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

Optimal. Leaf size=49 \[ \frac {i \text {Int}\left (\frac {\tan ^{-1}(a x)^{5/2}}{x (a x+i)},x\right )}{c}-\frac {2 i \tan ^{-1}(a x)^{7/2}}{7 c} \]

[Out]

-2/7*I*arctan(a*x)^(7/2)/c+I*Unintegrable(arctan(a*x)^(5/2)/x/(I+a*x),x)/c

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Rubi [A]  time = 0.11, 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 = {} \[ \int \frac {\tan ^{-1}(a x)^{5/2}}{x \left (c+a^2 c x^2\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(((-2*I)/7)*ArcTan[a*x]^(7/2))/c + (I*Defer[Int][ArcTan[a*x]^(5/2)/(x*(I + a*x)), x])/c

Rubi steps

\begin {align*} \int \frac {\tan ^{-1}(a x)^{5/2}}{x \left (c+a^2 c x^2\right )} \, dx &=-\frac {2 i \tan ^{-1}(a x)^{7/2}}{7 c}+\frac {i \int \frac {\tan ^{-1}(a x)^{5/2}}{x (i+a x)} \, dx}{c}\\ \end {align*}

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Mathematica [A]  time = 0.66, size = 0, normalized size = 0.00 \[ \int \frac {\tan ^{-1}(a x)^{5/2}}{x \left (c+a^2 c x^2\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

[Out]

sage0*x

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maple [A]  time = 0.47, size = 0, normalized size = 0.00 \[ \int \frac {\arctan \left (a x \right )^{\frac {5}{2}}}{x \left (a^{2} c \,x^{2}+c \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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