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

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

[Out]

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

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 1.45, size = 0, normalized size = 0.00 \[ \int \frac {x^m \tan ^{-1}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{m} \arctan \left (a x\right )^{\frac {3}{2}}}{a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^m*arctan(a*x)^(3/2)/(a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2), 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)^(3/2)/(a^2*c*x^2+c)^2,x, algorithm="giac")

[Out]

sage0*x

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maple [A]  time = 2.88, size = 0, normalized size = 0.00 \[ \int \frac {x^{m} \arctan \left (a x \right )^{\frac {3}{2}}}{\left (a^{2} c \,x^{2}+c \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {x^{m} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}{a^{4} x^{4} + 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**m*atan(a*x)**(3/2)/(a**4*x**4 + 2*a**2*x**2 + 1), x)/c**2

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