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

Optimal. Leaf size=27 \[ \text {Int}\left (\frac {x^m}{\left (a^2 c x^2+c\right )^3 \sqrt {\tan ^{-1}(a x)}},x\right ) \]

[Out]

Unintegrable(x^m/(a^2*c*x^2+c)^3/arctan(a*x)^(1/2),x)

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Rubi [A]  time = 0.06, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {x^m}{\left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 1.98, size = 0, normalized size = 0.00 \[ \int \frac {x^m}{\left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{m}}{{\left (a^{6} c^{3} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} + c^{3}\right )} \sqrt {\arctan \left (a x\right )}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m/(a^2*c*x^2+c)^3/arctan(a*x)^(1/2),x, algorithm="fricas")

[Out]

integral(x^m/((a^6*c^3*x^6 + 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 + c^3)*sqrt(arctan(a*x))), x)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m/(a^2*c*x^2+c)^3/arctan(a*x)^(1/2),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 3.34, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\left (a^{2} c \,x^{2}+c \right )^{3} \sqrt {\arctan \left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m/(a^2*c*x^2+c)^3/arctan(a*x)^(1/2),x)

[Out]

int(x^m/(a^2*c*x^2+c)^3/arctan(a*x)^(1/2),x)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m/(a^2*c*x^2+c)^3/arctan(a*x)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {x^m}{\sqrt {\mathrm {atan}\left (a\,x\right )}\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m/(atan(a*x)^(1/2)*(c + a^2*c*x^2)^3),x)

[Out]

int(x^m/(atan(a*x)^(1/2)*(c + a^2*c*x^2)^3), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m/(a**2*c*x**2+c)**3/atan(a*x)**(1/2),x)

[Out]

Timed out

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