3.714 \(\int \frac {x^5 \sqrt {\tan ^{-1}(a x)}}{(c+a^2 c x^2)^3} \, dx\)

Optimal. Leaf size=27 \[ \text {Int}\left (\frac {x^5 \sqrt {\tan ^{-1}(a x)}}{\left (a^2 c x^2+c\right )^3},x\right ) \]

[Out]

Unintegrable(x^5*arctan(a*x)^(1/2)/(a^2*c*x^2+c)^3,x)

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Rubi [A]  time = 0.07, 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 = {} \[ \int \frac {x^5 \sqrt {\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 5.45, size = 0, normalized size = 0.00 \[ \int \frac {x^5 \sqrt {\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*arctan(a*x)^(1/2)/(a^2*c*x^2+c)^3,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*arctan(a*x)^(1/2)/(a^2*c*x^2+c)^3,x, algorithm="giac")

[Out]

sage0*x

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maple [A]  time = 4.19, size = 0, normalized size = 0.00 \[ \int \frac {x^{5} \sqrt {\arctan \left (a x \right )}}{\left (a^{2} c \,x^{2}+c \right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*arctan(a*x)^(1/2)/(a^2*c*x^2+c)^3,x)

[Out]

int(x^5*arctan(a*x)^(1/2)/(a^2*c*x^2+c)^3,x)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*arctan(a*x)^(1/2)/(a^2*c*x^2+c)^3,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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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {x^5\,\sqrt {\mathrm {atan}\left (a\,x\right )}}{{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5*atan(a*x)^(1/2))/(c + a^2*c*x^2)^3,x)

[Out]

int((x^5*atan(a*x)^(1/2))/(c + a^2*c*x^2)^3, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {x^{5} \sqrt {\operatorname {atan}{\left (a x \right )}}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*atan(a*x)**(1/2)/(a**2*c*x**2+c)**3,x)

[Out]

Integral(x**5*sqrt(atan(a*x))/(a**6*x**6 + 3*a**4*x**4 + 3*a**2*x**2 + 1), x)/c**3

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