3.9.41 \(\int \frac {(c+a^2 c x^2) \text {ArcTan}(a x)^{5/2}}{x} \, dx\) [841]

Optimal. Leaf size=25 \[ \text {Int}\left (\frac {\left (c+a^2 c x^2\right ) \text {ArcTan}(a x)^{5/2}}{x},x\right ) \]

[Out]

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

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

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 2.11, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c+a^2 c x^2\right ) \text {ArcTan}(a x)^{5/2}}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 1.26, size = 0, normalized size = 0.00 \[\int \frac {\left (a^{2} c \,x^{2}+c \right ) \arctan \left (a x \right )^{\frac {5}{2}}}{x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*(Integral(atan(a*x)**(5/2)/x, x) + Integral(a**2*x*atan(a*x)**(5/2), 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)*arctan(a*x)^(5/2)/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 {{\mathrm {atan}\left (a\,x\right )}^{5/2}\,\left (c\,a^2\,x^2+c\right )}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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