3.1055 \(\int \frac {x}{(c+a^2 c x^2) \tan ^{-1}(a x)^{5/2}} \, dx\)

Optimal. Leaf size=41 \[ \frac {2 \text {Int}\left (\frac {1}{\tan ^{-1}(a x)^{3/2}},x\right )}{3 a c}-\frac {2 x}{3 a c \tan ^{-1}(a x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.05, 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 = {} \[ \int \frac {x}{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 1.55, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

[Out]

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

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [A]  time = 0.37, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (a^{2} c \,x^{2}+c \right ) \arctan \left (a x \right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x/(a^2*c*x^2+c)/arctan(a*x)^(5/2),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/(a^2*c*x^2+c)/arctan(a*x)^(5/2),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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mupad [A]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x}{{\mathrm {atan}\left (a\,x\right )}^{5/2}\,\left (c\,a^2\,x^2+c\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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