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

Optimal. Leaf size=45 \[ \frac{2 m \text{Unintegrable}\left (\frac{x^{m-1}}{\sqrt{\tan ^{-1}(a x)}},x\right )}{a c}-\frac{2 x^m}{a c \sqrt{\tan ^{-1}(a x)}} \]

[Out]

(-2*x^m)/(a*c*Sqrt[ArcTan[a*x]]) + (2*m*Unintegrable[x^(-1 + m)/Sqrt[ArcTan[a*x]], x])/(a*c)

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Rubi [A]  time = 0.0805591, 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 = {} \[ \int \frac{x^m}{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(-2*x^m)/(a*c*Sqrt[ArcTan[a*x]]) + (2*m*Defer[Int][x^(-1 + m)/Sqrt[ArcTan[a*x]], x])/(a*c)

Rubi steps

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

Mathematica [A]  time = 0.700901, size = 0, normalized size = 0. \[ \int \frac{x^m}{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Integrate[x^m/((c + a^2*c*x^2)*ArcTan[a*x]^(3/2)), 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} \left ( \arctan \left ( ax \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^m/(a^2*c*x^2+c)/arctan(a*x)^(3/2),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)/arctan(a*x)^(3/2),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{x^{m}}{{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{\frac{3}{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)/arctan(a*x)^(3/2),x, algorithm="fricas")

[Out]

integral(x^m/((a^2*c*x^2 + c)*arctan(a*x)^(3/2)), 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)/atan(a*x)**(3/2),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 )} \arctan \left (a x\right )^{\frac{3}{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)/arctan(a*x)^(3/2),x, algorithm="giac")

[Out]

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