3.6.33 \(\int \frac {x^m}{(c+a^2 c x^2)^{5/2} \text {ArcTan}(a x)} \, dx\) [533]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.09, 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 )^{5/2} \text {ArcTan}(a x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.41, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^m}{\left (c+a^2 c x^2\right )^{5/2} \text {ArcTan}(a x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.20, size = 0, normalized size = 0.00 \[\int \frac {x^{m}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \arctan \left (a x \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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)**(5/2)/atan(a*x),x)

[Out]

Timed out

________________________________________________________________________________________

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)^(5/2)/arctan(a*x),x, algorithm="giac")

[Out]

sage0*x

________________________________________________________________________________________

Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x^m}{\mathrm {atan}\left (a\,x\right )\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________