3.6.4 \(\int \frac {(c+a^2 c x^2)^{5/2}}{x \text {ArcTan}(a x)} \, dx\) [504]

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

[Out]

Unintegrable((a^2*c*x^2+c)^(5/2)/x/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 {\left (c+a^2 c x^2\right )^{5/2}}{x \text {ArcTan}(a x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 1.19, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c+a^2 c x^2\right )^{5/2}}{x \text {ArcTan}(a x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 1.94, size = 0, normalized size = 0.00 \[\int \frac {\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{x \arctan \left (a x \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}{x \operatorname {atan}{\left (a x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a**2*c*x**2+c)**(5/2)/x/atan(a*x),x)

[Out]

Integral((c*(a**2*x**2 + 1))**(5/2)/(x*atan(a*x)), x)

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

[Out]

sage0*x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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