3.698 \(\int \frac{x^m \sqrt{\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0639933, 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 = {} \[ \int \frac{x^m \sqrt{\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.774744, size = 0, normalized size = 0. \[ \int \frac{x^m \sqrt{\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.572, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m}}{{a}^{2}c{x}^{2}+c}\sqrt{\arctan \left ( ax \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*arctan(a*x)^(1/2)/(a^2*c*x^2+c),x)

[Out]

int(x^m*arctan(a*x)^(1/2)/(a^2*c*x^2+c),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*arctan(a*x)^(1/2)/(a^2*c*x^2+c),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{x^{m} \sqrt{\arctan \left (a x\right )}}{a^{2} c x^{2} + c}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*arctan(a*x)^(1/2)/(a^2*c*x^2+c),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*atan(a*x)**(1/2)/(a**2*c*x**2+c),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*arctan(a*x)^(1/2)/(a^2*c*x^2+c),x, algorithm="giac")

[Out]

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