3.1082 \(\int \frac{(c+a^2 c x^2)^{3/2}}{\tan ^{-1}(a x)^{5/2}} \, dx\)

Optimal. Leaf size=25 \[ \text{Unintegrable}\left (\frac{\left (a^2 c x^2+c\right )^{3/2}}{\tan ^{-1}(a x)^{5/2}},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0364513, 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 = {} \[ \int \frac{\left (c+a^2 c x^2\right )^{3/2}}{\tan ^{-1}(a x)^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.63012, size = 0, normalized size = 0. \[ \int \frac{\left (c+a^2 c x^2\right )^{3/2}}{\tan ^{-1}(a x)^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.678, size = 0, normalized size = 0. \begin{align*} \int{ \left ({a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}} \left ( \arctan \left ( ax \right ) \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)^(3/2)/arctan(a*x)^(5/2),x)

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: RuntimeError

________________________________________________________________________________________

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

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}}}{\arctan \left (a x\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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