3.1090 \(\int \frac{1}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}} \, dx\)

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

[Out]

Unintegrable[1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2)), x]

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Rubi [A]  time = 0.0361965, 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 = {} \[ \int \frac{1}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2)),x]

[Out]

Defer[Int][1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2)), x]

Rubi steps

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

Mathematica [A]  time = 0.798492, size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2)),x]

[Out]

Integrate[1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2)), x]

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Maple [A]  time = 0.962, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \arctan \left ( ax \right ) \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{{a}^{2}c{x}^{2}+c}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/arctan(a*x)^(5/2)/(a^2*c*x^2+c)^(1/2),x)

[Out]

int(1/arctan(a*x)^(5/2)/(a^2*c*x^2+c)^(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(1/arctan(a*x)^(5/2)/(a^2*c*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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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(1/arctan(a*x)^(5/2)/(a^2*c*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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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(1/atan(a*x)**(5/2)/(a**2*c*x**2+c)**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a^{2} c x^{2} + c} \arctan \left (a x\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arctan(a*x)^(5/2)/(a^2*c*x^2+c)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(a^2*c*x^2 + c)*arctan(a*x)^(5/2)), x)