3.870 \(\int \frac {x^m \tan ^{-1}(a x)^{5/2}}{(c+a^2 c x^2)^3} \, dx\)

Optimal. Leaf size=27 \[ \text {Int}\left (\frac {x^m \tan ^{-1}(a x)^{5/2}}{\left (a^2 c x^2+c\right )^3},x\right ) \]

[Out]

Unintegrable(x^m*arctan(a*x)^(5/2)/(a^2*c*x^2+c)^3,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 = {} \[ \int \frac {x^m \tan ^{-1}(a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 1.97, size = 0, normalized size = 0.00 \[ \int \frac {x^m \tan ^{-1}(a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{m} \arctan \left (a x\right )^{\frac {5}{2}}}{a^{6} c^{3} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} + c^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^m*arctan(a*x)^(5/2)/(a^6*c^3*x^6 + 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 + c^3), x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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maple [A]  time = 3.51, size = 0, normalized size = 0.00 \[ \int \frac {x^{m} \arctan \left (a x \right )^{\frac {5}{2}}}{\left (a^{2} c \,x^{2}+c \right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^m*arctan(a*x)^(5/2)/(a^2*c*x^2+c)^3,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*arctan(a*x)^(5/2)/(a^2*c*x^2+c)^3,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 \[ \int \frac {x^m\,{\mathrm {atan}\left (a\,x\right )}^{5/2}}{{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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