3.6.29 \(\int \frac {x^m (c+a^2 c x^2)^{3/2}}{\text {ArcTan}(a x)} \, dx\) [529]

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

[Out]

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

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Rubi [A]
time = 0.08, 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 )^{3/2}}{\text {ArcTan}(a x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.50, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^m \left (c+a^2 c x^2\right )^{3/2}}{\text {ArcTan}(a x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.26, size = 0, normalized size = 0.00 \[\int \frac {x^{m} \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{\arctan \left (a x \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)^(3/2)/arctan(a*x),x, algorithm="maxima")

[Out]

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

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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)^(3/2)/arctan(a*x),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x^m\,{\left (c\,a^2\,x^2+c\right )}^{3/2}}{\mathrm {atan}\left (a\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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