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

Optimal. Leaf size=29 \[ \text {Int}\left (\frac {x^m \sqrt {\tan ^{-1}(a x)}}{\left (a^2 c x^2+c\right )^{3/2}},x\right ) \]

[Out]

Unintegrable(x^m*arctan(a*x)^(1/2)/(a^2*c*x^2+c)^(3/2),x)

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Rubi [A]  time = 0.12, 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 = {} \[ \int \frac {x^m \sqrt {\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 1.12, size = 0, normalized size = 0.00 \[ \int \frac {x^m \sqrt {\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a^{2} c x^{2} + c} x^{m} \sqrt {\arctan \left (a x\right )}}{a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*arctan(a*x)^(1/2)/(a^2*c*x^2+c)^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(a^2*c*x^2 + c)*x^m*sqrt(arctan(a*x))/(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)^(1/2)/(a^2*c*x^2+c)^(3/2),x, algorithm="giac")

[Out]

sage0*x

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maple [A]  time = 3.36, size = 0, normalized size = 0.00 \[ \int \frac {x^{m} \sqrt {\arctan \left (a x \right )}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*arctan(a*x)^(1/2)/(a^2*c*x^2+c)^(3/2),x)

[Out]

int(x^m*arctan(a*x)^(1/2)/(a^2*c*x^2+c)^(3/2),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)^(1/2)/(a^2*c*x^2+c)^(3/2),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 = -0.03 \[ \int \frac {x^m\,\sqrt {\mathrm {atan}\left (a\,x\right )}}{{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]

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)^(3/2),x)

[Out]

int((x^m*atan(a*x)^(1/2))/(c + a^2*c*x^2)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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