3.9.87 \(\int \frac {(c+a^2 c x^2)^{3/2} \text {ArcTan}(a x)^{5/2}}{x} \, dx\) [887]

Optimal. Leaf size=29 \[ \text {Int}\left (\frac {\left (c+a^2 c x^2\right )^{3/2} \text {ArcTan}(a x)^{5/2}}{x},x\right ) \]

[Out]

Unintegrable((a^2*c*x^2+c)^(3/2)*arctan(a*x)^(5/2)/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 )^{3/2} \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)^(3/2)*ArcTan[a*x]^(5/2))/x,x]

[Out]

Defer[Int][((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2))/x, x]

Rubi steps

\begin {align*} \int \frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2}}{x} \, dx &=\int \frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2}}{x} \, dx\\ \end {align*}

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Mathematica [A]
time = 1.53, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c+a^2 c x^2\right )^{3/2} \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)^(3/2)*ArcTan[a*x]^(5/2))/x,x]

[Out]

Integrate[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2))/x, x]

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Maple [A]
time = 0.88, size = 0, normalized size = 0.00 \[\int \frac {\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \arctan \left (a x \right )^{\frac {5}{2}}}{x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2*c*x^2+c)^(3/2)*arctan(a*x)^(5/2)/x,x)

[Out]

int((a^2*c*x^2+c)^(3/2)*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)^(3/2)*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)^(3/2)*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 [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a**2*c*x**2+c)**(3/2)*atan(a*x)**(5/2)/x,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3878 deep

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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((a^2*c*x^2+c)^(3/2)*arctan(a*x)^(5/2)/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 = -0.03 \begin {gather*} \int \frac {{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^{3/2}}{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)^(3/2))/x,x)

[Out]

int((atan(a*x)^(5/2)*(c + a^2*c*x^2)^(3/2))/x, x)

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