dx &=\int \frac{\left (c+a^2 c x^2\right )^{5/2}}{x \tan ^{-1}(a x)^3} \, dx\\ \end{align*}

Mathematica [A]  time = 2.53549, size = 0, normalized size = 0. \[ \int \frac{\left (c+a^2 c x^2\right )^{5/2}}{x \tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(c + a^2*c*x^2)^(5/2)/(x*ArcTan[a*x]^3),x]

[Out]

Integrate[(c + a^2*c*x^2)^(5/2)/(x*ArcTan[a*x]^3), x]

________________________________________________________________________________________

Maple [A]  time = 0.77, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ( \arctan \left ( ax \right ) \right ) ^{3}} \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)/x/arctan(a*x)^3,x)

[Out]

int((a^2*c*x^2+c)^(5/2)/x/arctan(a*x)^3,x)

________________________________________________________________________________________

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 )^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*c*x^2+c)^(5/2)/x/arctan(a*x)^3,x, algorithm="maxima")

[Out]

integrate((a^2*c*x^2 + c)^(5/2)/(x*arctan(a*x)^3), x)

________________________________________________________________________________________

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}}{x \arctan \left (a x\right )^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*c*x^2+c)^(5/2)/x/arctan(a*x)^3,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)/(x*arctan(a*x)^3), x)

________________________________________________________________________________________

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}}}{x \operatorname{atan}^{3}{\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)/x/atan(a*x)**3,x)

[Out]

Integral((c*(a**2*x**2 + 1))**(5/2)/(x*atan(a*x)**3), x)

________________________________________________________________________________________

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 )^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*c*x^2+c)^(5/2)/x/arctan(a*x)^3,x, algorithm="giac")

[Out]

integrate((a^2*c*x^2 + c)^(5/2)/(x*arctan(a*x)^3), x)