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

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

[Out]

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

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Rubi [A]  time = 0.094579, 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 = {} \[ \int \frac{x^m \sqrt{c+a^2 c x^2}}{\sqrt{\tan ^{-1}(a x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.797557, size = 0, normalized size = 0. \[ \int \frac{x^m \sqrt{c+a^2 c x^2}}{\sqrt{\tan ^{-1}(a x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 1.248, size = 0, normalized size = 0. \begin{align*} \int{{x}^{m}\sqrt{{a}^{2}c{x}^{2}+c}{\frac{1}{\sqrt{\arctan \left ( ax \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(a^2*c*x^2 + c)*x^m/sqrt(arctan(a*x)), 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)**(1/2)/atan(a*x)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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