3.7.86 \(\int x^m (c+a^2 c x^2) \sqrt {\text {ArcTan}(a x)} \, dx\) [686]

Optimal. Leaf size=25 \[ \text {Int}\left (x^m \left (c+a^2 c x^2\right ) \sqrt {\text {ArcTan}(a x)},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.02, 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 x^m \left (c+a^2 c x^2\right ) \sqrt {\text {ArcTan}(a x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 1.43, size = 0, normalized size = 0.00 \begin {gather*} \int x^m \left (c+a^2 c x^2\right ) \sqrt {\text {ArcTan}(a x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 1.72, size = 0, normalized size = 0.00 \[\int x^{m} \left (a^{2} c \,x^{2}+c \right ) \sqrt {\arctan \left (a x \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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)*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.

________________________________________________________________________________________

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)*arctan(a*x)^(1/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} c \left (\int x^{m} \sqrt {\operatorname {atan}{\left (a x \right )}}\, dx + \int a^{2} x^{2} x^{m} \sqrt {\operatorname {atan}{\left (a x \right )}}\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*(Integral(x**m*sqrt(atan(a*x)), x) + Integral(a**2*x**2*x**m*sqrt(atan(a*x)), x))

________________________________________________________________________________________

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)*arctan(a*x)^(1/2),x, algorithm="giac")

[Out]

sage0*x

________________________________________________________________________________________

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

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),x)

[Out]

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

________________________________________________________________________________________