3.1093 \(\int \frac{x^m}{(c+a^2 c x^2)^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx\)

Optimal. Leaf size=28 \[ \text{Unintegrable}\left (\frac{x^m}{\left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^{5/2}},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.118238, 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 = {} \[ \int \frac{x^m}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.989379, size = 0, normalized size = 0. \[ \int \frac{x^m}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 1.246, size = 0, normalized size = 0. \begin{align*} \int{{x}^{m} \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(x^m/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(5/2),x)

[Out]

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

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

[Out]

Exception raised: RuntimeError

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

[Out]

integral(sqrt(a^2*c*x^2 + c)*x^m/((a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2)*arctan(a*x)^(5/2)), 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**m/(a**2*c*x**2+c)**(3/2)/atan(a*x)**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m}}{{\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(x^m/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(5/2),x, algorithm="giac")

[Out]

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