3.10.89 \(\int \frac {x^4}{(c+a^2 c x^2)^2 \text {ArcTan}(a x)^{3/2}} \, dx\) [989]

Optimal. Leaf size=93 \[ -\frac {2 x^4}{a c^2 \left (1+a^2 x^2\right ) \sqrt {\text {ArcTan}(a x)}}+\frac {8 \text {Int}\left (\frac {x^3}{\left (c+a^2 c x^2\right )^2 \sqrt {\text {ArcTan}(a x)}},x\right )}{a}+4 a \text {Int}\left (\frac {x^5}{\left (c+a^2 c x^2\right )^2 \sqrt {\text {ArcTan}(a x)}},x\right ) \]

[Out]

-2*x^4/a/c^2/(a^2*x^2+1)/arctan(a*x)^(1/2)+8*Unintegrable(x^3/(a^2*c*x^2+c)^2/arctan(a*x)^(1/2),x)/a+4*a*Unint
egrable(x^5/(a^2*c*x^2+c)^2/arctan(a*x)^(1/2),x)

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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 {x^4}{\left (c+a^2 c x^2\right )^2 \text {ArcTan}(a x)^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(-2*x^4)/(a*c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) + (8*Defer[Int][x^3/((c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x]
)/a + 4*a*Defer[Int][x^5/((c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x]

Rubi steps

\begin {align*} \int \frac {x^4}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{3/2}} \, dx &=-\frac {2 x^4}{a c^2 \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}+\frac {8 \int \frac {x^3}{\left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx}{a}+(4 a) \int \frac {x^5}{\left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx\\ \end {align*}

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Mathematica [A]
time = 3.71, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^4}{\left (c+a^2 c x^2\right )^2 \text {ArcTan}(a x)^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 1.00, size = 0, normalized size = 0.00 \[\int \frac {x^{4}}{\left (a^{2} c \,x^{2}+c \right )^{2} \arctan \left (a x \right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^4}{{\mathrm {atan}\left (a\,x\right )}^{3/2}\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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