3.8.82 \(\int \frac {\text {ArcTan}(a x)^{3/2}}{x^4 (c+a^2 c x^2)} \, dx\) [782]

Optimal. Leaf size=61 \[ \frac {2 a^3 \text {ArcTan}(a x)^{5/2}}{5 c}+\frac {\text {Int}\left (\frac {\text {ArcTan}(a x)^{3/2}}{x^4},x\right )}{c}-\frac {a^2 \text {Int}\left (\frac {\text {ArcTan}(a x)^{3/2}}{x^2},x\right )}{c} \]

[Out]

2/5*a^3*arctan(a*x)^(5/2)/c+Unintegrable(arctan(a*x)^(3/2)/x^4,x)/c-a^2*Unintegrable(arctan(a*x)^(3/2)/x^2,x)/
c

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Rubi [A]
time = 0.14, 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 {\text {ArcTan}(a x)^{3/2}}{x^4 \left (c+a^2 c x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[ArcTan[a*x]^(3/2)/(x^4*(c + a^2*c*x^2)),x]

[Out]

(2*a^3*ArcTan[a*x]^(5/2))/(5*c) + Defer[Int][ArcTan[a*x]^(3/2)/x^4, x]/c - (a^2*Defer[Int][ArcTan[a*x]^(3/2)/x
^2, x])/c

Rubi steps

\begin {align*} \int \frac {\tan ^{-1}(a x)^{3/2}}{x^4 \left (c+a^2 c x^2\right )} \, dx &=-\left (a^2 \int \frac {\tan ^{-1}(a x)^{3/2}}{x^2 \left (c+a^2 c x^2\right )} \, dx\right )+\frac {\int \frac {\tan ^{-1}(a x)^{3/2}}{x^4} \, dx}{c}\\ &=a^4 \int \frac {\tan ^{-1}(a x)^{3/2}}{c+a^2 c x^2} \, dx+\frac {\int \frac {\tan ^{-1}(a x)^{3/2}}{x^4} \, dx}{c}-\frac {a^2 \int \frac {\tan ^{-1}(a x)^{3/2}}{x^2} \, dx}{c}\\ &=\frac {2 a^3 \tan ^{-1}(a x)^{5/2}}{5 c}+\frac {\int \frac {\tan ^{-1}(a x)^{3/2}}{x^4} \, dx}{c}-\frac {a^2 \int \frac {\tan ^{-1}(a x)^{3/2}}{x^2} \, dx}{c}\\ \end {align*}

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Mathematica [A]
time = 2.72, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\text {ArcTan}(a x)^{3/2}}{x^4 \left (c+a^2 c x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[ArcTan[a*x]^(3/2)/(x^4*(c + a^2*c*x^2)),x]

[Out]

Integrate[ArcTan[a*x]^(3/2)/(x^4*(c + a^2*c*x^2)), x]

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Maple [A]
time = 1.51, size = 0, normalized size = 0.00 \[\int \frac {\arctan \left (a x \right )^{\frac {3}{2}}}{x^{4} \left (a^{2} c \,x^{2}+c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctan(a*x)^(3/2)/x^4/(a^2*c*x^2+c),x)

[Out]

int(arctan(a*x)^(3/2)/x^4/(a^2*c*x^2+c),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(arctan(a*x)^(3/2)/x^4/(a^2*c*x^2+c),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(arctan(a*x)^(3/2)/x^4/(a^2*c*x^2+c),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*} \frac {\int \frac {\operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}{a^{2} x^{6} + x^{4}}\, dx}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atan(a*x)**(3/2)/x**4/(a**2*c*x**2+c),x)

[Out]

Integral(atan(a*x)**(3/2)/(a**2*x**6 + x**4), x)/c

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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(arctan(a*x)^(3/2)/x^4/(a^2*c*x^2+c),x, algorithm="giac")

[Out]

sage0*x

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\mathrm {atan}\left (a\,x\right )}^{3/2}}{x^4\,\left (c\,a^2\,x^2+c\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atan(a*x)^(3/2)/(x^4*(c + a^2*c*x^2)),x)

[Out]

int(atan(a*x)^(3/2)/(x^4*(c + a^2*c*x^2)), x)

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