3.11.93 \(\int \frac {x^m}{(c+a^2 c x^2)^{3/2} \text {ArcTan}(a x)^{5/2}} \, dx\) [1093]

Optimal. Leaf size=29 \[ \text {Int}\left (\frac {x^m}{\left (c+a^2 c x^2\right )^{3/2} \text {ArcTan}(a x)^{5/2}},x\right ) \]

[Out]

Unintegrable(x^m/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(5/2),x)

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Rubi [A]
time = 0.08, 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 = {} \begin {gather*} \int \frac {x^m}{\left (c+a^2 c x^2\right )^{3/2} \text {ArcTan}(a x)^{5/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[x^m/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)),x]

[Out]

Defer[Int][x^m/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)), x]

Rubi steps

\begin {align*} \int \frac {x^m}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx &=\int \frac {x^m}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx\\ \end {align*}

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Mathematica [A]
time = 0.74, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^m}{\left (c+a^2 c x^2\right )^{3/2} \text {ArcTan}(a x)^{5/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[x^m/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)),x]

[Out]

Integrate[x^m/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)), x]

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Maple [A]
time = 1.59, size = 0, normalized size = 0.00 \[\int \frac {x^{m}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \arctan \left (a x \right )^{\frac {5}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(5/2),x)

[Out]

int(x^m/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(5/2),x)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(5/2),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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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(5/2),x, algorithm="fricas")

[Out]

integral(sqrt(a^2*c*x^2 + c)*x^m/((a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2)*arctan(a*x)^(5/2)), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m/(a**2*c*x**2+c)**(3/2)/atan(a*x)**(5/2),x)

[Out]

Timed out

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(5/2),x, algorithm="giac")

[Out]

sage0*x

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {x^m}{{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m/(atan(a*x)^(5/2)*(c + a^2*c*x^2)^(3/2)),x)

[Out]

int(x^m/(atan(a*x)^(5/2)*(c + a^2*c*x^2)^(3/2)), x)

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